The Generation of Gravitational Waves Citation Kovács, Sándor János, Jr. (1977) The Generation of Gravitational Waves. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/VHQW-W277. https://resolver.caltech.edu/CaltechTHESIS:08252017-093756825 Abstract This dissertation is written in three tracks. Track 1, (Pages ii-vi) is intended for those many many people who don't know any physics but who do know how to read English and a bit of Hungarian perhaps. Track 2 is intended for those one hundred or so people in the whole world who do know classical physics and have serious interests in, or are experts in relativity theory. Papers 1, 2 and 3 dealing with relativity are entirely Track 2. The remaining paper is Track 3 and is for the benefit of even fewer people who have a simultaneous interest in relativity and quantum field theory. I will not discuss the abstracts of the individual papers here - since each paper is preceded by its own abstract. Suffice it to say that this dissertation is a collection of papers dealing with the theoretical aspects of how gravitational waves may (or may not) be generated by gravitational bremsstrahlung, and in Paper 4 Walter and I try to show that some classical relativity problems may be solved with much greater ease via a quantum approach. Item Type: Thesis (Dissertation (Ph.D.)) Subject Keywords: Physics Degree Grantor: California Institute of Technology Division: Physics, Mathematics and Astronomy Major Option: Physics Thesis Availability: Public (worldwide access) Research Advisor(s): Thorne, Kip S. Group: TAPIR, Astronomy Department Thesis Committee: Unknown, Unknown Defense Date: 27 May 1977 Funders: Funding Agency Grant Number NDEA Fellowship UNSPECIFIED NASA NGR 05-002-256 NSF MPS75-01398 NSF AST75-01398 A01 Record Number: CaltechTHESIS:08252017-093756825 Persistent URL: https://resolver.caltech.edu/CaltechTHESIS:08252017-093756825 DOI: 10.7907/VHQW-W277 Default Usage Policy: No commercial reproduction, distribution, display or performance rights in this work are provided. ID Code: 10390 Collection: CaltechTHESIS Deposited By: Benjamin Perez Deposited On: 25 Aug 2017 20:23 Last Modified: 10 Mar 2020 23:39 Thesis Files Preview PDF - Final Version See Usage Policy. 33MB Repository Staff Only: item control page

THE GENERATION OF GRAVITATIONAL WAVES Thesis by I I I Sandor Janos Kovacs Jr. In Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy California Institute of Technology Pasadena, California 1977 (Submitted May 27, 1977) ii Materiem superabat opus. Publius Ovidius Naso. iii ACKNOWLEDGMENTS The fact that this disserta tion exists at all is very much dependent on the fact that I exist. The fact that I exist however, has been reliably shown to be due to my father and mother - S~ndor and Julianna Kov~cs. They had the strength and motivation to start anew after the tribulations of World War II and the Hungarian Revolt of 1956 . Their decision to emigrate to the United States was one in which the future of my sister and me carried significant weight. It is to my parents to whom I owe my deepest gratitude and greatest r e spect, it is to them that this work is dedicated. I am grateful to my sister Marika for her generosity,friendship, understanding and love. Her intellectual comprehension of, and deep intuitive insight into people's behaviour has been a treme ndous asset and our d i scussions on these and other topics consumed innumerable pleasant hours. I wish to thank my fri e nd, teacher and advisor Kip Thorne for his guidance, support and trust during my years at Caltech. His deep physical insight and formidable intellectual skill coupled with enormous patience and dedication made it a pleasure for me to be his student and colleague. I owe much thanks and deep gratitude to my close friend and associate R.B.Meil - whose relen t less and ded icated emotional support (i.e. love) greatly helped to stabilize the (sometimes hi ghly variable) environment. To Walter De Logi - my friend and collaborator - goes my appreciation for his sense of humour, philosophical advice, rena issance attitudes and sense of culture. As a pleasant a ddition to the above, Walte r happens to be a first rate scie ntist as well. In the non-academic domain the f riendship and support of Linda Na g y (Mrs. Dennis), Warren Dennis (WD-40),Jiuuny Odriozola(The '0 ' ), Stephen Metcalf, Ron Gregg ( The Moose), Bob Hanunond, John Herb and particularly Jim and Betty Mayer (and family) have been very help f ul. In general it is not possible for any human bei ng to be always iv helpful, especially in an intellectual and emotional endeavor that spans several years. Nevertheless I would like to thank the following people who, (in a time averaged sense) helped to inprove the local environment; Charles Alcock, Charlie Barnes, Francis Buffington, Bill Burke, Roger Blandford, Phil and Judy Callahan, Carl Caves, Steven Detweiler, Richard Feynman, Ricky Flarrunang, Willy Fowler, Jim Gunn, Tom and Patty Ganz, Stephen Hawking, Dave Mikkelsen, Ken Nordtvedt, Don Page, Richard and Josette Price, Paul Schechter, Bernie Schutz, Steve Slutz, Saul and Roz Teukolsky, Scott Tremaine, Bob and Lynn Wagoner, Cliff and Leslie Will, Alan Lightman, David and Ellen Lee and especially Rudi and Janja Husar. It is well known but not frequently articulated, that Kellogg Lab and all work within its realm would come to a screeching, grinding halt were it not for the unique talents and fantastic skills of Evaline Gibbs Jan Rasmussen, Marty Watson, Gloria Laird, Paula Dunn and Mike Robinson who keep the internal parts well tuned and lubricated. Special thanks go to our groups' own Polly Crandmonta gne and Ann Freeman - who keep our end of the machinery in tip top shape, and in times of crisis - are faster than a speeding s electric, more powerful than the administration, and capable of overcoming 10,000 dollar deficits in a single bound. Ruth Stratton,Gloria Laird and Jan Rasmussen did an outstanding job of technical typing the papers - without their skills the clarity of our publications would suffer greatly. Kiilonosen h~las va gyok Tokes Zoltan, Csepregi Las zlh, Bagi Gyorgy I I I I I • I I , I es Revesz Peter barati tamogatasaikert. I am thankful for support from an NDEA Fellowship for part of this research. Subsequent financial assistance was obtained from the National Aeronautics and Space Admini s tration ( NGR OS-002-256 ) and the National Science Foundation ( MPS75 -0139 8 ) and ( AST75 -01398 AOl ). v ABSTRACT This dissertation is written in three tracks. Track l,(Pa ges ii-vi) is intended for those many many people who don't know any physics but who do know how to read English and a bit of Hungarian pechaps. Track 2 is intended for those one hundred or so people in the whole world who do know classical physics and have serious interests in, or are experts in relativity theory. Papers 1, 2 and 3 dealing with relativity are entirely Track 2 . The remaining paper is Track 3 and is for the benefit of even fewer people who have a simultaneous interest in relativity and quantum field theory. I will not discuss the abstracts of the individual papers here since each paper is preceded by its own abstract. Suffice it to say that this dissertation is a collect ion of papers dealing with the theoretical aspects of how gravitational waves may (or may not) be gener atei by gravitational bremsstrahlung,and in Paper l~ Walter and I try to show that some classical relativity problems may be solved with much greater ease via a quantum approach. vi A Limerick to Gravitational Waves There once was the theory of relativ i ty Which serious studen ts pursue with humility, Whether gravitat ional radiation Is real, or just imagination To them, became a matter of facility. Oh you sly, sneaky, ancient gravity waves You're putting experimentalists in their graves! If you really exist Why are you missed? You even make theorists hide in their caves. Some argue it's a ma tter of lucky detection, Or perhaps of a di ffe rent research direction. But we'll all be elated If waves are reall y ge nerated, 'Cause then Nature will verify our prediction. Einstein thou ght God was not malicious I agree, but say Nature is capricious. She will not simply reveal Or make it easy to steal, The secret treasures that make Her delicious. When at l a st the darn radiation is found New explanations and theories will abound. In the zeal of celebration Let's recall the quotation "It's so sad that Einstein won't be around." Pasadena April,19 77 vii TABLE OF CONTENTS Part I. INTRODUCTION - entirely Track 1. 1 II. Paper 1. The Generation of Gravitational Waves. I. Weak- 4 Field Sources in collaboration with Kip S. Thorne, published in the Astrophysical Journal, gQQ, 2(1975)). Paper 2. The Generation of Gravitational Waves. III. Derivation of Bremsstrahlung Formulae 23 [ in collaboration with Kip S. Thorne, to be published in the October 1, 1977 issue of the Astrophysical Journal]. Paper 3. The Generation of Gravitational Waves. IV. Bremsstrahlung 98 [ material which will ultima- tely become a joint publication with K.S.Thorne]. III. Paper 4. The Gravitational Scattering of Zero-Rest-Mass Plane Waves in collaboration with Walter De Logi, to be published in the July 15, 1977 issue of Physica~ Review D ]. 153 1 I. INTRODUCTION During the course of the past several years while I've been a graduate student at Caltech, the hope that somewhere,somebody 's gra vitational wave detector will actua lly detect gravitational waves by the middle of the 1980's has improved considerably.( However, I should remark that the 11 • • • hope that gravitational waves will be detected within the next ten years 11 is a statement that has been heard for at least the past ten years!) The work we've done in Papers 1, 2 and 3 was intended to provide an add itional theoretical tool by which gravitational radiation sources may be analyzed . Any tool, regardless of how sophisticated it is or how much promise it offers, is worthless unless applied to solve a problem. We applied the mathematical tools we constructed to gain further insight into the bremsstrahlung problem, and had some success. The interesting results are contained in Paper 3 , but we paid a price of some rather deta iled and tedious prelimLnary work. It showed that the 'post-linear' formalism was us able and effective however it is safe to say that it did not provide a calculational shortcut to the final results. Since the contents of the papers speak for themselves, my remaining remarks in this section are intended to shed some light on how the work we did actually came to pass. In 1972 K.S. Thorne ( henceforth cited as Kip) and I decided to analyze in detail 'The Gravitational Bremsstrahlung Problem' with the hope that we could remove the previous r estrictions on the ratio of the masses and any constraints on the velocities. Of course we only held out hope for the weak-field case. After about a year of probing ( up several blind alleys) we realized that our initial approach using flat-space propagators was inadequate. We kept getting divergent volume integrals for the gravitational stress. The desire to solve the problem and find an answer (i.e. our personalities) forced us to return to the mathematical foundations of radiation prob l ems and devise a scheme that would work. The result of a ll this hard work and fancy mathematics is Paper 1. The interesting physics was still to come. By early 1975 Kip and I had the 'tools' to actually 11 Plug-In- 2 And-Grind " away at the physics problem we set out to solve. The results of this somewhat tedious ta s k (usual polite understatement) are laid out in Paper 2 of the thesis - Paper III. of the published series. We never intended to have a Paper 2 just with formulas and a separate paper just with results - but the leng th and complexity of the details ( and the pa ge limitations of the Astrophis i cal Journal) forced us into it. While all this was going on Walter De Logi ( henceforth cited as Walter ) and I started looking into the bremsstrahlung problem from the quantum point of view using diagrams. This approach was partially motivated by Richard Feynman's questions and suggestions during my oral candidacy exam in February 1975 . Naturally our first results using diagrams had nothing to do with the problem we originally set out to solve. Walter and I digressed ( after a s eminar given at Caltech by Phil Peters ), when we realized we could do plane wave scattering problems fa ster and easier using dia grams - with the added excit i ng bonu s of being able to scatter wave s from a scatterer with angular momentum - something that Peters could not do. The results of a few minutes of creative insight and several months of work is what turned out to be Paper 4. Concurrent wi th the work with Walter, the analytical and computer work for Paper 3 was going on. I spent several months looking for a theoretical and/or programming mistake while trying to get the numerical results of our Paper 2 to a gree with the results published by Peters - I finally concluded (correctly) that yes, there was a mistake somewhere. Peters had a previously undetected typographycal error in his paper and the discrepancy nicely vanished after a simple correction. At last, by the spring of 1977 we had the results to 'THE Gravitational Bremsstrahlung Problem' which is inc luded here for the enjoyment of the reader. I think it contains the most interesting material in the whole thesis, and as it is written here is material which will ultimately be published in a joint paper with Kip. For completeness, i.e. a desire to beat the bremsstrahlung problem to death from all directions, Walter and I tried , but could not find 3 sufficient time to complete the problem using a quantum approach. Unfortunately that piece of work is not part of this thesis but will be published seperately, as soon as i~s finished. l+ PAPER 1 5 Tru: ASTROPHYSICAL ]OUl<NAL, 200: 245-262, 1975 Septc:mbc:r I ·,:. I n5. The American Astronomi~t1 Society. All rights res~rvcJ. rrintc::ll in U.S.A. THE GENERATION OF GRAVITATIONAL WAVES. I. WEAK-HELD SOURCES* KIP S. THORl'iF. AND SA't'DOR J. Kov.Acs California Institute of Technology Received 1974 December 9 ABSTRACT This paper derives (§§ Il-IV) and summarizes (§ VJ) a new "plug-in-and-grind" formalism (i.e., an algorithm) for calculating the gravitational waves emitkd by any system with wc:ak internal gravitational fields. If the internal fields have: negligible infiuence on the syskm's motions, then the new formalism reduces to standard "linearized theory." Whether or not gravity affects the motions, if the motions are slow and inkrnal stresses are weak, then the new formalism reduces to the standard "quadrupok-moment formalism" (§ V). In the general case the new formalism expresses the radiation in terms of a retarded Green's function for slightly curved spacetime, and then breaks the Green's function integral into five easily understood pieces: direct radiation, produced directly by the motions of the source; iv/111mp radiation, produced by the "gravit3tional stresses" of the source; transition radiation, produced by a time-changi ng time delay ("Shapiro effect'') in the propagation of the non radiative, l/r field of the source; focusing radiation, produced when one portion of the source focuses, in a time-dependent way, the nonradiative field of another portion of the source; and tail radiation, produced by "backscatter" of the nonradiative field in regions of focusing. Subject headings: gravitation - relativity I. INTRODUCTION a) Introduction to This Series of Papers Thanks to the piurieering work of Joseph Weber (1960, 1969), "gravitational-wave astronomy" may he a reality by 1980. Although Weher's "events" may turn out to be nongravita tional in origin, second-generation detect.ors of the Weber " resona nt-bar" type, with amplitude sensitivities roughly I 00-fold better than today's bars, are now u;ider construction (Braginsky 1974; Fairbank a nd Hamilton, as described in Boughn ct al. 1974); and thirdg::neration detectors a re being discussed. The third generation should be able to detect and study the gravitational·xave bursts generated several times per year by supernovae in the Virgo cluster of galaxies. Detectors with other de5igns may succeed in detecting waves from pulsars (see, e.g., Braginsky a nd Nazarc:nko I971) and from nearencounters of stars in dense star clusters (gravitational bremsstrahlung; see, e.g., Zel'dovich and Ponarev 1974). And, of course, totally unexpected sources may be detected. (For reviews of the prospects for gravitational-wave 2monomy see Misner 1974; Rees I974; and Press and Thorne I972.) ln preparation for the era of gravitational-wave astronomy, our Caltech research group has embarked on a r.ew project: We seek (I) to elucidate the realms of validity of the standard wave-generation formulae; (2) to devise new techniques for calculating gravitational-wave generation with new realms of validity; a nd (3) to calculate the waves generated by particular models of astrophysical systems. Throughout this project we shall confine o urselves to general relativity theo ry. \'lost past calculations of gravitational-wave generation use one of three formalisms: (1) "linearized theory" or its quantum-theory analog; (2) the "quadrupole-moment formalism"; (3) "first-order perturbations of station2ry, fully relativistic spacetimes." "Linea rized theory" is the formalism obtained by linearizing general relativity about flat spacetime (see, e.g., chapters 18 and 35 of Misner, Thorne, and Wheeler 1973-cited hencdorth as MTW). It is also the unique linear $pin-two field theory of gravitation in flat spacetime- and as such it has a simple quantum-theory formulation . (For references and. overview see, in MT W, § 7.1, box 7.1, and part 5 of box 17.2). Linearized theory is typicnlly u::.ed to calcul•tte wave generation when the so urce's self-gravity has negligible influence on its motions (e.g., waves from spin ning rods and from electromagnetic fields in a cavity). In this paper we shall devi~e a new wave-generation formalism valid for any system with small but nonnegligihle self-gravity; and in Paper Ill (Kovacs and Thorne 1975) we shall use that formalism to calculate the gravitational bremsstrahlung produced when two stars fly past each other with large impact paramc::ter, but with a rbitrary relative masses and velocities. The" quadrupole-moment formalism" (in which the wave a mplitude is proportional to the second time derivative of the source's mass quadrupol•~ moment) dates back to Einstein (1918), and has been canonized by Landau • Supported in part by the National Aeronautics and Space Administration (NGR 05-002-256) and the National Science Foundation [~1PS75-01398 ] . 245 6 THORNE AND KOY Acs 246 Vol. 200 and Lifshitz (1951). The derivations of this formalism which we find in the literature are valid only for systems with slow internal motions and weak (but nonnegligible) internal gravitational fields (see, e.g., the post"12 -Newtonian derivation by Chandrasekhar and Esposito 1970. the matched-asymptotic-expansion derivation by Burke 1971 , and the de Donder gauge derivation by Landau and Lifshitz 1951 as made more explicit in chapter 36 of MTW). However, a detailed analysis given in Paper 11 (Thorne.: 1975) shows that only the slow-motion assumption is needed: the quadrupole-moment formalism is valid for any slow-motion system, regardless of its inkrnal field strengths. Paper 11 also extends that formalism to include the radiation produced by all of the source's other moments (both "mass" moments and "current" moments) ; and it derives formulae in terms of the moments for the near-zone fields, the radiation field, the radiation reaction, and the energy, momentum, and angular momentum carried off by the waves. In a forthcoming paper Thorne and .Zytkow (1976) will use the extended formalism of Paper II to calculate the "current-quadrupole " gravitational waves produced by torsional oscillations of neutron stars. "First-order perturbations of stationary, fully relativistic spacetimes" is a technique that has been used extensively in recent years to analyze waves from "fast-motion" oscillations of black holes and neutron stars, and from particles moving in the Schwarzschild and Kerr gravitational fields. (for reviews, see Press 1974, Ruffini 1973, and§ 36.5 of MTW; see also the recent paper by Chung 1973.) It is not yet clear whether our project will delve into this technique. b) 01:erview of This Paper In this paper we confine attention to systems with weak internal gravitational fields. Section II rewrites the exact Einstein field equations in a non-covariant form ("de Donder form") that is amenable to weak-field approximations. Section HI gives a systematic account of approximate, weak-field formalisms based on the exact de Donder form of the field equations-including the accuracy of the various formalisms and their relationships to each other. Section lVa applies the analysis of§ III to astrophysical systems, and concludes that, when analyzing their structure and evolution, one must typically calculate the stress-energy tensor 2 pv and gravitational field 1 /i'"' with accuracies: E =(typical value of ii inside source) - (mass of source)/(size of source) . 00 1 Section !Va also concludes that the external gravitational field 2'i"' must typically be calculated to accuracy j(error in 2'i"")/2'i,00 j ~ E 2 if one desires reasonable accuracy in the radiative part of that field. Section I Vb presents a "postlinear" formali$m for calculating a system's structure and evolution ( 2 T"' and 1 h"v) to the desired accuracy; and § IV c derives a formula for the higher-accuracy external field ( 2 h"'), which contains the radiation. Section V shows how the resulting formalism, when applied to slow-motion systems, reduces to the standard "quadrupole-moment formali sm." We recommend that, before tackling the rest of this paper, the reader peruse § VI. That section summarizes our postlinear formalism and our formula for the external (radiation) field. The "guts" of this paper, in terms of complex calculations, reside in the Green's function manipulations of §!Ve. Our particular way of handling the Green's functions is motivated in Appendix A, and has bec::n influenced by the following papers: DeWitt and Brehme ( 1960) (exact Green's functions for scalar and vector wave equations in curved spacetime) ; Robaschik (1963) (exact Green's function for tensor wave equation in curved spacetime); John (1973a, b), Bird (1974), and especially Peters (1966) (Green's function s in weakly curved spacetime). Although these papers had much influence on us, our specific manipulations are so different that we have found it impossible to trace the details of that influence in our writeup. II. EXACT GENERAL RELATIVITY, REWRITTEN IN "WEAK-FIELD LANGUAGE" We begin by writing the exact, nonlinear Einstein field equations in an arbitrary coordinate system in the form (§ 20.3 of MTW; § 100 of Landau and Lifshitz 1962) Ht,i:..•t.a3 = 16rr(- g)(T"' + tt-:L) (l) where "L-L" means "Landau-Lifshitz," and where =9uvga" _ ga'g"" ; g"' =( -g) g"', ( - g) = -detJJg"vll = - detils"'ll Hf,':..'l 1 2 ' (2a) (2b) tf:'!_L :;: (lfur(-g)] - l{ga8,AgAu,U _ g"A,Ag8",ll + !gaDgAugAv,0 gpu,v _ (g"Ag,,.gB',pg"P,A + gBAguvS"',Pg"P,A} + K>.ug'"gaA, vgBu.P (2c) + i(2g"Ag"" - ga8if'")(2gvpgt1, - gl>l7g..)gY•,AgPl7·"}. (2) 7 No. 2, 1975 247 GENERATfON Of' GRAVITATIONAL WAVES The equations of motion fo r the material stress-energy tensor T''v follow directly from the fi eld cqoations (1) and can be written in the equivalent forms po;o = 0, [( - .r;)(T"ll + ti'.~ i.)J. o = 0. (3) [Here and throug hout this series of papers we use the notation and sign convention s of MTW; in particular c = G = I; gccn and g" 8 are the componen ts of the metric; commas and o's denote partial derivatives, Y," = oaY = eY/ ox" ; semicolons denote covariant derivatives with respect to the metric ga 0; <u1d our sig nature is ( - + + + ).] Now, and henceforth in this paper, we impose three restrictio ns on our analysis: (1) We confine attention to systems with "weak internal g ravity"- i.c. , to systems throug hout which one can introduce nearly Lorentz coordinates. (2) We confine ourselves to" isolated systems " - i.e., to system s tha t are surrounded by a region ("l ocal wave zone"), much larger than a characteristic wavelength of the emi tted waves, in which all waves an: o utgo ing an<l in which external masses have negligible influence on the gravitatio na l field . (3) W e restrict our analysis to the interior of the source and its local wave zone, and throughout these reg io ns we use nea rly Loren tz, asymptotically flat coordinates, specialized to satisfy the de Donder gauge condition. Mathematically, these restrictions state that the "gravitational field" (4) has the properties lil"vl « 1 everywhere , lli"'I ,. . , 1/r as r-+ co ,. (Sa) where r = (x 2 + y + z )t' , 2 2 2 (Sb) fi"v is devoid of incoming waves at r-+ oo, (Sc) fiuv,. = 0 (Sd) (de Donder condition) . With these restrictions, the exact Einstein field equations (I) take on the form o,Tzuv = -)67T(-g)112(pv -(- tf."._L) - ( -g)- ll2fjua,lz•O,ct • (6) Here O, is the wave operator for scalar field s in the curved spacctime described by the metric gc:o: D. =(_g) - 1120a[( _ g)ll2gao00 ] . (7) [Appendix A, which is best read after one has finished reading the rest of the paper, explains why we write the field equations in terms of D. rather than in term s of some other wave operator such as 0 1 = ri«ll(,',/\1 (the flatspace wave operator) or Di (the curved-space wave operator for tensor fi elds) .] E quations (2b)-(7) are the exact, nonlinear equations of general rela tivity for any isolated, weak-field systembut they are _:vritten in a very special coordinate system rather than in generally covariant form. _ Because lh"v l « I, we can express each quantity in our formali sm, except T''v, as a power series in h"v. When writing down s uch a power series, it is convenient to raise an<l lower indices of h"v with the Minkowski metric ri"8 = rJ"P = diag ( - I, 1, I , I): ii =liua , etc. (Sa) l t is also convenient to <lenne a "trace-reversed" gravitational fi eld 1zuv by 1iuv :=. fiuv _ -}hri"' , (Sb) and to raise and lower its indices, like those of fiuv, with the Minkowski metric. Note that equation (8b) implies (Sc) To derive the explicit power series expansions for g"V, g"', g,,., etc., one can proceed as follows. Equation (4) is the desired expansion for g"v. It contains only two terms : (9a) The expansio n for the metric determinant ( - g) is obtained by inserting exp ression (9a) into the second of equations (2b): ( - g) = -det 119"'11 = - det ll11"v - j/uvll = 1 - li + J [(h) 2 - h1111hao] + O[(Ji)3] . (9b) The contravariant components of the metric are then obtained by inserting (9a, b) into the first of equations (2b) : guv = ( _ g) - 112guv = ri"' _ (fiuv _ Viri"v) _ }lifiuv + h"'[(/i)'.! + 2i/aOj/ao] + O[(Ji)3] ; (9c) 8 248 THORNE AND KOVACS Vol. 200 <:!nd the covaria nt components a re obtained as the matri x inverse o f these co n trava ria nt components: g,.. = -ri. . + Ii,,. - -}n71". + li",)i''v - -}lili,.. + tr1J,.(li2 - 211" 0lia8) + O[(ii}1] • (9d) The co nn ecti o n coefficients l'"afl• which appear in the usual expression T"·:· = 0 for the equations of motion, are obtained hy in serting expansions (9c, d) into the sta ndard fo rmula r""n = tg"Y(g,a,8 + gy{l,a - gap,y) = t(li"a,O + li"o,a - liap'") - i(S")z,p + S"ii.a - Y/aali·") + O[(ii)2 J. . (9e) Sim il a rly, the sca lar-wave operator D. is obta ined by inserting expa nsions (9b, c) into e4uation (7), and using the d;: Donder gauge condition (5d) to simplify: Ds = 71"11 oafJfl - (li" 13 - -}fi71a 11 )8A1 + O[(li) 2 ] ; (9f) <!nd the components of the Landau-Lifshitz pseudotensor a re obtained by inse rting expansions (9a,b, c, <l) a nd the d;: Donder condition (5d) into equation (2c): tc:fl L-L = (167T) -l{ 1..,,all.,, li/\v , µ fiM ,v + n 2·1 ·1 Au n•o/la/\ .v /lPu . P _ ·1 Jt.u ·1 ( n., " /\n /lBv·" "I U\J fi"fJ .}\. + nD/\n fi<rv .oliuo • J ., "/U \I (9g) H enceforth in this p aper we shall regard g"•, ( - g). g"•, gl<V• r"ao.Os, and tr«.'!.. r. as shorthand notation for the i1:finire power series expansions, whose first few terms are shown in equations (9). Given these expansions, the full co nten t of general relativity is embodit:d in the equations of motion for the matter (lOa) and the Einstein field equations Dsh'" = - l67T(-g)ll2(T''v + l/;':L) - (-g) - ll 2Ji"".t/i•/J,a. (!Ob) H.:ncc:forth we shall not impose the gauge conditions ii"• .• = O; rather, we shall regard them as cons<.: qucnces of the equations of motion (lOa) a nd the field equations (I Oh). Ill. Al'l'ROXIMAT!O N FORMALISMS FOR WEAK-FIELD SYSTEMS1 Th:! formulation of genera l relativity e mbod ied in equatio ns (9) a nd ( 10) is a n excellent sta rting po int for d erivatio n:, of weak-field approximation formalism s. To get a formalism of d esired accuracy, one can simply truncate each infinite se ries appearing in equations (9) a nd (10) at the appropriate point. \Ve shall des..:ribe the acc uracy of a fo rma lism in terms of its "errors" (i.e., the deviatiom of its solutio ns from exact solutions of the exact eq uatio ns [9] a nd [IO]). In cli~cussing errors, we sha ll use the small dimensio nless pa ram<:ter £ (characteristic size of [iuv inside the system) . (I I) = If the system is a dynamically changing lump of matter with mass Mand linea r size L (e.g., a pulsating star o r a n exploding atomic bomb), then £ ~ M/L. If the system is several lumps with masses m and sizes /, separated by dis tances b » I (e.g., a binary star system or two stars flying past each other), then £ ~ m/b if o ne is interested only in the relative motions of the lumps, £ ~ m/l if one is a lso interested in the internal structure a nd dyna m ics of the lumps. W e sha ll charaderize e very weak-fi eld approximation formalism by two integers nr a nd 1111 • These "order indices" tell us the magnitude of the errors made by the fo rma lism :2 !(errors in T'")/T00 i ~ £"r (I2a) !(errors in li"•)/Ti00 i ~ £"n. (12b) This secti on is closely related to the Havas-Goldberg (1962) analysis of approximation. fo rmalisms for equations of mot io n of point masses. 2 Note that all of the IP 'I an~ :G T 00 , a nd consequently all of the lli"'I a re ;( li00 . This fa<.:t dictates the fo rm of equations (12). 1 9 ~o. 2, 1975 GENERATION OF GRAVlTATlONAL WAVES 249 For example, a formalism of on.l::r (nr, nh) = (I , I) makes fractional errors of order" in both the stress-energy r;;nso r and the:: gravitational field, while a formalism of order (2, 1) makes fractional errors ,;2 in T"" and" in ii••. Errors in h"V, when fed into the equations of motion (!Oa), produce errors in P"; and similarly, errors in T 11 •, ''hen fed into the field equations (!Ob), produce errors in fzu-'. This feeding process places constraints o n the order indices (nr, n11 ) of any self-consistent approximation formalism. The constraints are revea leel explicitly by an orderof-magnitude analysis of equations ( IOa, b) : Consider a weak-field system with characteri stic field strength e and characteristic length-time scale /. Below each term of equations (IOa, b), write the orde::r of magnitude of that term: (13a) (TOO//) (13b) {e/12) Equation (l 3a) shows that fractional errors e"h in fiuv produce fractional errors Enh + 1 in pv; i.e., enr ~ e"h + l; i.e., (l4a) Equation (13b)- together with. the orcla_:-of-rnagnitude field equation T 00 ~ <i//2-shows that fractionill errors e":- in T"v produce fractional errors EnT in h"v; i.e., e"h ~ enr; i.e., (14b) Equatio ns (14a, b) can be restated as the foll owing constraints on the order indices of any self-consistent ,::pproximation formalism: n11 = llr - I or (15) In other words, the order (nr, nh) of any approx imation formalism must be either (n, n - 1) or (n, n)for some integer n. Suppose that a specific system has been analyzed using an approximation formalism of order (n, n ·- I). Denote '.:ly c.T ~' (x") and <• _ 1/iµ •(x") the explicit expressions obtained in that analysis for the system's stress-energy tensor 2.!Jd gravitational field. From these expressions it is straightforwilrd to generate an" improved" gravitatio nal field _J;-' (x" ) with fractional errors "n· The key to_doing this is the structure of the field equations (I Ob): In these field ::q ua tions, fractional errors of order e" ·- 1 in 1zu v produce fractional errors of order f" in both O. and th e expression -167T(-g)ll2(Tl<V + tt',':L)- (-g) - ll2Jiua, 8 fivtJ,a· Hence, Ji"v(t, x) satisfies the differential equation (n,n-1) L - L) - ( -g ) - 112Tzua l ,e TzvtJ l ,a ] • 0 s n lil uv _- (n,n-1) [ - 167T( - g )lf2(T"v + 1uv (16) Here the prefix (n, n - I) means that a quantity is to be calculated, with fractional error en, using nT"• and <n- llfiuv. Th is inhomogeneo us, linear wave equation for nfiuv can be solved using the retarded scalar Green's function for c urved spacetime (DeWitt and Brehme 1960): , ~-1>G(.3>', Y") = [ (the retarded scalar Green's function for the curved spacetime with the metric <•-l>g,,v] · {17) of the (n, n - 1) approximation-a Green's function with fractional errors en The result is (18) Tb is paragraph can b_e summarized as follows: Any approximation formalism of order (n, 11 by equation ( 18) for ,.huv, becomes an approx imation formalism of order (n, n). 1), when augmented Special relativity and linearized theory provide a simple example of the a hove remarks: Special relativity is the 2pproximation formalism of order ( 1, 0) which one obtai ned by the extreme truncation process of setting Fzuv = O i:i equations (9) and (10) : ofiuv = 0 ' (19a) 10 250 THORNE AND KOVACS Vol. 200 The retarded scalar Green's function for a space with metric oS:~v = 11,,v is 0 1 ( •\ '" G(.rm' 7 ;r , •~/J) -= (4 r.) - 10~ 1·ct [ 21/00 xo')(·\• - · x"')] · (19b) • (H ere S,.• L i-; zero if :lf' li es in the causal pa st of !:Ji', and it i" the Dirac delta function o therwise.) Hence, equatio n (1 8)-by which o ne must augment specia l relativi ty i11 order to obt:1in a formali~m (1r order (1, 1)- h:.i.s the form J1 1 iv = 4 J 1 Pv(r?')S1•• t[~17 00 (x 0 - x 0 ')(x" - x 0 ')]d 1x' :...= 4 J iT"'(xol.: ~xx1 x'I, x') tf'x'. (20) or The resulting (I, I) formalism (eqs. [19] augmented by cq. [20]) is the "linearized theory gravity" (see, e.g., § 7. 1, box 7.1, and chapter 18 of MTW). Newtonian theory and the "quadrupolt:-moment fo rmalism for wave gcnaation" arc another example. Newto nia n theory is the weak-fidd formalism of order (2, I) which one obtains by not 011ly truncating each saies that a ppea rs in equations (9) and {10), but by also imposing the sl\)W-motion and small-stress assu mptions IT''/Tool ~ "' (size of system)/(characlt!ristic time sca le o f changes) ~ E (21) 112 • Equati on (18), by which one augments Newtonian theo ry in order to obtain a formalism of o rder (2, 2), has the form, when evaluated in the radiatio n zo ne /iiT(r, x) = (2/rfi'! l(t - r) = (gravitational radiation 11dd) . (22) Here 111 is the red uced quadrupole moment of the so urce, and TT d eno tes "transverse-traceless" part. This is the sta ndard wave-generation formula of the quadrupole-moment formalism; sec chapter 36 of MTW. IV. WAVE GEt"ERATION UY A WEAK-HELD SYSTEM a) Motii:ation Weak-field systems are of two types: thos..: with neglig ible self-gravita tional forces (rota ting laboratory rods, microwave cavities, etc.), and those whose inkrnal motion:- a re s ignifican tly influenced by scll'-gravity (pu lsating stars, binary star systems, etc.). For a system with negligible self-gravi ty, special relativity gives a fairly accurate description of the interna l motions; and, consequently, lineari zed theory [the ( 1, I) fo rmali sm obta ined by attaching eq. (18) or (20) on to special relativity] g ives a fairly acc urate description of gravitational-wave genera tio n. For most weak-field astrophysical systems, self-gravi ta tional force~ arc importa nt. In this case, when analyzing a sys tem' s internal mo tion s, one must use a forma li s m of orclt:r (2, 1); and wh en calculating the waves th ose motions genera te, one must augment the (2, I) formali~m hy equation (18), thereby raisi ng its order to (2, 2). 3 lf the system has slow interna l motions and weak internal stre::.ses, Newtonian theory [order (2, I)] will s ufiice for analyzi ng its motions, and the quadrupole-moment forma lism [o rder (2, 2)j will suffice for wave generation. Howeve r, for analyzing fast-motion syskms (e.g .. two stars fl ying p ast each o ther with high veloc ity and dt:flecting each other slightly-the rela tivistic bremsstrahlung problem), one need s unrestricted (2, I) and (2, 2) forma lisms. The objective o f the nex t two sections is to derive such formali sms. b) The Postlinear Formalism A weak-field formalism of order (2, 1), unrest rict..:d by any constraints on velocities or stresses, can be obtained by trunca ting equations (9) and (10) at the appropria te order: (23a) (23b) ig'" = 7J"' - i 1 /i"v (where 1 h"' = /i"' - J)i17"v), (23c) 1g11v = 17uv + ihuv, (23d) r"a11 = -H1'1" cr,o + ilt""·" - ilta(J·µ) • iD. = (11"0 - i'1"11)8,/J11, (23e) (23f) 3 In very spc:cial cases scconJ-urdc r gravitational forces may be .1s important, ror the system's m oti o n~. as first-order forces . An example is a radially pulsating, weak-tidd star wit h adiabatic indt:x very near 4/3 (Chand rasek har 1964); see also the discussion accompanying equations (61) below. Wht:n analyzing such systems, on.: needs forma lisms of order (3, 2) and (3, 3). 11 251 GENERATION OF GRAVITATIONAL WAVES No. 2, 1975 1 8 1tf,'!..L =-~ (l6rr) - {·}7/"J17Au /i"' ,o /i"",v + 1}A,.17"" )z"".v 1h ".1> -- (r!'"' 17uv /in•.1> 1li•1>,/\ + 11°·' 11tiv /i''".o Jillf',/\) + ~(2·'1"''170" - 11"n17"")(2'l]vp1/ai - 'l] p"11.,)Ji",/\ JiM,,.} ' 2T"',v = - iI'"av ~T"' - iI'"av 2T"", (23g) (24a) 17" 1\li"'.a.~ = - I 6tr 2 T"' . (24b) We shall refer to the formalism describc:d by these equatio ns as the "postlinear formalism." To analyze: a system u:<ing the postlinear formalism, o ne mu~t fir st spec ify the fun ctio nal dependence of the stress-ene rgy tensor 2 7"'" on the system's nongravitation':!_I variables (e.g., density, press ure, velocities, electromag nc:tic field tensor, ... ) and on the gravitational field 1 /i'"; a nd one mu st then so lve eq uations (24a , b) simultaneo usly for the sy~lc: m 's motions ( 2T'" accura te up to fractional errors ~ e ~) and for the gravi tationa l field (/1"' accurate up t o fractional errors ~ e). Paper Ill will carry out such a calculation fo r the motion of two stars or arbitrary relative masses and Ye!ocities, which fly past each other with la rge impact parameter. c) 111e Postlinear Wave-Generation Formalism Having calculated a system' s internal structure and motions using the postlinc:ar formalism , one can the n c.:alcul ak the gravita tional waves the system emits, ii'", by evaluating expression (I 8). In evaluating (18) o ne needs an explicit expression for the retarded Green's fum:tion 1G(f1"', f~;) associated with the metric 1 g.,, = 17 11 , + 1 '111 ,. In the next subsection(§ l Vc[i]) we derive 1G(Y'', 81'); th en in§ IV c( ii) we pl act: constr<_!.i nts on our system which simplily 1 G(Jil', P ); ·and finally in§ JVc( iii) we use 1 G(!1'', .OJ') to evaluate the wave field 2 /i "'. i) The Green's Function 1 G(.?'', .9 ') Wr: s hall obta in 1 G(PI'', f!I') by taking the weak-fi eld limit of the exact Green's function G(& ', &') for a spac.:e described by an exact metric ~11 ... The cxac.:t G rec:n 's function is formally rather simple, so long as the congruence ,Jf geode~ics that emanate fro m the sourc.:e point. ~' does no t get foc.:uscd so strongly along the future light cont: of Y ' that geodesics cross. Henceforth we shall assume "no cros~ ing or geocle~ic.:s o n the light cone." Late r (eqs. [·l8], [48'], [48"] below) we shall exam im: the constraints plac.:ed o n the radiating system by this "no-crossing" a ssumption . Dt:Witt and Brd1me (1960) have derived the exact Green's function G(o/', ({I) for the case of no crossing . Thei r Green's function consists of a "direct part" and a "tail., G(9 ', .9') = GdlrocL + G tilll • (25) The direc.:t part is no nzero on ly if :!!' lic:s on the future light com: of o/' [denotc:d J' (:JP')l. Ily virtue or the " nocrossing" a s.-;u mption, when !1" is n ear } +(!if'') there is a unil[UC geodes ic.: from :/'' t o [!fJ with a unique squared lc.::ngth .Q(..1:', .0") ;:=.; ("World function," see Synge (1960)) = t( - 1 for timelike geodesic, + I for spacclike geocbic)(prop er distance a long geoc.ksic) 2 = a in n otation of DeWitt aml Brehrne (1960). (26) Because } +(::J' ') is characteriLecl by n = 0, Gdlrect must have the form (27) where ore t is the Dirac delta function o n and nc:ar } + (~~'),and is zero o n and near the past light cone [1 - (& ')]. The quantity ~(9', 9') is an amplitude fac.:tor which would be unity in flat spacetirne, but in cur ved spacetime is given by fl.(& ' & ) = -~1et ll o U/ox"ox · 11 . ' jg(::?')g(.0"')11 / ~ 2 0 (28) We sh all use an expression for the tail different f'rom, but equivalent to, tha t given by DeWitt and Brehme. To derive our expression we in sert equations (25) and (27) into the wave equation LJ,G(&'', &) = - [g(.:r')g(&> ')J - ll-IS(x0 - x 0 ')8(x 1 - x 1 ')8(x2 - x 2 ')S(x3 - x 3 ') =: - The result i ~ [g(.0")g(&'')) - 114 01(X - x') . (29) 12 Vol. 200 THORNE AND KOVACS 252 where 15' and o" are the fir st and second derivatives of the Dirac del ta function for.<};' on and near } +(fll'), and are zero for fl' on and near J - (:1''); V is the 4-dimensional gradient ojJerator; and vK is thl.! covariant dl!ri vative along the 4-vector (30b) K = vn. (Here and below we suppress the subscript" rct" on 8(0).] We then manipulate expressio n (30a) using the relation s QS"(Q) = -2o'(n), (VD) 2 ~ 20, o.n - 4 = GJt.- 1VKt.. (30c) (The first of these is a standard identity for Dirac delta functions; the second a nd third are eqs. (1.11] and (1.63] of DeWitt and Brehme 1960.) The result is 0.ctau = -(47T) - 1(0,Ll1'2)o(G). (31) We then use relations (30c) and the relation Ll(.9', &') = l to rewrite this in the form (32) Equation (31) tells us that ctan jumps from zero outside the light cone to a finite value in side the cone, witho ut having any singularities on the cone. Equation (32) allows us to write (restoring the subscript "ret") if & ' rt r-(&) = -(47T) - 1 f (33a) [In Ll(.9>', IJl'")J.".[Q(.9'', IJl'")]·""/)'rot[Q(:J'', &'")]G(.9'", .9')[ -g(&'")] 112 d~x" if .9'' E 1- (f!l') . ' (33b) Here 1- (.9') mean s "the interior of the past light cone of .'/'" ; and condition (33a) suppresses the unwa nted lightcone part of (33b) [i.e., suppresses (47T) - 1 ( 1 - Ll 112)/5(0)]. Equation (33) is the form of the tail which we shall use. This form was suggested to us by the work of Peters (1966). \Ve now specialize the above equations for the retarded Green's fun cti o n to the case of a weak gravitational field 1 g~v = 71,.v + 1 hµv. beginning with equation (26) for the world function. Let,\ be an affine parameter alo ng the geodesic linking & ' to &': <G'( .\) =geodesic with coordinates g"(,\) ; "6'(0) = &' , l/S'(l) = .9' , 0 :::; .\ ~ 1. (34) The equation (26) can be rewritten in the form (cf. Synge 1960, p . 47) n(.9'', &) = i tgµv(dgµ/d>-)(de/d>-)ci>- . (35) '(/ The right-hand side is actually an action principle for the geodesic equation (cf. MTW, box I 3.3). Therefore, if we evaluate the integral along the " straig ht line" (36) (see Fig. I), which differs by a fractio nal amount of O(e) from the true geodesic 167(.\), we will make fractional errors in Q of O(e2 ). Such errors a re acceptable in 1 G([:JP', & ) , since its fractional errors are also O(e 2); cf. equation (17). The result of integrating expression (35) along the slightly wrong curve 0 <C(.\) is iD(&'', &') = 0 U(&', f?i') + y(&', f?i'), where =-!Xa X 7/a11 , y(&'', &') = }Xa X J iha11d.\, oO(f!l'', :JI') 11 8 (37) (38a) (38b) o'if (38c) Equation (37) is the desired expression for the world function. Turn next to the amplitude factor ~(&', &'). Either by direct calculation from eqs. (28), (23b), (37), (38), and (812), or by invoking equati on (95) on page 63 of Synge (1960), one arrives a t the expression (39) 13 No. 2, 1975 253 GENERATION OF GRAVITATIONAL WAVES p' (al (bl FIG. 1.-The points 9 , G'', tr used in evaluating the postlinear Green's function 1 G(.'.1", 9) and in calcu lating the postlinear gravitational-wave field 2fi" ' (fJ' ). Pa rt (a) shows the parame trized straight-line curves 0~(,\) and 0'6'"(>.) linking G', ?, and .9* ; part (b) shows the 4-vecto rs X", x•·; and X• linking them. Here a(.o/'', &) =!XaX f R., .\(I - ,\)d,\, 8 1 (40a) 0 o'lf where 1 R a11 is the Ricci tensor, accurate to first order in 1 huv: iRa8 = -t ihao,/ · (40b) In equation (39) we have simplified no tation by using a prime to deno te quantities evaluated at 9 ', i.e., 1 /i' ;::::; 1 '1(9 ') while 1 /r = 1/i(.o/'). Henceforth we shall reserve primes for this purpose-except that o' and o" are still deriv:itivcs of Dirac delta functions. Turn next to the" source term" (In t.) , a· ~}a " for the tail (eq. [33]). The tail itself is of 0(£) co mpared to the direct part of the Green's functio n; therefore we can permit fractional errors of 0(£) in the tail-which means we can use the zero-order value of Q·a" in the source of the tail : [on(&', .9'")}·"· = x"" - x"' =xa·. (41) By combining this with equation (39) and by using equation (B7) of Appendix B, we bring the source of the tail into the form (42) where {3(&', &") "" xa· X 8 " I iRap"A2d,\ (43a) o'lf" and 0rf" is the "straight line" from r!Jl' to r!J!" (see F ig. I) 0 <(?": ~a= xa' + ,\X"". (43b) Turn, fin ally, to the propagator G(r!Jl", & ) and the volume element ( ·- g") 112 d~x" which appear in equation (33) for the tail. Because the ta il is of 0(£) compared to the di rect part of the Green's functi o n, we can ignore all curved sp ace corrections in the amplitude of the propagator (but no t its phase), and in the volume element: (44) All of the pieces for the first-order Green's function are now at ha nd. By com bining them (eqs. [25}, [27}, [33}, (39}, [42}, and [44}) we o btain the following result : 1G(Pl'' . ' r!J!) = 1Gdlrect + 1Gtl\11 ' (45a) 1Gdtrect = (4rr) - l[l + a(r!Jl', &)JSret[iQ(&'', .9')}, iGt"u = 0 = - (47T) - 2 J{3((1,P', &") Sret[1f2(&'', .9'")Wret[1Q(.9'", .9')]d 4 x" (45b) if .9'' ¢ 1-(9 ) if 9' c: 1- (:JP). (45c) 14 Vol. 200 THORNE AND KOVACS 254 JJere 1 0(Y ', !11') [and similarly 1 D(.'9"', Y"') and 1 0(.Y", :i')] is given by equation s (37) and (38), et(.9', .11') (the "focusing function") is defined by expre~s ions (40), and {3(.1)', ::!'") (the "tail generator'') is defined by expressions (43). ii) Co11s:raints Designed to Simplify the Green's Function Expression (45) for the Green's function is valid o nly if geodesics emanating from Pfl' fail to cross o n and near ) "(?J"). Crossing would be caused by gravi tational focu sing: and at any crossing point, the exact amplitude factor ~(3'' , !::I) would diverge. Thus, the criterion for no cro~sing is finit..:ness of~ along l +(JI''). Consider our first-order expression (39), (40) for 1 ~. Evaluate it in the mean rest-frame of the source, with coordinates centered on [~' so that X" = rn", n° = I , n = (unit spatial vector pointing from & ' to ;'7J), (46) r = (spa tial dista nce from .?i'' to Pfl) , rf r =(frac tional distance from & ' to g>) ; and invoke the first-order field equation 1 Ra8 = (1 Ta11 - h«s 1 T). The result is ,\ = 1 ~(&'', .9) = 1 + 2a, a = t f (n"n8 1Ta0 )i'(I - i/r)di. (47) This expression for 1 ~ can never diverge if the sourcl! is bounded, because once the integra tion point r gets outside the source, 1 7;, 11 vanishes and 1 ~ stops increasing. However, if the focusing function a approach~s unity inside the source, then second-order and higher effects will come into play. As one moves out into the vacuum beyo nd the source, those second-order effects will be essentially tho~e of the "focusi ng" or "Raychaudhuri" equation; they will produce a divergencl!. Thus, the constraint CONSTRAINT : a(:J'', .9 ) « I for all .' ¥' and (48) is necessary for the validity of the first-order analysis, and simultaneously protects us fro m "geodesic crossing." ~·or a system that is roughly homoge neous with mass Mand linear size L, equation (47) gives a ,..., (Al/L) ,..., "« I ; (48') so there is no problem in satisfying the constraint (48). However, for a highly inhomogeneous system (lumps of mass m and size /, separated by distances b » /),and for rays originating in one lump a nd passing through another, equation (47) gives ct ,..., b(m/13)1 ,..., (b/l)(m//). ln this case the constraint (48) is significant: it says that to avoid too much ray focusing, the lumps must not be too far apart: (48") The Green's function· (45) would be much easier to use if, througho ut it, we could replace the first-order world functio n 1 U by its zero-order approximation 0 Q = ·l'rJ,..X" XV. Let us examine 1 fl (eqs. [37] and [38j) in the rest frame of our source, for points Y' on or near l + (.~'): 1 D{&'', .9) = 1(X0 + X)[-X 0 + X + 2y([~'. &')f(X 0 + X)]::::::: X(-X 0 + X +Ms), where X =IXI = (distance from source to field point), M s= y(.9'', & )/X = ("Shapiro time delay"). (49a) (49b) (49c) For field p o ints .9' far outside the source, the dominant contribution to the Shapiro tim e delay is the asymptotic "l/r" field of the source. It produces a huge delay of A =2M In (X/L) = (Shapiro time delay due to asymptotic field of source) , M = (mass of source) , (50) L = (characteristic size of source) . This delay is time independent and is independe nt of where inside the source f?J'' is located (aside from a neglig ible piece of size ,..., 2lvfL/ X); therefo re its o nly effect o n the radiation is to delay the arrival time at a given radius. Henceforth, for ease of calc ula tion, we shall remove this con stant de lay from the argument of our Green's function. 15 No. 2, 1975 GENERATION OF GRAVITATIONAL WAVES 255 We can always reinsert it at the end of the cakulation if we wish. With this constant delay removed, we can rewrite 1 n as iO(&'', r3') ~ X[-X 0 + X +(Ms - A).ll'·,?] ~ oD(~ ', &!') + [y(& ', !~) + AXuUµ] for (!)' inside the source and O' far outside it , (51) U" = P,,fM = 4-velocity of source; X " Uu = -· X . (52) where The remaining "internally produced" delay b etween fl'' a nd O', b.t8 the total dday between two internal points &' a nd JI" : (Ms - A)11·rP ,...., (t.t 8 - A)y ·y ,...., (Ms)fN~· ,...., J A, is of the same order of m agnitude as '1 dr,...., M for homogeneo us source 1 00 across source ,...., m In (bf I) for Jumpy source. (53) Henceforth we shall assume that this internal time delay is small com pared with the characteristic time scale o n which the source changes-·i.c. , small compared with the characteristic reduced wavelength i\ of the radia tion emitted, (54) CONSTRAINT: (Ms)1aterna1 ,...., m In (bf/) « i\. [example : If i\ is I 00 times larger than the Schwarzschild radius, 2m, of a lump, then bf I can be as la rge a s exp ( 10) ~ 2 x 10~ without causing problems. Another example: If i\ ): h (which is the case for bremsstrahlung), and if I » m (which is required for fields to be wea k), then the conditio n b ): I (separation of lumps bigger than size of lumps) guarantees that constraint (54) is satisfied .] The constraint (54) a ll ows us to expand our delta fun ctio ns S(1 Q) in powers of the internal time delay. Di sca rding terms that are quadratic and hig her-order in (b.t8 ) 1nternai/i\, we obtain for the Green's fu nction (45) Y') = 1 Gdlrect + l Gtall ' (55a) 8 8 11 iGdlroct(.9', JI>) = (477) - {oral1XaX 7]a/l) + a(.?i'', 9")8rel }XaX 17,,11) + [y(Y'', .o/') + AX"U,,]8~eliX"X 71,,8)}; (55b) iGt"n(& ', # ) = 0 if .9 '' ¢ J-(fJ/) 1 G(.9', 1 = - (477) ·- 2 I {3(& ', &'")o' rol }X". x r 11c:a)OraltX" Xn7J«8)d4x" (55c) Jn these equations (56) sec Figure 1. Equations (55) are our final form for the scalar Green's function in a space with linearized metric 1 g,, ., = 71, , + 1 huv· T hi s Green's function has fractional errors !(errors in 1 G)f1GI ,...., M aximum of {E 2 , rtE, [(ut3 ) 1ntorna1fi\]e} in general, 2 ,...., E for most sources ; (57) and it has been stripped of its asymptotic time delay (eq. [50]). iii) The Grm:itational-Wm:e Field , fiu• By inserting expressions (55) for 1 G((:f.i'', .c!Jl) into equation (18) we obtain the following expressio n for the gravitatio nal field far outside a weak-field source: }iµv = } i't,v + i i't.v + 2T11fr.. + }iW + 2li!ft ; (58a) JOret (-!X"X 11aa)2T ' v(.9'')[I - ili(&')]d x', ii~v = 4 J a(&', .9')8,etUX" X TJaohT"' (P ')cf" x', ii¥ri = 4 J [y(r3"' , &) + AX"Ua]o'r•tUxaxn17,.8)2T"'(.9'')d x ', zii~vv = 4 J oret(}X"X 71a )[itC.L + (1677) - )z"",a hv". 1at5"·d x', }i'Dv = 4 8 4 1 1 8 1 8 }i~.r, = ( - lf77) 1 11 1 f)y·e ( r-w> 4 0 {3({31J', .9'")8' r•t(}X". x o-11,,11 )8mUX" :p.,.1«a)~Pv(Y'')d~x·d 4x'. (58b) (58c) (58d) (58e) (58f) 16 256 Vol. 200 THORNE AND KOVACS Here .0/1' and .0/1" are source points with coordinates x"' and ,r;z· (cf. Fig. I); the field point !3' has coordin ates xa; 11 2 T v, 1 tf. ~ 1 ., and 1 fiuv are the stress-energy tensor, the pseudotensor, and the gravitational field obtaint::d by a postlinear analysis (cqs. [23] and [24)); 8ret is the Dirac delta functi on on the future light cone of the source and zero on the past light cone; 8'r•t is the derivative of 8ree with respect to its argument; X" , J<a, and xa· are (59a) ex, fJ, and y are defined by integrals of the first-order Ricci tensor 1 Ruv and of the metric perturbation 1 li11 v along the straight line between two points ex(&',&) = !X" X 8 L fJ(9'', (}'") = xa· xr fo iRap(x"' + °AX"")>. d>.' 2 (59c) y(9'', fi') = tXaX 6 f ihao(x"' + AX11)d>. ; (59d) -1 1Raix11' + ,\X"),\(l - >.)d>., (59b) 1 - J\Xaua is that portion of y which is produced by the asymptotic, l/r, external field of the source - J\XaUa = J\( -X«Ua) = (Shapiro time delay produced outside source) x (distance from source point to field point) (59e) (see § IV c[ii] above); and &' E 1- (&) means that the integration (58f) is performed over field points .9 '' that lie inside but not on the past light cone of & . Each piece of the distant gravitational field }i•• has its own physical origin and significance: The first piece ii'bv is the "direct field." It is produced by the stress-energy 2 pv and propagates as though spacetime were flat. It includes the zero-order, nonradiative, " I /r" field of the source, and also that portion of the radiation produced "directly" by the source's motions. If the internal gravity or the source has negligible influence on the source's structure and evolution, then all other parts of 2 Tz•• will be negligible compared with the direct fie ld ("linearized theory"; cf. cq. [20] and the associated discussio n). The second piece 2 /ij!.• is the "focusing field." It is the amount by which the direct field is augmented due to focusing as it passes through regions of nonzero Ricci curvature (nonzero stress-energy). The third piece 21i¥P. is the "transition field" (first discovered in the equations of general relativity by Chitre, Price, a nd Sandberg 1973, 1975; analog of " electro magnetic transition radiation," Ginzburg a nd Frank 1946). It is the: amount by which the direct fidd changes clue to Shapiro-type time delays within the time-varying source. The fourth piece 2'iv/ is the "whump field." It is the field generated by "gravitational stresses" 1 t{!, ~ L + (l6rr) - 1 1 /i•0 " 1 /iv".o· We have given it the name "whump" because in our minds we have a heuristic image of gravitational stress€?s linking various pieces of the source, and going" whump" (i .e., quickly rising in stren,gth and then quickly falling) as the pieces of source move past each o ther. The final piece 21i¥t is the "tail field." It is generated by the direct field in tho se regions where focusing has deformed the geometry of the direct wave fronts. · Although it is useful, heuristically and in calculations, to split 2 Jiu v into these five pieces, one should not attribute too much physical significa nce to each individual piece. For example, no individual piece satisfies the E instein field equations or the de Donder gauge condition. Howeve r, the five individual pieces combine in such a way that their sum does satisfy the field equations and gauge condition; see Appendix C. V. SLOW-MOTION LIMIT OF THE WAVE-GENERATION FOR~ULAE Consider a weak-field system which has slow internal motions a nd weak internal stresses. Characterize it by the following parameters: L =(size of system) ;\ =(characteristic time-scale of system) = (reduced wavelength of radiation) M = (mass of system) v = (jT '1fT )max = (maximum internal velocity) 0 S2 00 =CIT"j/T° )malC = maximum of (stress)/(density). 0 (60) 17 GENERATION OF GRAVITATIONAL WAVES No. 2, 1975 257 Chapter 36 o f MTW derives the quadrupole-moment formalism for gravitational wave generation under the following assumptions (eqs. [36.18] of MTW) L/A « l which implies v « l ; (6la) M/L « L/A, S 2 « L/l'.. (6lb) Constraint (6la) is the standard slow-motion assumption-the only assumption truly necessary for validity of the quadrupole-moment formal ism (see Paper II). Constraints (61 b) say that the motion mu st not be too slow if a weak-field calculatio n is to yield the quadrupole-moment formalism. in terms of the characteristic frequency w = 1/A this "not too slow" assumption says w 2 » (M/L)(lvf/L3 ) , w 2 » S 2 (S/L) 2 • (61 b') A violation of these assumptions occurs, in dynamica l systems, only when the gravitational and stress forces counterbalance each other so precisdy that second-order gravity, 2 fiµ•, can affect the motion significa ntly (cf. C handrasekhar 1964). In this case an analysis based o n the postlinear approximation cannot possibly give a correct description or the radiation. It is instructive to see how the postlinear radiation formulae (58) of this paper yield the quadrupole-mo ment formalism, wht::n applied to a system satisfying constraints (61). We begin by combining the direct and whump fields (58b, e) and then breaking them up again, differently: (62a) (62b) f 2fif:;°&2 := (1 f4TT) o,.tC-}X" X 017afl)[1hµp,a ifiM,PJat 9.d 4 x' . (62c) We thl!n evaluate expression (62b) in the rest frame of the source =4 fjuv 2 ow1 J ( - 1g)(2P• + 1tCL)rot d~x' . Ix - x'I , (63) and by carrying out the analysis of 4 MTW § 36.10, we bring the spatial transverse-traceless part of thi s field into the form (64) H ere l 1k is the "reduced quadrupole moment" of the source, and ff;! is its transverse-traceless part. This is the standard quadrupole-moment formula for the radiation field. An order-of-magnitude analysis shows that all other parts of our expression (58) for 2 fi1k are negligible. In particular, by using the following relations valid for the source's interior 1 fjOO,..., MJL , l fia/J ,J ,..., · fjall/L I ' fi"'M/L, (65a) as well as the relations a,..., M /L, (y + Axau"),..., rM, (65b) we obtain for the ratio of each other part to the" DWI " part (eq. [64]): l2fib\v2/C2ilL\1rrT1,..., (1)(v + ft:)2 « i, (66a) (ft:) « 1, (66b) (ft:) « l . (66c) j 2 /i~k/(2h£kw1?TI ,..., ( 1!~) !zlif~/(2fibkw1)TII ,..., I2lift/C)ibkw1)TTI ,..., ( ~) •Note that (- ,g)(2T"' + 11/'.". 1) here plays the same role as T •• + t"' in MTW. The key properties which they share a re (i) vanishing coordinate divergence; (ii) same role in retarded integral for 2li"'. 18 258 THORN E AND KOVACS Vol. 200 VI. SUMMARY Of OUR "PLUG-IN-AND-GRINL)' fORMALIS~l FOR WAVE GF.l"ERAT!ON Our postlincar formali sm for wave genera tio n can be summarized as fo llows: R egime o.f z;a/idity . The forma lism is va lid for any system satisfying these constr:iints: (i) The gravitational field must be weak everywhere: IJi~ vl « 1 everywhere , (67a) and the source must be isolated (see discussion preceding eq. [4J). (ii) Gravitational and nongravitational forces must not babnce each other so precisely as to enable second-order gravity to influence the syst-:m's motions signific:intly. (iii) The source must not focus substantially light rays emitted from within itself. ~lathcmatically this constr:iint says ia(.9'', 9')1 « l for (!J' any event inside the source, 9' any event on the future light cone of &'' , (67b) where C! is defined by equation (5%). For further discus~ion of this co nstraint, sec the first half of§ IYc(ii). (iv) The "Shapiro time delay" for light propagation within the so urce must be small compared with the characteristic timescalt: A for inkrnal motions of the source. Mathematically this constraint says that in the mean rest-frame of the source (67c) (Llts)1ntoroal y(.fl'', .9)/lx - x'I « A· = Here x and x' are spatial locations of events &' and r!J'' that lie inside the sou rce, .9' is on the future li ght cone of £11', and y is defined by equation (59d). For further disc ussion, see the second half of§ l Yc[ii]. Calculation of the system's motion. For a system satisfying these con~tra i nts one calculates the internal stru cture and dynamics by using the postlinear formalism of§ IVb (eqs. [23) and [24)). Calculation of tlze distant field. To calculate the g ra vitation::il fidd li"v in the radiation zone, far from the source, one takes the result of the postlinear analysis, plugs it into equations (58) and (59) , and grinds. In Paper 111 we shall use this l'ormalism to calculate gravitational bremsstrahlung radiation. APPENDIX A WHY USE THE CURVED-SPACE SCALAR-WAVE OPERATOR? 'I In laying the foundations of our analysis (in and near eq. [6]) we write the Einstein field equation s in terms of the curved-space scalar wave operator D •. We choose to do this because the obvious alternatives (the flat-space wave operator O, or the curved-space tensor wave operator 0 1) would ultimately lead to complications or dangers in our analysis. The flat-space operator Dr treats the field propagation from the ou tset as though it were o n flat-space characteristics (straight coordinnte lines). Because the true characteristics suffer the Shapiro lime delay which involves a logarithm of distance, the use of Dr would lead to logarithmic divergences in the radiative field at large r. If one were sufficiently careful, one could remove those divergences without serious crror--but that is a dangerous enterprise. Even if one succeeded, one would be left in the end with the interesting effects of focusing, time delay ("transition radiation"), and tail all lump-:d into the whump part of the field. We prefer to keep them separate. Consider next the curved-space tensor wave operator (Al) (cf. MTW cq. [35.:.64]). Because the true propagation equation for very weak gravitatio nal waves on a curved background is 11zuv = 0, it is tempting to formu late our analysis in terms of 0 1 rather than D •. By using 0, we push into the "whump" part of ii"v an important physical effect: the curvature-ind uced rotation or po larization. In effect, part of our whump field corrects the error in our direct field's unrotated polarization. Had we used Di rather than Os, pola rization rotation would have shown up in§ IVc(iii) as a separate piece of the radiation fie ld. The ten sor wave operator has a disadvantage which, for our purposes, outwc:ighs the above advantage. Suppose that one constructed a tensor Green's function for 0 1 n (/\2) or for any other wave operator with the form Dotherh'" = fiuv; a" + (any "background" field)"v c:oii"8 • (A3) 19 No. 2, 1975 259 GEN ERATION Of G RAVITATIONAL WAV ES That Green's function would have a first-order tail 1 Gr;fi' 6 "(&', &')with "sources" f3'"""8 ' involving the P-.iemann tensor (cf. cqs. [43a] a nd [45c]). Such a tai l wo uld orif!inate everywhere on the light cone of . f , wht.:rt:as the tail 11 1 Gt" fo r o ur scalar Green's fun ction or iginates on ly o n rays th at have passed th rough matter. In practical calculatio ns in vo lving lumpy so urces-sec, e.g., Paper 11 !- that tail would be as di fficu lt to calcula te a<; the ,,·hump part of the fide\ . We preft:r our scalar tail because of ib greate r simpl ic ity. By using O. we dump all serious calculational complexities, for lumpy sources, into the whump part of the fi~ld . APPENDIX B LIN E-INT EGRAL IDENTITIES The weak-field Gree n's function 1 G(.o/', .9) used in this paper involves tlm:e integrals a, {3, y along "~traight lines." ln this appendix we take the line of integration to be (Bl) The three line integrals are 1 a= f xv 1 1 y := tX" x•· (B2) 1 1zuvdA, t X" X'' R.,vA(J - A)dA, (B3) R.,, A2 dA, (B4) 1 1 f3 ::: X" 1 where 1 '1.,, is assumed to satisfy the de Da nder condition /i.,v•v = ihuv'v - } ih,u = (85) 0 and the Ricci tensor is therefore given by (136) and where th e index notation used is that of a Lore ntz frame in fiat spacetime. Below we list a numher of useful identities lin king the line integra ls a, {J, y, their derivatives a t poi nt ~>, a nd the va lues or 1 '1u v and 1 R., v at .rJI: (B7) X 0 a,/J = -}{3 ' X " Xcra, "a = -{3 + }X11 xa 1 R 0 a, (BS) X"y,0 = y + }X"Xa1 h0 a , (89) X 11 xaY.oa = X " xal ,,IJIJ + -}X 0 x aX'1 '1pa,t ' (BIO) (Bl I) Similar idc:ntities invo lving derivatives at PJ>' and mixed derivatives at {J' ' a nd &'can be derived fairly easily. For example, (Bl2) = = where 1 '1 1 /t(.o,JJ) and 1 '1' 1 h(PJ>'). The deriva tio ns of these idc:ntities are quite straightforward. The nccc:ssary techniques are illustrated by the following ch:rivation of identity (B7) : By d ifferenti a tin g definiti on ( 133) a nd making use of equations ( 131 ), we obtain a .0 = X " = X" = X" f f f R.,11 A( I - A)d,\ + -}X" xv f (cRuvfc~")(cgaj(Jx"),\( I R., 0 .\( I - A)c/,\ + -} X"X" f Ru0 A(I - A)dA + -}X11 X• f (Ruv,a)(Ao"0 )A( I - A)cl1\ R 11 •• 0 A2 ( l - A)dA. - A)d,\ 20 260 Vol. 200 THORNE AND KOVACS Here R,.v,o =-• c:R,.v fug 0 is the derivative of Ruv at the integration point <C(,\). Whe n multiplit!d by XP this cxpre:<>sion gives XPcr.,P = X",yv = X" xv f f Ruv,\(l - ,\)c/A + -}X"Xv f (Ruv,oXP),\ 2 (1 - ,\)d,\ R,,v,\(I - ,\)c/,\ + JX" xv s: (dRuvfd,\),\ 2 (1 - ,\)d,\. Ily integrating the last expression by parts we obtain XPcr.,o = xu x· f R,.,[..\(l - ,\) - 1(d/dA)(A2 - A3 )]d,\ = -}XU X " f Ruv'\ 2 d,\ = tf3 ; QED. In this case the integration by parts gave no endpoint terms; but in other cases (eqs. [B8]- [Bl2]) nonzero endpoint te rm-; are obtained. Jn manipulations of our weak-field Green's function 1 G(9', .9') and of our second-order gravitational field 2 fiuv (see, e.g., Appendix C) two other identities are useful: {JS' = (a8),/ - a,/8, (Bl3) (yS' + cell)./ = a,/o + 1 fiP 0 S.1> 0 • (B14) Here o is the flat-space propagator bt:tween & ' and &' 0 == sc.l}XP X"ri(JO), (Bl 5a) which is related to the 4-dimensional Dirac d elta function (Bl5b) by (BJ 5c) and S' is the derivative of the propagator (HI Sa) with respect to its argument. The absence o f primes o n indices and on h's in (813) and (Bl4) indicates that all d eri v:1ti ves and endpoint terms are taken at &'; none are at .'l''. The identities (813) and (Bl4) can be derived with some labor from the identities (87)- (Bll). APPENDIX C PROOF THAT THE "PLUG-IN-AND-GRIND" FORMULAE FOR 2 Ti"' SATISFY THE FJELD EQUATIONS AND GAUGE CONDITION Here we briefly sketch the proof that our second-order gravitational field (eqs. [58]) satisfies the second-order Einstein field equa tion (eqs. [16] with n = 2) and the de Donder ga~ge condition :~r,~v_v = 0. As part of our proof we sha ll derive expressions for the amount by which each piece of 2 1zuv fail s, by itself, to satisfy the field equa tion and gauge condition. A preliminary step in our proof is to rewrite the" tail " and" transition" fields (58f) and (58d) in new forms. Although expression (58f) for the ta il seems optimal for practica l radiation calculations, the restriction & ' E 1- (fJJ') makes it nasty for formal manipulations. To get rid of this restriction we take expression (Bl3) for {38', in it we replace .9' by .o/l", and then we insert it into expression (58f). The result, r:,,, = (I/ ) 2'11'L 7T ff [ "' > ~ Ci1'")] ,o• p-~0 r ct(I·2 X""Xlr 7'Ja8 )~°rot\'t ( LX-a'\V8 TJ.,/J ) 2 r uv(0. 7>')r./·l X "C./1X , , a (0 .:r .:r (Cl) is an expression which gives the same value for 2'i¥1. whether one imposes or omits the restriction :?!'' c= 1-(l!I'). One way to see tha t (Cl) is oblivious to the restriction :'/,fl' E 1- c;::J) is thi s : Take the so urce equation (31) for the ta il of the exact curved-space Green 's function; calcul:ite its lowest-order form Gt"ll./ = -(4rr) - l[cc(fJJ'', 9 )],PoS(1X"XPTJap); invert this using a flat-space propagator; use the resulting Gt" 11 to calculate 2 Ti¥L the result will be expression (Cl)and nowhere in the derivation did one need to impose the restriction r!fJ' E 1- (& ). 21 GENERATCON OF GRAVITATIONAL WAVES No. 2, 1975 261 Expn:ssion (58d) for the transition field involves a " time-delay function'' y(:ll'', Y') from which the loga rithmic, "external time delay" A(- X "Ua) has been remo \ed . A straightforward subtral'.ti0n of the external time dday is well suited to practical calculations, but poorly suited to formal manipulations of /1"'. In the formal manipulations of this appendix we shall perform the truncation in a "smoother" manner : We surround the source by a (hypo thetical) <.:loud of negative-mass material, with total mass, - kf, equal in magnitude to that of the source, +M. We put the cloud far enough from the source (e.g., at radius 2! ~ lOOL) that it is \ ery diffuse. and thus contributes negligibly to the line integrals u and {3; but near enough that the Shapiro time delay 2 .\/ ln (.fl'/L) in going fro m source L to cloud.Pis small compared with the time scale;\ of the sourl'.e·s internal motion s. The cloud automatically removt:s the external Shapiro time dt:lay; no artificial trum:ation of i' is needed. The ~ecl1nd-orda gravitational field is then given by equations (58) and (C l) ei·ery1r/ien.' (inside the source and out), except that \\ t: must remove the artificial truncation from (58d) : (C2) Turn now to the proof that our second-order field satisfies the second-order Einstein fidd equation. We begin by applying the first-order wave operator (C3) to each of the five pieces of our second-order field. By applying 1 0s to the direct fidd (cq. [58b]) and by using equation (Bl5c) we obtain 1Ds2lil'i• = - 16rr(l - -} 1h)2P• - 4Ji0 ao0 oa f Orel}X"X8 7Jao) 2 P •(f )d 4 x'. (C4a) Dy applying 1 0, to the whump field (eq. 58e) and by using (BJ5c) we obtain (C4b) By applying 1 Ds to the tail fidd (eq. [Cl]) and by using (B15c) we obtain l '" -.... - 4 D s2 IiITL J[ ';;µ •• ;;'}r/J)] .o 00:, O7/Cl(J) 2 T"V(r,Jl') a ( er Orct(.lX"X 2 · f ( /I,.' •' • (C4c) By applying 1 0 , to the focusing field (58c), and by using (Bl5c) and the relation et(d', d') = 0 (cf. eq. [B3]) \\e obtain l - 4 D s2Iii•• F 4 - J [a(q,4' !JI>)] "o ' .D , 1 07 rot( -Lxax Z 1a fi ) 2 T'"(Y ')d· x ' 8 7/ )]· 0 T""(d'')c/"1 \-' + 8 J. [cl(Cf'' ' ' •CJ!)] .fJ [ort!L(lX"X .? ad 2 · • (C4d) By applying 1 0, to the transition field (C2), a nd by using (814), (BJ5c), and c1(PI', d') = 0, we obtain 1 0 s2li¥h = 4Ji0 aa0 aa f s... t0-XaX117/ao)2P •(G'')d4x' - 8 f [a(Y'', !Jl>)]. 0 [oret(-}X" X 8 7J.,n)J· 0 2P•(.9')d 4 x'. (C4e) By adding up all five pieces (C4a)-(C4e) we obtain (C5) which is the second-order Einstein field equation (16). Turn now to a proof that our field (58) sa tisfit:s the de Donder gauge condition 2 h"•.• = 0 except for fractio nal errors of 0(£2 ) . From (58b) and the relation a. f orct(! XaX 07/ao)f(•'?')d 1x' = Jo,• .(}X" X 0 7J"n)o,.f(&'')d4 x' , (C6) valid for any function /(!JI>'), we obtain 2h'i!.v = 4 f oret(JX" X 87/ao){2P •(Y' )[l - 1h([jl>')]} .•.d 4x ' . (C7a) 22 262 THORNE AND KOVACS From (58e), (C6), and J1°a,a == 0 we obtain 2h~/,v = 4 f Oret(-}X"X871::8)[1t ['._ L,v + (l67T) - 1 1h"0 ,av iTz•a.oJat9·d·1x'. (C7b) We now add (C7a) a nd (C7b) an<l use the postlinear equations of motion (24a) rewritten in the form bP•(l - 1h) + it"•J.v = 0 (cf. cq. (3)) to obtain (.)ir:i• + 2h~,;'),v = (47r)- 1 f Orell-X"X 871ao)1h"P(.1i''),a•v• J1'~(&'), 0 .d*x' . We then use 1 h" 0 (r!J''), 0 • = 0 together with (C6) and a relabeling of indices to obtain (2li't/ + 2li*'>.v = ( 4rr) - 1a. J cSrel!X" X 871a11)1li"•(&'),"·0• 1fi0a(9 ')d 4x' . We then give :1'' tht: new name f/" and rewrite 11i"•(.?J'") as a retarded integral (the solution to eq. (24b)); the result is (ii'A• + .}im .• = (l/7T)a. II Orol~-°f"X 11c:s)iii a(Jll")[or.tO.X"" x 8 0 0 ·7/ao)l.o·a· 2T"\t!l'')d*x'd*x". (C7c) Dy applying a. to expression (Cl), adding it onto (C7c), using identities (B14) and (Bl5c), and integrating by parts, we o btain (ii~·+ 2h~v + 2h!{lJ.v = -48. - 4ov J J y(.9'', .9)S'r.llX"X 871ap}iP•(:1'')d 1 x' (C7d) a(3'', &)Srot(-}X" X 137/ccllhP•(.9'')d4x' . Comparison with expressions (C2) for iiHi. and (58c) for 2!ii• shows that C!i't:i• + 2'i~.: + ii1r. + /i¥rt + 2!i~» .• = o ; (CS) i.e., our total second-order field does satisfy the de Donder ga uge condition . REFERENCES Dird, J. !vi . 1974, preprint (Cambrid ge University). Boughn, S. P., Fairbank, W. M., McAshan, M. S., Paik, H.J., Tabe::r, R . C., Bernat, T. P., Blair, D. G., and H a milton, \V. 0. l 974, in Pmc. IA U Symposium 64, Gravitational Radiation a11d Gravitational Collapse, ed. C. Oe\ViccM ore tte (Dordrecht: Reidt:I) . BraginsJ..y, Y. B. 1974, in Proc. IA U Symposium No. 64, Graritatio11al Radiatio11 a11d Gravitational Collapse, ed. C. De\VittM orctte (Dordrecht: Reide l). Braginsky, V. B., and Nazarenko, Y. S. l 97 ! , in Proceedi11gs of t/r(' Co11fere11ce 011 J~'xperime11tal Tests of Grm:itariu11 Theories, JPL Technical Me::mura ndum 33-499 (Pasadena: Caltech Jct Propulsion Labora tory). Burke, W. L. 197!, J. Math. Phys., 12, 402. Chandras..:khar, S. 1964, Ap . .!., 140, 417. Chandrasekhar, S., and Esposito, F. P . 1970, Ap. J., 160, l 53. Chit re, D. M., Price:, R. ll., and Sandberg, V. 1973, Plrys. Rev. L etters, 31, 1018. - - - . 1975, Phys. Rev., in press. Chung, K. l'. 197:1, Nuow Cime11to, J4H, 293. DeWitt, B. S., and Brehme, R. W. 1960, A1111. l'lrys. (USA ), 9, 220. Einstein, A . 191 8, Preuss. Akacl. Wiss. Berlin, Sitzber., 1918, 154. Ginzburg, V. L, and Frank, l. M. 1946, Zit. f :ksp. Teor. Fi;., 16, 15. Havas, I'., and Goldberg, J. N . 1962, Phys. Re11. 128, 398. J ohn, Von R. W . l97.1a, Amr. Phys. (Germany) (7], 29, 15. -- - -.. 1973b, ibid., 27. SANDOR J. Kovacs, S. J ., and Thorne, K. S. 1975, in preparation; cited in text as Pape r 111. L andau, L. D., and Lifshitz, E. M. 1951, Tire Classical Theory of Fields (Londo n : Per;;amon Press). - - - . 1960, ibid., second edit ion. Misner, C. W. 1974, in Proc. I A U Symposium 64, Grm:itatio11al Radiatio11 and Grm;itatiomil Collapse, ed. C. DeWittMo rette (Dordrecht : Rddel). Misner, C. W., Thorne, K. S., and Wheele r, J. A . 1973, Grmiitation (San Francisco: Free::man), cited in text as MTW. Peters, P. C. 1966, Phys. Rei:. , 146, 938. Press, W. M. 1974, Ann. N. Y . Acad. Sci., 2U, 272. Press, W. M., and Thorne, K . S. 1972, A1111. Rev. Astr. and Ap., 10, 335. Rees, M. J. 1974, in Ondes et Radiatio11s Gmvitatiomrel/es, Proceedings_ of /11tematio11al Col/oq11i11m No. 220 of the C.N.R.S. (Editions du C.N.R.S.). p. 203. R obaschik, V. D. 1963, Acta Phys. P o/on., U , 299. Rulfini, R. 1973, in Black I/oles, ed. C. De::Witt a nd B. S. DeWitt (New York: Ciordon & Breach), p. 451. Synge, J. L. 1960, Relativity: Tire General Theory (Amsterdam: North Holland). Thorne, K. S. 1975, in preparation; c ited in text as Paper II. Thorne, K. S., and Zytkow, A . N. 1976, in preparation. Weber, J. 1960, /'hys. Rev., 117, 307. - - -. 1969, Plry.I'. Rev. Letters, 22, 1320. Zcl'dovich, Ya. B., and Polnarev, A.G. 1974, A str. Zhur., 51, 30 (English Transl. in Sor:ir!t A .1·tr.-AJ, 18, 17 (1974)). Kov.Acs : Kellogg Radiation Laborato ry 106-38, California Institute o f Technology, Pasadena, CA 91125 KIPS. THOR NE: Kellogg Radiation Laboratory 106-38, California Institute of Technology, Pasadena, CA 91125 23 PAPER 2. THE GENERATION OF GRAVITATIONAL WAVES III. DERIVATION OF BREMS STRAHLUNG FORMULAE* SANDOR J. KOVACS a nd KIP S. THORNE California Institute of Technology, Pasadena, California 91125 *Supported in part by the National Aeronautics a nd Space Administration [NGR 05-002-256) and the National Science Foundation [AST7 5-01398 AOl]. 25 ABSTRACT Formulae are derived describing the gravitational waves produced by a stellar encounter of the following type: The two stars have stationary (i.e., non-pul sating), nearly Newtonian structures with arbitrary relative masses; they fly past each other with an arbitrary relative velocity; and their impact parameter is sufficiently large that they gravitationally de0 fleet each other through an angle small compared to 90 • 26 I. INTRODUCTION A:~D MOTIVATION The gravitational interaction of two stars flying past each other not only de~lects their trajectories; it also produces gravitational waves. Those waves are the analogue of the electromagnetic bremsstrahlung radiation produced when an electron flies past an ion. For thi.s reason they are called "gravitational bremsstrahlung radiation." Formulae for gravitational bremsstrahlung radiation have been derived by many researchers in a .variety of different contexts, using a variety of different methods and a variety of different ~pproximations. Everybody agrees on the spectrum and angular distribution of the radiation for the rather trivial case of low stellar velocities and large impact parameters. However, at high velocities different methods give discrepant results (see Paper IV for details and references); and at no velocity, low or high, has there been a detailed study of the amplitude as a function of tL~e, angle, and polarization. The time dependence of the amplitude is important because that is what wottld be measured by a broad-band gravitational-wave detector--for example by Doppler tracking of interplanetary spacecraft (Estabrook and Wahlquist 1975; Thorne and llraginsky 1976). In this paper and its sequel (Kovacs and Thorne 1_977; "Paper IV") we sha1.l attempt to give a definitive and coraplete treatment of the bremsstrahlung problem for the case of large impact parameters (i. f;!·, for small deflections of th·e stellar trajectories). We shall place · no constraints on the stellar velocities or on the ratio of the masses of the two stars. This paper will b~ confined to a derivation of the amplitude as a function of time, angle, and polarization. Paper IV will examine the details of that amplitude, will use i .t to compute other features of the radiation 27 including spectra, angular distribution of radi ated energy, total energy radiated, low-velocity limit of va rious quantities, and high-velocity limit. Paper IV will also compar~ with the r esults of prev ious comput ations and will attempt to r esolve their high-ve locity discrepanc.ies. In computing gravitational bremsstrMlung a t high velocities , v 'V c, one must not use either "linea rized theory" or th e "quadrupole-moment formalism." Lineari ze d theory makes fractional errors of o rder unity at low velocities or high velociti es ; and the quad rupole-moment formali sm makes fra ctional errors l arger than unity at high velociti es . e.g.,§ IV.a of Paper I.) (For discussion see, By contrast, th e "post-linear f ormalism" constructed in Papers I and II (Thorne and Kov~cs 1975 ; Crowl ey and Thorne 1977) makes fractional errors of order (gravita tional r adii of stars )/( imp~ct paramete r) << 1, whether the encount er vel.ocity is high or low. Th is paper will utilize the pos t-linear formalism. This paper is di vided into eight sections. Section II gives a post-lin ear analysis of the stars' trajectories and "fi r s t-order" gravit ational fields -µ\I 1 h ; §II also i ntroduces the notation to be us ed throughout the paper (see its third paragraph). Section III introduces th e pa ramet ers and functions in t er ms of which the radiation ficl.d will :ie expressed ; and it presents a number of formul ae and geometric relationships to be ·used in the derivation of the radiation field. Section IV presents formal post-linear expressions for th e second-order gravitational field ~\) h , whi ch cont a ins th e r adia tion. 2 Th at field is broken up into five parts: "direct," "focussing," "transition," "tail,· ~ and "whump ." Section V evaluates the direct field in t erms of the parameters and functions of §III; §VI evaluates th e sum of the fo cus ing, transition, a nd tail fields; and §VII evaluates the whump field. Section VIII 28 adds all the pieces togethe r to yield the full second-order f ie ld -°"iI\I 2 h (eqs. 8.1 and 8.2); and in the res t-frame of one of the s tars it projects out the radi a tion field (transverse-traceless . part of 'hjk) . 2 Th e fina l radiation fi e ld ( eqs . 8. 11 and 8.13) is expre sse d in terms of two timeand angle-dependent amplitudes for two orthogonal states of pola rization. POST-LINEAR A.~ALYSIS OF THE STELLAR ENCOUNTER II. Consider two stars A and B with arbitrary masses mA and ~· which undergo a near encounter. Le t the initial rel a tive ve locity of th e s tars (velocity of A as measured in rest frame of B, and velocity of B as meas ured in rest frame of A) be v ; and l et the impact parame ter of th e· e ncounter . be b. Require that the s tars' radii rA and rB be · small compared with th ei r impact parameter (2. la) so that one can igno-re tidal interactions between the s tars and tidal gravitation~l radiation (Mas hoon 1973) . On the other h and, r e quire tha t the s tars be .sufficiently large (2.lb) that th'e ir internal gravity is nonre lativis tic, and the pos t-linear formalism i s va lid. Also r equire tha t ~ : . (mA + ~) I (bv 2 ) << 1 (2. le) so that the angles of deflection of the stellar trajectories a r e very small. , We shall use the dimensionless quanti~y I:; as an expansion parameter in analyzing the encounter and the emi tt ed radiation. In particular, we shall calculate the stars' trajectories , velocities, and accelerations .only 29 up to first order in ~ . This will be suf ficicnt to give· us fractional errors in the gravitational-wave field of order 2 (error in wave. field) I (wave field) "' r,; Y (2.2) where (2.3) (see D'Eath 1977). To obtain greater accuracy than (2.2) at high veloci- ties one would have to perform a post-post-linear calculation or use an entirely different type of approximation scheme-e,g,, that of D'Eath (1977), We shall analyze the stars' motions using the post-linear formali.sm of Papers I and II. That formalism is couche.d in the language of flat Minkowskii space time, even though space time is actually curved. We shall perform the analysis using frame-independent Minkowskian language and notation. Greek indices will run from 0 to 3, Latin indices from 1 to 3; sans- serif letters (print ed version) or letters with arrows over them (manuscript) will denote 4-vectors and 4-tensors; bold-face letters (printed vers ion) or letters with squiggles under th em (manuscript) will denote 3-vectors and 3-tensors; subscripted conuucis will denote partial derivatives; geometrized units (c .., G == 1) will be us e d; and the ., ignature of the metric will be -t++ • Each star carries with itself a clock that reads proper time--TA for star A and TB for star B. The clocks are synchronized to read T A at the moment when they a re nearest each other. = TB == 0 We shall denote the world . lines of the centers of the stars by , zAa.(TA) Here = zAa.(TA)+"QAa.(TA) ' and are the spacetime coordinates (2.4a) (in an 30 arbitrary post-linear co"rdlnate f rame ) of star A when its clock reads a time TA, and of star B when its clock reads TB; zA and zBa ~re the s tars' unperturb ed coordinates (coordinates in the limit ~ _., 0); ?A"a and }B a are the first-ord e r corrections to the unperturbed coordinates. We shall de- note the 4-ve locity of star K (=A or B) by = u..a + ...Ka --- dz a/d"" + d"!..._a/dT UKa --- dZKa/d""K • K w K 'K IK K (2. 4b) and we shall denote its gravitationally-induced 4-acceleration by (2. 4c) z The unperturbed world lines a 8 (T ) are straight lines 8 in our (arbitrary) post-line~r coordinate frame; see Figure l. Define the "impact vector" = vector separating the t wo stars ) at thei r moment of closest approach · ( in the. case of unperturb ed world lines • (2. 5) Of course, b a is orthogonal to the unperturbed world lines of both stars (2.6) 0 Call the center point of .... ... the "collision event" and denote :!.t b .... .... _., terms of xc, b, and the unperturb ed 4-velocities uA,uB .... x c In the unperturb t!d world lines are described by (2.7a) - -1 The moving stars generate a post-line9r gravitational field lhµv· trace reversal (2.7b) Its 31 h (2.8a) 1 lJ\I i s related to the first-order metric of spacetime by (2 . Sb) (See eqs. (I,23c,d)--i.e., eqs. (23c,d) of Paper I; and r ec:al.l that in keep:!.ng with our flat-space notation all indices are raised and lowered ~s ing the Hinkowskii The gravitational field stress-energy tensor 2 TlJ\I --µ\I h is generate d by the pos t-linea r 1 = Tµ\I (we s hall drop the pre fix 2) through the flat- s pace wa ve equation (I,24b) 0 h11\) (2.9a) f 1 This equation has the solution ( 2.9b) 4 x and x' d e note fi e ld event and int e gration event; d x• i s th e fl a t- Here space 4-volume element · d4x' :: dxO' dxl' dx 2' dx 3' and 0 G(x,x') is the flat-space propagator of e quation s (I,19b ) a nd ( I I,6b) 0 c(x, x ') In this propagator for (2. 10) (l/4n) o [-1 (xa-xa' )(xS-xS' )nNal . r et 2 v.µ (2.11) oret · i s the retarded Dirac delta function--which is zero x in the causal past of x' 1 and is the ordinary Dirac d e lta function otherwise. Equation (2.9b) ·shows th a t - µ\I h obeys a linear s u perpos ition law·. this 1 allows us to split it · into a field due to s t a r A plu.s . a fie·ld due to s t a r B 32 (2.12) Here 1' llV is the post-linear stress - energy t e nsor. of star K. K This stress- energy t ensor satis fi es e quation (I,24.a ) T µv K where rµ 1 av •V - rµ l av 1' cxv - lrva · V T µa K (2.13) K a re the firs t-order C.1uistoffel symbols comput ed from t he firs t- order metric (2.8b). Equations (2.12) and (2.1 3) are a coupled set of equations which de termin e : (i) the internal structures of the stars , (ii) th e gravitational fields of the stars, (iii) the gravitationa l deflection of the stars' world lines, and (iv) the gravitational field of the s ystem. Because the stars are smal l compared to th eir i mpact pa r ameter , their internal s tructures are influenced by th eir self-eravity , but not by each other ' s gravi ty--i. e. , tidal interactions can be n eglected. We shall t ake the int ernal structure of each star as given, and s hall focu s attention exclusively on the s t ars ' r e lative motions. Our n eg l ect of tidal interac tions implies tha t each s tar's s elf field h as no influence on i t s trajectory. Hence, the motion of star A is gove rned entirely by the fi eld of star B, and conver sely. By using this fact and performing a 3-volume integral of equation (2.13) over the inte rior of s tar K, one can r eadily v erify tha t th e cent er of s t a r B moves a long a geodesic of the metric gµv = nµv + hAµv, and the cente r of s t a r A moves along a geodes ic of the me tric gµv ~ nµv + hBµv. The proof is identical to that found in standard references-e.g., Exercise 39.lS(d) of Misne r, Thorne , and Wheeler (1973 )--cite d henceforth as MTW.] TI1e resulting geodes ic equation for sta r K--accurate to first orde r in the deflection parameter C--is (2.14) Here the symbol K' means "th e oth er star", i.e., "not K"; and 1 hK'aB is the field of s t a r K' at the location of K. In calculating 1hK'a$ for use in this geodesic equation , we can treat star K' as a point source (tidal int er ac tions negligible), which moves along the undefl ected world line a errors in 8K ~ are of orde r • The r es ulting fract ional Th e s tress-ener gy t ens or for s tar K' a t event x' i s th en TK. 0$ (X t) = f Q ~·~• ~ 1 a 4 x - zK' TK' l' u ( I ( ) ) d (2.15) TK' where o is the 4-dimen s ional Dirac delta function 4 2 o(x0 ) 6(x l ) o (x) 6(x 3) 6 (x) 4 ( 2.16) By putting this s tress- ene rgy tensor into equation (2.1 2) for integrating over - \c, iiv and 1 a4x• we obtain (2.17) The integr al ove r TK' can be performed using expression (2.11) for the propagator. Th e res ult i s ( 2.18) .... He re .XK, is the unique past-directed null vector reaching from th e field point x to the unperturbed world line of s tar K' . The corresponding tra ce- 34 reversed gravitational fi e ld (metric p e rturbation) is ( 2.19) He re we ha ve us ed the f act that the unpe rturbe d 4- velo city "K' h as unit length relative to the unper turbe d me tric: uK,µ uK,v nµv = - 1. In evaluating the 4-acceleration of s tar K we s hall n eed the gradient of 11<, ~ 1 We can evaluate th e gradient by l ett ing the field point (x). _., change in - 1 ->- x x ->- ch ange by an arbitr a ry, small amount to x + E, • sion ( 2 .19) on + The depen dence of expres-+ i s contained entirely in t he null v e ctor .X K'; thus the hK , is (2. 20a) ->- whe re 'ii-+ denotes the gr adien t along[, ( nota tion of MTW). . E, "t ->'J(K' The qu an tity can b e calculated u s ing th e re l ations ( cf . e q. 2.7a) -+ -+ ~· - x - -+ x ·~·o < o 0 which det ermine .KK' and ( 2 . 20b ) TK' in terms o f the field point -+ x • In particu- l ar , V-+ -+ ~ J(K' ->- - 'ii-+ x + E, -+ ~· \l+ TK' E, ->-+ -t;, +UK' 'ii+ TK' [. (2. 20c) from which we can ob tain ..... ... ..... J( ) 0 =.K'K' • ('V-+ E, K I = -.KK' .... . t + (-+t1i<_1 •. J(K,) ~ TK' (2.·20d) 35 Vl Ti, and insRrting back int o ( 2.20c ) w~ f ln<l Solving ( 2.20d) for (2.20e) wh ere ++ n is the Minkowskii metric. r ecalling that -+ ~ Combinini-.: this result with (2.20a) and is arbitrary, we obtain (2.21) We are 1101o1 in a position to evaluate the Ii- ac celeration of particle K. ++ By inserting express i on ( 2 .21) for the gradient of hK' into equation (2.1 4), we obtain -~ ~· ->- (~,. ~·) 2 (2 .22 ) Ile re ->- -+ (2.23) y - -uA • uB is the s pe cific en ergy of star A as seen in the r0.st frame of n--and also the specific energy of B as see n in the rest .frame of A. Al~o -~ , is z/ CTK); i.e., it is the unique past-directed null 1 vector reaching from the " field point" xa = ·zKa(TK) to the. world line of K' • ... We now derive an expression for ~· as an explicit fun ction of 'K' the proper time of the accelerated star. In our de rivation we can treat both stars as though they moved along the straight lines zK(TK) and zK 1 (TK,); by doing so we make fractional errors in ~ a . of order "(2.7) for the unperturbed world lines we have ~ • From equations 36 _, 11<_, = ["K,]x=zK(TK) _,.. E: KI b -r -.-. + UK' T KI - l\cTK (2. 24a) The time TK ' , at which the field r.lUst be " emitted" in order . to reach star _, K at· time TK , i s de t e rmined by the condition t hat kK' be null (2.24b) 0 Solving for TK' we obtain (2. 24c) where i is defined by [b 2+ 2 2 V y T 2 1/2 K ] distance from K' to K, as m0.asured by star) K' at the moment, in his frame, when the ( clock of K r eads TK · (2.25) By combining express ions (3.24c) and (3.24a) we obtain the relation (2.26) and by inse rting this into express ion (2.22) · for 3Ko. and using t h e rela- tions .... 2 .... 2 ~ = ~· -1 , -y we obtain (2 .27) The spatial part of this acceleration, as seen in the rest frame of K', is depicted in Figure 2. 37 _,. Th e perturbation wK of the 4-velocity of star K is t h e integral of + ~ over time TK, with initial v a lue tion gives + WK = + wK(--<0) = 0. Straightforward integra- I UX· y 2 (l+ v 2 ) Notice that the total deflection angle of K as seen in the rest frar~ e of K' is !spatial part of ~K( 00)I Ispatial part of ~I 2y = - 2 2 <1 + v ) UX, I (yvb) 2 2(l+v ) v2 - Yv D\c• b (2. 29) whi ch agrees with equation ( 25 .119) of ~ffW--whet"e the d ecivation require d tI]z << "J<•• (Our derivation i s valid for arbitrary relative masses of K andK'.) _,. If we wished, we could int eg rate ~K(TK) to obtain}K(TK), the pertur~ bation in the world line of star K due to gravitational deflection. We shall n ot do so because our calcu l.::t.lon of th e bremsstrahlung will require _,. no explicit knowledge of aK(TK) • The chief r es ults of this section are: (i) express'ions (2. 4) which split into perturbed and unp e rturbed pacts the world lines , Ii- velociti es , and 4-accelerations of the two stars ; (ii ) e xpression s ( 2.7) for the unperturbed world line of star K; ( iii) expression. ( 2.27 ) for the 4-accele ration of star K; (iv) expres s ion ( 2.28) for the perturbation of the 4-velocity of star K; and (v) expressions ( 2.18) , (2.19), and ( 2.2 1) for the first -order gravitational field of star K-' at points far from the s urface of K' (dis- 38 GEOMETRY AND NOTATIO~ FOR THE BREMSSTRAHLUNG CALCULATIONS III. We t u m now from the de t a i ls of the s t e llar e ncounte r to a calcul a tion of the bremss trahlung it produ ces . In this section we introdu ce th e no ta- tion and ·g eometric r e l ation s h ips to b e used in the calculation. Most of our analysis wi ll b e fra me -independent. It will oft e n utilize an expansion of v ectors and t e nsors in t erms of four "b asis v ec:tors " + (3.la) uA = ( unp e rturb e d 4-ve locity of s tar A) + u f ~ ( unpe rturbed 4-velocity o f star B) + - .b/b "' + q :: ~A , (3.lb) ~B) ( unit v ector, orthogon a l to a nd and pointe d a l ong im~act+directi~~ (3 . lc) ( ·unit v e ctor, o_rtho_~on ·' \ to+uA, uB, and Xi , ) a nd s uch th a t uA, u B, ~ , § yfo'?/1 a right- h anded syste m, i. e ., caSyo uA ~ b q > 0 (3. ld ) · The s calar products of these ve c t ors are [ cf. e qs . ( 2 . 6) and (2. 23 )) ->- 2 UA c .. 2 UB + . ->- ... + UA. UB = -y -1 ->- + ->- • J) "' uA • q = UB UA . ->-2 ):; .. ..... . + + + J:> = UB • q "'p • q ->-2 = q := +l • (3. 2) 0 The obs e rver is locat e d f a r fr om the r egion of space wh e re th e s tars encounter each oth e r. His sep a r ation from the e ncounte r r e gion i s describ ed by a pas t d irect ed null v ecto r k ( pas t -directed null v ec tor reaching from the - observe r ' s wo r l d lh1e t o the "collision event" The scalar produc t s of k ; c ) (3.3) with th e 4-ve locities of th e stars will b e denot e d by RA - k. 'tiA .+ + ~ - .k . '-'B = (distance from e n count er r egion to observer as seen in r es t fr a me of star A ( dist an ce from e ncounter r e gion · t o_ observe r as seen i n r est frame of star B ) ). (3 . 4 a ) (3. 4b) 39 We shall s omet i mes v ork in the rest frrune o f star A. we s hall use Min k01Jskii coordinat e s with In that fram e _,. -> p along th e y direc t ion, q along the z di r ect ion, a nd th e motion of 8 along the x direct Lo n (Fig. 3): . In this frame the null vector .... .... .... p· > e q (3.Sa) z is k .... (3. Sb) k unit spatial v ec tor pointing f rom) n ~ ( collision event toward distant observer 2 2 + f3 + 6 "' 1 + ->- ->- cxe +f3e + oe x y z (3. Sc) (3 . Sd) cl We shall r efer to (cx,f3 , 6) as the "direction cosines" of the observe r. We shall als o sometimes vork ~n the rest frame of star ll using coo rdinates t , x,y ,z with ->- + u A "' .... YCO - vye;z _,.. _ .. _,. 0-- 0- q y 0 .... 6- z (3.6a) _,.. .... k + .. - -1\<eu + n) (3 . 6b) _,. n --+ . ,.... .... = Cl(}-x +-Se-y + 6e-z a2-·~ s2• + v (3 . Ge) 1 ( 3.6d) Using equations (3 . 2) - (3. 6) one can r eadily derive th e following relationships among th e quantities th3t r e fer to star A and those th at r e f e r to star B: RA = y ( l + <Xv>\ a .. f3 .. \ = y ( l - cxv)RA (3. 7a) v a "' Cl - v ( 3 . 7b) l+av 1 - (XV Cl + B y ( l +av) ' - B B ~ y ( l - av) ' (3 . 7c) J~o (3 . 7 d ) y ( l - Clv) r<1+av) Note that the scalar products of th e ob server's null v ect or th e fundam e ntal b asis vectors -~ ~,.. + -~ UA, UB, l5 , q k with ar c -~ -+ k .'.J' ,,.-SR =-'Jfa A B (3. 8) As the ob serve r moves along h is own world lin e , h e sees a time -changing gravitation.al field ( radiation). We s h all express the t ime dep e nde nce in terms of two parame ters, TA and TB, which are fun ctions of the observer ' s proper time T Th ese par ame ters are defined as follol.ls (:.; ee Fig. l1a) : 0 At proper time T the obs erver is at a specific event 0 ..... Z (T ) along his 0 0 -+ world line • . Th e past light cone from Z (T ) int e r sec ts the ~erturb e d world 0 0 line of s tar K at a proper t ime TK = TKo a l ong that wo rld line. - We s h al l refe r to TKo as the "re tarded t ime for s tar K", a nd we shall de fine (3 .9 ) Th e distance from K' to Kat time 'Ko' as measured in the res t fr P-Jne of K' we s hall denote by (3.10) (cf. eq. 2.25). A simple calculation in the r es t frame of ei ther star A or star ' B yields the follow i ng r e lationships between the tw·o r e t a rded times: )~ 1 T Ao = (1 - av )yT llo - Sb (1 + av )yTAo. + Bb 'no (3.lla ) or, equivalently, 1' (3 .llb) A Figure 4b s hows several Li-ve c tors which wi ll play important role g in our d e riva tion of the radiation fi e ld. The fi r g t o f these i s the v e ctor r :::::- uBTBo - uATAo + b :: (vetimectorTAr eaching from s t ar a t) to star at time TB ->- A ·>- B 0 Note tha t the squared leng th of . (3. 12a) 0 ->- S (3.12h) i s positive , e xce pt in the s pecia l case th a t on e stnr is precis ely in th e ->- ->- "sh a dow" of the other [ S para ll e l to k a nd th e refore null; 6 = ab - 7frA = ab - $Tll = O] ; in th.a t case s 2 v a nis h e s . o "' 0 , Note also th at ·>- S has the following proj e ctions on the Ii-velocities of the stars: (3.12c) and th a t these proj ections satisfy 52 + s 2 52 + s 2 Note also that S is orthogonal to k A (3.12d) B 0 (3.12e) The next vector of interest is ->- ·>- ->- -+ Y = S + SA uA = (projection of S orthogonal to A) · >- vector perpendicular to uA and r eaching) • ( from world line of A to B nt TBo (3 . 13a) Note that -+2 ·>- + y • k y (3.13b) Another useful vector is M=S + (SB- 9.A) -:;B = (prast-dBirected null ve1.dcto1 ir reacfhAing ) • rom at 'no to wor ne o (3.14a) ->- I ts scalar product with \IA , ..... u A (3.14b) = is used in the construction of another vector ., (vector reaching orthogonally from) world line of A to tip of (3.15a) M Note tha t (3.lSb) Note also that in the rest frame of A yY .. b • J x • y(T - vt ) A· A lxl a .e. B (3.16a) !~I = MA= y(tA- vTA). . (3.16b) 43 A us eful che ck on our final ans wer for the r adia t iooi fi e ld l s provided by the following demand: If we i nt erchange s t a r s A and B, and r eve r s e the + sign of b so it once again r ea ch es from A to B a t th e r.omen t of closes t approach, and pre cisely r ev er s e the s pati.a l di r e ction of th e observer as seen in e ith er frame A or fram e B, the n th e obs e rver s houJ<l s ee pr e cisely the same radiation field a s be fore. In other words, the r adia tion field should be unchanged by t h e conver s Jons + ... u... + u .lj R A t + -$ A ' ... + q + -q + (3.17) 6 + -o a+ -a a+ -a IV. $ + -8 FORMAL EXPRESSIONS FOR GRAV I TATIONAL FIELD IN WAVE ZONE In Paper I we de rived a "plug-:l n-and-grind" formula for the g r avitational field hf1\I 2 f a r outs ide any post-linear sys tem. field satisfi es the Lor entz-gauge conclition ~\I 2 errors of O( E: 2 ) , where source. "' 0 and has fra ctional is the magnitude of the. internal fi e ld of the E: By contrast, th e • \I Thi s g r avitationa l 1111\1 of §II has fractional errors of O(E:). 1 The formal expressions for 1111\1 (§rv.c.iii of Paper I, with minor 2 changes in notation) involve the following biscalars--i.c., the following scalar functions of a s ource point x' and a field po i nt x: (i) the flatspace propagator 44 e 1 1 et a' B' G(x,x') - 4TT 6 ret [-2 (x - x )(x - x )n~ ..,.6 J O =- (4.1) [cf. eq. (2.11) and associat ed dis cus sion); (ii) the de riv a t i ve of th e flat1 a a' B B' space prop agator with r espe ct to the interval 2 (x - x )(x - x )nee : 1 1 a a' B B' G'(x,x') = - 6 ' - 41r ret [-2 (x-x )(x-x )n~..,.6 J O 1 a G(x x') "" - --0--0..,-'- - 0 0 , (x - x (iii) the focussing function ) ax (4.2) (eqs. I,40a and I,59b) l 0 01 a(x,x') = t<x - x f 1RaB[x'+ A(x-x')) A(l-A)dA 81 )(x8- x ) (4.3a) 0 (iv) the tail generator (eqs. I,43a and I,59c) 1 B(x,x') = (xa - xa' )(xB- xB' ) J (4.3b) 0 and (v) the time-delay function (cqs. I, 38b and I, 59d) 1 1 (xa - xa' )(xB- xB' ) y(x,x') ~ 2 f (4.3c) 0 Her e haf3 is the post-linear me tric perturbation of §II, 1 Ricci curvature t ensor calculated' from the post- linea r metric to first order is a flat-space affine parameter along the straight line reaching from x' to x (cf. eq. I,36). In place of the full time-delay func- tion y(x,x') we shall be using a truncated version of it: T·( ') ~yxx ( ') - (contribution produced by source's) Y xx stationary, external, "l/r" field ' ' (4.3c') The truncation process is discussed at length in §IV.c.ii of Paper I; and we shall make it explicit in §VI below. 45 The external field ~1\1 2 consists of five parts (eqs. I ,58): a "direct field" -;-i.1\1 l.61r J 0 G(x,x ') Tµv(x')[l- 2hD (:c) /1 h(x')] d x ' 1 ( 4, 4a) a "focusing field" . {4.4b) a "transition field" -µv h.fR(x) 2 = 16n J yT (x,x') 0 c'(x,x') Tµv (x') d4x' (4.4c) a "tail field" where the integral is over source points x' inside and not on the past light cone of x; (4.4d) and a "whump field" (4,4e) The external field is the sum of these five parts h11\) 2 h11\) + 2hl-'v + 2~T\)R + h;lv + h11V F 2 rL rw 2 D (4. 5) The sources of the external fiel~ are tl.G post-linear stress-energy tens·o r Tµ\I and the gravitational "whump st r esses" (eqs. I,58e and I,23g) (4. 6 a) where _ 1 { 1 pv ap [3cr YA + µa \IA ( ycr [3p + Ba yp) ,MµvaSyApo = lGn 2 n r1 n n n n n Tl ;1 n _ Cni1C1n van8Anyp + Tlµan wnypTlSA) + .!8 (Zn PYTl \IO' -n µvn yo) (Zn aATl 8p -n APn a~)} (4.6b) l~G For some purposes it is useful to express th e sum of the focusing, transition, and tail fields in the following form ( c qs. II,36 and II,37c): (4. 7) l[ere "oµ - "/" o oxµ an d "oµ' - "/"xp' o o In order to use the above formalism, we need explicit expressions for the post-linear s tr- ess-energy tens or TlJ\I and gravi t a t ::.unal field 1~\I. The stress-energy spl i t s into parts TAµ\I and TBµ\I associated with the two stars (4. 8) In its own inst a ntaneous r es t frame , each star is assumed to b e a static Newt onian sphere 1 with stress-energy t ensor that h ns physicai components 11·1 ' . i s nee d e d f or case o f expos i tion . i n §VI • IIoweve r, Crow l ey 11.s assumption and Thorne showed in Paper It that the bremss trah lung rad iati~n is i ndepe ndent of the internal structures of the stars. The s tars need on ly satisfy the constraints of equa t ions (2.1). "" T 00 K pK(r) , T Oj K 0 T jk << pK(r) K for (4.9) ; T aS .. 0 K 0 < r < rK for r > rK r - (x2+ y2~ z2)1/2 47 The center of the star (origin of reference frame for cq. (4. 9)) moves along the world line ZKa(TK) which was compute d in §II. At most stages of the calculation we can ignore the internal structun!S of the stars, a nd treat theiii as point masses with the following stress-energy tensor (valid in ~post-linear ref e rence frame): Here ZKa(TK) and · uKµ(TK) are the world line and 4-vclocity of star Kat time 'K' as given by equations (2.4), (2.7), and (2.28); and [l + '21 lh] is an approx i mat i on to ( -g)-112. Like the stress-energy tensor, the post-linea r grav.:.tatioual field h11V 1 splits into two parts produced by the two stars (4. lla) [cf. eq. (2.12)). For field points far from star K we can write (4.llb) where .... uK .... is the unperturbed 4-velocity of star K and ,K K is the unique past-directed n~ll vector from the field point x line of K; see equation (2.18). r; • - µv This expression for i1l< gravitational deflection of the stars' tional errors of order to the unperturbed world trajectories~i.e . , (x 1 ) ignores the it makes frac- It is easy to write an e xpression for - µv hK 1 (x') which , takes account of the deflection of trajectories, and which also takes account of the int~rnal structure of star K--but we shall not need such express ions. 1~8 The largest component of our external field is (1+.12a) The radi a tive part . of the fi e ld will be (4.12b) The fractional errors in our formulas for 111" are O(E 2 ) 2 Consequently. 2 2 any t erms which are cubic in the masses (e.g •• mA ~/RAb ) or of higher orc!er can be shoved into the error and ignored. throughout the computat ion. ' ' direct" part, T ~(see lJV We shall do this ruthlessly For 0xample, in all parts of 2~v . except the appears mul tipli ed by a quantlty propr-:tion<ll to mA or eqs. [4.4b 1 c,d)) . In such c:ises we nee <l keep only those parts of Tl.IV which are linear in mA or ~--which mear.s that in place of equation (4.10) we can use the stress-energy tensor of an undeflected star T lJV ( x 1 ) (4.13) K V, ·When 1' µv and DIRECT FIELD -pv , h are split up into "A+B' contributions, expression 1 (4.4a) for the direct field reads (5.1) The second lin~ is of quadratic order in the masses--and at quadratic order it is ignorant of the gravitational deflection of the stars 1 trajectories; 49 i1v only the first-order (undeflected) TK and - 1 hl\. enter. An a r es ult, the second line has nothing whats oeve r to do with the r adiation. It is sfa1ply a nonlinear contributor to the stationary self field s of s tars A and B. Because we are not interested in the nonlinea rities of th e s elf fields, we shall rlrop the second line and write (S.2) Here the summation is over K"' A and B, and K' means "not K". Since the field point x is far outside the source, we can use the point-mass stress-energy tensor (4.10) in evaluating e.xpression (S.2). By inserting it into (5.2), int eg rating over x' to remove the o tegrating over 'K to remove the orct·of 0 ... Here <KK 4~ ul.luv K_ _ K_ (l _ .! + . ·>- KK • UK ·v of TK~ , in- c (cq. 4.1), and omitting nonlinear self-fields os above, we obtain t l [ K 4 l h ) 2 1 K' (5.3) ret is the unique past -directed null vector reaching from x to the + II perturbed world line of K, and "ret It means t 'evaluated at the tip of ,KK • Because we are interested only in the "l/r" (radiative) par t of the field, ... we can replace .KK by the vectdr + k of §III and Fig. 4; s uch a replace2 ment produces time-dep endent errors of 0(1/r ). by splitting -+ UK ~y making this replacement, into its unperturbed and perturbed parts {e q. 2.4b), and by dropping terms cubic in the masses, we obt ain (5.4) 50 The first line is the non-radiative "Coulomb" gra vit a tiona l field of the source. it. It has nothing whats o ever to do with r a diation, s o we sh a ll drop The second is the "dire ct" part of the radiation field. · Be cause it is quadra tic in the masses, we make a negligibl e e rror by evaluating i t not at the~ retarded time of the de fl e ct e d tra j e ctory, but rather at the re- tarded time TKo of the unperturbed trajectory (cf. Fig. 4a): 1{1DV{x) "' r; 2 K The field l-r4:. k • uK [2uK(µwKv) - UKµ. v uK . (kk •• ~\~K + 2 h )JI (5.5) l l K' at 1~ 1 [zK(TK0 )] can be evaluated by taking the trace of expression (11 , llb) and by invoking (2.24a) -4~, + + - ~, ·~· and by then using expression (2. 26) to evaluate + 11<' The result is (S.6) [cf, eq. (3.10)], By inserting this e xpression into (S.S), and by inserting expression (2.28) for wK(TK), and by invoking equations (2.7b). (3.lc), (3.4), (3.8), (3.9), and (3,7a.c), we bring the direct field into the final forn. ~\) 2 D ---= 4mA~ 51 (5.7) As a check on this result, one can verify that it is invariant under the transformation (3.17). VI. FOCUSING, TRANSITION, AND TAIL FIELDS There· are two methods for calculating the focusing, transition, and tail fields: that of Paper I {eqs. l1.4b,c,d above) and that of Paper II (eq. 4. 7 above). Method l has the disadvantage that it requires a very special treatment of "shadow regions" (i.e., of observation events x from 1.'hich one st.ir is viewed as lying partially in front of the other). How- ever, outside the shadow regions method I has the advantage that the Dirac delta functions in the integrands cause all volume integrals to collapse into line integrals. Method II, on the other h and, permits one to treat shadow regions on an equal footing with non-shadow r egions; but its volume integrals do not collapse into line integrals. As indicated in Paper II, method II is powerful for proving theorems, while method I is powerful for carrying out explicit computations. In this section we first use method II to show that the sum of the focusing, transition, and tail fields, -µv 2hFfRTL(x), has a form in shadow regions which is the analytic extrapolation of its nonshadow form. We then use method I to calculate analytically the nonsh adow form of hl~RTL' and 2 we assert on the basis of the method-II analysis that this analytic result is valid also in shadow regions ; 52 For the method-II analysis pick a specific event z on the world line 8 of the cen ter of star B, m1d construct the future ligh t cone J+(Z ) in the 8 rest frame of star A. Figure 5 is a spatial diagram of J+(Z 13 ). In thl\t figure the s hadow r egion is the extrapolation to l ar ge distances of the cone The ha lf-angle of the s hadow cone, 05 marked " s hadow region ." , is (6.1) In §V of Paper II Crowley and Thorne showe d that throughout the method-II computation one can use a monopole, point-mass idealization of the stars' structures without producing any fractional errors larger than yrA/b and yrB/b • This is equally true in the shadow r egions and outside the shadow regions. 111is can only be true if the shadow has no influence on the form --vv of hFTRTL- -i.c.' i f ;:i.IV hFTRTL in the (very small) shadow regions is obtain- able by analytic continuat ion of its form in the adjacent non-sh adow regions. Having established this import ant fact, we now compute the nons hadow -1.JV form of hFTRTL using me thod I. In nonsh adow regions the focussing field vanishes , so we need compute only the transition fi e ld and the tail field, and then add · them toge ther, a) Transition Field In expression (4, 4c) for the trans it.ion field we can split the. timedelay function yT .and the stress-energy tensor Tµv into parts due to each star. T YA As with the direct field ( §V), products involving the same star--i.e., µv TA and yB T µv TB --lead to nonradiative , nonlinear corrections to self- fields; they are of no interest to us, so we drop them. then becomes Expression (4.4c) 53 2HTR,K(x) = 16rr f oG' ( x,x ')yK 'T ( x , x ' )TKj.J\) ( x ') d4x I (6.2) Let us calculate YAT ( x , x') in th e r est fr ame of s tar A. for x the lo- cation of th e dis tant observer and for x and x' s uch that the ray j oining them does not pass through star A ( no s h a dow at x); s ee F1gure 6. We begin by calculating the untruncat e d time-delay functi on of equa tion (L1. 3c): I 1 1 (xex-xex' )(x S- x $' ) YA(x,x') = 2 (6.3) 0 The post-linear field hAexl3 produced 1 sta r A in its own rest fr a me ( Fig. by 6) is (6.4) cf. cq. ( 2 .19). lie re x" i s the s pat i a l part of the v ec tor x the or igin ( cente r of star A) to e v e nt x". ex" rea ching from This exp r ession i.s vnlid not only at event s f a r from star A, but also arbitrarily close t o A' s surface , because A is spherica lly symme tric (cq. x ex - x and footnot e 1). 11.9 De fine ex' (6. 5) and combine with equations (G.3) a nd (6.4) to obt ai n 1 mA[ (x0)2 +I~ I 2JJ (6.6) 0 A straightforward integration yields m~[(X0) 2 +1~1 2 1 yA (x,x') l~I R.n 1~11~1 + x•X..: - - - -·- - - -1~·11~1 + x' • X (6.7) The truncation process, as d escribed in §IV.c.ii of Paper~. cons i sts of 54 dropp i ng the contribution to the delay function (6. 6) which comes from outside the immediate neighborhood of th e source--e .g., the contribution from outside a radius of ~lOb. Performing this truncation, we get 0 2 2 _mA_r_cx__)_ _+_1_~_1_1 9,n ( l~I lOb ~I r ) l~'ll~l+x'·x . (6. 8) 2 We now manipulate ~~,B(x) (eq. 6.2) keeping in mind equa tion (6.8) T for YA and using expression (4.2) for 0 c• Because [ ( 1 /X0 )yAT (x,x')] depends on x0 only through th e quantity 0 X0 = x0 -x O' lxl ~ ·R = (dist ance to obs e rver), A and because the propagator forc es X 2 the differentiation on the last line of (6.9) causes that line to be o: l/RA (afte r 0 c has been integrated out), rather than 0: 1/RA - -i.e., it makes the last line nonra<liative. For this r eas on we can drop the l as t line and write (6.10) T By inserting expression (6. 8) for y A , expression (4 .13) for the "undeflected" TB µv , and expression (4.1) for functions we obtain 0 c(x,x'), and then integrating out the delta 55 [(X0)2+1 ~12 J x 1~1 0 9,nl lOb I~ I 1~'11~1 + x' •xi- J-x' =z-U ( Tno )J . (6.11) At the evalua tion point, x' Xo c x o - z o(T B Bo ) = I-~ I - n ' z (T - -B Bo ) ; (6.12) n - x/lxl Consequently, to order l/~ (radiative order) (6.13) here dTB /dxO follows from cqs. (6.12), (2.4b), and Figure 3: 0 - n • u = y(l - av) -B - n • The qua ntiti es I_:BI and ~ • ZB are conveniently evaluate d by reference to Hgure 4 an d the ass ociat ed discuss ion i _n §III: .:B(TB 0 ) ca~ be thought of as a 11-vector which is purely s patial i n the rest frame of A. of thus it is identical to the 4 -vector with th e relation n = -k/R - A + Y of Figure ~. Wh e n thought Thi s fact, together a nd equations (.3 .13b), implie s that (6.15) By combining this result with expression (3.12 c ) for SA, sion (3.10) for R.B , and with. expression (3.9) for T (d/dr 8 C!~BI + n • z )) -= vy(a + TB/R.B) - - B TBo 8 with expres - we obtain (6.16) 56 By combining equations \6.1 5), (3.12d), and (3.12c) we ob tain [ '~·' : n•z.l - - Bo 1 1 = SA B tB+ SA iB - CXTB - f3b 52 sz (6.17) Finally, by inserting equations (6.14), (6.16), and (6.17) into (6.13) and invoking equations (3,5d) and (3.10) we obtain -:µ\I 2h.fR,B "' (G.18) -µv The analogous expression for zh.fR,A can be derived either by an analogous calculation in the rest frame of B, or by invoking the invc.riance 7:1.J\I 7:1.J\I The total transition field Czh.fR,A + 2 hTR,B) is then relation (3.17) . 2vb (ab 2 srA> - RAtAS (1 +av) +u µ (6.19) B s 2 + 0 --i.e., as S Note that this transition field diverges as becomes null--i.e., as + S + b e co.mes parallel to k --i.e., as the observer moves into the shado•,., region; cf. Figure 4 •.. It is easy to verify that the divergence has the form h11" z·"TR,K imp act parameter of ray from star K~ db a: _l_ ~ b ray dx 0 b = as it passes star K' on its way to ray ( observer at x (6.20) (cf. eq. 6.13), below. We will discuss the significance of this divergence in §VI.c 57 p) Tail Field By discarding terms that repre sent nonlinear corrections to self fields, we can rewrite the tail field (4.4d) as (6.2la) 2~~,K(x) = -161T JJ 0 G(x,x") SK (x",x') 0 G'(x",x') T/v(x')d x d x• • 4 11 4 1 (6.2lb) Here the x' integral is over events x' inside· and not on the past light cone of x. Let us calcu_late 13A(x" ,x') in the rest frame of star A. In doing so we must take account of the internal structure of star A, because SA is zero uqless the straight line from x' to x" passes through A. ~ression 8ir(TaS - (4.3b) for t By combining ex- S with the post-linear field equation _ lR~G ~ Tyynae) and with express ion (4.9) for TAaS'. and by using spherical polar coordinates centered on the event x' (Fig. 7a), by expressin~ the affine parameter in tenns of radius and r as ). r/r" , we obtain r" c(t")2 + (r")2] BA(x" ,x') I 41TpA (r/r") 2 dr/r" (6.22) 0 Let us now perform the integral over x" in expression (6.2lb), using express ion (4.2).for 0 c•: . . J ~ 1 OG(x,x")e A (x" ,x') oG' (x" ,x I) d x" ., f OG(x,x") ..,A x ,x ')[-1 Cl ttt -<lt" OG(" x ,x ')] d 1," x a (" (6.23) 2 ~+ - f-ClG(x x" )JS (x" x')dO"] G(x" x')dt"r" d" ··at" [l_ t" 0 ' CL A • ' · 0 CL' r 58 In the third line we hnve used integration by part s , plus . the f«c t th ;it, as long as the points x and x' are very far outside of star A and x" is in the shadow region where SA 1 0, pendence on angle (8" ,¢"). to pull the 0 0 G( x ,x") and 0 G(x",x') have negligible d e - This negligible angular depend e nce a llowe d us G's out of the angular int e gral and evaluate the m on the ''center line" of the shadOlol x" CL = ( t", r" , 0, 0) (6.24) The angular integral in the third line of (6.23) can be performed using equation (6. 22) J r" 2 2 3 SA(x",x')<ln" = [(t") + (r") ](r")- J J 0 .. l11Tm [ (t") A Here 2 4rrpA (r,n")r d'"' ,. d r all angles 2 + (r") 2 ] (r")- 3 H(r"-r ) . (6.25) s ll(Z.:) is the Heaviside step function z.; < 0 H(z.;) (6. 26) z.; > 0 and r s is the distance from the- c e nter of star A to the source point x' . (see Fig. 7a). By inserting expressions (6. 25) and (11.l) into (6. 23), by integrating first over t" and .then using the fact tha t 0 function of x - t", we obtain 0 c(x ,xCL) is a 59 "'4nm A · f ~"(t") 2 + ")(r")~ H(r"-r) G(x,x"}_!__ 6 (- .!. 3 s 0 CL ret 2 <lt t "( r ~1- ~2 J G(x,xCL)+ t(t") +t" (r s 2 ·= mA "° H(r"-r ) J · (r") 0 3 00 =- -2mA = -2m (IQ. "ox 2 11 11 ) 2 2 J 2 ] .... a~" 0 c(x,xa) 0 (t") 2 .t" +.!.r" )dt"r" dr" l111 r"dr" r"=t" H(r" - r ) J--,~ (0 G(x,x'CL' )) t". ..r" dr" r 0 _a_ Joo [OG(x,~xCL)J A <lxO r r dr" (6. 27) t"=r" s We now change our origill of spatial and temporal coordinat es so it i s centered O' # 0 , and use the relations (Fig. 7b) x on st a r A with l~'I + a , r" l~I~ x= (6. 28) O" XCL - ·x O' = t" 1~· I + a , r" . II ~CL= -an' plus expression (4.1) to write J .OG(x,x")eA(x" ,x I) oG' (x" ,x I) i'x" 1 "' _ mA 21T "ox0 mA 0 O" 2 _a_ J"" 6 ret [- Z(x -xcr) a 1 " 2 + Z (~ - ~CL ) ] da I~ I + a I 0 H[(x. - ·x 0 O' I I +~ a) - [ - -l (x0 -xO" ) 2+. -1 (x-x .. - 21T 0 ax { [ ( x I - J )-d~'I+ l~I)] da 2 .CL . . 2 - II ;:cI.. ) 2.1 ] 0 x -x 0I I I. - x -a I al~I+~·~· (6.29) } • 60 By invoking expressions (6.28),. perfonning the d ifferentiation with respect to a, setting xO - and using x 0 l~I = l~I "' (re tarded time as measured by A) = TAo , (6. 30 ) RA (aside fr om fractional errors 'V l/RA), we bring this into the form a(l + n • n') =TA - x - - O' 0 (6.31) We now evaluate .,-i.t\I 2 h'fL,B by inserting (6.31) and (!1 . 13) into (6.2lb), and int egrating out the o (x"-zB). 4 Th e result is (6.32) In t:~e rest-frame of A the zero-order world line of B is (cf. Fig, 3 and cqs. 3.5) (6.33) By putting this into (6,32). 3 and integrating out TB • using the geometr~c relationships of Figure we obtain 61 I-B z I (6. 34) 0 where TB Cx ) is defined by 1 (6.35) Notice that expression (6.34) for the tail of B is identical to expression (6.13) for the transition of A --except that it is evaluated at a diffcrent time TB! Just as we evaluated (6.13) using 4-vector techniques, r.o we shall evaluate (6.34) using 4- vector techniques. We begin by noting that relation (6,35) says that to reach TBl on the world line of n. one should (i) begin at the field point x, (ii) go b a ck along the past light cone of x until it hits the world line of A at zA (TA ) , (iii) go back along the past 0 light cone of zA (TA0 ) until it hits the world line of B at 'Bl' Comparison of this prescription with Figure 4 shows that TBl occurs at the tip of the -+ -+ vector M --and that therefore (since J is purely spatial in the rest frame of A) (6.36) J By com.bining this relation with (6.33), (3.16), (3.Sc). and (3.7a,b) we obtain 2 2 . x (ZB ) Iz I + ~B • ~ "' vy .rz:T + n x v y TB -B --B (6.37) .,, () L and by combini ng wllh (Ci . 35), .(6 .3 3), (3.16b), and (2.J) we obt a in (6.38) ..... In the r es t fr mne 6 f s t a r A where J is pure ly s pa tia l, equat ions ( 3. Sb ) a nd ( 3 .12)-(3 .15) imply RA(.11.A+ Sn) 1 1 Iz13 I + : 13 • Q l)i S2 "' RA [£.A -(ctl.'A+--~-~? ~ J (6.39) 52 By combining equat ions ( 6 .37) - ( 6 .39 ) with ( 6 .3l1 ) we obt ain -µv 2h.fL ,B (6.40) The a na logous e:xpres s ion f o r .,.-ll\I 2 hTL,A can b e derive d e ith e r by a n a nalo- gous c alculation in the r es t f r ame o f B, or by invoking t he inva r i a nce r e l a t i on ( 3 .17). The t o t al t ail f ield i s t h en (6.41) This tail field, like the trans ition field ( eq. 6.19), diverges as the obs erve r moves into the sh a dow r egion-- and in just th e same manne r ( eq. 6.20). G3 c) Sum of Foc usin g , Tat!_1_~md Trans ition Fie ld s J,et us add the tail and tran s ition fields (6.111) and (6.19). In doing so, by invoking equations (3.llb), (3.7), and (3.Sd) we can show that the second term in · each {}of ( 6.41) cancels the s econd term in ea ch { } of (6.19). As a result, the sum of the t ail fi e ld, trans i t ion fi e ld, a n d (va nishing ) focusing field in non-shadow regions is (6.4 2) By the argument a t the beginni n g of §VI, this formula is also valid in shadow regions. Because of its validity in shadow regions, expres ,.ion (6.4 2 ) must not contain any of the n e ar-shado;o1 dive rge nce (cq. 6. 20) that characterizes · the transition and tail fiel ds individually. The dive rge nce in the tail must precisely cancel the di v e ree nce in the trans ition. That thi s is indeed true one can verify using e xpre ss i ons (3.7), (3.10), (3.llb), and (3,12b) in the neighborhood of the shadows --i.e., at lol«1 , lab - BTBI << b . (6. 43) 111is kind of cancellat i on between transition f i eld and tail field has shown up previously in · computations of . electromagnetic radia'tion reaction for charged bodies moving in weak gravitational fields ( Rudolf 1975) . We suspect that the c ancellation may be a rather g e neral f e ature of computations with Green's functions in we a kly curved space time. the' warning at the end of §IV of Paper I: This stren gthens One should not attribute great physical significance to the individual pieces of a weak-field Green ' s 64 function or to any single one of the five pieces of our gravitational field ~v zh . . Only the full Green's function and full radiation field are physically meaningful. VII. WllUMP FIELD a) \..'hump Field Expressed in Terms of Whul!l.P Integrals The whump field (4.4e) is an integral of products of first-order fields. Only cross products (field of A tic1es field of B) contribute to the radiation; "AA" products and "IlB" products are nonlinear corrections to self fields and can therefore be ignored. Using this fact, we can rewrite the "whump stresses" (eq. l1.6a) as · ( 7 .1) By combining this expression with equation (4.llbJ for lhKaB and then inserting it into equation (4.4e) for the whump fi e ld, we obtain .... RK .... - rspatia l distance from x' to unperturbed = ~(x') =.KK' uK - .\world line of K as measure d in rest frame of K (7.2c) i.ivaByJ..pcr By invoking expression (/+.6b) for H , per.forming the contra c tion of indices with the u's and. using equations (3 . 2) for the scalar products of the u's, we bring (7.2a) into the form + 2uA<l1uB">r;·e· ~A-yePPJ-nµv[y<~B·e·~A) +tczi- l)ePPJ} . (7. 3) ) TI1e "whum? intcgr.::ls" (7. 2b ) satisfy two important identities. Because. in the r es t fra me of star K they s a tisfy + u A • (7.4) 0 Through much of the rest of this section we shall work in the rest frame of star A. where 1 , In this res t frame ·he eoa = o ~A= 0 , (7. 5) yve -x id~ntitl es (7.l;) become -v e (7. 6) ax The bas is vectors of thi s coordina t e system are _,. e (7.7) By combini ng equations (7.11), (7.6) and~with -~ y •-• (7. 7) = }' cxB + e = e + 'ca ® e~ we obtain the relations (a UA f3 ) UB (7 .Sa) 66 (7.Bb) (7 .Sc ) a e a ( 7. Sd) e xx + e YY + e z z =e j j lly inserting these express.ions into (7.3) · we ob t a in + uA 2 2 Ci1 . v)[-xcz-v )(1-3v ) eY.X UB 2 21T v 2 /1 y ~] 211 4 + Zq(µu v) [ ·- /C2-Jv -v ) ezx _ /c1-Jv 2 ) A 2v 21r 2v ·(e,,.z -211ezx)~IJ (7.9) b) Trick for Evaluating Whump Integr als Our task .now is to compute the s patia l whump int e grals e jk them into e qu a tion (7.9). and insert In comp ut i ng Cjk we s h a ll u se a trick s u ggested G7 by the method of Peters (19 70 ). We now de rtve the trick: As in Papers I and II, de fine t he Green's function for the space time with me tric (7 .10) by ' 1D s cA ( x, x ,) ( ae n - l a!3) G - l 1A A, a$ Split GA(x.x') into a !lat-space part plus a perturbation due to the field lhAcx$ G A = (7 .12) and invoke the !lat-space equation na!3 G o ,ae c -6 (x-x ') 4 to bring (7 .11) in to the fonn (7 .13) Now apply the differential operator ay + ay' to both sides of this e quation and use the fact that 0 G is a function of xy- xy' (so [a+ a ,] c ~ O) to . y y 0 obtain (7 .14) Next apply -aA, to both s ides and u s e -aA, 0G = aA 0c to obtain (7 .15) 68 Then multiply both s ides by 161' TBlJ\l(x') and integrate ove r x ' using (7 .16) to obta in h aa h lJ\I 1 A ,y 1 B ,Aaa llf{-167T f 4 [((\+Cly,) a )., 6GA(x, x ')) TBll\l(x ')d x •} (7.17) Then inve rt thi s equation u sing the flat-space Green' s funct ion to obtain (7.18 ) Here the primes on the indices on the l eft-hand side indicate th at the are evaluated at x ', not x. Next rewrite the left-hand side using the gauge condition 0 + (negl igible tenns quadratic in masses) and using integrations by parts and using a ()a' 0c = .-aa 0c , to ob tain f (lhA- a'a' ,y • -hBµ'\)' , >, a · a· 0 c(x,x')d'•x' .I 1 A· 4 - µ'v' - a'l3' AI) aa,aa, 0 G(x,x•) d x • (lhA ,y' lhB .. aa a13 J • h"• s 1 69 1ben invoke the relation ( e qs. 4.l l b and 7.2c) (7.19) and combine with equations (7. 2b) and (7 .18) to obtain (7. 20) Then ins ert the stress-ene rgy tensor for B ( e q. 4.13) and integrate out the x' on the right-hand side to obtain 4iT 2 f [(Cl+ Cl ,)Cl" liGA(x,x')) '= · y y . I\ . x ("' )dTB' • ZB 'B (7. 21) ' Finally~ specialize to the rest f rarae of A and thereby obtain (7. 22) the perturb a tion in the Green's function i s easily d e t e rmined hy comparing equations (7.12) and (l1. l) with eq u a t ions (55) and ( 5 6) of Pap e r I: -If l Joi I- (x) \ i f x' ' 1 G(x , x") SA(x" ,x') c• (x" ,x') i'x'l r if X E I-(x) r 0 0 (7. 23) Here x'e 1-(x) means . that x' li es ins ide and n ot on the past light cone of x. We shall res trict ourselves to field points x wh ich .are · not i n shadow r egion s . (As with the focu s ing-plus -transition-plus tai l f iel d (§VI), the analysis of Paper II guarantees that the W'hump f i e ld in s h a dow regions is 70 the analytic extension o f that in n earby non s hadow r e gions. ) The r es tric- tion to nonsha dow regions allows us to set a A(x,x') = 0, and to invoke equations (6.8), (11.2), (6.5), (6.31) a nd (6.30) to obtain ___!. 6GA(x,x') U:A O' 1 + 2rr ao [ ll(TAo- x (TA - x o• l~'IJ - J- (7. 24 ) + x' •n)RA 0 - - In the ·first line we can bring the differentiation all the way to the left by invoking the fact that (l/XO) a XO = O(l/R ) = ( quant~ty that can be dropped ~ithout) 0 A cha,..ging radiative part of field • (7. 25 ) We can also u se the relations (cf. Fig . 6 and e qs. 4.1, 6.5, 6.30) 2 0 2 G(x,x') = .!_ o [- .!.cx ) + .!. lxl J 41T ret 2 2 O 0 X - XO "' lx_I "' x0 - I~- I-x O' + x'•n "'. T Ao -xO'· +x'·n (7.26) l~I "' l~I " RA +(fractional e rrors ex: l/RA) Ix' I !xi + x' • X = R. <Ix' I + x' • n)+ (fractional errors A- ct: l/R.) A to bring (7. 24) into ·the form O' 21T I - M (x x) = m A • A o(-rAo - x -ao { . H ( T Ao - x (TA - x o +_ :::_'_· -~)- 9, RA O' O' - Ix' I) } + x' •n)R - A We can then us e the fact that Il:'I+:::'. ~ I lOb n (7. 27) 71 -n + n. j J 0 plus relations (7.26) to writ e o("C Ao - x O' + x' • n) RA(lx' I+ '.:: ' •n) j (n + n ., J ) (7. 28) and we can combine this with (7.22) to obtain J{ j j ' O' a,,[-(n~+~n~)-o~(-'~Ao~-~x~+~x_ '_·_n_) RA<l::;'l+x'•n) 11. (Note that it does not matter whether we put our spatial indices up or down: sj = sj for any ~.) This is our "trick equation" for evaluating the whurup 'int egral s. Its power lies in the fact that it involves only a single line integral along the world line of star B. whereas the original whump inte grals (cq. 7 .2b) involve extremely diffcult 3-dimensional inte grals over the obsE!rver'.s past light cone . c) Evaluation of Whump Integra l s For the s pecial case of ejx = -v -1 ejO we can use the fact that the [ J quantity in eq . (7.29) is a function of x 0 - x O' to rewrite (7.29) as "' 1 ·a a ( ~) 21r .oo v 0 0 J{ j (n+n . j' -x )o("C Ao O' +x'•n) - RACl:::'I + x.' •n) 72 We can then integrate out the delta functions and wipe out the a a on 0 0 each side of the equation (the inter.ration constant i s zero bec aus~ ejx vanishes in the limit T 0 zB - z -B • n] - Ao ~ -+ 00 ) [T and we can use the rel at ions , 1 B ~' -= ~B li es at the Bo] <"~~ [ tip of ~ (cf. eqs . 6.12. 6.30, and 6.35), th~reby Sin FJr,. 4 x' = z lies at the -B. ~ [ tip of M jn Fig. 4 B Bo J (7.30b) ) J + nj' (jx' j + x' • n)dTA /dTBl A o nj (7. 30a) ohtaining x' .. z (T - [ vR J (/.31) For the more general case, expression (7.29) reduces to j I j (n + n ) + R (T A Ao k' O' o(TA0 - x O' - x + x' • n)' 11 - By integrating out the delta functions and wiping out one this into the fonn 3 we bring 0 73 ~[ ojk- n j' n k' (7.32) Notice that the trace of this e qua tion, after int egration over time , reads (7.33) In equations (7.31), (7.32) and (7 . 33) it is h elpful to not e that, for TB= 'no or TH= TBl and f or ~· = ~B (TB) • nx+ nx' 1 dT Ao /dT B ( I~' I + ~' • ~) _ <lTII · _ d_ 2 - dT dT Ao B [ •1/vy Ix' I + x' • n_. - (cf, eqs. 6.30 and 6.33), and that similarly J 74 J • · j' ox·J - n x' n j ' (1/+ nx ) (nJ+ n· ) _d_T_/~d-T- [ Tx' I (Ix' I + x' •n) (I x ' I + x' • n) 2 Ao R _ _ 1 (7. 34b) In particular, by combining (7.34b) with (7.32) and integrating over time, we obtain e2j_ _ [ 21f- n j (nx+ n x' ) + (n j ' + n j)/ 1J.::!J__ RCix'l+x'•n)dTA/<lT RA(jx•j+x'•n) A _ o Bo - J x'=- zB(TB ) 0 (7.35) In the first bracket d(zB 0 - :B · ~) I dTB x y(l - vn ) (7. 36a) in the second (7. 36b) (cf. eqs. 7.30 and 6.33). -[ Combining these relations with (7.3 5) , we obtain x'= z (T D Bl (7.37) ) Equations (7.31), (7.32 ), (7. 33) , and (7.37) are somewhat formal expressions for the whump integrals. We now evaluate them explicitly in terms of --. 2 the same parameters (a, 8,o ,a, 8,o ,v,y,b,TA,TB,£.A,£.B,S ,RA,~) as appear in 75 equ ations (5. 7) an<l (6.11 2) for the <li::ect , focusing , traP.s it lon , and tail fiel els • T~e whump integrals Cxx' Cyx' and Czx can b e evaluated hy inserting the vect ors of Figure 4 into exp r ess ion (7.31). The relevant quantiti e s for ins e rtion are For n x TB= TBo --i.e., for x'= zB at tip of S B = a n x y' z "' (7. 38a) y y x b • z' 0 • 1~· I -+ Iy I = 9,B ; (7. 38b) n y' b/ 9.B = n z' (7. 38c) 0 (7.38d) · cf. cqs. ( 3.Sc), ( 3 .16a), and (7.36a ). -+ TBl --i.e ., for n x x' B• Ct x . :: J n . y' n dTA/dTBl y[l + vn A y' b • M = y(t - vTA) A A x' = Jx/M n (7, 39a) x l~I = zB a t t i p of M z x I~' I x' = Jy/MA • x' ] n z' tA/MA cf .. eqs. (3.Sc), (3.16b), and (7.36b). x z' 0 (7. 39b) 0 (7, 39c) (7. 39d) 76 By inserting th ese quantities into expression (7. 31} and ci1e n invoking cqs. (3.7), (3.10)-(3.12), and (3 .16) we obtain the following: e b(ab-BTB)' b(ab - 2 vS ~ iB vS 2 "Br A) (7. 40) RA iA TB (ab - BTB) ~ = - -""---___.;::- 2TT vS 2 ~ iB 2 2 2 y{$(l+av)TA + [vS - a(l+av))bTA - aSvb } 2 (7. lil) vs~ tA (7.42) By combining relations (7.31) and (7.37) we obtain (7.43) By then inserting expressions . (7. 38) a nd (7. 39) into (7 .113) and by invoking (3.7) 1 (3.10)-(3.12), and · (3.16) we obtain l'-xy _yx = (ab-BTB)( aTB+ Sb) 2TT S2~ R,B (7.44a) 77 exz - ezx in We next insert expressions (7.38), (7.39) ancl (3.7a) :1.nto (7.33), obtaining ~ 2Tr = eXX+ eyy+ e Z7. We next turn to e 0 ~x /dTAo (7.45) 2iT zz According to (7.32), with n z' 0 and = 1, it is given by (7. 46) The integral that appears h e re cL!n be evaluated using n (a,13,o) and Ix' f CnYnx' - nxn y' ) (7 .47) .R bvyC!x'l+x' •n) A By combining this with (7.46) and invoking (7.38) 1 (7.39), (3.7), (3.10)- (3.12), and (3.16) we obtain 78 2 [-(1- av)( ab - $T ) + 0 bv] B (7. 48) By combining this result for with (7.110) for ezz with (7.45 ) for exx+ eyy+ e zz, exx'and with (3.12b) for s 2 we obtain (ab-BT 8)TB[TB- v(aTB+ Sb) l (7.49) ~£.BS bv 2 (ah-ifrA)T [T + v(aTA+ Bb) l A A We turn f:i.n ally to 0 dx /dTAo = 1 eyz+ Czy (7. 49) , From e qua tion (7 . 32) wi t h n z' and = O we have (7. 50) The integral th a t a ppea r s h ere can b e e valuat e d us ing n = (a, 8 ,6) a nd 79 x x' x' y 2 x y y' 2 I~· I [n + n - n (n ) + n n n ] RA bvy nzCl;::'I (7. 51) + x' •n) By combining this vith (7.50) and invoking (7.38), (7.39), (3.7), (3,10)- (3.12), and (3,16) we obtain (7. 52) 11iis complet es our evaluation of the vhump integrals. &_f_i_nal Answer for the 1-lhump Field By inse rting e xpressions (7.110), (7.41), (7.42), (7.114), (7.45), (7.48), (7.119) and (7.52) for the whump integrals into equa tion (7.9) for the whump field, we obtain 80 +)) (µ v) { uB 2 - + . 2 2 v S RA.2.A [- 2 4 2 BTA-a b][(2-3v ·- v ) TA+ v(l-3v )(aTA+ Bb)] · 2 (BTB-ab][2(1-})TB - v(l+v ) 2 2 v S ~.2.B q(µu v){ A (~T8+8b)]} 2 2 2 [2(1-v )tA +v(l+v )TA(aTA+Sb)] y"'6 v2S2RA.2.A (7.53) R. A + qµq vy 2 (l+v 2 ) { - 2 · [-(l+av)(STA-ab ) + -2 6 bv] bvS RA 1 + B 2 bvS ~ [ (l-av)(ST -ub) + 6 2bv) } 8 . 81 TOTAL FIELD /u~D GRAVITATIONAL--WAVE AMPLITUDES VIII. The total gravitational field h11v in the radiation zone, with the non- 2 radiative "coulomb" part removed, is obtained by adding together the di rect field (5.7), the. focusing-plus-transition-plus-tail field (6.42), nnd the whump field (7.53). The result is: (8.1) The coefficients which appear here a re as follows: 2 2 2(1-v ) + av(l+v ) H M = 2 v (l+av)RA£.A (8. 2a) 2(1-v2 ) - av(l+v2) ·2 v (1-av)Ril'B (8.2b) 82 ( 8. 2c) 2 2 - - -'--- (Ctb-arAJ[z (l-v ) TA+ v(l+v ) (UT A+ab)] 2 2 v S RAiA + 2 i2 2 4 2 [cxb-RTB][(2-3v -v )TB- v(l-3v ) ( ctl'B+Rb)] (8.2d) v S P'B.2.B (8.2e) (b.2f) (8.2g) 83 {ab-BT_../rA "' X2 (l+v2 ) { bv l\b 2 [TA+ v(uTA+ ~b )] S RA'),A (ab -$Tl'l)TB + [TB- v(aTB+ 13b)]} (8.2h) S21);'),B _ y2 (l+v2)o s ~ 2 {b (aTll+f3b) y2(l+v2)'l,A H qq (8.2i) '),B [(1-Kiv) (ab-BTA) 2 +62bv] bvS RA 2 2 y (l+v )'l,B 2 - - - - [(1- cxv) (ab-Wr ,) - 6 bv] 1 bvS 2 ~ ' H n "' -2bvy ( 8. 2j) 2 (8. 2k) All of the quantities that appear in these coefficients are defined in §III. One can easily v e rify that the gravitational field ( 8 .1) possesses the star-int e rchange proper ti e~ of e quat ions (3 .17). In the radiation z one where this f ield exists , the post-linear gauge condition -µv = 0 h 2 '\) reduces t ·o (8.3) where ...,. k is the propagation vector of equations (3. 5) and (3.6). It is straightforward but tedious to verify that our gravitationa l-.,.•ave field (8.1) satisfies this gauge condition. 1~e verification utilizes eq u ations (8.1), {8.2), (3.8), (3.7), (3.12b), and the following relationship [which is derivable from equation (3.S a) or (7.7), and ++ ~ .... ...... -~ -~ + _. + n ""-eo®eo+e x ©e x +ey ©e y +e z ©e z ]: 84 (8.4) It is also straightfor..rard, but tedious to take the slow-motion limit of expression (8.1) in the rest frame of star A and verify ·that its spatial components, 2.6). 'hjk' agree with the slo·..r-motion field derived in Paper IV (eq. 2 In taking the slcrw-motion li.mit the following approximate relationships [derivable from eqs. (3.Sa), (3. 7), and (3.10)-(3.12)] are helpful: ->- TJ "' c y a - a vet$ , ...q ->- e 7. 6 - 6 - vctti (8. 5) Here the quantities on the right-hand side of each expression can be interpreted as iA + .tB ' (8.6) · with fractional errors of O(v). In any Lorentz frame the rodiation fiel.d is obtained by projecting out the transverse-traceless part of hjk (see Box 35.1 of MTW). Le t us perform 2 such a projection in the rest frame of star A--which has basis vectors (8.7) (cf. eq. 7, 7). For the purpose of the projection we shall introduce spherical polar coordinates (r,0,~) centered on star A with the polar.axis along the direction of motion of B (x direction) 1 and with (0,~) a (~/2,0) 85 along the impact direction (y direction): x = r cos e , r s in 0 cos ¢ , y z = r sin 0 sin <!> ( 8. 8) The orthonormal basis vectors associated with these s pheri ca l coordinates are (8 . 9a) -+ -> -+ e,.. _ n r e cos x ..... e + ......e y sin e cos- <P + ·>-e z sin 0 s in <I> (8.9b) - >- 4- -e sine+ e cos e cos¢+ c cos 0 sin <I> -> x y -+ -> y z (8.9c) z e,._ c -e sin <I>+ e cos <I> <I> The ob se rve r, locat ed in the ( 8.9d) -> -+ n direction, uses the basis vectors r to characterize tra nsv e r se direc tions; and he uses their tensor -> and c,._ c" <I> products .... -·~ +> ex .. c,._ @ e,._ - a o (8.10) as the polarization basis states for the gravitational waves: ...... -TT 2h (8, 11) By applying the standard tr ansv erse- trac el~_ss projection operation to \ijk 2 (Box 35 . 1 of MTW) • we obtain the following expressions for the wave amplitudes A+ and Ax : -> A .. 21 (ea + 1 . 2h . e +-> -+ .._.. -> e,.. - e,._ <I> 2h . ..e,..) ....... . . . ~,;..)e 2h . e~ + e ,.. 2h 0 Ax .. 2 (;,.. ..- > -+ (8.12a) <I> -+ <I> (8.12b) 86 These expressions are mo::it easi ly evaluated in the r est frame of star A + + + . -~ ++ (basis vectors eo, e x , . f~ y ' ez), using C'quations (8.1) and (8. 7) for h and 2 ->- a> equations (8.9c,d) for e~ and e~ • The result is (8.13a) where RA is the distance to the observer in the rest frame of star A (which we are using). and a+ and ax are dimensionless amplitudes given by - cos OsinOcoscj>vyllbll - cosOsin8sincj>vyHqB ( 8.13b) (8.13c) The dimensionless ampli_tu<les a+ and ax are functions of the ve:.o- city v and energy factor y = (1-v2 ) -1/2 of star B, the impact parameter b, the observer's angular location (8,¢), and · the observer'~ proper time One can express a+ and T • 0 ax explicitly in terms of these parameters by making use of the direction-cosine relations a "' cos e • 6 .. sin 0 cos cj> , 6 "' sin El sin ¢ the equations of §III, and expressions (8.2) for the H's. (8.14) 87 In Paper IV we shall examine the detail s of the radiation field (8.13) and its associated spectrum; and we shall compare it with the various special cases that have been co~puted by other r e searchers. While carrying out this research we relied heavily on insights gained from close scrutiny of Peters (197U) bremsstrahlung calculation. Peters' formalism and approximation scheme were different from ours; but they were sufficiently similar to give us insight. 88 REFERENCES Crowley, R. J. and Thorne, K. S . 1977, Ap. J., in press; also available as Caltech Orange-Ai.d Preprint 450; cited in text as Paper I I. D'E.ath, P. 1977, paper in preparation at DANTP, Cambridge, England. Estabrook, F. B. and Wahlquist, H. D. 1975, Gen. Rel, Grav., Kovacs, S. J. and Thorne, K. ~· 439. s. 1977, ~·· in preparation; cited in text as Paper IV. Mashoon, B. 1973, ~., ~~~· 83. Misner, C. W., Thorne, K. S., and Wheeler, J. A. _1973, Gravitation (San }'rancisco: W. H. Freeman and Co.), cited in text as HTW. Peters, P. C. 1970, Phys. Rev. D, ~· 1559. ,,. Rudolf, E. 1975, Ann. Inst. Henri Poincare, ~~· 113. Thorne, K. S. and Braginsky, V. B. 1976, Ap • .l. (Letters), 20 4 , LL Thorne, K. S. and Kovacs, S. J. 1975, ~·· 200, 245; cited in text as Paper I. 89 FIGURE CAPTIONS Figure 1. (a) Spatial diagram of unperturbed stellar encounter in center- of-mass frame. Figure 2. (b) Spacetime diagram of same encounter; The spatial part of the 4-acceleration of star B0 as seen in the rest frame of star A. The equation describing ~B is [cf. eq. 2.27] Since yvTB is distance travelled by B, the acceleration is directed toward the focal point shown in the figure. In the limit v << 1 the focal point coincides with the location of A; but for y >> l it is far on the other si('·· of A. The vertical component of the acceleration is proportional to the inv.erse cube of th~ distance from A to B. Note that as y ~ ~ the acceleration becomes more and more orthogonal to the trajectory of n. Figure 3. Spatial diagram in the rest frame of A showing the direction n to the observer with its direction cosines a, $, 6; and showing the vector b that reaches to the point of B's closest.approach. and the spatial part of the velocity of B, ~B • Figure 4. (a) Definition of re·tarded times TAo and TBo. -% ~ (b) Definition of ->- several vectors :s, M, J to be used in th~ derivation of the radiation field, Figure S. A diagram showing the conical region of spacetime where star A shadows star B. 111is di~gram is confined to the future light cone J+(ZB) of a specific event ZB on the world line of B. In other words, th~ one spacetime direction suppressed from 'this diagram is the direction lead- . ing off J+(ZB). The coordinates x,y,z used in the diagram are the spatial coordinates of an inertial frame of star A. 90 Figure 6. Spatial diagram of the geometry us e d i.n cor.iputing the time-delay function yA(x,x') in the rest frame of star A. The origin of the spatial coordinates is at the center of star A. Fi.gure 7. f 0 The spatial geo::ietry used in C!Valuating BA (x'' ,x') and G(x,x") SA(x",x') 0 c'(x",x 1 )l x". 1 In equations (6.22)-(6.27) we use the coordinates and parameters of diagram (a). switch to the coordinates of diagram (b). Thereafter we 91 T,B =0 2 .Us b x ~ ,_.c 3 2 O=rA (a) (b) Fig. 1 92 0 2. 3 b focol point Fig. 2 93 z --------------A'I n y O' a Fig. 3 x world line of observer unperturbed world line of A (a) B (b) 95 Fig. 5 96 -L Q) > L (1) (f) ..0 0 c 0 ( /) -0 0 .___.. -L x~ ~t 111 x~ Q) > L a.> (/) . ..Q 0 c 0 ( J) ·-0 - 0 .___.. Fig . 6 97 --. ..0 II xl _J :::::u .._ -->< >< ---0 Fig. 7 98 PAPER 3. 99 THE GENERATION OF GRAVITATIONAL WAVES IV. BREMSSTRAHLUNG This is material that will ultimately be a joint publication with Kip Thorne . *·~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ Supported in part by the National Aeronautics and Space Administration [NGR 05-002-256 ] and the National Science Foundation [AST75-01398 AOl]. 100 ABSTRACT This paper uses the mathematical formalism and equations derived in the first three papers of this series to give the results of our analysis of gravitational bremsstrahlung. We develop some new insights about the detailed structure of the waveforms and compare our results with those of other investigators. The key features of the analysis include a) the need to use not one, but two naturally occurring time scales, b) the symmetry property in the center-of-velocity frame, that reflection through the origin leaves the magnitude of the wave amplitude~ unaltered, c) fairly simple analytic formulae for the amplitude as a function of time, direction, and polarization, d) proof that in the relativistic case there is no logarithmic dependence on r in the total energy radiated, e) analytic formulae for the power spectrum in the extreme forward direction, valid for arbitrary velocities and £) expressions for the zero-frequency limit. 101 I. INTRODUCTION We anticipate somewhat optimistically perhaps, that the de tection of gravitational waves will become a reality by the mid l 9BO's. In this spirit we have embarked on this series of papers dealing with the generation of gravitational waves. Our aim is to elucidate the realms of validity of the present computational techniques and to try to devise new techniques valid in new realms. Our ultimate purpose is to be able to calculate the waves generated by models of astrophysical systems. The literature is replete with references by many researchers on the topic of gravitational radiation and a thorough review of computational techniques has been given by Thorne (1975 ). The foundations for this paper have been laid down by the first three papers in this series (Thorne and Kov~cs 1975, "Paper I"; Crowley and Thorne 1977, "Paper II"; Kov~cs and Thorne 1977, "Paper III"). In this paper we give the results of our detailed analysis of gravitational bremsstrahlung. The context in which aspects of this particular problem have been investigated previously has varied widely. The various approaches in- clude the use of Newtonian theory, Turner (1977), Hansen (1972 ); postNewtonian theory, Wagoner and Will (1976), Turner and Will (1977); the "quadrupole-moment formalism," Ruffini and Wheeler (1971); perturbation of the Schwarzschild geometry, Peters (1970), Misner (1972 ); quantum gravity, Barker and Gupta (1974), Smarr (1977), Feynman (196 1); the method of virtual quanta, Matzner and Nutku (1974 ); the method of colliding plane waves, D'Eath (1977). 102 Nowhere before has there been a detailed analysis of the wave amplitude as a function of time, angle and polarization valid for arbitrary velocities and unrestricted mass ratios. Our analysis is valid for weak fields only an<l also requires that the angle of deflection b e sma 11 compare d to 90 0. There is no hope of detecting gravitational radiation from any astrophysical system by surrounding the source with detectors covering 4n steradians. Therefore we focus little attention on the total gravitational wave luminosity LGW or the total energy radiated per hertz dE/dv - instead we concentrate on the structure (i.e., time, polarization and frame dependence) of the amplitudes since the amplitudes are the quantities that gravitational wave astronomers are seeking to detect (see MTW, chapt. 37). This paper is divided into six sections. Section II gives the well known low-velocity limit and defines our notation and coordinate system. Section III discusses the frame dependence of the amplitudes, the symmetry properties in the center-of-velocity frame and the need to use two time parameters TA,TB to describe the radiation. Section III concludes by giving expressions for the wave amplitudes, [eqs. (3.17) and (o. 18)) and their symmetry properties [eq. (3 . 20 )], in the frame of star "A" as a function of time and angle. Section IV analyzes the high velocity limit of the amplitudes and breaks the radiation zone into three overlapping regions of interest: the "forward-region," the "intermediate-region" and the "backward-region" [eq. (h.l)]. Section V is devoted to the frame and time dependent features of the wave forms and includes a comparison between our work and that of Peters (1970). 103 Section VI contrasts our results with those of other authors and gives an expression for the specific flux (eq. ( G.9)], valid for arbitrary velocities in the extreme forward direction. Section VI concludes by giving the zero-frequency-limit of the radiation. 104 II. LOW-VELOCITY LIMIT Consider two objects (stars, planets, black holes ••• ) which undergo a near encounter. Require that their interaction be entirely gravita- tional; and for specificity, call them "stars." Require that the stars' sizes rA and rB be small compared to their impact parameter b rA << b, rB << b so that tidal interactions can be ignored. (2.la) Also require that they move with low relative velocity v, and that their impact parameter be large enough to guarantee only small deflection of their orbits: (2 .lb) The assumptions v << 1 and b >> (mA + ~)/v 2 >> (mA + mB) ("slow motion" and "weak relative gravity") guarantee that the orbital motion can be analyzed with high accuracy using Newtonian gravitation theory, and (by virtue of 2.la) using a point-particle description of the stars. This is true even if one or both of the "stars" is a black hole (D'Eath 1975 ). Moreover, these same assumptions guarantee that the gravitational waves emitted can be computed with high accuracy from the Newtonian motion using the standard "quadrupole-moment formula" h j kTT :::: ( 2/ r) Yj k TT ( t- r) (eq. 36.20 of Misner, Thorne, and Wheeler 1973 ). ( 2. 2) Again this is true even in the case of black holes (Thorne 1977b). The computation of the orbits and waves is so straightforward that we shall not give the details here, we give only the final result for the radiation field. 105 We shall des cribe the final result using Cartes ian coordinates (t,x,y,z) in which star A is initially at rest at the origin; the moment of closest approach is time t = o, the orbit of star B lie s in the x-y plane with initial velocity in the x-direction, the unit spatial vector pointing from the origin toward the observer is n = cos $ e + sin e cos CD e + sin e sin Cfl e 1 ...x ' -y -z (2 . :5 ) and the observer is in the radiation zone, a di~tance r from the origin. (See Figure 1, and note our unconventional relationship between Cartesian and polar coordinates; a more conventional approach would make the notation change e · -y e - -x e , -z e - e"'Y .) -e x - -z' We shall decompose the radiation field into two orthogonal polarization states with basis tensors x e :::: e . . . e ® e-qi +e-qi ® e-o (2 .4a) where ~f) and :cp are unit vectors in the e and Cfl directions :e = r -10I ce = - e-X sin e + e. . .y cos e cos Cfl + e...z cos e sin cp ' ( 2 .4b) e - (r sin e)-cp 1 o/ocp = - e -':I sin Cfl + e_z cos cp • (2 .4c) In describing the time dependence of the radiation field we shall use retarded time t - r, measured in units of the time b/v for s tar B to travel a distance equal to the impact parameter: T o= (t - r) v/b: ( 2 . 5a) and we shall use the function £ ( T) - (1 + ·r2) 1/2 ' (2 . 5b) lOG which equals the distance between the two stars at "time" T, measured in units of the impact parameter. In terms of the above quantities, and to first order in the deflection parameter c, the gravitational-wave field (transverse-traceless part of the metric perturbation) is given by (2.Ga) (2.6b) r J cos e sin e cos cp ' + 3'l' + 2T + 1 ... £ a, ~ ~3 cos 0 cos 'I' sin 'I' - [~3 + 7+ 1J (2.6c) sin e sin 'l' • Herc A and A arc the amplitudes of the two polarization states. + x (2.6d) Note that their magnitude is l~mAmibr, which has the physical significance an<l size r = = _ relative gravitational potential energy of stars at moment of closest ap roach distance from stars to observer gravitational potential at observer's ) 4 (Newtonian location produced by stars' relative potential energy (2. 7) Here mA and mB are measured in solar masses, !vb= the impact parameter b is measured in solar radii R : and distance to the source r is measured 8 in units of the distance to the center of our galaxy. Perhaps it is not totally hopeless to think of detecting such radiation in the 1980 1 s (Zel 'dovich and Polnarev 1971~). 107 The time dependence and angular dependence of the radi a tion field are contained in the r enormalized amplitudes a and a . + order unity for !Tl ;S 1. x They are of Equations(2 . 6c,d) express them in terms of impact parameter, b, velocity times retarded time, v(t -r) =T, distance 2 1/2 between stars, £:: (1 + T) , and observer's angular location (B,cp). Notice that, the radiation pattern ( 2 . 5 ) is totally insensitive to the relative masses of stars A and B: it depends only on the product of their masses and the amplitude is not frame dependent in the low-ve locity case. Moreover an observer at e < rt/2 (toward whom star B moves in a frame where star A is at rest) sees precisely the same radiation features as an observer in the opposite direction (toward whom A moves in a frame where B is at rest). This fact is embodied in the parity properties of a+' a)(, and the polarization tens ors: a If (e , cp) ·> (rt-e , n+cp): -> + e ->- e a+ ' -+ -+ (2 . 8) e -> -e ax -> a>< ' ->< . ->< Figure 2 displays the time behavior of a and a for several loca+ )( tions of the observer. Notice that, by choice of convention, the ampli- tudes begin at zero for T ->- -ro, but, except in special locations, they do not tend to zero for T -> +co! Since the displacement of a free-mass detector is directly proportional to the amplitude of the wave, bremsstrahlung radiation will produce a permanent change in the distance between the test masses of a free-mass detector. This angle dependent effect occurs because the stars' gravi tational interaction permanently .. changes the magnitude of I jk• No such effect occurs for radiation from sources (e.g., stellar collapse) which begin in a stationary state and end in a stationary state. 108 Free-ma s s (broad-band) detectors respond dire ctly to the amplitude of a wave and have no mechanical re s toring forces. On the other hand, resoncmt detectors (Weber-type bars , supercooled monocrys tals of sapphire, etc.) can be tuned to yield information about the spectral conte nt of the incoming waves. For our case, the spectral characteristics of the radiation are given by the Fourie r trans form of the renormalized amplitudes a+ and ax. These transforms are defined by; +oo +oo J a+ e -iWt (2:rr) x -ro 1 dt = .! 1/2 v J +x -co a e - i (w/v)T dT • (0"- • 9 a ) For the amplitude ( 2 . 6c,d) these Fourier transforms are a +w = 2 2 2 ~ (vb) [-(K0 sin e + uK 1 ( sin e + sin cp ( ;c )l/2 2 J, + i( 2[K + uK0 ] cos e cos e cos cp} 1 (2 .9b) 2 -l ( b\ [tuK cos 0 cos q> sin q> } Xw -( 2 :rr)l/ 2 v / 1 a - + i( ·(K1 + uK ) sin e sin cp} 0 J, (2.9c) where u = w/vb, K = K {u), K ~ K (u) , 0 1 1 0 (?. . 9d) and where K and K arc modified Bessel functions. 0 1 The specific flux carried by the waves in the two polarization states is (see p. 102 7 of MTW) 4m m )2!wa 2 ~+ (er~ s) = _! ( ~ v cm2Hz 8 4 br 1 ( mAmB) br = 16 2 u +w1 2 2]' 2[ , . 2 2 2 2 (K0 s1n 8 + uK (sin e + sin cp - cos e cos cp)} 1 ( 2 . lOa) 109 where v ~ w/2 n is the frequency of the radiation. The total energy radiated per unit frequency is s t u.-+ .,..x) r 2 clO 1.J< ·I· .;• -dE = dv v v '2 64 (mAmB) = 5 b and the total energy radiated is f:i.E = f dvdE dv 2 = 371c 15 (mA~) b v (2.12) b • To gain insight into the low frequency domain of the radiation one a+w or a should not concentrate on ')( W J they diverge as w _. o,· rather, one must look at the real phys ical energy carried by the waves and consider <lE/dv as w _. O. The zero frequency limit of dE/dv may be read off from eq . (2 .11) since the term in square brackets -> 1 as w .-.o . The result is, 2 dE = 64 (mA~) dv 5 b (2 .13 ) Alternatively it may be obtained by noting that for w - t 0 +CX> 1 wa = +w (2,r) 1/2 x = J . a (T)dT -00 l (2,r) 1/2 + x J +oo = 1 ( 2 rr) 1/2 -CD +oo 1 dO+ •dT ::: a dTx (2,r) 1/2 +x -CX> r:i.a ~ (2.14) where ti.a denotes the change between the early and late limiting values + x of a Using eq. (2. 6c, 2. 6d) we can read off these values for ~a+ ,· + )I x t:i.a + = 2 cos e sinG cos cp and using (2.11) we get: ti.ax = - 2 sin e sin cp (2.15) 110 dE I dv w=O ( 2 .16a) which when integrated gives 2 dE = 64 (mAmB) dv U)_ =O 5 b • I (2.16b) As a closing remark in this section we note that either for the nonrelativistic or the ultra-relativistic . case the w ~ 0 limit of dE/dv may be attained by two methods - the tedious method of actually Fourier transforming the specific amplitudes then taking the w ~ 0 limit or the rather trivial approach of looking at the net change in hjkTT before and after the "collision." The above amplitudes and spectra have been previously derived by Ruffini and Wheeler (1971). Utilizing a quantum mechanical approach, Smarr (1977)obtained the same results for the w ~ 0 case. Turne r (1977) used a purely Newtonian multipole expansion formalism to calculate the amplitudes and spectra for various bound and unbound orbits as a function of eccentricity. 111 III. ARBITRARY VELOCITIES a) As sumetions Turn now to the case of a stellar encounter with arbitrary relative velocities v: but still insist that the stars be small comp ar ed to the impact parameter b (neg ligible tidal interaction) a nd tha t the impact parameter be large enough to guar~ntee small deflection angle: (3.1) In Paper III we used the post-linear wave-gene ration formalism of Paper ~ I and II to compute the gravitational-wave field for s uch an encounter to within fractional accuracy 2 (error in wave field)/(wave fie ld) = o(cr ) . (Our analysis was not able to say whether the error was C, 3 Cr , (3 . 2) Cr, Cr2 , 2 The Cr error was deduced by D'Eath (1977) u sing a mathe- matical formalism very diffe r e nt from ours.) Unfortunately, our computa- tion relied on the assumption that the s tars are large e nough that their internal gravity is nonre lativistic. Thus, we have no guarantee that our results are valid if one or both of the objects are black holes or neutro.1 stars. On the other hand, we know that the weak-interval-gravity assumption ( 3 . 3 ) is unnecessary in the low-velocity case v << 1, where the "quadrupole-moment formalism" is valid (Thorne 1977b ), a nd a lso at high velocitie s r >> 1, where the "colliding planewave " formalism is valid (D 'Eath 1977). In both the se limit s the waves are inde pendent of the stars' internal structures so long as tidal 112 forces can be neglected (rA << b, rB << b): black holes give the same radiation as normal stars. There is no apparent physical reason why structure-independence should break down at intermediate v e locities. Thus, it seems reasonable to assume that our results are valid independently of the assumption ( 3 . 3 ). b) Description of the Waves Recently Thorne (1977a) has proposed that the results of gravitational wave calculations be expressed in terms of a complex scalar field A;:::; A + (s.4a) + iA· , x which has the property that at a fixed event in spacetime and in any reference frame the spatial, transverse, traceless part of the metric perturbation is h TT = (3.4b) jk Here ( 3 .l+c) and ~l and ~ 2 are vectors with the following properties: purely spatial in the chosen reference frame; (ii) (i) they are they are orthonor- mal and are orthogonal to the propagation 1~-vector k of the waves e • e = C -a -b ab' e • ·-~ = 0 : -a (3 .4d) together with any future-directed 4-vector ~O' the triad (iii) 2 ~l' ~ , ~ forms a right-handed system (3 .4e) 113 (here Eis the Levi-Civita tensor): (iv) !l lies in some chosen fiducial plane (2-space) which passes through the propagation vector k. For our bremsstrahlung problem we choose our fiducial 2-space to be the one spanned by the propagation vector ~ and the difference ~B-~A between the stellar 4-velocities (3 .5) fiducial plane In writing down our formulae for the gravitational-wave field A we shall find it convenient to use three different reference frames: the rest frame of A (denoted S), the rest frame of B (denoted g), and the center-of-velocity frame (denoted 8). The rest frame of A is the same as that used in the low-velocity problem (Figure 1). In this frame the propagation vector is~ = ~ +~, 0 where n is the unit radial vector pointing toward the observer (eq. 2.3); and u -u = (y-1) e + vye • - B -A -0 -x Consequently the spatial part of the fiducial 2-surface is the n I\ e plane. -x The unit vector ~l must ·- lie in this plane and must be perpendicular to ~; thus, up to sign (which is of no importance), ~l is the unit vector pointing in the O direction: e • ·-~l = -o Conditions (o.4d,e) and the demand that ~ purely spatial then guarantees ! 2 2 be =e : -er in frame g (rest frame of A) ~l = ~e' ~2 = =~ Comparison (3.6) of equations (3.4b,c) with (2.4a) and (2.Sa) reveals that the above choice of fiducial 2-surface (eq. 3 .5) leads to polarization base states which are the same as those used in our low-velocity analysis. 114 The r e st frame of B (frame ·g ) is related to that of A by a Lorentz boost with ve locity + ve , and a translation be de signed to pla ce the -x -y origin of coordi nates on star B: t -· r(t - vx), x = r(x-vt), y : :; y - b, z == z • r = r(l - o:v)r, t o: ::= cos e r x/r, (3 . 7ti) t-r+f3b = r( 1-o:v) ' (3 . 7b) f3 :;; sin e COS · cp ::.: y/ r (3 . 7c) • ~ 2 2 2 1/2 - - -2 - 2 - 2 1/2 Here r = (x + y + z ) and r = (x + y + z ) are distance from source to observer as measured in the two frame s; t-r and t-r a re retarded time; and we assume lt-rl << r, lt-rl << r r adiation zone). (observer in It is straightforward to verify tha t in frame ~ the spa tia l par t of the f iducial 2-surface is the n A e -x direct i on (where ~ is the unit spatial vector pointi n g toward the observe r), and tha t consequently (3 . 8 ) in frame g (rest frame of B) c e _, - 1 = -e ) are related to (e, e, ~ )in the same way as (e ., e ) --x - -y --z - e -r.!) are re lated to ( e , e , e ); s ee equati ons ( 2 .l~b,c) and Fi gure 1. - x -y - z where (e ~e , e - Cfi - The center-of-velocity frame g moves with speed + v e r elative -x to 3 , and with spe ed -v e _ r e lative to g, where -x v = rv/(r+1), (3.9) The spatial origin of g is situa ted ha lf-way be tween the trajectori es of the two star s . t = r( t- ~), Thus x = y(x-vt)' z z ' (3 . lOa ) 115 r = y(l-av)r, t-r + 1'2 pb t-r , (3 .lOb) y(l-aV) and similarly ~ r = r('t+vx), ;:c =1(x+v't), r = y(l+Ct'V), ~ - y= y -z = -z , 1 + ~, - = 't-r + 1/2 ~b , t- r (3. lOc) (3. lOd) y( 1+6.V) o. = cos e = :;.1;, (3. lOe) In frame g the basis vectors for the polarization tensors are: e -2 = e~ (3.11) ~ cp Here (e- , e-) are related to ( e-, e-, e-) in the same way as (e , ~cp) -x -y -z -8 - 0 -cp are related to (e e , -z e ),· see equations ( 2 .4b,c) and Figure 1. -x , -y In Paper III we derived the full wave-zone gravitational field (the trace-reversed metric perturbation) h µv language (equation III-8.1). , in frame-independent From that result one can compute the scalar gravitational-wave field A by proje=ting out the spatial, trans- - verse, traceless part of h µv in any frame one wishes, and then equating it to expression (3.4b). The gravitational waves of equation (III-8 .1) have symmetry properties similar to those which we encountered in the low-velocity limit: The radiation field is totally insensitive to the relative masses of the stars A and B; it depends only on the product of their masses. Consequently, in the center-of-velocity frame, whe re the encounter would have f orward-backward symmetry if the masses were identical, the radiation a lways has forward-backward symmetry: llG A+(t,~,e,~) = A+(t,~,n - e,9 + ~) Ax(i,;,~,~) = - , Ax(t,;,n - ~, ; + n) . (3.12a) This is the arbitrary-velocity analogue of equation (2.o); in the language of Paper III it is the synunetry property (III-3.17). Of course, the waves are also syrranetric under reflection through the plane of the encounter A ('t,~,o,~> =A + + ('t,;,-e, - ;> , (3 .l2b) c) Analytic Expressions for the Amplitude Previous computations of bremsstrahlung (Peters 1970; Matzner and Nutku 1974) have been carried out in the rest frame of one of the stars. To facilitate comparison with those results, we shall express our amplitudes in terms of the coordinates (t,r, e ,~) of the rest frame of star A (Figure 1). In our expressions we shall use the direction cosines of the observer, as seen in the rest frame of A a = cos e, f3 = sin e cos_ o , 0 = sin e ·sin <p ; (3 .13a) and also the analogous direction cosines, as seen in the rest frame of B - O'..-V a = cos e == l-av' 'Z'I - - f3 ~ = sin e cos ~ = r(l-av) ' 5 ·- sin e sin <p = r( l~o:v) (3 .13b) Whereas in the low-velocity case there is only one charac teristic timescale associated with the radiation, 6t = b/v = (time for star B to move a distance equal to the impact p3rameter), in the general case 117 there are two timescales. The first is the analog of lit :~ b/v: Star B, as it passes star A, exerts a gravitational acceleration on A. Because of the "relativistic pancaking" of B's Coulomb gravitational field, A's acceleration has duration 6t - b/(vy) rather than 6.t - b/v. Because the moment of closest approach is t = O, this will produce radiation which is naturally described by the retarded-time parameter (3.l4a) TA:=vy(t-r)/b. The radiation should have a bump at time TA : : : O, of duration 6.TA - 1. The second characteristic timescale is associated with the acceleration of star B. As seen in B's rest frame Sits own accelera- tion has duration 6.t = b/(vl). The radiation produced by this acceleration is doppler shifted, as seen in the observer's rest frame (A's frame): there i t has duration 6.t = l(l-civ) 6.t. The strongest radiation is emitted at the moment of closest approach, t star Bis at (x = o, y = b, z ~ 0). = o, when The observer sees this radiation emanating from B at retarded time t - r = - ~b. Thus, it is reasonable to introduce the time parameter (3 .14b) and to expect the radiation to contain some sort of bump at time TB - O with duration CITB- 1 • In writing down the radiation field it is convenient to introduce the following functions of our time parameters: £ = (l T 2)1/2 A+ A ' (3 .15a) 118 S 2 ~ (o: - f3 TB) 2 + 5 The two expressions for s (3.13b) and (3.14b). 2 2 2 - ' 2 -2 2 £B = (o: - f3 TA) + 5 £A • (3.15b) are equivalent by virtue of equations 3 The functions l/£A 3 and l/2B describe the rise and fall of the gravitational accelerations that the stars exert on each other (eq. III-2.27), while S 2 is the squared length of the 4-vector which reaches from star A at the moment it crosses the past light cone of the observer to star B at its passage through the light cone (eq. III-3.12). Note that TA' TB' £A' £B' and s 2 as defined here are renormalized by a factor b from the same quantities in Paper III: (3. 16) In terms of the above parameters the scalar gravitational-wave amplitudes are given by (3.17a) ~~ = G1 sin2 o + c2 (sin2cp - cos 2 e cos2 cp) + c3 cos e sin e cos ~ 2 + c cos e sin e sin~+ c (1 + cos e) cos cp sin cp , 4 5 A x :::: Gr) cos e (2 cos C\) sin cp) c. (3 .17b) G sin e sin 9 + c sin O cos ~ 3 4 . 2) + G cos e( cos 2 cp - sin cp • 5 (3.17c) (3.18a) (3.lOb) 2 [ - 2-Yv 2 J [a... aj. . £A - ( ) 2 2(1-v ·)TB - v(l+v ) (CY.TB+ 13) 1 f3TA 82 f)T B y 1-CN £B ' (3.l8c) G l.~ yB 2v( l-0.V) S 1~-A [(2-3v2-v4)£A2 + v(l-3v2)TA(O!rA+~)J 2 x. (3.l8d) 2 Ge._ = ~) y(l+v )5 2(1-o.v)s (3.l8e) 2 In expressions (3.17) we have introduced the same renormalized amplitudes a and a as for the low-velocity case (cqs. 2.Sa,b). + (Note that )I because of the factor r in (~.17a), a and a are not scalar fields.) + x Equations (3.17a,b,c) for a+ and ax are the same as equations (III-8.13b,c) and (III-8.2), with b replaced by unity because of our 2 renormalization of TA' TB' £A' £B' S , and with some algebraic modifications that utilize equations (III-3 .7) and (III-3.llb). It is straightforward, using equations (III-8. 5 ) with b = 1, to take the low-velocity limits of a and a and show that they agree + x with equations (2.6c,d). 2 In expressions ~.18) the dependence on the function s is somewhat delicate. For observation events at which one star is in the "shadow" of the other, s 2 vanishes. However, the quantities which multiply l/s 2 120 in c , ••• ,G 1 way that 5 also vanish in the shadow - and they vanish in such a c1 , ••. ,c5 change smoothly and not sharply as the observer moves into and then out of the shadow. For further discussion see § Vlc of Paper III. The forward-backward symmetry (3. 12a) does not show up very clearly in expressions (3 .17), (3 .18) for A and A because those expressions + x are written in the coordinates of A' s rest frame. By transforming ( 3 .12a) from center-of-velocity coordinates to A's coordinates (eqs. 3:loa and 3.9) we find that the amplitudes must be the same at the (3.19) In terms of spherical polar coordinates r,e,q) and the retarded time parameter TA (eq. :5 .14a) this relationship says - A )C (3 . 20a) where T A2 (3 . 20b) Note that the direction cosines and A-B time parameters associated with these "equivalent" events are related by 121 a.2 = l\, t32 - a:l, t32 = - 131' 0:2 = - al, i\, TA2 = TBl (3.20c) 52 = - 51' TB2 - TAl (3.20d) 52 - - cf. equation (3.13b), (3.14b), and (3.20b). Before displaying graphs of the wave amplitudes and discussing their spectral features (§ V below), we shall derive high-velocity (r >> 1) formulae for them. IV. HTGH-VELOCI1Y LIMIT In the high-velocity limit, y >> 1, it is useful to split the radiation zone into three overlapping regions:.· the "forward region," the "intermediate region," and the "backward region." In terms of angles measured in the center-of-velocity frame g, these regions are -e << re/2 , forward intermediate backward ;,-1 << e ; --1 r << re - e , re - e << re/2 (4.1) When one transforms to the rest frame of star A using the relations cos e = - cos e + v - y >> 1 ' l+v cos e ' (4.2) one finds the following locations for these three regions: forward intermediate backward -1/2 e << r ' r-1 << e << 1<I 2 , e >> r-1/2 (4.3) 122 We sh a ll now derive limiting forms for the gravitational-wave amplitudes (3.17), (3 .18) in the forward, intermediate, and backward regions. a) Forward Region In the forward region we introduce D'Eath's (1977) angular variable (!~ .Ii.a) which has values of order unity at the angles of greatest interest, and which satisfies ~ << 'Y 1/2 (4 .4b) ' throughout the forward region. c ;2 cos cp ~ cos cp, We also introduce the notation s ~ sin cp =sin cp • ( 4.l~c) The two time parameters TA and TB are related to each other and to retarded t i me (eqs. 3 .14b and 3 .t::>a) by T A 2)' (TA + = y(t-r) b ' ~ cos cp) u~ . 52 1 + *2 We expect the waves to show structure on the timescal es 6TA - 1 and t:{fB - 1, which correspond to times t as measured by the obse rver in A's r est frame (4 . 6) Careful scrutiny of equations (3 .17), (3 .18 ) reveal s that, when viewed on the slow timescale ~TA - 1, A+ and Ax have disconti nuous time derivatives at the time TA = - y cos cp (i. e . TB = O); but the values of 123 and A are continuous there. A + x When viewed on the faster timescale 6TB - 1 near TB = O, the time derivatives vary smoothly but there is no significant change in the amplitudes themselves. This behaviour permits us to focus attention exclusively on the slow timescale 6T A - 1, but suggests that we shift the origin of our time parameter TA to the moment TA = - "' TA ::::; TA + ·~ cos cp = * cos ~ of discontinuity: r( t-r) r('t-r) 1 b + '¥ cos ~ = _b___ + W cos C? • (4. 7) 2 To derive the high-velocity limit of expressions (3.13)-(3.18) we now replace e throughout those expressions by v/r, we replace cos c.p "' by c ancl sin cp by s, we replace TA by TA * cos c.p, and we expand in powers of t ::::; l/r, keeping careful track of the magnitudes of the errors at each step. The results for various parameters appearing in (4 .Ba) r(l-o.v) - 2 1 =.: 2 e(l + (,0 ) , 1-~ (1+.Bb) 2 a= --2-, (1~.8c) l+* (1~ .8d) and also "' 2 £A = [ 1 + (TA - ye) J1/2, (4. 9a) 124 2 In all these expressions the fractional errors are of order E - except l~ v and a, (eq. l~ . 8a ), where they are of order e:. The result for A and + A is )( A + ·1- iA x :::: 4mA~ l.~mA~ (a + ia ) = (a + ia ) , br + >< + x br I' (l~ .9b.) (4.9c) a :: + 1 + 2.A [ 2 2 2 2 "' 2 "' 2 c(l+* ) - 2((l+c ) + c * ) ~TA + 2c(l+~·~ )TA- - 2V~A3] j (4. 9d ) These formulae for A and Ax have fractional errors of order maximwn + -1 -1/2 ) "' : they produce ( "Y , W"Y • Notice the absolute-value signs on TA the discontinuities in the time derivatives of the amplitudes discontinuities which would actually be smoothed out if one looked on 2 ~ (1 + * )/r A the shorte r timescale ~~ Equations (1~ .4), (l-~. 7 ), and (4. 9) are our final, forward-direction, high-velocity formulae for the gravitational waves in terms of either center-of-velocity coordinates (t, r, G , ~) or r e st-frame-of-A coordinates (t,r,O,cp) . We shall discuss these formulae in § V below. 125 b) Backward Region Formulae valid for the backward region can be obtained from the above forward-region formulae by invoking the synnnetry r e lations (3.12) - or, equivalently, (3.20) and (3.12b): A + . _(l+*2) 4mA~ (a + 1A x - b 2 !' + r + ia ) x = l~mAmB (a + ia x ) , + br !' a = [expression 1~.9c], ax= [expression 4. 9d "] ' + where the quantities appearing in (4.9c,d) are given by (4.10b) (4.10c) c = - cos cp = - cos cp, = s = + sin cp = + sin ~0 , r(t-r) b 1 - -2 ~c ' (4.lOe) ( l~ .lOf) (expressions 4 . 9a) • c) (4. lOd) Intermediate Region In the intermediate region, y -1 << e << ~ I 2 c--1 y << -o, -y -1 << n -e), it is convenient to introduce the new angular variable x = 0 (y/ 2 )1I 2 = er =( 1 - cos~- )1/2 , 1 + cos e (4.lla) which has values of order unity at the angles of greatest interest, and which satisfied -1/2 !' << x << !' 1/2 ' throughout the intermediate region. - --1 << x << !' !' (4.llb) We also introduce the notation 126 c :..: cos cp = cos cp, ~ (4 . llc) s = s in cp = sin cp As we have seen, in the forward (and a l so in the backward) direction, A+ and Ax vary on the slower of the t wo timescal es TA and T • 13 In the forward direc t ion the s l ower timescale is TA = TA + c onstant; in the backward direction it is TB = TA + constant . In the inter- mediate region it t urns oat that there is a continuous transition from v ariation on timescale TA to variation on timescal e T • 13 To make this t ransition apparen t it is convenient to introduce a timescale defined by c +2 (4. 12 ) llere the relationships between the various timescales are derived by insertin g equation s (4 .11) i n to (:3.lOb ), (3 .7c ) , and ( 3 .ll~ a,b), and by 1 2 ignoring frac t ional correct i ons of order ( 2/y) / • Notice that at x .... (2 y)- l/2 ( the forward edge of the intermediate region), 1 .... ( 2y)-l/ 2 (the backward edge of the intermediate t . . TA +~ ; at x 1 region), t e = -rrf 2 ) , t = ( t-r )/ b . TB - ~; and a t x = 1 c -> To derive high-velocity, intermediate-regi on formulae for A. and T Ax we now replace e throughout expressions (3 . 13 )-(3 . 16 ) by (2/ y) l/ 2 x, we replace cos cp by c and sin cp by s; we replace T and T by A B 1 2 ( 2 y) / x ( j + c/2 ) ( eqs. 4 . 12 ); and we expand in powers of Tl ;::: ( 2/y)1f2-, kee ping careful track of the magnitudes of the errors at each step . The resu l t s for various parameters appearing in A and A are + x 127 v==l- 1 4 11, 8 (4.13a) 2 y(l-av) = x , - -2) a = - ( 1 - 2L..2 'll X - ' - ~ ::: 1lx.-1 c' 5 = 1\x.s , (4.13b) -1 5=1\x s, (4 .13c) {l+.13d) -1 -1( r TB ::: 211 X t+~ 1 ) ' (1~ .13e) ·2 s :: s 2 + 4t 2 . (4 .13£) 2 In all these expressions the fractional errors are of order 11 or 2 8 except in v (fractional error of 11 ), a (fractional error 11 x. ), an.: a (fractional error 114 x.- ~). The final results for (1\x) 2 or (Tl/x.) 4 4 A + and A x are + iA ::: + x A 1 l~mAmB b (a r 2 + iG ) = (l+x. ) + x 2y Iii (2 )3/2 a = --'Y--x + x. a x = - (2r) 3/ 2s x x. 4m m A B (a + ia ) + x ' hr if Iii > ~ lei if ltl < ~ lei , s2+4 t2 {s2+4t2)-l if i < - ; Icl 1 if - ~ lei < j< ~ lei 2 - (s2+4l 2)-1 i f ~ lcl < 1 · (4. ll~a) (4 .14b) (4.14c) 128 As in the forward and backward directions, so also here, a and + ax are continuous functions of time, but their time deriva tives are discontinuous. The discontinuity at 1 = - passage of T through zero (eqs. 4.13d,e). 8 1 c is caused by the 2 If one were to look on the rapidly varying timescales TA and TB rather than the slower timescale TA' one would see a smooth change of derivative at l - ± ; c. Formulae (4 .14) are embarrassingly simple considering the tortured route by which we have arrived at them. There must be some easy way to derive them; but we have not yet tried to find it. V. DISCUSSION OF THE WAVE FORMS The pioneering, and most definitive previous work on thi s problem, was that of Peters (1970 ). He studied the same type of encounter as we do, but with the added restriction that one star (say B) is a test particle with infinitesimal mass , and the other star is ma ssive: (5.1) His method (first-order perturbations of the Schwarzschild field of star A) was not amenable to the comparable-mass case. However, our analysis has shown that the radiation is independent of the mass ratio By comparing Peters' equations (2 . 23 ) and ( 2 .21~) with our (3 .4), (3. 6) , (3 .14a), and (3.15a), one can derive an expression for the r znormalized ampli.tudes Q.+ and ax in terms of Peters' metric functions A,B,C: 129 da + = _!_-[A sin2 o + B(cos2 o cos 2 cp - sin2 cp)- · 2C cos o sin e cos cp] , 8vy dTA (s. 2a) dax 1 dTA = 4vr [- B cos e cos cp sin cp + C sin e sin <.,,] • (5.2b) Peters gives horrendously complicated expressions for his functions A,B,C in his Appendix A. The fundamental quantitie s which go into those expression~ rewritten in terms of our variables, are BPETERS = ~US = sin e cos cp, ~PETERS ~ vUS' EPETER.S = aus = cos e ' (5.2c) (5.2d) rPETERS = 'i'us' We did not have the fortitude to check analytically Peters' results against ours; so we checked them numerically instead. When we failed to get agreement, we asked Peters for help; he rechecked his formulae against his original 1969 computations and found a misprint in the paper; and after correction of the misprint, his formulae ( 5 . 2 ) and ours [analytic time derivatives of eqs. (5 .15) and ( 5 .16 ) ] gave identical numerical results. Peters' misprint was the omission of a factor 2 of (l+~ y ) in the denominator of the w term in his expression for A. 2 2 Peters never discussed or graphed his wave forms, so we shall give a thorough discussion here: Figure 3 shows the wave form A (t) in the forward and backward + direction (e = 0 and ~, cp ~ 0) for various velocities ranging from v == 0.0001 to v ~ 0.9999. Plotted vertically is (A x vanishes in the plane of the encounter . ) a+ 2 for e -· o, 'Y cf. equations (3.18b). 130 brA 1/2 a _____+_ _ = (-i~~) ~ for 6 = n; 3 4mA~'Y (1-v) y (5.3a) Note that in the rest frame of A, which we are using, at a fixed distance r from the source the backward amplitude is 2 smaller than the forward amplitude by the factor [(1-v)/(l+v)l/ • Plotted horizontally is f e o TA = vy(t-r) b or = ' (S. 3b) cf. equations (3.14a,b) and (3.20c,d). Note that the wave form in the backward direction changes more slowly, by the doppler factor 2 (l+v)y = [(l+v)/(1-v) ] 1/ , than the wave form in the forward direction. The wave forms of Figure 3 were computed from equations (3.17), (3.18). Figure 3 shows clearly the transition from the low-velocity regime [uppermost curve; eq. (2. 5 )] to the high-velocity regime [lowennost curve; eq. (4.9)]. Notice how, as v increases, the slope of the wave form becomes discontinuous at TA = TB ~ O. Notice, moreover, that the discontinuity is smoothed out when one examines it on the ''fast" timescale (6TB =- 2yTA - 1 in forward direction; 6TA ~ 2y6TB - 1 in backward direction). These phenomena were discussed analytically in § !Va. Figure 4 shows the wave forms A (t) and A (t) in the directions + x 0 = n/2, ~ = o, n/2 (directions which are transverse in the center-ofvelocity frame). Plotted vertically is 131 brA + Or )<. br y A a +or x • + Or)I' r cf. equations (3 .9) and (3 .lOb). '6/ 2 , ( 5 .l~a) Plotted horizontally is t =-- (l\ 2y )1/2 TA + 21 -- (12y )1/2 TB 1 2 (5.J+b) Figure 4 shows the transition from the low-velocity regime [uppermost (2.6)] to the high-velocity r egime [lowermost curve of each set; eq. curve; eq. (4. ll~) ]. Notice, again, the development of the disconti- nuities in slope at high velocities, as discussed in § !Ve. Figure 5 shows the wave form A (t) at ver y high velocitie s, y >> 1, + in the extreme forward region backward direction n - e ~ r-1 ( or, equivalently, in the ex treme e $ 1~2) and i n the plane of ·the encounter, ~ = O. The curves are labeled b y values of the angular parameter * = er = er/2in forward directions 112 = (11 +- cos e) cos e = (n - o)y/2 in backward directi ons [cf. eqs. (4 .4a) and (!~ ~ lOc) J. brA + ( 5.5a) Plotted vertically is brA + = ---~-3l6mA~r brA + in forward directions, ( 5 . 5b) in backward direc tions (5 . 5c) brA + 132 [cf. eqs. (4.9b) and (4.lOa)]. TA A T + ~ = A r Plotted horizontally is _ r (tb- r) __ _ Ctb- r__ ) _ 1_ ,Y1,1• = 1+¥2 2 Notice how, for 1\£ (5. 5d) in forward directions, 2 t-r r('t-r) b b in backward directions. ( 5 . 5e) > 0.5, the overall amplitude drops with decreasing v, and the characteristic timescale increases. As ~ gets somewhat larger than 1, this behavior is described by the intermediate-region formulae a+ o:: 1/ ·" 'I [cf. eqs . (4.12 ) and (4.14)]. A ' .0.TA a:: * As ~ increases into the domain 0 ~ 1, one can see the wave form beginning to approach the "inverted me sa" shape (Fi g . l+ a) characteristic of the intermediate region. VI. a) SPECTRUM AND ENERGY Total Energy Radiated The total energy carried off by the waves, as seen in the r es t frame of star A, is 6E = 1~1( +oo re 2rc I IJ [ 2 2 2 (c\A+) +(c\AJ ] r dcpd8dt; -oo 0 (6 .1) 0 see, e. g ., equation ( 35 . 27) of MI'W. Pet ers (1970 ) evaluated this triple integral numerically for various encounte r velocities v. r epeated his computations, but we h ave confidence in them. graph one can read off the high velocity limit We h ave not From Peters' (6.2a) where the factor of 20 is uncertain by a few tens of percent. Matzner and Nutku (1974) used the method of virtual quanta to compute b.E in the high velocity limit, and obtained 2 2 ?i 2 6E ::.: 256 (mA~I b ) by~ £n (4y ) • (G.2b) Smarr (1977) evaluated the zero-frequency limit of the spectrum (see §c below) and, assuming (incorrectly) that dE/dt1Jd(l is roughly constant up to an angle-independent cutoff frequency wc ==Ky/b with K - 1, he obtained (6.2c) in agreement with Matzner and Nutku. It is straightforward for us to compute 6E at high velocities to within a factor of order unity: We first compute the energy radiated 1 per unit solid angle, dE/dn, in the extreme forward region e .:5 y- , in the intermediate region rregion e ~ ~/2. 1 << e << ~12, and in the extreme backward Then we integrate over solid angle. In the extreme forward region the amplitude has magnitude (6. 3a) [eq. (4.9)], and it varies on a timescale /'::;.~A"' 1 corresponding to 6 t,...., b/y [eq. (l~.7)]. (6.3b) Thus, in this region (6.3c) 134 The region encompasses a solid angle 6n ,..., y -2 , so it contributes an amount (Lill) extreme forward ,..., ( mAmB)2 3 y 7 (6 .3d) to the total energy. In the intermediate region (which Smarr presumed was responsible for the logarithmic term) ( 6 .l~a) [eqs. (4.ll~a) and (4.lla)]. frequency values. These amplitudes agree with Smarr's zero- The characteristic timescale in this region is 13. "t ,..., 1, corresponding to -1/2 ,..., be 6t,..,, bxr (eqs. (4.12 ) and (4 .lla)]. ( 6 .4b) Thus, in this region dE 2 dn ,..., r A+ )26 (6t t (6.4c) = y -1 to near 0 = rr/2, so it contains The r egion extends from near £) a total energy 42 (6E)intermediate =J ~~ 2rr sinede y (6.4d) -1 Almost all of this energy is at the extreme forward edge of the region. An analogous computation gives for the energy in the extreme backward region e ~ rr/2 135 2 ( 6 E)backward - ( 7mA~) 2 (6.5) Y • Thus, the total energy radiated in the rest frame of "star'.' A 3 [ ( 6.3d ) plus ( 6.l~d) plus 6.5)] varies as y, as found by Peters ( 1970), and not as ·l £n (4r2 ) as found by Natzner and Nutku (1974) and inferred by Smarr (1977). While we understand why Smarr's method failed (incor- rect assumption that the cut-off frequency is angle independent), we have not dug deeply enough into the Natzner-Nutku method of virtual quanta to understand its failure. b) Spectra To obtain expressions for the specific flux ,+x or the energy v radiated per unit frequency in the arbitrary velocity case we have to Fourier transform the wave amplitudes (3.17), as we did for the low velocity case [eqs. (2.8)-(2.11)]. Unfortunately, the Fourier transform of eq. (3.17) cannot be expressed in closed form for arbitrary values of (O,cp). However, in the precisely forward direction (e=cp=O) and pre- cisely backward direction (e = rr, cp = 0) the Fourier transform of the amplitude [eq. (3.17b)] can be done analytically. Using eqs. (3.17b,c) with O:=O:=l, ~ =f3= 5=5=0, (A>< vanishes) we obtain for the "+" polarization a (e = o,cp = o) + 2 - ( 1 + v) ( 2TA + 1) ] I.A (G.6) It suffices to discuss only the forward amplitude a (o,o) since the back+ ward amplitude a (rc,O) is obtained by using the Doppler shift relations + 1 2 (4.10); see also eq. (5.3a,b), i.e., a (rc,O)=[(l-v)/(l+v)J / a (O,O). + + l3G The Fourier transform of (6.6) is defined by 00 b a+w f (6.7) -00 since t - r = TAb/vy. Performing the integral we obtain 2 y(l+v)b[ a+w == 2 1/2 (1 + v) K2(x) - (1 - v) K2(y) v (21!) J' (G.8) where ~ denotes a modified Bessel function with argument x = wb/vy or y = y(l - v)x. Using equation (2.9a), we obtain for the specific flux in the forward direction for the case of arbitrary velocities 4 ~ + = ( mA~ v br ) 2 4 2 /' ( l + v . 2 ~ [ ( 1 + v) K ( x) - ( 1 - v) K l61!V2 2 2 2 0 <- ( y) J • (6.9) The low-velocity limit of (6.9) may be obtained by setting x = wb/v, r + 1 and utilizing the recursion relations for the derivative of K2 (x). The result is for v << 1. (6.10) This agrees precisely with the low-velocity results of §II if we set e = ~ = 0 in equation (2.9a). Figure 6 shows the specific flux in the forward (e = 0) and backward (e= 1!) direction for velocities ranging from v = 0.0001 to v = 0.9999999. Plotted vertically is 137 for 0 = 0 or (6.lla) for e = re Plotted horizontally is loglO [~~] for e = o, or 1 oglO 1/2] ( [ wb 1+v vy 1 - v) for e= n: (6 .llb) since for the backward direction the frequency is redshifted by the 1 2 Doppler factor 1/-y(l+v) = [(l-v)/(l+v)] / • The structures of the curves may be understood in terms of TA,TB and Figure 3 . For low velocities only one timescale T = TA:::::'. TB is relevant, so the spectrum of Figure 6 shows only one characteri s tic frequency (w"' v/b). At high velocities y >> 1, the two timescales TA and TB are no longer the same, so the spectrum shows two characteristic frequencies; that associated with its peak (w"' vy/b; 6TA ,...., 1) and that associated with 2 the beginning of its exponential fall-off ( w ,..., vy /b; i\TB,...., 1). .r <'r, · v ex: The v-2 behavior between peak an d exponential fa1 1-off d ue to the tcm- poral discontinuity of A at high velocities in Figure 3 . In the limit v = 1 (y = oo), a true discontinuity would exist in Figure 3 , and the 'J. v ex: v-2 behavior in Figure 6 would continue forever. The behavior, ~ v ex: v- 2 , is easily understood by noting that a discontinuity in cA+/ct 2 produces a behavior a +w fore ~ v ex: I wa+w 1 2 ex: 2 1/ w in the Fourier transform of a , and there- ex: 1/ w cc + 2 1/ v • 138 c) Zero-Frequenc y Limit To obtain formulae for the zero frequency limit we proceed by calculating the net change in A+ as we did for the low-velocity case in x As TA' T §II. B -> -a> the amplitude s a and a -~ x -+ 0 since a ll the CO- efficients G1 through G -> O, cf. equation ( 3 . 18). 5 only c 1 and c 3 For TA' TB -> +oo are nonze ro, and we can write 2 m [ 2 cos G - v ( 1 + cos e )] , t;,.a == y(l+v ) sinecos 2 + t;,.G ( 6 .12a) ( i - v cos e) 2 ·- -2y( 1 + v ) sin e sin CD x - (1-vcos fl ) ( 6 .12b) ' where t;,.a+ denotes the change between the earl y and late limiting values x of a+· The energy radiated per unit frequency interval per unit solid an ~.; le for either polarization i. s given by e quati.ons ( 2 .13 ) and ( 2 .14 ), x "+x) _-1-2 (4mA~) (t;,.G+J2 -2rr ( v b 32rr r 2 2 ';J - (6 .13 ) x Use of (6. 12a) and (6 .12b) in (6 .13 ) give s us our final answer for the ze ro-frequency limit dwdn (~) 2 2 2 4 2 2 2 _ _ 1_ ( mAmB) ·./( 1 + v ) sin 0 cos cp [ 2 G (1 e)J -b cos - v + cos 4 2 + 321c (1 - vcos o) (6.14a) (~) dwdn x 2 . 2 cp . 2 e sin sin 2 (1-vcose) 'Y2( 1 + v 2 ) (6 .14b) 139 Thi s r esult agrees with e qu a t ion r SCH' (2.12) of Smarr (1977) i f we l e t 2mA - 2 m = m and o == (1 + v )rSCH/bv . B ' (There is an overall fac t or of 1/2 diffe rence du e to diffe r ent conventions .) We recove r our pre vious (2.13), i f we i ntegrate (6.14) ove r solid a n g l e r esu lts, cf. equation and t ake the low-ve locit y limit . Because of the sin 2 e term in equation ( G. ll+) there i s no zero fre - quency radiation emitte cl in the prec i se l y forward ( 0:;: 0 ) direc tion. How- ev er, there i s forward beaming due to the ( l - v cos o) f ac tor i n the de nomina tor for nonzero values of e within the usual " forward cone," i. e ., for 0 < EJ -:5 l/ y . To determine the high-velocity limit of (6. 14) in th e " f orward r egion" we r e introduce the notati on we used in §rva, and make use of equations (!~ . 4a), ( 1~ . 4c ), ( 1~ . 8a ,b). A c a r eful expansion in powers of E a l/y gives (6.15a) (~) dwdn (6.15b) x Note tha t thes e expre s sions for the ene rgy rad iated in the " forward r egion" ha v e the same y 4 d ependence as e quation ( 6 .9) for e n er gy radi- at ed in the precise l y f orward ( e = 0 ) direct ion. The symmetry prope r- tie s discus sed in §III t ell us that dE/dwdn for the backward r egion (in frame A) is g iven by ( dwdn +x ~)backward (~)forward (.!.__~) 1 v d wdn +x -1 (6.16) where (dE/dwd n)forward is equation ( G.15 a orb). To evaluate the energy radiated through the "forward cone" we integrate equations (G. 15) over solid angle, where our angular variables are ~ and cp with 0 :;;; ~ ;:;; 1 and 0 ;:;; ~;:;; 2rc . The expression for dE/dw is 2 l+ ( -mAmB) b- · ( ddEw) _, == ~ [ 5/l+ + .en(2) Jr 2 ( G.17a) (6 .17b) We now turn our a ttention to th e energy r adiated at zero frequency in the "intennediate r egion." We use e quations (4. lla), (l1- .13a), and ( l+ .13 b) in e quation ( 6 .14a) and (6 .ll+b) to obtain ( ~) dwdi1 (~) awan 2 Note that since 2/ T] = = + ~( 4mAmB \2 L [1 2 b ) 2 nfi 1( X 'I 2 s 2 6 x T] )( ±] 2 2-·X 2 ( G.18 a) ( 6 . 18b) r these expressions have the ·r3 Ix'?. dependence we expect from e quations (4 .14 b,c) and (6 .13) . To ge t an expr es sion for the energy emitted into the "intermediate region" we integrate equations ( G.18) over solid angle. The angul ar variables are x and cp, with limits (2y )-l/2 ~ x ;:;; rc/2, 0 ;:;; ~ ~ 2rc. The final expressions are give n by ( 6 .19a) (6.19b) which are consistent with equation (2.13) of Smarr's (1977) zero frequency analysis. In The agreement is more transparent when we note that [-t ~J = [(i ~ ~r J = 2 In [ (1 + v)rJ - 2 ln[2y] 2 2 In 2 where we have neglected terms of order 1/y . To get an order of magni- tude estimate of relative amounts of energy radiated into the "forward cone" vs. the "intermediate" region we use (6.17), ( dE/ dw) forward + or x - - (dE/dw)intermediate + or x (6.19), an<l get 1 £n(2y) • (6. 20) This shows that in the zero frequency limit, the energy radiated into the "intermediate region" dominates by a factor of,.., £n(r) the energy radiated into the "forward cone," in agreement with Smarr. Because the cutoff frequency of the spectrum is a function of angle (e,~), this effect is no longer valid when we consider the total energy radiated. (See remarks in §VIa.) 142 REFERENCES Barker, B. M.' and Gupta, s. N. 19 74, Ph ys. Rev. D, ~' 334. __ , D'Eath, P. 1975 , Phys. Rev. D, 12 2183 . DI Eath, P. 1977' Phys. Rev. D, in press. Crowley, R. J.' and Thorne, K. s. 1977, A2. J.' in press; cited in text as Paper II. Feynman, R. P. - unpublished letter to V. F. Weis skopf, January4 to February 11, 1961. Hansen, R. 0. 1972, Phys. Rev . D, ~' 1201. I Kovacs, S. J., and Thorne, K. S. 1977, AE. J., in press; cited in text as Paper II I. Matzner, R. A., and Nutku, Y. 1974 , Proc. Roy. Soc. London, ~~§ , 285 . Misner, C. W. 1972 , Phys. Rev. Le tters, g~ , 994. Misner, C. W., Thorne, K. (San Francisco: s., and Wheeler, J. A. 1973 , Gravita tion W. H. Freeman and Co.); c i t ed in t ext as MTW. Peters, P. C. 1970, Phys. Rev. D, !, 1559. Ruffini, R. R., and Wheeler, J. A. 1971, in Rela tivistic Cosmology and Space Platforms, Proceedings of the Conference on Space Physics, E.S.R.O., Paris, France, p. 132 . Smarr, L. 197 7, Phys. Rev. D, (to appear April l S, 1977). Thorne , K. S. 19 75, Lectures presented at the International School of Cosmology and Gravitation, Erice, Sicily, March 1975 . Thorne, K. s. 1977a, in preparation. Thorne, K. S. 1977b, in preparation (Paper VI of this series) . Thorne, K. s., and Kov!cs, S. J. 1975, Ap. J. , ~QQ , 24 5; cited in text as Paper I. Turne r, N. 1977, Stanfo rd Universi t y pr e print. Turner, M., and Will, C. M. 1977, in prcp ~ ration . Wagoner, R. v., and Will, C. M. 1976 , Ap . J., g19, 764 . Ze l'dovich, Ya. B., and Polna rev, A. G. 1974, Astr. Zhur., ~!' 30; Engli s h translation: Sov. As tron. - A.1, 17 (1 971~ ). FIGURE CAPTIONS Fig. 1. Spatial diagram in the rest frame of A showing the trajectory of "star" B and the orientation of the unit vectors !:e and ~cp· Note that the polar angle e is measured from the x-axis. Fig. 2. The time-evolution of the renormalized wave amplitudes for slow-motion bremsstrahlung in the rest frame of "star" A. The location of the observer is determined by the values of O and cp. (See Fig. 1.) Note that for an observer in the plane of the encounter (the x-y plane), cp = 0 and the "x"-polarization vanishes. Fig. 3 . 2 Plot of the renormalized wave amplitude a /y + the forward (in frame A) in (e ~ o) direction for v = 0.0001, 0.1, 0.3, 0.5, 0.7, 0.96, 0.9999 (uppermost to lowermost). Note that as v increases the slope tends to become discontinuous at TA== TB= 0. This plot also describes the amplitude in the backward direction (e = rr); see eq. (5.3). Fig. 4. a) 3 2 Plot of the renormalized wave amplitude a jy f + for e rr/2, ~ = 0 for velocities from v = 0.0001 (uppermost curve) to v = 0.999999 (lowermost curve). Note the discontinuity in the slope I = O. 2 Plot of a+or x /l/ for e = JC/2, ~ = 1f/2. as v increases is synnnetric about b) velocities is the same as in a) above. The range of Note the development of a discontinuity in the slope of a and in the second derivative + of a at t = 0 as v increases (lowermost curve in each set). Fig. 5. Plot of a /y + 2 in the x-y pl~ne for v = 0.9999 and for various values of the angular parameter * = ey. For ~ << 1 we are in the extreme forward region an<l the curves are similar to the lowermost curve in Fig. 3. As ~ increases and we move away from the x-axis we enter the forward most part of the intermediate region and the curves begin to resemble the ''inverted mesa" shape of Fig. 4a. These plots are also valid for back- ward directions; see eqs. (5 .5). Fig. 6. Plot of the specific flux eq. (6.lla) in the precisely forward direction for various velocities as a function of frequency. As w increases the first "bump" is near ~,...., v!'/b (for D.TA,..., 1) followed by a 1/ J dependence (which on the log scale gives a 2 slope of -2) until at w ,..., vy /b (for ATB"' 1) the curve drops off exponentially. 146 x .. .. ~ ~ C- . -s:Jc'J' : ~o t ./;_) ~. 0 ml 'O I ~1 .... 21 I .: . : ~ ·f~b I --:;iA~ y : --- ~ __ cp ___)'····... .. :.. .;· .. Fig. 1 . .. . ·.: z 147 8 = rr/4 ¢ =0 8=0 <f> =O -4 -3 -2 2 -I 3 A+ 4 -=1i===i==+===--+--·i----t---r--1----i-- T - 4 -3 -2 -I 2 3 4 -1.0 e = rr/2 e = rr/6 <f>=O cf> =rr/6 I. -4 -3 -2 -I 2 2 3 4 3 4 -1. 8 = Tr/2 <f>= 1r/2 8 = rr/4 ¢ = rr/4 -4 Fig. 2 -3 -2 q (\j LO LO (j <! ~ -0 w :? ~ LO 0 I 0 - - I L() -1 0 (\j I LO ......_...___,_~_.__~..____._~........_~...___,~_,_~_.___, (\j 0 . r0 LO d (j I I I'-- d OJ d I -I r0 - I zA/+lf Fig. 3 LO - f'-: - I I OJ -: I (\j I 149 0 N LO ci -+--> 0 w 2 rLO d I 0I -I 0 N I LO N LO 0 LO 0 0 LO 0 I 0- LO I 2!£)1+v Fig. 4a -I 0 N I LO N I Q' r0 I l SO 11 I I I 1111 I 11 I I I 11 I I I 11 I I l JI I I \ 11 I I I ff I I I q /11 I I \\ 111 I I \\ 0 N JI I I I \\ II I I I \\ II I I I \\ II/ I I \\ II I I I I/ I I I I I I I I I I/ I I/ I I/ I It I II I I I I \ ,,, \\ \ \\ \\ I I; I I; I 0 \\ \\ I\ LO cj II I ~I I ~ I .;-/ / I I cf( \ \ ~ I \ \ \ \\ \ LO z \\ \ \\ \ \\ \ \\ \ \\ \ 0 II \\ \ \\I \\I I 0 ' 0::: - o_J _ tO + CL II x u I 0 -I 11 I I I LLJ 0 I I I 0 I I I 1 _J - 0- - 0 <1 II ' I I - <1 0::: N <1 -- II If u f- N II II I \\I II I ~6 <1 - · II II II II // \\ I ci I- , _J 0 <t 0 N (f) I (f) LO "--~-J.-..........-"-"~~-'--~-'-~__,_~~...L.-~~ c\J LO LO LO LO C) r0 <;J" LO ' tO I I I I Fig. 4b 15 1 0 (J) co f'l{) (J) ~ 0 0 II II ~ ~ 0 0 0 (\J II II ~ ~ f'- l.O LO ~ r0 (\J ~ o w 2 -I f.. (\J I r<) I ~ I - LO I lO I f'I coI (J) I (\J 0 z,..{/+v Fig. 5 -I 0 (\J- I I 1S2 <;j" <;j" 0 <;j" II > (.() r<) N (J) (J) (J) r0 (J) (J) (J) (J) a: II co N a: > II ~ > (\J 0 N l[) .....----...... _QI>-3> ~ II (.() > 0 (J'I 0 0 N 0 q > II co ci <;j" ci 0 0 <;j" ci I co ci I N - I 0 N I <;j" I U) I co I 0- I [~] 01601 Fig . 6 (\J -I <;j" -I .....J 153 PAPER 4. That gravity should be innate , inhere nt, and essentia l to matter, so that one body may act upon another at a distance through a vacuum and without the medi ation of any thing else, by and through which this action and forc e ma y be conveyed from one to another, is to me so great an absurdity, that I believe no man who has in philosophical matters a competent faculty o f thinking, can ever fall into it . Newton 1G4 The Gravitational Scattering of Zero-Rest-Mass Plane Waves )(- ' ' WALTER K. DE LOGI and SANDOR J. KOVACS, JR. W. K. Kellogg Radiation Laboratory California Institute of Technology, Pasadena, California 91125 ABSTRACT We have used the Fe ynman-diagram t e chnique to calculate the differential cross s ec tions <ln /da for the scattering of zero-restmass plane waves of spin 0, 1, and 2 by linearized Schwarzschild and Kerr geometries in the long-wave length, weak-field limit (wavelength of incident radiation >> radius of scatterer >> mass of scatterer). We find that the polarization of r ight (or left) circularly polarized electromagnetic waves is unaffected by the scattering process (i.e., helicity is conserved), and that th~ two helicity (polarization) states of the photon are scattered diffe rently by the Kerr geometry. 'This coupling between the pho- ton helicity and the angular mome ntum of the scatterer also leads to a partial polarization of unpolarized incide nt light. For gravitational waves, on the other hand, there is neither helicity conservation nor helicity-dependent scattering; and the angular momentum of the scatterer has no polarizing effect on incident, unpolarized gravitational wave s . * Supported in part by the National Aeronautics and Space Administration [NGR 0~ -00 2- 256 ] and the National Science Foundation [AST76-80801]. 155 I. INTRODUCTION Recent observations by Harwit et al. 1 have placed an upper limit on the difference of deflection between left and right circularly polarized radio beams passing near the limb of the sun. Whereas pre- vious electromagnetic tests of general relativity (light bending near 2 the sun, Shapiro time delay of radar signals, gravitational redshift ) probe only the geometric optics limit of electromagnetic-gravitational coupling, this experiment goes beyond geometric optics. 1be deflection is independent of polarization in the geometric optics limit; but for real, physical waves the helicity of the wave should couple to the angular momentum of the deflecting object ("magnetic-type" gravitational effect) to produce helicity-dependent deflection~ helicity dependence which, for the sun, is below the accuracy of Harwit et al., but which should exist nevertheless. A number of recent papers have used general relativity theory to investigate this helicity dependence and other aspects of the interac. i ng b o d y. 3-13 tion between incoming waves an d a gravitat Gradually the full picture of such interactions is emerging; but there remain as yet a number of gaps in the picture. 1be purpose of this paper is to fill in one of those gaps: the full details of the long-wavelength limit for rotating and weakly gravitating bodies (wavelength) _ 2rr/u.i >> (size of body) .::; L >> (gravitational radius) ~ M for scalar and gravitational waves as well as electromagnetic. ( 1. 1) 156 In the regime 2rr/w >> L >> M it is better to speak of a "scattering" of th e waves than a "deflection"; and it is most useful to calculate the amplitude Tfi for scatte ring of an incoming plane wave Ii) into an outgoing (final) plane wave If). derive everything of From this scattering amplitud e one can int e rest ~ the explicit form of the scattered wave; the differential scattering cross section dcr/dn; the amount of focussing; i.e., the deflection angle in the regime whe re it has meaning, (wavelength)<< (impact parameter); etc. We, like some others b e fore us, lli.-15 have found the Feynman-diagram technique to be far more powerful than partial-wave analyses for studying the long-wavelength limit of classical scattering. Historically the Feynman technique was first used in conjunction with quantum electro- . dynamical processes. 16-18 Its efficiency as a problem-solving tool soon led to its widespread use in many aspects of quantum interactions, . including quantum gravity. 19- 2 1 However, since classical scattering is the long-wavelength limit of quantum scattering, one can perfectly well us e the technique to solve our type of classical problem. Our paper is in six sections. Section II gives the Lagrangians, verte x rules and diagrams needed for each type of wave (scalar, electromagnetic, and gravitational), as well as the formula for the differential scattering cross section in terms of the transition amplitude . In sec- tions III, IV, and V we treat the scattering of scalar, e lectromagnetic and gravitational waves, respectively. Section VI discusse s and con- trasts our results with those of other authors. 157 II. FEYNMAN DIAGRAMS FOR SCATTERING Th e classical problem of the scatterin g of a massless field propagating in a slightly curve d spacetime may be treated by quantizing both the gravitational backgr ound and the scattered field. In this scenario both fields e volve in a Minkowski spacetime and couple according t o the Feynman v er t ex rules. wor k This approach may be contra ste d to the {: p e t ers, 13 in . wh.ic h t h e gravitationa . . 1 b ac 1~gr oun d is . cons i. d ere d o~ to be a passive nondynamical entity, whos e influence on the propagat- ing field is e mbodied in a curved space time Gr ee n function. In thi s section we summarize the r e l e vant Feynman rules. The wave e quation for source-free scalar waves 0 lj! - uR\j! =0 ( 2 .1) may b e obta ined from the -Lagrangian density ..ts - - ~ (gat3,,,~ ,,,~,A + uR'\I 2) ,a 2 ( 2 . 2) I-' where u is a constant, R the curvature scalar, g = det For u = 1/6, 'Ir llgat311 and represents conformally invariant waves. . Following F e ynman 19-21 and Gupta, 22 and since we r e quir e that !h°f31 << 1 everywhere, we e xpand the g rav itati onal field about th e flat Minkowski background: 0!3 - 9 ( 2.3 ) whe r e the gravitational coupling co'."lstant A -- ~ and we use units in 158 which G = ti = c = 1. Indices a re raised and lowe red using t he Minkowski me tric ~0(3 = ~0(3 = dia g (-1, 1,1,1), commas denote partial de rivatives, semicolons de note covariant derivative s with r es pe ct to the me tric and hot3 is th e trac e reversed metric per turbation. The de t erminant f a ctor ,/":g, g0f3 and R now become infinite series in "A, (2.4) g ot3 = ~ot3 - 2"A(h- ot3 - 12 -h~ ot3 ) + O(A2 ) ( 2. 5 ) (2. 6) - µ where the trace of th e metric perturbation is deno ted by h = h • µ Expanding (2.2) in powers of A we find that: ( 2.7) whe re (2. 8 ) ( 2. 9) The free (i.e., noninte r a ction) La grangian~ describe s the free propagation of the sca lar field ~ in Minkowski space, where as the terms proportional "\ "\ 2 to ", " , e tc. r e presen t the interaction part s of ..!.., i. e . , they dete rmine how the gravitational fie ld hot3 c ouples to the scalar field ~ . In this formalism, quant ization of th e Lagrangian density is equivalent to treating hot3 and ~ as quantum field operators. From;('. we may derive th e ampli tude T for a transition of the 21 1 159 scalar field from an initial plane wave state with wave-vector ("momentum") lka to a final state with "momentum" 2 ka while absorbing a graviton with momentum qa and polarization eaf3 (Fig. 1) (2.10) Here we have used the notation A(aB~) = ~ (AaB~ + Bcf~) and the super- script 1 (2) denotes the initial (final) state. Conservation of 4- momentum requires that, ( 2.11) In this calculation we shall limit ourselves to interactions propor2 tional to A , (single graviton exchange); in other words, we shall calculate the scattering cross sections in the first Born approximation. In the classical limit for the scattering of waves with angular frequency w by a mass M with angular momentum J, this corresponds to 2 calculating at first order in the dimensionless quantitie s Mw and Jw • Since our interest is restricted to a gravitational background geometry generated by classical energy-momentum distributions which are not affected appreciably by the scattering process, we may replace the virtual graviton by an external field. 23 In particular we consider only static fields; hence in the vertex rule (2.10) eaf3 stands for the 3-dimensional Fourier transform of haf3 and the graviton 4-momentum is pure spacelike (qo = 0). The transition amplitude T above has been normalized by the 21 definition (2.12) 160 where s 21 is the S-matrix connecting the initial to the final state. With this normalization for r 21 , the differential cross section for the scattering of a zero-rest-mass wave with frequency w into a solid angle dn is: (2.13) dcr where D denotes the density of final states: D 2 = w ( 2rc) df! 3 (2.14) Turn now to the scattering of electromagnetic waves off a slightly curved background. The manifestly covariant photon Lagrangian density, obtained by minimal coupling to gravity, is: ..!' EM ( µa v~ ) = - -41 "Jr--g g g F F µv Of3 (2.15) where F is the electromagnetic field tensor computed from the µv Maxwell vector potential A µ F µv by: = A v,µ - Aµ,v ( 2. 16) From (2.15) and (2.16) one obtains the field equations for the sourcefree electromagnetic field: F + F + F µv;A v\;µ \µ;v F v µv; = 0 0 ( 2. 17) (2.18) We expand the photon Lagrangian density in powers of A according to (2.7) and obtain: 161 _ µa v~F 1 4J - - 4 Ti Ti F µv 0!3 (2.19) (2.20) After proper permutation of the photon labels, .e provides the graviton1 photon-photon vertex rule {see Fig. 1) Here 1 a l a k and E are the 4-momentum and polarization vector of the ingoing photon, whereas the outgoing photon. 2 a 2 a k and £ denote the respective properties of In accordance with the external-field approximation eaf3 denotes the Fourier transform of hat3. Note that the transition amplitude (2.21) is invariant under a gauge transformation of the form: i where l e: (i = 1,2) a+ (2.22) is an arbitrary scalar. Finally we turn to the scattering of gravitational waves by the gravitational background. One arrives at the matter-free Einstein field equations R µv 0 (2. 23) by varying the Lagrangian density -;fG (2.24) 162 UV Taking for our basic fields g r-· 'LV -.:-gg' andrJ - :J ~l V =- « I ,--: ~1 V 1 0 '1-g , rathe r than the metric itself, we can express the Einstein gravitational Lagranginn density (2.24) in the particularly convenie nt Goldberg 21.;. form; (2.25 ) After \·7e expand (2.24) in powers of A, the components of ;f'-C become: (2.2G) .t Tio.e, -T1µ v (Tl 0;'3, ~L 1 'f\1 ,/ o: - 2 T1 T1 !3, c1, VI-' µf3, 0 v + 2 h \.tCX V - .!. h ·r; ) + h' O'h µv,o: (2.27) ,p. ,v 2 The interaction part ;1. , appropriately symmetrized with reo;pect to the 1 graviton labels, provides th~ expression for the thr.:'e - graviton vertex (see Fig. 1): -µv{ - 2(1-* l k 2k .e:· : 2·-· c .• 'Ae =· 1-· If( e = ~l 1 f3 2 a 2- ·>:· µo: e v~ k ,0 ) + I~[ l· k • ·k ·e - µ0 2 1- k( c - wllere e 2--:-:-o_p e µv 13 2- ·:.:- - q ( e q 1 0: µv kA ·!· ,, l_cx 1 · _ 1-;.:- 2e_:f:J: 2k ) ] "' µv. c', v µex c ...,n. k v 2 .... n. 2 .. 1 s c ~+ _q • k ... ~·.-.e - f3v Pv µ 2 --Y.- n. 1 1 ·e f-' - _q_ • k + llµ)~l. 2k + ( k 1- v v 2 - ·;>i 2--lEe + e µv ~ ~L 1- ] fJ 1-1 1-2... r. e) - q • k e c -· . µv ?- 2-r. 1- + _9_ • -k e 1-o~ e ; µv ~. lk)lg :2~\- , lk 11 1-2-X- -1- T1 ( <i 2kCl 2e-·);\~0..9 - q lk 1(;2e;-•:•:o;3))} µ 'v e e ·1µv a p a: P lko: · , c 2ko:, 2-o3 e · ; and qo:, eo;3 rE:>fcr to the momenta and polarizations of the eravitons and ·1 ··- 2~= ~ denotes the tensor inner (2.28) 163 product. Unl i ke th e graviton-photon-photon transition amplitude ( 2 . 2 1), the three-graviton trans ition amplitude is not invariant unde r the analogous gauge trans formation, which in this instance is of the form; i-Of3 e + i-Qf3 e + ika 13 X + ikf3 a X (i = 1, 2 ) ( 2. 29) a represents an arb it rary vector. where X In general, the gauge invariance of the amplitudes is guarantee d by the Feynman-diagram formalism as long as all the diagrams of the same orde r in the coupling constant a re include d. Due to our ignor a nce of the propaga tor for an objec t of mass M and v e ry high qu a ntum mechanical spin, we omit all diagrams but the graviton-pole diagram. (Thi s difficulty in formulating the quantum problem could probably be avoided by a classical analysis.) In the ex t e rnal-fie ld a pproximation (no r e coil of scatterer) the amplitude corre spondi ng to this dia gram i s given by (2.28 ) where ~µ v stand s f or the 3-dimensional Fourie r transform --µv of h • The ex t e rna l-fie ld a pproximation s e rves to simplify the alge- bra but the effe ct of the omitte d diagrams is to yie ld a n amplitude ( 2 . 2 7) that is not gauge invariant, and is valid only for small s cattering angles. III. SCALAR WAVES Since the waves have wave l e ngth much l arger than the scatterer, they cannot probe (at first orde r) either the scatte rer's internal structure or the quadrupole and higher-order mome nts of its gravitational field. For this reason, a nd because we calculate only to lowest 164 order in A, we can approximate the scatterer's gravitational field by the linearized metric for the exterior of a spherical body endowed with angular momentum: g 2M oo = -( 1 - - ) , r Here M is the mass of the body and Ma = J is its angular momentum. The Fourier transforms of the hat3 are given by, e e 'AM 00 =2 oj - q iAM e. JO = 2q2 (~ x q). - J (3. 2) ejk = 0 where q is the (pure spacelike) momentum transfer q = 2 1 ~ - ~ (q 0 = O). Permitting the angular momentum per unit mass~ to vanish in (3.1) or (3. 2), we recover the linearized Schwarzschild geometry. (2.10), (2.13), (2.ll~), Using Eqs. and (3.2), the differential scattering cross section becomes 2 = {(1 - 2u sin 4 sin G/2 M 2 G/2) 2 + w2(~ • ( 1EX 2 £)) 2 } · t h e angu 1ar f requency o f t h e sea 1ar wave, In t h e a b ove w is (3.3) 1 "'k and 2 "'k are unit 3 -vectors along the propagation directions of the incident and scattered fields respectively, and G is the angle between and 2 k. 1-. ~ and Allowing ~ to vanish (linearized Schwarzschild geometry) one recovers the result previously obtained by Peters: . sin I (1 - 2u sin G/2 2 13 G/2) 2 (3.4) 165 Due to the r -1 dependence of the Newtonian potential, for the case of 4 minimal coupling (u = O), Eq. (3.5) reduces to the usual l/sin Rutherford-type cross section. G/2 For non-minimal coupling (u f O), the cross section still exhibits the Rutherford-type angular dependence, but only for G << 1. This is not surprising, since it is the scalar curvature R which gives rise to u-dependent terms in the cross section. Considering that R is nonzero only along the worldline of the scatterer, we see that for large impact parameters (i.e., small scattering angles) the scalar curvature cannot significantly contribute to the differential One may rewrite the scattering cross section for cross section. rotating bodies (3.3) in the suggestive form: (~~)SCllW. (dcr) dn + M2 a 2w2 sin 1 2 . 2g sin2 cp sin a SJ.n (3.5) G/2 with a, G, and ~ as shown in Fig. 2. Equation ( 3 .5) shows that the effect of angular momentum is to add a positive semi-definite term to (dcr/dn)SCHw· For small scattering angles this angular momentum term is negligibl e with r e spect to (dcr/dn) cHW· 5 This can be easily understood by noticing that for large impact parameters the r dominates the r -2 -1 00 dependence of the magnetic-type gravitational field h . , which is the source of the angular momentum term. oi - depende nce of the Newtonian potential h Another inter- esting feature of (3. 5 ) is that the scattering in the backward direction is finite and independent of the angular momentum a: (3.6) lGG IV. ELECTROMAGNETIC WAVES Theoretically more interesting and of possible observational importance is the gravitational scattering of electromagnetic waves. We choose the polarizations of the photons to be purely spacelike [ 1~ = (o, 1:), 2 £ = (0, 2f)) and use Eqs. (2.13), (2.14), and (2.21). The result for the scattering of electromagnetic waves with initial polarization 1 ~ into some polarization 2f is: ( dcr) = _ _il~- dn 4 sin4 G/2 iw(2<1E x 2B) . :<1: . 2_:*) + (<2E - l~) x 2~·*). :<2E. i~) + (<2E - iE) x \) . ~<1£ . 2f*)] 2 (4.1) .+ 1 For linear polarization ( ~ and 2 : real) the contribution of the angular rnornP.ntum a to the cross section (4.1) will be proportional to a 2 2 w, 1 2 whereas for circular polar iza tions ( : and E complex) the contribution will include an aw-term. We first consider circular polarizations (i.e., pure helicity states) and we choose for the photon basis states: 1 R ~L = _1_ (e .f2 -x ± i ~y) 2 R = _1_ (e ± i ~J ~L .f2 -G .... e where e ' -y' -x (l+.2) e-cp are unit vectors in the x, y, G, and cp directions. After some algebraic manipulations (4.1) yields: (l+.3) 167 2 [cotg G/2 ± 2aw cos G/2(cos a cos G/2 2 I 2 2 . 2 2 G . 2 + sin a sin QI 2 cos cp) ) + .~ a w sin a cotg sin Cf' 2 (4.4) where the first (second) subscript denotes the initial (final) polarization and the upper (lower) sign in (4.4) refers to the RR (LL) case. linearized Schwarzschild geometry by Peters: For the (4.4) reduces to recent results obtained 13 dcr SCHW. dcr SCHW. (--) - (dn) d!l RR 4 2 M cotg G/2 (4.5) LL In the circular polarization basis the scattering matrix is diagonal, which explicitly shows that helicity is conserved by the scattering process. 1his is not restricted only to our situation, but rather is a general property of electromagnetic wave propagation in any orientable spacPtime manifold. 11 25 ' Moreover, for the Schwarzschild geometry the scattering cross section is helicity independent, whereas for a rotating scatterer it is helicity dependent. This results in a differential gravitational deflection of right and left circularly polarized electromagnetic radiation by a rotating object. For a given impact parameter b of the incident beam, we define the angular splitting as: 5 = (angle by which R heliciti photon is scattered minus) angle by which L helicity photon is scattered We then solve the inverse scattering problem 26 (4.6) and find, to lowest order in aw: , 4M 3 [ b 3] 0 = 2 a w cos a (b) ln(2M) - 4 (4.7) 168 To obtain this result we have used the constraint that; 4M 0 << b << 1 (4.7) It must be stressed that so far we have only discussed pure helicity states. For any linearly polarized or unpolarized incident wave the scattering cross section summed over final polarization states becornes: (~~)total = M2 {cotg 11 Q/2 + 4 a 2 2 2 w [cos G/2(cos a cos G/2 + sin a sin G/2 cos ~) 2 + (sin a cotg G/2 which for the Schwarzschild case reduces to (4.5). (4.8) We therefore con- elude that all linearly polarized incident beams are deflected through the same angle. However, since the diagonal elements of the scattering matrix in the circular polarization basis are unequal, linearly polarized incident waves become elliptically polarized. For an unpolarized wave packet, on the other hand, the paths of different helicity photons are split by an amount given by (4.7). In addition, the angular momentum a induces a partial polarization of the scattered waves. We define the amount of this polarization by: p = (4.9) and we find to lowest order in a: p = 4 a wlcos a cos G/2 + sin a sin G/2 cos ~!sin 0/2 tg 0/2 (4. 10) In concluding this section we note that independent of ~ or the initial polarization, the cross section for scattering in the backward direction vani sht-~ s. 169 V. GRAVITATIONAL WAVES Using (2.13), ( 2 .14), and ( 2 . 28 ) we compute the differential cross section for the scattering of gravitational waves from an initial g into some final polarization ~: 1 polarization 2 M 4 sin (cos 2 2 1 G + w [ ( k x 2 k) . a )2)11-~: 2-*,2 ~ (5.1) G/2 This r e sult was derived in the transve rse-traceless (TI) gauge . 2 Although the transition amplitude (2.28) is not gauge invariant by itself, (5.1) yie lds reliable results for small momentum transfers, i.e ., for small scattering angles. By analogy with the photon case, we choose for the graviton basis stat e s the circular polarizations given by: 1- R ~L -1 [ ~e ~ e 2 -x-x e e ± i(e- x e- y + e-y-x e )J - y-y (5.2) Substitution of the initial and final states into ( 5 .1) yields dcr) (dcr) (dn RL = <ln LR = 2 M 2 4 16 sin 2 2 . 2 (cos G + aw sin 2 2 a sin G sin ~)(l - cos G) G/2 The nonvanishing of ( 5 .3a) clearly illustrates that here, unlike the electromagnetic case, helicity is not conserved. Moreover there is ne ithe r differe nt scattering of opposite helicity state s ( cf. 5 .3) nor pa rtial polarization of unpolarized incident gravitational radiation. 4 ( 5 .3a) 170 The latter is e asily seen by noting that the scattering cross section for either h e licity state is given by [adding (5.3a) and (5.3b)]: 2 2 2 . 2 . 2A . 2 ) ( cos G + a w sin a sin ~ sin cp . 4 sin Q/2 . 4 ) ( cos 2G + 81 sin G (5.4) Similarly, for the scattering of orthogonal linear polarizations denote d by 11 e == - :::::+ ~2 1e ~x (e-x-x e - e-ye- y ) 1 {e-x-y e + e-y-x e) -- ~2 (5. 5) one finds, after sunnning over the final polarizations a nd use of (5.1), da <an)+ M2 ·- sin .i. 2 2 2 2 . 2g sin . 2 cp) {cos g + a w sin a sin Q/2 1 4 2 2 {cos g + . sin G cos 2cp) 4 do ( dn)x M2 ( 5 . 6a) 2 2 2 2 . 2cp) . 2g sin - --:-4- (cos G + a w sin a sin sin Q/2 2 1 . 4 2 (cos Q + 4 sin Q sin 2cp) ( 5 .6b) For unpolarized incident gravitational waves [i.e., averaging over cp in ( 5 . 6a) and ( 5 . 6b) and summing], the differential sea ttering cross section is given by ( 5 .l~). Allowing a + O, we recover Peters' results 2 apart from a factor of cos Q: da SCHW. 2 (dcr )SCHW. (dn)THIS paper = cos O d{l PETERS ( 5 .7) 171 For small angle scattering there is good agreement. One may recover Peters' r es ult exactly by calculating the scattering of gravitational waves off a massive spin-0 meson. Inclusion of all the relevant Feynman diagra ms then leads to a gauge invariant transition amplitude. Actually, for the choice of the TT-gauge only the graviton-pole and seagull diagrams survive, and one obtains Peters' results exactly, i.e.: 1- 2-*,2 1 !:_: ~ I 4 - sin G/2 (5.8) As a concluding remark, we note that independent of the polarization of the incident gravitational wave and the angular momentum ~'the cross section for backscatter is nonzero. Whereas the exact dependence of (do/dn)G=JC on the angular mome ntum ~ cannot be inferred from the cross sections derived above (they are valid only for small scatt e ring angles), one finds from ( 5 .8) that the gravitational backscatter in a linearized Schwarzschild geometry is given by: (5 .9) In addition, if the incident radiation is in a pure helicity state, the backscattered radiation must have the opposite helicity. VI. SUMMARY AND CONCLUSIONS The differential cross sections for the weak-field gravitational scattering of long-wavelength scalar, electromagnetic and gravitational waves have been calculated using Feynman perturbation methods. For the linearized Schwarzschild geometry, we have recovered the 172 results obtained by Peters, malism. 13 although he used a Green function for- In p3rticular, forelectromagnetic waves helicity is conserved, whereas for gravitational waves it is not. Endowing the scatterer with an angular momentum a, leads to helicity~dependent effects in electromagnetic wave scattering. Although the photon helicity is still conserved, the coupling between this helicity and the angular momentum of the scatterer results in a) different scattering of right and left circularly polarized photons and b) partial polarization of unpolarized incident electromagnetic radiation. The high-frequency limits of these 5 11 effects have been discussed before by Mashhoon. ' Whereas in the high frequency limit (wM >> 1), the angular split 5 [defined by (4.6)], and -1 polarization p [defined by (4.9)] are proportional to aw frequency limit (wM << 1) they are proportional to aw. , in the low- This confirms the belief that the magnetic-type gravitational field of a rotating body distinguishes between the helicity states of a photon only in the diffraction limit, i.e., when the wavelength of the incident photon is of the same order as the Schwarzschild radius of the scatterer. Gravitational waves do not exhibit any of these angular-momentuminduced effects. As a final conunent, we note that this method may easily be applied to the gravitational scattering of non-integer spin or massive fields. ACKNCMLEDGMENT We appreciate many stimulating discussions with Richard Feynman, which we enjoyed greatly during the course of this research, and we thank Kip S. Thorne for pointers regarding our literary style. 173 REFERENCES 1. M. Harwit, R. V. E. Lovelace, B. Dennison, D. L. Jauncey, J. Broderick, Nature _?49, 230 (1971+). 2. See any textbook on the general theory of relativity, e.g., C. W. Misner, K. S. Thorne, J. A. Wheeler, Gravitation (Freeman & Co., San Francisco, 1973). 3. W. W. Hildreth, unpublished Ph.D. thesis, Princeton University (1964); and R. A. Matzner, J. Math. Phys. ~ 163 (1968) have studied the scattering and absorption of sc~lar waves by a fully nonlinear Schwarzschild black hole using partial wave analyses. 4. C. V. Vishveshwara, Nature 227, 936 (1970) studied the scattering and absorption of gravitational waves by a Schwarzschild hole using the Regge-Wheeler formalism and a partial-wave analysis. 5. B. Mashhoon, Phys. Rev. D7, 2807 (1973) and _!Q, 1059 (1974); and R. Fabbri, Phys. Rev. Dl2, 933 (1975) studied the scattering and absorption of electromagnetic waves by a Schwarzschild hole using partial-wave analysis and the "dielectric formalism" of Ref. 6. 6. A. M. Volkov, A. A. Izmest'ev, G. V. Skrotskii, Zh. Eksp. Theor. Fiz. 3 7. 1254 (1970); Sov. Phys.-JETP 32, 686 (1971). P. L. Chrzanowski, R. A. Matzner, V. D. Sandberg, and M. P. Ryan, Phys. Rev. Dll~, 317 (1976 ) have constructed a formalism to study the scattering and absorption of 11 Coulomb-distorted11 plane waves by a rotating (Kerr) black hole. This formalism is valid for all types of waves (scalar, electromagnetic, gravitational); it uses the Teukolsky equation and partial-wave techniques. 1~ 8. A. Einstein, Ann. Phys. (Germany) 49, 769 (1916) studied the deflection of waves by a spherical, nonrotating body in the geometric optics limit for large impact parameter (b >> A, b >> M; weak-field limit). Of course, this led to his famous prediction of the deflection of light. ~ 9. G. V. Skrotskii, Dokl. Akad. Nauk SSSR 111~, 73 (1957); Sov. Phys. - Doklady 2, 226 (1957) studied the deflection of electromagnetic waves by a rotating body in the large-impact-parameter, geometric-optics limit; his study revealed the rotation of the plane of polarization caused by coupling to the body's magnetictype gravitational field (angular momentum). 10. T. C. Mo and C. H. Papas, Phys. Rev. ~ 1708 (1971) used a combination of dielectric and Debye formalisms to restudy the same problem as Einstein (b >> M >>A), and discover an increase in electromagnetic wave intensity due to gravitational focussing. 11. B. Mashhoon, Phys. Rev • .!?..!:..~, 2679 (1975) and Nature 250, 316 (1974) used the dielectric-Debye formalism to study deviations from the geometric-optics limit for high-frequency electromagnetic waves and large impact parameter (b >> M >> A), and for rotating spherical bodies . His study revealed the dependence of deflection on helicity. 12. P. J. Westervelt, Phys. Rev. D3, 2319 (1971) used flat space wave equations to calculate the scattering of plane electromagnetic and gravitational waves by the Newtonian gravitational field of a point mass. 175 13. P. C. Peters, Phys. Rev. Dl3, 775 (1976) gave expressions for the differential scattering cross sections for the scattering of long-wavelength, plane scalar, electromagnetic and gravitational waves by a weak Schwarzschild scatterer. His method utilized Green functions in a weakly curved spacetime. 14. B. S. DeWitt, Phys. Rev. 162, 1239 (1967) in this third paper of a series on quantum gravity, gives expressions for the scattering • of gravitons by scalar particles by the use of diagrams. 15 . P. J. Westervelt and L. F. Karr, Il Nuovo Cimento 66B, 129 (1970) studied classical photon-photon and photon-particle cross sections using the general relativistic geodesic equation and compared their results with those obtained by quantum field theory. 16. R. P. Feynman in Quantum Electrodynamics, edited by J. 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Newton; Scattering Theory of Waves and Parti cles (McGrawHill, New York, 1966 ) , p. 140. 177 FIGURE CAPTIONS Fig. 1. Tiie graviton-zero rest mass field-zero rest mass field vertex. Tiie wavy line repre sents a graviton. Tiie solid lines repre- sent either scalar, electromagnetic, or gravitational quanta. Fig. 2. The spatial orientation of the angular momentum a and the scattered direction 2~ k relative to the incident direction 178 qa , -a{3 e Fig. l 179 z 'k " rv 11 I I I I I I I I I I I I cp "" I . ""-J x Fig. 2 """' y