On the Dynamical Space-Times Which Contain a Conformal Euclidean 3-Space Citation Robertson, Howard Percy (1925) On the Dynamical Space-Times Which Contain a Conformal Euclidean 3-Space. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/4RWV-4C94. https://resolver.caltech.edu/CaltechETD:etd-12072004-142200 Abstract No abstract. Item Type: Thesis (Dissertation (Ph.D.)) Subject Keywords: Mathematics and Physics Degree Grantor: California Institute of Technology Division: Physics, Mathematics and Astronomy Major Option: Mathematics Minor Option: Physics Thesis Availability: Public (worldwide access) Research Advisor(s): Bateman, Harry (advisor) Epstein, Paul Sophus (advisor) Thesis Committee: Unknown, Unknown Defense Date: 1 January 1925 Record Number: CaltechETD:etd-12072004-142200 Persistent URL: https://resolver.caltech.edu/CaltechETD:etd-12072004-142200 DOI: 10.7907/4RWV-4C94 Default Usage Policy: No commercial reproduction, distribution, display or performance rights in this work are provided. ID Code: 4823 Collection: CaltechTHESIS Deposited By: Imported from ETD-db Deposited On: 08 Dec 2004 Last Modified: 03 Oct 2019 23:38 Thesis Files Preview PDF (Robertson_hw_1925.pdf) - Final Version See Usage Policy. 1MB Repository Staff Only: item control page

California Iustitute of Dechnolosy Pasadena cOslifornia ON DYNAMTCsL SPACH-TILUS WHICH CONTAIN A COLFORLiAL HUCLIvwan o-SrACw Thesis for Degree Doctor of Philosophy Howard bercy Robertson 1925 ON DYNAMICAL SPACS-TIMES WHICH CONTAIN A CONFORMAL HUCLIDEAN 3-SPACH, Hinstein's theory of gravitation requires that phys-~ ical space-time be a four dimensional manifold whose line element is determined by an invariant quadratic form (0.1) de > z Gey Ax PAK” revel 4 whose coefficients g,,satisfy the ten differential equations (0.2) Gevt dew in points outside of matter. In these equations ) is the so-called cosmological constant and G,.y the contracted Riemann-Christoffel tensor S ZB ea S where BS = EG Lreebfen ed pend} Sa eeat Blew Leva} being the three index Christoffel symbol of the second kind. Using this value of B ue , anda introducing g, the determinant whose elements are the g,,, we may write 2 xe Pax Gey = & The functions =) vv re | Wee Dew | Kee Aes (1) AN] are cakled the components of Riemann curvature; if they are constant and equal to sach other the manifold in question is a hyper-sphere. A material particle in a field defined wy ecuations (0.2) is assumed to move along a non-minimal geodesic, the equations of which are “y # < Ax Ay PL a Bye A beam of lisht follows a minimal geodesic, i.e. a geodesic along which (0.31) dar 0 It is the purpose of this thesis to restrict the functions g,, a priori, to solve the equations (0.2) under such restrictions, and to give a geometrical or, if possible, a physical interpretation of the solutions. In particular, we examine dynamical orthogonal manifolds which contain a conformal euclidean 3S-space. Tie line element of such a world must be of the form Bar 2 Fr ra) Tha ed ede |e Ge ees) Aa where f, is really a function of the variable x*. If we interpret x* as the time coordinate, we see from (0.31) that the velocity of light in this manifold is independent of its direction at a point. Physically, then, we propose to find all dynamical space-times which admit of orthogonal coordinates in which the velocity of light is isotropic. I The Lgquations. The proolem may be stated: To determine the functions X(r4z2¢\ and Bdxyrt) in Gt), dar = ©” LAdxredurs darfaePae® so they satisfy the cosmological equations (0.2). For con- venience we have here written ue mh u . 1 ‘ a +. {se F2etF i oe ne Or ee ies oe 3 Letting Ayo 3X Mat 3 yt etc., we may write the components of the tensor Gy, ee Guy ad Aye Ae Ks -*. BL- x. BL 4 PBB, a BL, a ( Kia - Xa B, | v L La -B)¢ ( \ G = Pa Pe Oa OOO AE eT +e \ nat Me smart VW : (p-\ G - 2) ag + he a= .\} + e- Sa Bie Page PSA) + Blac R) 4 Alms a 44 =~ the other six components are obtained from the auove by the eyclic permutation (123). The equations (0.2) then become G4 5 bre = Gra > 0 A Cc D> CK 8! Le G+ Guise Gay - Gag & yo LA E Gas - re? or M4 ~ 4 B= 0 ete. ~ R—- KY ~La@ 4 Bre FB, = Ka- KO -tse. RL «+ Ads Bt ete, u clea) “a Wh Re Hy) az 4K ee 4 Fe Putz ye 2 A S ag MH HBL MS Lt RBS + Biy so etc, - | D a ee ee ee eee ~e) ox mF { eg t a (wa ~Pa)} >Re On solving set A we find G3) ge begr,- Ang fit) where f is an arbitrary function of t. Suustituting this ‘value of SB in the remaining equations, an. letiing x > - Lrg e we Rass Cay -_ _ LL — = (a) ete D bs e C Qua — bSv 2. Sava LC Vee 2 eta &. Q Rs g DH te eae Re Ra) BERET REE BEC) quation His not independent, for it is a third order differential equation of which D is the first intesral. Equations B and © give on intezration Rly = BALK YL) ete. Rar kaye PALES E) etc, ete., where a, 1 etc. are arbitrary functions of x, y and az. We may now state: A manifold whose metrical relations are determined by Ayre dyredrt at) dar = ~ 4 yt Cuyrt) d&™ Rox grt) igs an Binstein field (i.e. satisfies the equations (0.2)) of the required type, provided ¥ and @ satisfy B eye Malasey Wy, = SHOX Se) Qe Cheer] (\.4) . Cc. Re Ray er LMxsz) Ray- 8a = eam CK 42) R,,~ Par * e m( xr 4a) D HL CR eae Sas) 4 SCO MOLL et) Tae S KLE) where a, b, c, 1, m ani n are so far arbitrary, functions of x, y and z, except for the identity 2xrwmim+o?. II The Conditions of Intesrability. Wwe must uow find the which insure the existence B, C and D above. The first three sets conditions of intesrabilit, of an integral of equations of conditions are found oy dif- ferentiating the equations. First, since °° AR x4 OC. ~ i) ash eX ay oh set 3 requires that \ (a) Ras Lear Qbe TRE = RcetLecy Next, 2 Cea Bs) = Ue ~ 2S an anu whence from B and ¢ aes. Ure ae b ax 24 ae The two other euyuations of C yield gimilar conditions; this set may be written ~c x b,)\ = 9 i Cam. - a+ C\ > 0 a) em- Sax eo * i g be RA HO CMe Brey = © » The third sat is obtained from equation D in conjunc- tion with B and ¢. Since Ley = Cau + ey - el an \ D may be written -el Rive Py) ~ 5 ‘ A gr Comemye @raetaey = B-€°04) Differentiating this equation with respect to x and sub- 8 {ue stituting Qe ere RB, eb we find e> \ - by~ Cu * 5 Came ans \t = 0 The eyclic permutation (abe)(1mn)(125) gives two other con- ditions, corresponding to differentiation with respect to y and z, so \ (r.3) QC, + An ~_ \ Ar + b, = “he conditions (2.1) and (2.2) represent five equations vetween the logaritimic derivatives &, g Sans * of @ , which requires that the functions a, 1 etc. satisfy the two eliminants of the ecauations. we may then solve for the derivatives of @ , and demand that this set of f equations be consist differential ant with the original ones. It will ue found convenient to var; the vroceedure by whic:. this is aceomplisnhned, according to the form cof a, bd and ec. We here develop these additioual conditions for ti.e use alboc#O +; the corresvonding conditicns for the alter- native cases are of a modified form, and ii is of advantage to develop then at 2 later stuse. In the case abcz+o, then, we wa, soive (z.1) for any two of the lozarit.umie derivatives of e (vith respect to n terms of the other one; thus cs a ~ be RL & b * g bb > xX, y and z) Cr A,- 8s ‘eC Substituting those into the first equation of (2.2) and collecting coafificients of &@ and e& we find \ (ui) Ave, + Aexe where Al = bch - alcd-b*) A ~ LY be SB aces by) = a Cee) Eble cyt | Proceeding in this way vith the other two equations of (2.¢) we may throw them into the form A. es Ave ze A. Re, + Aye = 0 where 4) and a, ete. are obtained from a! and a, vy th @ a,cliec peruutation (ave)(1lmn)(123¢). 1 In order that our conditions be consistent with equations B we must have = ce by . Rise an Lee te SE] = wa Hliminating &, this becomes (4) | Bie, + Bezo where ( a, _ by , Abs \ a, > a b awe 5 L Ca- by L 1 ca be), - "2 ca,- bye Oa ey) C4, \\ Also RJ els Reso \ ‘ - By ey + R,e=°0 where tne new B's are again obtained by cyclic permutation. Finally we ast that & @ WSs | eh ‘wee 2 t = r WI i @ je ra C4 7D cle a 1, au 1s R- Ray ay Leet e oy Transformins this e:uation as in the previous set, we may write it " ! (1.9) Cc, ee Gk, + C0 zt 0 where Cee- bt) - 4 rary As +” x (a,-% 6.) - ae C9,~>%,) Cc Lb we \ pb Cae BY eA Ca bY C4 bY . = Ca,- Ss), ~ ee ¢4@,-c,\ 6 a-se,\4 Correspondingly u . f _ Cc. ge ~ Cy QL + CC. e —~ 9 Cc. es Cy ey + C,e> 9? + The 14 conditions which have been obtained avove are not independent, but we have vritten them in this form for Ga the sake of symmetry. III Solution of the Equations. In order to solve the ecuations (1.4) we divide the problem into four mutually exclusive and exhaustive varts, according to the form of the three logarithmic derivatives Ge Rs Rs € @ e ofe. The cases which arise are: (a) none of the three expressions e oo e are functions of *t (b) one and one only is a function of + (c}) two and two ouly are functions of t (ad) all thres are functions of + We now consider these cases separately. In the first case must be of the form Lab) a g- & G) w lve) LO in order that its logarithmic derivatives with respect to X, y and z be independent of t. ‘From A d,' ot) aL T™ draleys Ft) so 7T is a function of t alone. Substitutine the value of @ given by (3.1)a above into 8B we find Tray ts cta lk But k is really a function of t, whereas o and a cannot contain this variable; hence o., and a must vanish. Ap- plying this reasoning to the remaining equations 5 and ¢ we see that b, c, 1 and m must also vanish. ihe conditions of intesrability are satisfied in this case. Onus ti@ equations 8 and C become the solution of which is o = A Cx + arr) adn sdiye drake where tie d's are arbitrary constants. Hquation vb requires ay poe * that - ~c¢) el dort) O'+ a where pon Atedy sdf - addy il Bor case (a), then, the functions 7 and e are Ys ta JR - TR e- ht) (acheter) edix cdg edie eds | wa wnere k is an arbitrary function of t, and the d's are arbitrary constants. we next consider the case in which only one of the logarithmie derivatives, say x , is a function of t. Hence G,1\ 6 e2 & Cyt) oO% 4,2) where ~ is really a function of t. : a, , & . Since @ and @ as well as a, 1 ete. are functions of x, y and z alone, it must ve inpossisle to solve equations (2.1) or (2.2) for in terms of them. Hence a, b, ec & 8 and 1 must vanish. But then or Dr, HR 7 + Fue whence O.+0 . Similariy 7, +o . w@ conclude, there- fore, that e is a function of x and t alone. 12 IW oN is now a funstion of x alone. The con- ditions of intesravility require that i.ea@. %tm~? , But then n is 4 constant. The line element is thus defined tor tnis case by tek C3,.n\ b TS et €ce) where e is a Weierstrasse @-function (whose periods are dependent on t) satisfying UL 1: s SY to Oo Cn - ame 4 ¢ oO Ve now let * and 8 ve functions of x, y, 2 and t, S . “ and e a function of x, y and z alone. Consequently ee BCArtt) HCx yi) (2.1) recuires that b and c be zero, so since we must have re oc, + m1. > 0 an, whence ao+o . Hence here The conditions of intezravility (2.2) are Rem FAO = - Te Om,-2,) lye Ry m t+ Quran >- 4 @ OCm,44,) 13 Since it must be impossible to solve these equations for . , 8 < and e ren ~ OW NW. - A, ~ o a mM™ Ma OL or Oe AW mM + 6 (3,7. ¢ pn CA L* ms) we AC A,~ am _\ x 6 fo these may be added (2.3), i.e. IM & LN oad DA, CW ye Trays Myr be In order to intesrate the above equations on a, m and n, we write the second of set (3.2)c uA. ws ~ aaa on eliminating n by means of the first, assuming for the moment Ann ¢ 0 The general integral of this equation is a: aos pane + pal where i# is an arbitrary function of | . Vo evaluate F and complete tie intesration of (3.3)e¢ it is convenient to interchange the dependent and indevendent variaules m, n and y, %, assuming that the Jacobian 4+ 0 Our @ Ma auations then become where Ow, + 3 etc. Hliminating a and y from this om set by (3.2)e ana its integral, and solving for the two remaining derivatives 2 and z, we find { Lae 2 Cet) G rm = 74 ye me Baw moma rm) a ron ain am Oo am requires that " 7m Fr" (Bj)x~o id et of euuations a Wie may either intesrate the last : s or return to the original with y and z as independent var- jables; the latter proceedure is perhaps preferavle. Since w- A, - - (1-1,) no loss of generality is incurred by letting y,+24,+ oO. Equations (%3.2)¢ are now ~-a 2 : z INN L MM ~~ ” 4 | 7 ry a, Aa Using these values of m and n we may solve (3.2)ce for go and & . aS Oo. . + b-45* Ay 1 Ya-4Arr uw Saye yr Oo bh gh aat On intezration Con aA > nay) Cater) oS whence Twn > . Sf, IA > . Cate tt) Catt) fy 15 The condition (&S.1)e requires that Re RK e OLE i.e. @ is a homogeneous function of y and z, and of first degree. we take ee sachet) f- 5 Substituting the above values of e@ , a, m and n into (1.4) we find that Ac&e) must satisfy (23 \*- ws a ad . ae . q* rel a? 4) Ane a4 \* & ah i+ & > Gag) s ye & where C is an arbitrary constant (not zero). To intesrate this equation let ACK, &) + se ula, ) { = tom ¥ U(%t)is tien defined by y avy \" 2... & Se, 2®L etc) (2%) a i it fhe solution of this equation is { ~¢c Or- oF . & (\=4 C xircey | where & is a Weierstrasse WY -function whose invariants are 3 | Set ok S52 - Gs + £ (A&-€ce Vie now return to e and express it in eylindrical coordinates KK Ar RB wa wae BS vzAY JY er B \ - a + B (J=¢ WY « sco) Cy. 4\ ec where m(4\is an arbitrary function of t. The above solution has been obtained under the restrictions yom m\ Onnm4o ¢ O e 64 2) we now examine the cases in which these do not hold. First, assume a@mm-o, Then it is seen from (2.1l)e that A®Artmtn~o , tor otherwise it would be possible g to express ‘ or % in terms of x, y and z alone. LHouations (1.4) become Quy - Riot Qyy s ° Re eS = BL ec) whence Ve ALLS) Qt A, lt). 4 dale) — , te Cer ere). ~ do -4, Consider now the ease in which a Cm vn \ eee xz Oo a lA vw) i.e. Ql m m)~o, Hor this purpose we express a, mandn in terms of a parameter oc aA ~ ACe) man an Ce) M2~—~m Ca) Mhe equations (2.2)e and (3.3)¢ are then Ot ek AA mM EL oO W,rvwi\e oan] pea . (S \ Co,- ome SL.) oO (am'istt.m') rl - alo, 2 where ' indicates differentiation with respect tor . 17 (3.31)¢e requires that pa'+ Lm! - sa! ya! Lawl tm! for if this were not satisfiedso_ando,would vanish. But a, mand n would tren be constants and the determinant would be zero, contrary to our assumption. Thus one of the Ae eguations (%.21)c may be replaced by the eliminant of the two. Phe second of equations (3.21)c can only be satisfied s ma Me ren . * ~ et if Cu) =e or Oye Soy ; it can be shown that both possibilities lead to the same solutions. They are U here obtained by taking (me \zo, ran \! . yince ( = \ ~ 0 then Mm>y & MTN W where vy andv are constants satisfying the condition trio. The eliminant of (3.31)e¢ yields at Cprer) 2° - as oa 4. s Vo ~ i 4 bd 1 fhe first alternative, @ ~® , leads to the conelusion that a, mand n are constants. By (%3.1)c QL mm: QL A SO . 4 (sb) ¢ ee eC Re Re) Hecht) Ae ee Equations 3B and O are satisfied if aK \S. BK gr 3 Com a OA 18 z Finally, we may have yp+izo0 1.6. pri. Then pPozw>t et Tyr ery whence oz fn Cy+4e2r) . From (3.1)e e,2 cel sO Ge KCyeir,t) The solution of equations (1.4) is found to be (Cs 1) ¢ ge KO 4, & | Ke sree where “ satisfies any equation of the type aK ‘ , Ho ws WOE) eR ee and aK ~ at : ov “ee VO Four solutions (2.4-3.7)ce of the cosmological equations have thus been found in this case. Of these only the first is peculiar to (c), for the other three are (as are also those of (a) and (»)) special cases of solutions considered in detail at the end of this section. The only remaining case is that in which all the logarithmic derivatives of e with respect to x, y and gz are functions of +. We first dispose of the case in which one of the quantities a, b or c is zero, in order that we may then deal with the conditions of intesravility in the form applicable in the case abct#oO, 19 If one of the three quantities a, b or ¢c vanish the onditioris (2.1) requires that the other two do likewise, for in the case we are now considering it is not possible S etc. in terms of x, y and z alone. Simil- arly the conditions (2.2) demand that A>™+™=°9 , to express Equations (1.4) can now be integrated immediately, yielding ge Alt) Cee Dele d, C4) xc adiledy edyttirg Agee) C, t T= e ‘ £64) C3. O\ A where the d's are arovitrary functions of t, and au se Erect yr , Having disposed of the case Q®bc29 we may taxe as our conditions of intevrability sets (2.1), (2.21), (2.3), (2.4) and (2.5). Since a, 1 ete. cannot depend on t, | (2.21) and (2.4) can only be satisfied if all the coeffic- jents A, A'; 3B, B' vanish. \ \ f First A, = Av ad /\, ~~ oO so , QA ¢ i. yar 6 Cas ‘. . OC Bt v)\ ya je bc Cct~ br) AM Se OO AX~ os) ae a This makes C' in (2.5) vanish, so the remaining coeffic- jents C and ¢' must also. If wea now consider those conditions on the functions and G@ which are of first order in their derivatives, fee) a, 0 we find that they must satisfy 12 first order jifferential eauations However, of these but 6 are linearly independent. for these 6 we may take two of the first set, all of the second and one of the last; they are b ~ & oS a o. L Cc, = Ax ae 1 an < an) n ~ ob (saya Ay 424i A - 6, ah bo nr a b . qQ ~a,Cct- b*) + 6 ct Cy by 4 be (2,-Cu+ by) =o ~ b(a-¢) +t Cy ai - avert 4 ca Cmen-aacg ys oe qc~n ~* by 4 ( ~m, - a) where 1, mand n are given vy (3.1)a. It can be shown that the conditions of complete intesrability are not satisfied for this set of 6 equations in 3 dependent and 3 independent variables, so in order to find a, b and c we next consider the conditions on them which are of second order in their derivatives. First, since 3, > B.- Ryr ° we have ; Qa q 7 (a,- bi\y,s4 Ca,- b, \ ( Tet > - ea | 0 eta Now the ecuations (3.2)d may ve solved for all other deriv- atives in terms of those with respect to 2; this sveing done it is easily shown that a,- &> by S Ny 4 Ax _ 4 4, Ls oq Le , @ ON a A) S/o ty ' 4 yy _ 5. Jur, N Ow a) Cc ? ~ a = woere The equations above may aow ve intezrated ones, yielding “a 4 s wv 5 4 \7 ¥Y Csr) ad d,- by ~ oR? eC rN ? ¥Ou 5) Similarly 4 \y ¥ , b a Cy co b aq % GOV > N ° x Cy 7) 4 4; Ve i - > > WN Cy. ae COAG f& (4, x) Note that the arbitrary functions «, ® andr are simply con- nected with a, b and c oy the relation ACH) | @Cz* Ye Cx) =~ O a b C Again ey- a, _ 5 Ns + 4 Qa, 7 by n L Cs 4) "sy "y e oe ~ 4 5 G7 3.CUT 2 th 6 COL whenee ry ae _ @ Cx, 2) CS.2VV A ar lL ate err yt YoU TTT Likewise 4 ee ; ¥ Cx, 4) 33 Lohse NM 7 = Four other exoressions may oe ootained from these two by the £ ef of eyelic vermutation (abc)(«pa)(xyz). a In order to determine the so far arbitrary functions xiyr) , BC, x%\ and rO%44) we treat the second set of con- ditions in an analozous manner; they may be vritten 5 — v C t 5 Ca,- ob. ) ca ‘ Janey | ¢a,- b_\ No L bo a b “0° at x ~ 4 ~fy 4, + 6% 6) a \ deg Lca,-c,\ N SR rE oP “Atco CV.) A 22 sing the values of 4,- b, ete. given by (%.21)4 ary) , s@Cruy_ 4 Cy.is\a 24 — Correspondinely AZ alr 2) . 2 exo). oa OK QPOAN — BxKl4e _ 6 a OK ea. a) readily found; fF The solutions of these equations are C4,tA\A @ = A 2- Cr iby s B»r-Ay+t Cu where A, A, etc. are arbitrary coastants. Returning now to-equations (3.22)d we find with the aid of the above values ofa, @ and ¥ Oo ae a Cyr A pi -— 5 2 A (pert la x (Baer) 4 ads-Bar + Cx \ , My wc n” b i. where X is an arbitrary function of x. Two similar ex- cressions are obtained by the permutation (X¥Z)(abc) eta. (3.21 requires, as we have mentioned Aa & € + cm -—_ Oo wa b cc Applying this condition to equations (3.25)d wa find, on Bt ating coefficients of the independent variables, that KR > a Koa . A GS congt aque eta. Also Afar 3 Bat CC, -~ Oo enc Hence OY , hy a ; cape nN \- A (-egteot)4 X CBy+ C.)4G y-B,t+ <A| ray) Len) or | & ao a2 OA (4.6) A A > 5 v Ww S/ where < = te a -Vo_ : X ovw Cutav ew \ ER ~ Llab c\ N 7 Tt | _ = - 4 (eta sace) ae (Bat Ge) Cao Batt S&S A AA, * BBRa* CC4 > 0 These values of a, b and c satisfy the first order equations (3.2)a as well as those of second order, and represent the general solution of the conditions of intezrability (except for the case x*~f+%r-~° to be considered later) insofar as the conditions do not involvee. We must now return to set (2.1) and suvstitute the known values of a, b and c. Then VQ, -U RL TRO cane WL Vee KR TO v@eyewe, + FE+ O In order to effect the intezration of these equations we first throw tnem into vector form V dors, e x A + ea Q where a~ bey) Sf: (KX RT Now @€>AxrA *« B where we tne) Rt ARC SO H~CAR Ge, A-Dxo and IN ~ R= =. ire A nx A A+R 4AxD + SA ~~ If A7vo , it is possible to make a transformation of coordinates from M toA+% where ¥ is defined by 24 in this new system € and A secome — —— x _ ~ = — ee. AXA hie A nee AAA 4 = A me (2.1)' tells us that V teqge lies in the plane determined we by rand A . so the surfaces o(rk4reé)+ const. (t fixed) sare Surfaces of revolution whose ages are in the direction A. Changing to polar coordinates (n% 4) in which the direction oO ae) £ is that of A, the equutions (2.1)' are % uae Q~ 0 2 Age ( _ ¢% \n in oo 4 ° mv4 ce: L an “ 3 Lgwee | a7 04" whence g n*~ «* \ ns & ; a8 4 = _ ~ £ cement EV af If, then, tL. €S QG> eK. 4,3 + > a 4 an the conditions of intesrability are satisfied. The equat- ions (1.4) reduce to %y . os ~ 7 , § te NS Ch HE) > (24 | CQ nets UK 2 £) which is of the same form as the equation optained in sestion (c). If Gtoit may be written K yu 8 a : Wawes VL eee rs \- c= \ a €Se | a €? & hs er e ten a s oN whence StS bey! 4 scot _ q> yes (Se FR (%.r1\ a 20 Another type of solution arises if ¢:-0, for then cy. rv Bd ( 24, \ ~ nes Cae a» _ ££) xX >) au Ww is thus again expressible in terms of a Y -funetion. The above solutions were obtained on the assumption A#O, and are furthermore subject to the restridtions imposed in assuming that no relation exists between a, b le the algebraic solution of a and @ which renders impossi equations (3.2)d as required on p. 20, and to the tacit assunption that x Qatto, for if this condition were not satisfied it would be impossible to evaluate the functions A, Y and Z@ as on p. 22. The examination of the cases arising from the relaxetion of these conditions leads to 7 additional solutions, 3 of which can be reduced to those aneountered in previous sections, (In reducing solutions arising under this case to those of the preceding parts we must of course waive the requirement that all the logarithmic derivatives of e with respect to x, y and z are really funct- ions of t. there is, however, no ovjection to this »rocesedaure for we have divided the problem into these four parts verel, for convenience in solving it, so if we can show that a sol- ution arising under (d) is reducible to one of (b) or (c) no loss of zenerality is jocurred by SO doing.) we first consider those cases in which the processes leading to equations (3.26)4 are justifiable. If Ao the set (2.1)' may, on shift of origi pea reduced to or I Ol A) a ao —_— + Ol \) Q A (vieqe- 5 Hence e must satisfy Te. O+° A so VR: ca ' ce) oO 2) ing our axes so that D= (Ao,0) the above becone ae. An Sy > ae af ok. ie. © is a function of y, 2 and t which is homogeneous of first dezree in y and 2; this leads to solution (3.4)c. In this case we have tacitly assumed that BDto , as well as 040 (in this latter case x®erwould be zero, so we Jo not entertain this possibility at »resent.) Similarly we assumed on p. 24 that not only A+0 but also that Ato. As lonzs as we are dseling vith real vectors these restrict- —— — ions, i-gg O FO, AHO do not concern us, but if uc Fh wish to obtain the complete solution of the proulem ve ust consider the imaszinar, and conclex solutions as well. anal valine of this completensss will be evident in the next section, in which the relations between our solutions will Briefly, if A =e but Ato we may take A = (0, 8,¢8), “he inte-ration of (2.1) and the subseyuent limitation in- nosed by (1.4) are not as neatly eficcted here as in whe a3 PPESEGALINE Gace pis 2 $ 7 + on 5 Fant Se VC oy - 5 ~ Stree but es the reasoning is cecsntlially tne sane ve merely state the result: Yo this possibility there sorresponds the solution v n- = (3.3) 4 ez cseer) KC 4, &) {= 4 G40 : oa OL where | wie 24 a WE AC We RH HEI ras — = —_— 4 Again, if. A+o and +o but DO +othe problem cannot se reduce? to that of seetion (a) oy rotation of coordinates as on p. 26, for Db nay only ve reduced to the form (0, n» (B,) The solution of (2.1)'’ is then found to be . . Ate Oe is. AYA e = x «C4, t)\ $2 Se and the cosmolozical equations require that « satisfy > \ Next we concern ourselves with the case xps-o. XH, ¥ and Zin (3.25)a4 may no lonzer be obtained by the device employed avove, vut we may start with or Ribs WY Ces) (3.2 5\'d bp W4) wok Ar) cu 2ir\ Y Cx se) where X, Y, 4 and Y are to ua determined oy (3.2)d. It can v@ Shown that tne first five of these ecuations require either (1) &, Y and & be constant or (2) wan - — Ky ty fo these cases correspond the following solutions of the fundamental eguations: oD CO (1)> Here it may be shown that rotation of the coordinate system reduces e to a function of at most two of the varlabvles x, y and 2, and that the solutions then ovtained are (3.2)b, (3.6)e and (3.7)c. (2) In the sezond case a new solution is obtained. Prom (3.2)a we find that whenee by (1.4) caa\d R= win, +) where We now entertain the last possibility, i.e. that there exists between a, b and @ a relation which makes equations (3.2)d linearly independent no longer. Examination shows + . ~ « \ \ . sts : that if N= ae pet a? we are not justified in eoncluding (3.21)a. The solution of saquations (1.4) in- oO a cident to this case is of the type is bd ez (s+er\ u& €) b= aa where W satisfies any e.uation of the form 29 we have now obtained all solutions of equations (1.4). This has been accomplished by breaking up the problem into four parts and solving the e@.uations under the restrictions imposed in the several sections. It is found, however, that there is to some extent duplication of solutions. In order to avoid this repitition, and to collect our results, we here arrange our 14 solutions into 3 classes, cousisting altozsether of 10 sub-groups. In collecting we have further from the Q Simplificd the solutions ty choosing coordinate sroup of linear. orhtogonal transformations which allow us to express @ in the least number of variasles. For all these solutions LG at } 3 ee (1) Nae dU) TO + A. Ce) Yr AC) % wd Gla “+ da (4) where the a's are arvitrary functions of t and rm Ct) = Ave dds dS - Adda, (2) a e= G44) he we ct Satisfying any equation of the type REE Leth ey WES a 4 b ez Lae try Khe) Ls n* Ae where «lt,+) satisfies the equation in (a) above. 30 (3), a @ = Ror, &) satisfying = & of ue Se L \ (| -L Ran *e D\* b Q >~ ok «C&S, &) 4+ 2 where avg aK u 24 «WD aM ‘ - —~ K--4C — = (4) (4 +t) ugk a4 * (cher) % D(«] CG ez TKS 4+) 6 2 Ma oe where a) Ey gaae) 2 AO Dy, Pu) ad this solution is really a special case of (c) avove, arising on putting ¢+°o , but the auxilliary equation for w is integrated by means of a different substitution (ef. (3.28)d as opposed to (3.27)d). \ Ny where “ satisfies 5 a ens Oe e RX KER e\ {> Dees . . nAt- & f Vee CS*et) KEG, 4] i+ wire 6 & FO ne a L aA \e. “eo ~, in which ae OS . \ ~ AC K7?4 OK) g ez ex, &) where GLY = AC e? + OC) Lox) ram of tlLese solutions only {1) has teen expressed in int- eeral form. the seven cases arising under (3) have been defined by first order, second degree ordinary differential equations, all of which ma, be thrown into the form CAS \ 40 ks $ ¥o + Oe) $-tian dA by one or another of the substitutions suggested. the solut- ion of this eauation is reacily obtained in terms of a Wetor- strass'’ § -function whose argument is Cy , but the inter- « retation to be ziven in the succeeding sestion makes it te desirable to retain the differential form at present. } Only two of the ten solutions, those of case (2), cannot be senerally exvressed in integral form. TO obtain a solution of this type we must first assion to ¥§\ a definite form, intezrate the resulting ordinary differential equation in ¢, , aud then replace the two constants of integ- ration oy arbitrary functions of t. IV Interpretation. Now that we have solvcd the problem proposed at the beginning of this thesis, we turn to the interpretation of the solutions thus obtained. This may be done along either geometrical or physical lines; for this purvose we first ae Set out the appropriate tools and then proceed to the con- sideration of the sclutions themselves. Les 05 a) ® In ali cases and the seodesics (0. La @ <ece) era) wc ; ~ 2 & Ceearagr) —-( S42 _ fi) fee LLk & 5 aa Lt Tose ion ~ 4% with two. analogous eguations for » and az, and x - n ; e \ . e ' fe ga Syn) -( SB -& - sa \ ta i+u ‘4 -& a\ ee feeb) ° Q Ga g FUE T Ocal LL e e cat where te dot indicates difierentiation along tne curve. One integral of these euuations is, of cox Arse, . ea gyre KO tT ae 2 er 4°Ct \ so for our equations we ma, take this relation to-sether with any three of the four (Ck Y + R, _ ®iy Cy ym v ~~ —— & —~ 0 g @ Qs erce) ‘ we ) * er che) s (4-01 | ‘ ey, & bse Me Gores a g et ESCay AE \ te 2 Pa pe RU OL, Ry grin) ¢ ts erie) er <ercs) From these equations it is seen that if e does not contain one of the variables x, y or Z, s&y xX, then a particle which is at any time moving in a surface x+ const. always moves in that surface. Next we produce the components of the covariant Riemann-Christotfel tensor Boyee . hey are vee — CR Oe Oe coy | - ‘: ees) | Re Braves 5 ( Sw ~ Ry (or) Qa ( & xX xv e er ce) > veo eee) Vr& x . Ras . - Ran Wisie = e> Baad pr<rce | ‘9 Brave Bsases The remaining 8 non-vanighin=s components ve Outained from the first four above uy the syclie permutation (122). In the »revious section we used tresly any ilsAsar transformation of coordinates *, y end 2 which isit the euclidean 3-space eleuent sAvedye dis invariant, i.e. amy such tnat pore KL KEY s APSR PAD sut we may use a much more general class of transformations Which sends our line mont ads (1.1) into one of the same form. In particular, this is accomplished if dove PCLILT\ AZ for as” is then transformed into at. AZ TT. LAY BN ra) - > v ° In fact “pPCKEYZ+\ That T(XYZt) is of the Ty CRT E\~ igs a consequence of ( We consider the Kelvin (cf. Bateman "Electrical and Optical Yee WiieGnee jor ~ x CRLL eC x CETL), 4ISTS\ RFR) €) orrect form, i.e. yPCe Tad I) st PORT Ye | FEC class of transiormations due to wave \iotion"»y.é1) arras ro yee nu) Q- roXxeiLy Inversion, for which ak 2. eye ak ~ a eB AR NE may se oovtainsd by the double application of a transformation or the previous form. ad. at Henes we may state the importent theorem: If te ¢x14z+«) & solution of the cosmological eauations (1.4), then ao am (2 R-a Rte \ 4p Weck (4,04\ PELL +\ ~ ~ Y+eL” LeyreLy! celyeeLy? * is also. Similarly, inversion leads to a solution a (4,04 PORTE © Pe e(* ae a ae * | w8 now proceed to the consideration of the various types of solutions. If we substitute the valuss of @ and ~ given oy solution (1) p.29 into the cgom-onents of the Riemann-Christofiel tensor Byyee , we find that the manifold which they define is a hyversphere of Riemann curvature ~ 3 . Gnais solution gontains as & special case that of &. Kasner, ln which ett. (Ameriean Journal of latienaties, Vol. KLIII (192£5)). oe dsduced from tne results art (a) section III svove. Here 7 is, ss we have mentioned a fiunetion of t alongs, and a sSliget seueraligation of the work of that seetion ullows us to state: If "stationury" observers (i.@. obsorvers ata point &, y, 2 goustent) in an ortinozonal aynamical srasue-time find is indemendent of their rcsition in space, and - * ~ ma a s 2 ~ an? es a+ + ... aot tN ae ae tops es velocity of Licht is isotrorie at & point, then space-time ev i ta & hypersphere. If tee cosmelosical vonstent A is zero then © snace time of this tyve is Sclilean. if transform this solution by (4.04) avove we find PALE 4) = DEN RTE Oa) H+ OCT + O00) B+ DY C4) dredoady-Addg = De OF LOd 4D, Henee all transformations of ty ce (4.04) or (4.05) send this solution into itssalf. further, this is the only solution of our oroblem for wich all the components of wha galled tie Riemann curvature ere finite. we do not selieve that the two solutions (2) ». 29 Gan have any physical significance, for they ere essentiaily omplex. Again, if tie cosmologieal constant A is sero the line element ds degenerates. in order to illustrate the method of am iying the trausformation (4.04) for this and the sueceeding section, we here carry it out for the solution (£)b. Here ew ~ Lyaee K £ aK 2 yr O(t) - so there must exist a solution ps: Sx (- a Te | ) r ? c Lv _ }) os f al my r ariting this K (- hl>, “it } js By t~™ i me —_ and ey ot Kl 4+) = 3 Woe 2S) ahr — Mm 3 =~ Hence \WC2,4) satisfies : Yogi a eB 2. Races) S&S gs ; —~ But this ecuation is of the form ah . AT acs “3 UT whence (2)b goes into itself on applying (4.04). On per- forming various transformations of Kelvin's type on (2)a and (2)b we find they go into themselves or into each other, but not into a solution of any oth The seven solutions of group (3) offer the most iater- asting possibiliticas. We first examine the relations ). existing between the various solutions (a) - ( Qy As already mentioned, solution (4) may ba derived from (ce) by letting @-+°. we have avplied to all these solutions the transformations (4.04) and (4.041), and have fowd that in each case ve get a solution which hes already been placed ke is ct Na 5 cone w cg by ro) keg by fr K ct cr bs in) La] ct D Olutions naturally divide uy iuto two groups (a), (b), (¢) and (a), (e), (f£), (g) such hat starting with any solution in either zroup, tic remaining solutions of that sroup ma, be outained from it vy means of (4.04). hese reletions are sraphically shown on the accom- panying diagram: here a plain arrow Ge +(j) means that i gives 3 on transforming by (4.04), and @)——--+) thet j may be obtained fron ion inversion (4.041). i All solutions, with the exception of (d) and (2), go into themselves on inversion. In order to discuss this grous it is only necessary to consider (c), keeping in mind the possibility c-o. It is, however, more convenient to discuss two separate solutions, one from each of the sub-groups previously mentioned. In particular (a) and (g) are chosen, for they are of comparatively Simple form. (a) represents a spherically symmetric dynamical solution, & being ex- pressible in terms of a ©-function of r and t alone. 415 ois ceiling the case, there exist at any time an infinite nunoer of spheres each wolint of which is a sinsular point, ro and these spneres are not stationary ilu time. Similarly with (g), in which © is a @ -function of x and t, there exist at eny time an infinity of surfaces of singular points. we do not sés at present any physical significance in ti:ese solutions. In conelusion, then, we nave found that there exist gut three distinet orthosonal dynamical 4-spaces which con- tain a conformal euclidean 3-space and satisfy the cosmolog- a ical equations. The first of these is a hypersphere and Qy a another is deternined by functions wiose urgunents are essentially complex. The only case in which ve may hope to find a physical content is seen to resresent fields of 2 so far unrecognized type. I desire to nere exnress my appreciation of the int- erest which Dr. H. Bateman and Dr. Paul Epstein of the Instit- ute faculty have taken in this problem, and especiaily for the constructive criticism which Dr. Bateman has offered dur- 4 ins the course of the work. These acknowledsements would be incomplete did I not mention Dr. EH. 7. Bell of the Univ- ersity of Washington, who first interested me ii the Yheory of Kelativity.