Theory of Gravitational Spin Citation Cassen, Benedict (1930) Theory of Gravitational Spin. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/XKY5-NG34. https://resolver.caltech.edu/CaltechETD:etd-01302008-114056 Abstract No abstract. 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): Unknown, Unknown Thesis Committee: Unknown, Unknown Defense Date: 1930 Record Number: CaltechETD:etd-01302008-114056 Persistent URL: https://resolver.caltech.edu/CaltechETD:etd-01302008-114056 DOI: 10.7907/XKY5-NG34 Default Usage Policy: No commercial reproduction, distribution, display or performance rights in this work are provided. ID Code: 412 Collection: CaltechTHESIS Deposited By: Imported from ETD-db Deposited On: 30 Jan 2008 Last Modified: 03 Oct 2019 22:59 Thesis Files Preview PDF - Final Version See Usage Policy. 818kB Repository Staff Only: item control page

August 7, 1930 I\eccipt is hereby ackrmw1edged of two copies of a thesis fer the degree of Doctor of Phi 1osophy conferred in ,Jule 1930 by Benedict Gassen TH>~-ORY OF GR .vrv~:rIONJ\L SPIN. The~1s Benedtct by Cs.seen In 'art1al f'tltlllment ct the Requlrementa f'or the degree of Doctor of Phlloaophy California Inst1 tut.e of 'Ieeh.liology :Pasadena , ca11rcrn1a 1930 ·~ened let c~ seen. Dirac' B ' ell knovm H'1m11 tom1an operator 3 ,. " + ~Ao + deeortb1ng ~n electron by the potentl~l• Bd~ (pt- % A.) +<X.+>n~ -f- in an el ctromagnet1c field determined AoJ A, 7 A2./ A~ , when m·iltiplied by its conjugate operator, Fives a qu,q,drat1o Ha.mtltonlan operator - ( p +~A ~ (f " +~A c "~z. -1- LJ c i.) +-~ ±, a-;,/-/,, flgfic i d,lt 2 0 v {..C1 The lRst ttrm of this operator /, ... , ,. (.,/ shov-s thR.t the electron a.eta as if it had an imaginary rpin enere·y in an electric f1elc.1., or v:h~t amounts to the ~mag1na.ry e~u1e electric field. thinp, ~.·real sp1n energy in an In v1hat follo,· e ~ 1 t will be shtwn that an imaginary electric field h~a the same pr0;ert1es e a re:t.'-l !Jcoeler~t~on or gravitational f1eild r1nd that we may etti·"er say th8t &;~:gravitational field is an 1meg1nary etectr1c field or that an electric field is an lms.g1nAry grav1t'1t1onal fie~d. Fu~ther , by the introduation of 9 gravitational vector potent lal a.s an imagine. ~'Y electromagnet le vectctJr potential, 1 t is poes 1ble to describe electromagnetic a.ad grav1 tational .:~.henomens. by a simple un1f 1ed theory. Th, s theory h· s a.a 1 ts baste the known theory of the electrtmagb.et1c field, &~"""ftOt the ,1!;inete1n1an geometrical p1.ct11re of the gravi tatiom~l f 1el4. When an 1:nag1nary electric f1.elrt is 1nt:erpreted as a gravitat- 1onal f 1eld, then 8 D1raci3n electron, described by the above Hqmiltontr:ln operator, ""ill a.ct ae if it had a. gr'1v1tat1onal spin rn0 °r nt when in a gravitational field. Th ts aee~' to be (2) 1m·· ,~1nriry r·p1n ten1s the r1·al pl1yP109.l n ir~·niflcance of : ~1( ... h1c~~ are necess!1.ry 1n «he <'Pt!:'.\/lrattc f'ora of D\.r e's H'lmtl ton '.an opera.t·~r 1n or,,. r rho•1 ld be re lat 1. v1•t tcally 1~ t hn t 1nvar1°nt. A6 be the SC"l~r ~1ot0nt 1-a.l Let. nj A'/ Az..> A 3 ttie com)onente or the ve•t r potential of the elf'ctromacnet 1.c and ~denttf~r .l v:1th t~l ctrio oh~rr.e 1ene,~ty, J,,; h.J J; ";<1th the current dene1t1eo, then th' (1) nre the Mn.xvell- eqm1t1.ons Po1eeon eqnat1ona of the e ieotromn.gnetic fi~·:,ld. Vo be the . . cnlar potent tql 0f tn,o ·· ccel- LeJt v· ~th oom onents V; J }!; ) v~ • Jz// - 172u) Ct -- _L 4-7T (I cz. Ji! //~ J {,./- 0/ I/ z / 43 and 1d.ent1fy c6 l energy <lenet-ty, a.nd 1th the ma.es ~enstty or _J_ times the cz. c,) lz) c,, w1th the com)onents of {_z) the ~el"ton. -Poisson momentum. ,...e will t 1Y'n (rll equ'· tione In the came ,r·..,y tk!!t 1 hr: ~omponente nf' fo~~ee on ar+in1t electric chqrge "'re £ :: __(_ Jlie c L )f - d A-a i>X the components of ('·re.vi t~ t it:)Ml f.")rce -.-l- JL/;_ c rJt - It sr'eme !'1.eoessary td; introduce >f:!" '.mi t mru.ie are JI{ ~ d ylth1d r:ravit':lt.,_on~l vector (3) potential in -,:cclwr tha.t gra.v1 tational v~~ves might be quant- ized. Otherwtse,there would be no gra.vita.t6onal Poynting!s vector ~.nd. there wo 111 b- no qcttf")n ln the f teld th8.t 1 could be quanti~ed • Of cou: ae, equations (z.)a.re equivalent to four or Elnstein's ten equ~ttone for a ~ eak gravitational field where ~ is a linear function of the jt_,. i. Ii v ould be possible to consider a e,:aoe time manifold with a complex line element, thatt.1s, w1 th oom.~lex the real part of the the ueual way &nn f K ~/(, '1 and interpret J as grav•tational potentials 1n t:,• tmaglnary parts as electroma&net1c potentials. However, this proceedure would leat\ to ..._ 'ell known d11':t1oM:t1:ea, that the trne ~quatione satisfied by the tj.._5are non-11nertlt and only take the Poisson form for v,eak fields. Tie kno'tr from experience tha.t the electromagnetic eqnat1ons are llnea.tJ for strong f ie)ads 9.8 well as for weak f1elds. As w0 desire t6 eonsir1e:r the fra.v1tat1onal laws as being exactly an·'.l '."goua to the"f:el.ectromagb.et ic laws, 1nsteac~ of ~1elng the metho·:1s of th• general theorr of relativity, we v·1ll assume the grRv1ta.t1onal equations Do be linear and or the same form as the electromagnetic equat•one ae riven by equations (l) and (2°'. By def 1n1ng ~=J;rl7Cv Gt = /It -f 'r Vt l where 7 1e a (2) in the form J constant, we ca.n combine e u·- t1ons ( 1) and (4) Ti= .f-lT(_ _L/_L~_'7..:G> cz )i7 v t/ This combination can be made more than a merely formal one r •. particular n1u1erloal value. ~~e define r a.a the square root or Newton's constant J. r.·hich entere by riving into the Nev.-tonian law of grav1 tat1on. The accepted numerical value of - Jl so that is 7:; .:2.j~ -t' 10-+ (SJ em*/rYz..4-4::. Tft.e reason why we ta.ke this p~.rtt~ular value of 7 is illustrated in the simple limiting case of the repulsion ot two particles • For a particle To ::: L -1- {,, rlo =- e -ft rJ-'Yl where and iA the charge of the particle in elestrotatio un1ta e,,, Y>1 ie 1 ts ma.11 in grams. Then a particle of charge m and m".lSS e in an electro-grav1Atat1c f le.lei of potential is acted upon by a force (e+tr;..,) ~46 -ft ""j1-U4l Vo) "' -e~!l(J +r. .Mr~V: -t'1{m~4 u1 1.ud~) The field ot a ;>article at rest of m- SS /?1z and charge <?~ followe from (4) to be C. : e /f.h7 ../£ mt! (j 7--...... ) Therefore the Conlomb-Mevrton repulsion of tv:o partiolea is given by (5) I= c~ t,,, m~J 1 (ez.1i ?"fn-z) h~ e1 e"L _ 'ZzMt m, -2_-z. =i,r.. -f t 7@ l?'f -1- ez. 1#, J ~"2. The real part of this force ie the actual force to be expected with the correct relative e1gne for the electrical and the grav1tat1ona.l parts. 'I'hat is, the tact that masses ot like sign attract, follows from the fact that charr:es of like a1gn repel, 1f masa ·a considered to be aw. an imaginary charge. There does not seem to be any reason why negative mass should be excluded from existing. However, negative maaa would r.ave the property0of o?lleot1n.g i teelf together and being repelled by positive maeees. Therefore 1t would seem that 1t would have a te~er.,:y to remove itself tc/tnf 1n1 ty w1 th respect to the uni verse of poc lt 1ve mass. If any ner-at1ve M9S& 1a obrerva.ble by us, 1t will probably he found r-eceeding a.way from our universe of positive mass at ~n accelerq11ng speed! It should be noted that two particles euch FlS 1nucleus have and an electron, on the p eeent theory, would a~·eom)lex intera("'U_on Pnergy. T~e real part of the 1""teri.ot1on 1•sthe combined eleotrostatlo and gravitational interact ion. 'Ihe imaginary ··-a.rt h'l s no nesm ~ ng " i th r ·,SP'"' Ct to ~-•P previous theories. If we are to consl~er complex Hamtlton1an f,mct i - ne, v:e :nu et be able to tnterpret th(~ non-Herm1t1.an Hamiltonian operators o·btaine<l by means ot-:."'the correspondence principle, descrtbl?\g the correst)ond1ng qua;itize:1 cystem. (6) The Hamiltonian oper~tor must be of the type 1{nown as normal, that 1s, that if H is the oper'ltor and H"*' is its conjugate transposed operator then 1n order that 1 t may be transformed to the 'i ia.gonal oanon1oal form. A ncn-Horm:ltlan Hamiltonian operator, o(.1 the normal type, ,: hen trnne formed to the d.1a.gonal form, w ~ ll 1n e-eneral 1 hav9 complex component ·erms. According to the paralleli~m between electromagnetic and gravitational phenomena i11·hich we a.re following, and v'hich is exhibited in the following table, the real parts ______ ___...,,_............_....,...~ .............-----------~---··--··-------- .! rrm.g 1na.r;z electric charge -+ mass electric current -r !nag1narI electric f1elr1 7 e-rav1 t. field momentum ,. magnetic f1eld-,..grav. vector f1e of the eiganwerte of the Hamiltonian operator should constitute the spectrum of the ordinary energy, v1hich energy 1s proportional to the total mas a of the system represented by the Hamiltonian. The 1rnag1na.ry p..,rts should then const1tufle the spectrum of the imaginary energy, which energy must be proportional to the electric charge ot the system. If the total mass ~nd charge or the system are to be simultaneously measurable, and if f1:repreeents the mano operator ~nd [ the charge operator w~ have (1) (7) Then ·.r. a.nd E can be s1mul taneously transformed to the diRronal form and we wirt h~ve /-/ = M- rhere H ~ e the total Ha·rr.. ~ l tl\n 'Vrn. (' perflJ or. TH.e vr.·:.we a t1.on S th,, t changes H t transform- qt ts ~ ia.ron,11 ''o:c·m: therefore tra.ne- forme both v: ·:inn E to their diagonal forms s1mul tuously. So that in the cane thn.t 111?..s:o an!1 charge a.re e1mul:baneonsly me"lrmrAble, the waye functlons &f the system can be obta!nell. by just cone1,~ering the r~·1l part of the Haml.to:'l.i?.n f~mct1.on,tha.t 1.e, the Hermitian pa.rt of the Hamiltonian operator. In this case, the electric c'- arge 1 ie a constant of the motion. Thie corresponds +.o the faota 1n all ~ystems of which we are tam111ar. The potential energy in t' e ~ . , Ham1 iltonian function for an electron 1n '"': due to a pos1 tive nucle'"f' of ch~1rg- as should be exper: ted. Of ~ourse, centr~.l field cf force Ze and mass M is tht:. gr~\"lt~;d ional )art of V·1te 1nterqction is n,1mEn lcr:illy 1nsicn1f1cant in comparison ~1th the electroetat'c part 9n 1 h~e no effect in any ordinary measurements. It 1e hov·c'ver. n cessary for a complete theory to t91te into aaeount a.nd quantize the gra.vi. t at1 onal f 1eld. ( 8) ~:·e will quantize the gravitational waves by ( 1) t· e method developed by Heirenberg and Pauli r1) for t~e 5~, l, (1929) Zeits, f. Phyeik nugnt17at1on of the electromaghetic fielA. Or, to be more prec iee, "· 1.11 qua.n:.b1ze the w~ electromae:ne~ le and p-ra.vi i at- 1omal ~1elns tor~ther. G4 ~ l bo J -t'9 =tc t~nd let Fa/.' : : d ~ _ 0 G1-(a:)15) = I; 2 J ~,?f) ?J l-£ c) ~ If V'e put I then and {le) F+"" ~t(£ -!try h) ,, (l<-=~-z/~ ) r r rz3 r51. f-,2 ) J fleld. Also I.I, +".k7J"' are the componentsAoi: /( n~e complex magnetic f";_~ =- ~..t • Then for a apace free of charge and mass, ecpJ.attone ( 4) e.rE:; equ1 valent to These E:quation• are derivable from the variation pr1no1ple J where JL dfdf =a L = - ~ ~ &;~ r~/6 (13) where 1n this oaaec L is no t';,nger in reneral a real function. \s Heisenberg ?.nd Pauli have shown for the real case, the variables a.1- v-h1ch a e canonically conJug4te to the c;( a.re given by Ji (!~) J)G<(-- o)4 so tl::at ( A__ .:0 'J 2/:,) ) ;;:, 1 ::::. 0 . ( 9) and gravi tat1.~11.al waves jointly in vacw1m ~re then [ _G~) G~ '] = o [, F+i- ' ~i J .:-(J -Kc -, L F+k ) bx.,] : J',c ~ . --::::~ (-k ~I) 2-// --- )-~ / 1 v '1cre [Py 4J'] te an abbreviation of [P(x I) X 2 J X J) } Cls-) X) (vj~ 1 X ,l\J.) a.- 1 \.. ,5) 1 ,,_ ) 'tl. I) 11¢_/ 1 2 ./ form 1 1lation , the part1cul'1r values Accor,1 · ng to the renerAl . of ~ any fielde at different epa.ce- time points comm'1te wi'h each other, Thie if1 expressed by the fact that when )>., &'(A) k-')=o 1 A a.re different vectors, but S( A J :!:::-) =I 1 From equa.·tone (15) follow [(1-f" +t,?'J(J)> CHK.'+i1"Jk')] =-o [ ( El +v'r ~)) ( bl: I+ l "r t=k. I) J -: : 0 +l/Y Fv) ) ( f-/ <,~I fl/}" J -1-0J:;: 1 [ ( [ ( wher·e ) [l L - ,- ' ~ I I I J (. . t?' U(;I 'j....) l 7 -- l:::v.1-1 ..f (,, ?" f- <,.rr/ ) (•-fl Ac ~ o(&, k_..- I),, 'Ztrl ( 16) ) S(&l~'{e ~ X.'-1-:z- _,/ ox.r .. "n opertor ruch tha.t ~lf(-!;,)0:!::. ~ [ ~ !!:--:&! ~ ~ !Y-,~')1 u %~/o/ ~~ ( ) ( ( 5 Th(_., equatlaona (16) , on expansion gi:re ~J 1-1k 1 [Jv 1 J1c'J t q-- ( [1-41-1 J,/] +-[Jc. J 1-1K.'J) =o [ £1-, £1c'] - 'r1- [F" J F,,_'] tl-r ( [ £l J F~·J I- [Ft,} fk] )::co [ 1-1 1 2 ]- . (. [fI,) 1-l 4J -'F [Ft-) Ji,'+-1] -1- vr ( [t" di ~I J + iF;, HI Jl l & ,\ J x) '; u J i+ v- m,--·;> lo Xt..,.~ i··e no lonfer h~:ive eimple commnt.gt1on l ·we for the electro- map.netic :"' 1eld alone or for the ("r9.Vl tat i.onr.ll f iels alone. The laws given oy HeisAnbeg and Tquli :::;;:-ea i'irf;t approx1ma tion to these. B J.t all the methods use 1 by Heisenberg 1 ·-:>na Paull, es_ecially those used in their second p·•:pe1·( 2 ) / (10) are appl1cab~e 1n our case 1f we use complex p6tent1ala whel''e they uee real potent1,,ls. Our complex Lagrangian function LCs) for '.he electromagnetic and grav1ta.t1onal radiation 1s L (S) =: T'1e <13) correrrond1ng CO!ll[:'lex Ha.m1lton1an fo· the e:rav1-electro- magnetic r~d.1~tion ~s Htf>~ -fir+~JG"'_LrsJ:: -~ fte~-.L2r,, F+/<,+-Yi~rll<r;t /<..~f Jt+ Y.K_ 2. ~ k-; I c) . () K_:/ I ,,K.:1 For w~=mt of R. better Ham1lton1.an for t~e m~iterial ~>art we follow Hetsciiber,,- an1 "."''lttli ln 1.u~lng uir c 's c . ~srator ICite<l in tb'.,>bsrinnlnr: of this theste. by 'lSe of th tF operator ~::m1 T~';.e Ytlv·· eon tton obtained 1 ts .-ct .io ir~t. are obj:;ained by where we !ire now ue tng the complex potent1alfl of the or41nary and The y; ~ !'.l.re the ro~ t~e p~rti~lee v- e /t 0.A tnptead potent1~ls of the eLectromapnetic f&eld iclthe l complex charre of the )article. operators corrcspond~ng to th· wave f mctlons 1 and the ~ r ~ are the oper·a.tors a•; joint Ct&' ~ ( ll) by the use or t~e u/' ~'~ comp~e so ·~at w~ w1ll not ~1ve th:: :n here. S~ A current vector w l th components satisfying ~ J~ ::o 0 ~,A' /< is given by dL (n,J ,)6,A ::. t=-- e+t"fh/is complex, -the components of the current vector As 8 re re~-1 complex 8nd repre·aee.-'tr~momenta as well as current. Then the pan of' (21) 1e the equatton of cont.1nulty ror electric flow While U1e 1mag·tnary ?art is the ern19.t~on of continu:t ty or moment.um flow. - ~ ~ -.,// 1-16,,, j := z -:zrr r tr ~ i..1£.. L ''-') ~ 0 =2 ~l.4)4.r ~:t(Jk })f;. 1-hk. CJth?cz 2(d) )y_>f>F t°; 6~1 I lt -z:rTt 0it~ -r Ter 4-f'.r r ii' I k //; d" I i & :} f./' fo:- G.t. Emd the complex Hamil ton1a.m a.nalocous to t:1e real one used (z:y by Heir-enbe:rg anr1 P.<;tul1 1.s the ru:-2 of this 'md that for the radiation r·ven in (lB). T':1e 1nva.rla11ce o"' ,. of the aQ.rr~~,onl?n funct1..onql 6 ~ , 2( j an<i 7j ·'I-} •mder t 1e t''ans"ormat ·on 1 &~'~G +d_x i!/ /'( iJ/' . -~r;-·:ii.rx d°X~ ) 'Jfl -;: c he . .-·., )f>f·~'Pd~,,~ 'cf;) >//' ~ 'P (;" he or t'.-ie cor.,es ~.nnd in,-· ·: nf in 11ct1mnl transforma t tonA, in the rc·?j (12) C~Pe v·hen G+ i!' taken as zero ano Y ie n srb1tr9.ry runct1on or '!, 1 Yz., x~ 0 GI" I~ b'(, -f- s- ~ o Xi reqn1.r;·: the exlstenc:e of sn oper"tor Bommut ,_ng vri th the is } ;7/ ,'zj_ 4- -!- m2tf ll _L \ /- l (, oX S Gt hc /tJ '5 f;, ) JV The opere..tor con,i11gate to (;l 1e from (14) - F+t ; : - ~ ( £ f l 1h ) rrherefore z-,; .-----;- l Ee- -f t-~r I~ .J h The ope·rr;itor conjugate to ~ ie fro:n (20) I -::- ?//' ;f Therf ore Then (26) becomes :nann~'r, fo: '.'l.n (29) ts ~ constant of +11e motJ.on • A.s it ~.:ould. be~ so ~r-·ltra..ry <'ontlnu,,us funct'on c~ :p ~~: iE\ f L'I' t (Y~)',JX-s)tt"n ~ ~;: ++7T(eh'>'1??)%31fy; a. con8t--·~nt of th.e mot ion, If- C is "."J , l.t v ~ ll V:.c·n ( 13) im: g1m~.ry p8.rttt of (30) obta~ning 1: ~'l: -/- fTTe ~ ~X: 2f; :: o ) ( 31) f ~~ +~q l ~~47:: ~ 0 ... This aep0ratl.on is allo•a_:>le a.a ~E & and the df- I s oX ()'f.. ::Z "/;'J(~ com'autee with the f Ae the total Ham~ltontan /-/ (sJ.rf/tn,J,wi·:en G4 =-o then , is i=ra.u~e lnv!1rtant. 'Jn.de::- t.l:e trn.ne.for:·:uitiom. (21-t) t~e rel~tlon~ {31) hold in this C?P~. Tf t~ey hold tn gener·~l t 'en Gf -rn·rnt enter' lnto t'•e '..r-,r1llton·~.a71 a,e -:, ·.:;rod1wt t:~e Hamil t- 1 - .. ~.. . • Tha.ttthis 1 P the oase ts shown by Heiaenberg a.nii. Pauli for the H~ui!il tonia.ns involving rc'"'l . 0tent1als. It holds in an exa.ctl:· ei•l r -rray v-lth the Hainilt.on·ian involving complex ·)otentlals. We have s7· own then, t~" . 1 t - E ..-.~Jantum theory 0f the electromagnetic field ae develo:ed by Heisenberg ana Pauli c~n be use~ without dlff ~culty when the potent1°le ~re complex ~md the imagina:y 1arts ~~•are 1nterp::·eted ~s grav1ta.t1on0l £.:;otent1al~. Re turn lng to Di ':, ac ' s f-Ie.. milt o ni -:1 n for a (14) for tbe l&ne~r op ra·or 6. 554 x 10 --z.7 as usual, the real electric s:in ~om·nt 1e _:f-_li7 +ITC v~t~h ir extremely small, the retll p-ravi ta.tional epin moment 1.e ~ Ae/'r_ 4/T?J1C and i he Yeal epln mo·nent in a. grav1.ta.tiona . field is _:J:- ___ )t 1' 2-_ wh1.ch is ~~lso 4-trc f'Ytremely small. The fact th'· t ? particle hF.~s a. epin energy of J. .lier C 1n a grav1:at~onal field . :f-rrm~- r F lncl!c'::tes that an i(:_tcnse p:ravi tat~ona.l field sho 11ld c~•use a s :Jli ttlng of spectral t··rmE. Due to the fact that the spin m~tr1cee for .· I I He . ·the .~·ra.vi t-.t.1,..,na~- epin ..-::r(:: the .x 5· and not the o ..rnature ,( ·f t· ~· spj)it.t1ng vrill be entirely :1iee1milar effects. t~ th~'tt &ue t.o £~ 4 earth the value The numerkal V"l11_e of At 'the su:: ·face of ~he ls d>f 7 gravity d>f the earth is l.47x ia1° due~ to the normal so thst the maximum energy d1fferenc~~ in ergs for [-;;n electron sp1nnlnt:: Y1th -nd against the earthis gravitational fleld le :Z x 3, 13 k /tJ - 13 ergs 2x .196 electron xolts which ie very This ~s equ1V8lent t~ consid.erablG • 3ut the signs of the e~gem·erte of the matrices c:".g. d, [ 0 Oc; I tt2 (10 IO 0 Io o Io O o 0 00 ~ 0 0-t~ ' 0 l 0 0 l-t 00 () ccrnsponding to the mr:itr~cee r +I P(~ r 0() 16 o o o -I Io co \.. 0-1 ~o . Ei. nd are both of tt·:e B!'ime sign. An opposite grav1t.stlona.l epin will therefore pertain to one o~the tour f JWh1ch 1 is only very slightly excited under ordinary conditions. rr the grav1 ts.tiona.l eh1ft of he energy letels or Bn 9.tom are the same for ~ 11 of them, then no shift will occu1~ 1n the observed posl t~one of spectrr;l lines ae the shift 1n the final level will be the es.me as the ehU't in the initial level • Gravitational shifts of spectral lines could be produced only when there is a coupling between the grav1tat1onal spin and the other qua..ntum numbers of an electron in the atom. Besides the well knol'n commutation rules (16) r or the o j- and Uv: ~ 's ()[ l Wf) ~ t. ::. - vt.. k ·.:.\'.'._ . l c {!-A.-) c:r-;, u1~ .= - Ck u ,, ~ (t IK) I r ( (~~) ~, can ea.s1ly find those betY:~een the cl 'r Ml~ the x~ a.e er; ct l, -==- 02, OC k. ::. ()- - J t oc4 ~. l- ct;, - X kc;_· = ~';/ f- ~ (C:·::~z;3) 1 ( ) l; K~ IJ z; ~ / cf k )),>. (3 £ ) ) cl~ 0 -Z)?J _ The f 1, ,·Bt set of rielat1ona 1n ('36) spin cormautee with the grav1ta.t1onal spin 1n the same direction, ao tha.t 1 t le poaa1ble to s1mul taneo'.1al1 spec1t1 the two sptne in a given d1reot11on. If the excl~111on principle, excluding states ·of neeat1ve energy, ac recently auggested by Dirac, •.aU&•"4 1e 1n operatton.• then the relative prob.a.bill tJes or finding an electron Ti'1 th a riven magnetic sp1n having grav1.tat1onal sp1ns w1th or against a gravt tat1onal 1Ueld sudd.enly applied in some d1rect1on will b~ gre&,tly modified rro11 those expected. by the second set of relations ln C36f).