Propagation of gravitational potential and gravitational field-strength has been considered here on the basis of Rumer’s Invariant Theory of Gravitational Waves. Finally some special cases of weak field have been discusse

Krori, K D

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ON RUMER’S INVARIANT THEORY OF GRAVITATIONAL WAVES K. D. KRORIDltartment o f  P h y s i c s , C o t t o n  C o l l e g e ,  G a h h a t i  ( A s s a m )(Rcrdved, July 7, 1963 ; Resubmitted, Aj>ril 22, 1964)ABSTRACT. Propagation of gravitational potential and gravitational field-strength has been considiirod here on the basis of Kumer s Invariant Theory of Gravitational Waves. Finally some special cases of weak field have been discussed.I N T R O D U C T I O NRumer (1962) has constructed a theory of gravitation by breaking up the fourth-rank Riemann tensor into a sum of two 10-component tensorsand which have the symmetry of Riemann tensor (anti-symmetrical in ft, and v as well as in a and fi) :18FfifaP ui’ali (1 )Mfiy„p is called the matter-tensor and Ff,y^  ^ the gravitational field-strength tensor. The expression for F^ y^ P can be ^v i^tten asF^y"P =  R^yOP-M^y»P-  i  ... (2)where T is the source-tensor. The field equations can be expressed asi  [(Ty% ~(T^“ )y]~ i  {gyOT, ^~g%T,y) ... (3)which consists of 20 independent equations. [( )g indicates covariant differen- tiation.]The purpose of this paper is to discuss the propagation of the gravitational potential (defined below) and of the gravitational field-strength onthe basis o f Rumer’s theoryP R O P A G A T I O N  OF G R A V I T A T I O N A L  P O T E N T I A LConsidering the anti-symmetry of F^y„^ in a and f i , it can be written asi^ipaP — ... (4)190so thatO n Rumer^8 Invariant Theory o f Gravitational W aves 191Again, because of anti-symmetry in // and v, be expressed as^  U-vci —We consider that is a traceless symmiptric tensor of rank 2, obeying the divergence condition :=  0  ^ ... (6)Now, from (2) and (5), we have ;~  F/tg {F/lg i  ~or (Eddington 1957)... (8)remembering the properties attributed to in ((5), (8) gives the propagationof gravitational potential in matter.p K o p A a A TI o N iyn g r a v i t a t i o n a lF 1 E L D-S T H E N G T H F^ tvafi Prom (3), we have, by covariant differentiation.i  [(^ 0ti^ )/i6 (I'an)pd] i I'' r<s] ~  9^ ^^ {Tni>ap)y6-  9^^{{F,Pafihy+R 5-^  Uvat)== g'^P(FfiPa )^py+g^ (^F fivfifi)ay+9'^ {^R%y6l' cpofi+R\>yfiF +li^0y^Fnvae+(R%i0s^£va"\ R^ Q86^ nve+R\fi^ i^^ Mea~^ R%a{)^ £P^ '^ ^^Also, we havei  i  iUdiT, fitt~^ 9finl^  ^Pa\ === 9^ {^RiivHfi)ya=  9^^{(T A^6i»)a7+-®Vvtt-  ^e7'6^  +  -B%7ft  ^ nyaFnvtp-\-F^fiyoF••• (10)Subtracting (10) from (9){ i  —  ( ^ q m ) / ' ^ ] —  or 1  [ { ^ 6 f ) f i f ^  ( I ' s / t ) v a ]  i  [ 9 d P ^ i  H a-96.T , ,J } == g''P(F,,,e)y6~9'^^(f>^H$)y« =  □  F ,.,,+ g ‘^ HF%y6F,..fi " ^ R ^ p y ^ F f i p ^ g )  {R fiyaI^ et'fi0~^ R P'^ al^ He^ '^^ R dyaRfiPtPwriting G ^  nva^ )^ y^(11) gives tlie propagcation of field-strength ip matter.W K A K F I E L D  A P P H O X  I M A T I O N S :SOME S P E C I A L  C A S E SI f  wo oonsidor the field to he weak, then we m ay write (8) in the following form :freplaoing by 8^ )^□  <l>^a -  rr\<f>,a+8 '^P3r^^0 0,, ... (12)Considering a thin distribution of matter or free space. (12) takes the simple form :J92  K. D . KroriG 0 , 0 ( 13)This sliovvs that gravitational potential travels with fundamental velo(*ity through a very thin distribution of matter or free space.Under the weak fH‘ld condition, (11) comes to (replacing covariant differ­entiation by ordinary differentiation)- {  I ] -  i T.  ^ , ,]}nya^tv^P I' vyoL^ +  ^ '^ 6^ya^ fU'sP I" /4r(‘>F) +  eva~{~M ap(^ f^Jive~\ ^  vpf)^ ^^ juca jia6  ^ei/3 Pa6^ /^ i^ ivt va6 ^^ tifP) ••• (14)Further simplifi(nation can be made by considering va(mum. Tn that case, we get from (14)□  ... (16)This means that gravitational field strength also travels with fundamental velocity in vacuum.P L A X E  G R A V I T A T I O N A L  W A V E :  A N O T H E R  8 P E C 1 A L C A 8 EFollowing the idea of Weber (1961), we now consider the case of propaga­tion of plane gravitational wave. Tn such a case the field changes only along one direction in space; we choose for this direction the axis Now we have from (5)^  ^ vciP i^ fta)i>P i v^tt)fiP i^M>p)v{t'' i^ v^p)fi(x ••• (fi)It is evident for the case under consideration that the derivatives of and with respect to x^  and x^  vanish.On Burner''s Invariant Theory of Gravitational Waves 193  The 21 components of may be written down as follows froin( 5 ) :-^2424 ^  0 2 2 ’44 -^1224 -- ~ 0 2 2 ’ U ^1212 —  0J■^2434 ~  0 2 3 ’44 ^1234 ^ ““ 0 2 3 'U ^1213 0J■^3434 0 3 3 ’44 X  1324 — “ 02 3 ’ 14 P^1313 ^  0 ;^1334 ” 03 3 ’ ij|-P’1223 — 0 ^2323 ~  0-^1323 0 ' •P^ 2324 “  9^1423 ~ 0 ^2334 — 923M1(16)1^434 013'44 " 9^34‘ 14 ^1424 ^  0 1 2 ’44 ^24M4~  9^ 34 ’ U  ^ 13»14 -^1211  0 2 4 ’ 11~^ ^ ^ 12’ 14-^ 1414 — ^^ 11»44~"014’14 + 044'11(17)(18)In vacuum. =), (J7) and 1(18) :-^ 1424 ^ 0^24 -P^ 1214 =  0^1, TPX 3313-  ^^ 2424 - ^ 1 2 1 2-^ 3434 '^131313 1434F 3 4  —  t  —  ()1314' 1^212 “ = 001224 +  ^  i;^  23 “  2434"^ 4 4  —  ^ 3434+-^2424 +  ^  1414( 1 9 )■^ 1213 — 9o jIt is easily seen from (19) that ^^ 1434, ^ i424’ ^1214 ^uu  (^ )^zero. Then the components of that n^main are the ten components of (16)which are expresscxl in terms of only three potential components ?^ 33*It is evident from (16), (18) and (19) that0 2 2 ’ 44“t"033»44 ^  9or (on integration), ... (20)9^ 22 +  ?^ 33Here we have put the integration constants equal to zero since we are inter­ested in the varying part of tlie field. Thus, a plane gravitational wave is determined by only two quantities and 0^2 “ ?^ 33 transverse since it isdetermined by the potential tensor in plane only (Landau and Lifshitz,1962).R E F E R E N C E SEddington, A. S., 1957, Tlie Mathematicdl Theory of Relativity, Eq. (34.8), P-73.Landau, L* D,, and Lifshitz, E. M., 1962, The Classical Theory of Fields, P. 449-352.Rumer, Yu. B., 1962, Soviet Physics, JETP, 16, 402.Weber, J., 1961, General Relativity and Gravitational Waves, P. 90-92,

On Rumer’s Invariant Theory of Gravitational Waves

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On Rumer’s Invariant Theory of Gravitational Waves

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