Acta Physica Academiae Scientiarum Hungaricae, Tomus 47 (4), pp. 357--360 (1979)

NON EXISTENCE OF AXIALLu SYMMETRIC FIELDS IN ROSEN'S BIMETRIC THEORY

OF RELATIVITY

By

T. M. KARADE and Y. S. DHOBLE* DEPARTMENT OF MATHEMATICS, NAGPUR UN1TERSITY, NAGPUR-440 010, INDIA

(Received in revised form 3. IX. 1979)

In ROSEN'S bimetric theory of gravitat ion the non existence of the axial]y symmetric massive scalar meson field (together with electromagnetic field) is established.

I. Introduetion

A new theory of gravitation proposed by ROSEN [1] displays the attrac- t i re features of general relativity without singularity. This bimetric theory of relativity is based on two metric tensors gq (a Riemann tensor deseribing the gravitational field) and 7q (a tensor of flat space-time which describes inertial forees).

The field equations of bimetric relativity ate

w h o r e

and

Ki j = - - 8 ~ k T ~ j , (1)

1 K i j = N i j - - ~ Ng i j ,

N~ 1 ~ i = --~ 7 ~ (g~ ghj,~) lE, (2)

where bar (I) stands for eovariant differentiation with respect to 7ij and k = = (g/7) 1t2, and T i j i s the energy momentumtensor of matter of other non gravi- tational fields. In ah earlier work, the authors [2] have shown that nonexis- tenee of statie plane symmetric massive scalar field is established in bimetric relativity f o r a plane symmetric metric. This is not the case in respect of Einstein's general relativity. In this paper, we have pointed out that the non existence of a cylindrically symmetric massive scalar field is valid both in Einstein's general relativity and Rosen's bimetric relativity.

* Permanent address: Y. S. DHOBLE, Department of Agriculture Engineering, College of Agriculture, Nagpur-440100, India.

Acta Physica Academiae Scientiarum Hun$aricae 47, 1979

358

ds 2 = e2=-~ ( d T 2 - - d R 2) - - R 2 e-25 dcp2 - - e2~ d Z 2 ,

with the conventions

xi-----R, x 2 = ~, x 3 = Z, x 4~= T.

T. M. KARADE and Y, S. DHOBLE

Here we c o n s i d e r a n axially symmetr ic Eins te in- -Rosen space-time

(3)

The unknown variables ~ and fl are functions of R and T only. For the impor t - ance of this metric (3), one may refer to KARADE [3]. The f lat space-time corresponding to (3) is

da2 = d T 2 - - dRZ - - R2dcP z - - dZ2 ~- 7ii dx i dxJ . (4)

From (4), one finds easily tha t

711 = 7 3 3 ~ - - - 7 4 4 = --1 and 722 = - - R 2 "

The 7-Christoffel symbols of the second kind are defined as

1 7i¡ ( aTj~ OTkh O?'jk ) r ; k : - ~ I--a~x. + axJ ax.

From (4), we then obtain the non vanishing F-symbols as

1 1"22 F 2 - and F ' z 2 - - - R . 2 1 - - - - - -

R

The straightforward ealeulations for the line element (3) yield

K , ' - O. (5)

From (1) and (5), we derive

T~ = 0. (6)

II. Massive meson field coupled with ah electromagnetic field

The mat te r tensor for the massive sealar meson field is given by

M q = ~ Vi V J - - 2 gij( s Vs - m21/2) ,

a v where V i stands for . and the electromagnetic field is specif iedby the stress

Ox t tensor

E i j - - l [ - F i s F } - + - l - 4--~ ~gij Fsp Fsp] , (7)

Acta Physir Academiae Scientiarum Hungaricae 47, 1979

NON EXlSTENCE OF AXIALLY SYMMETRIC FIELDS 35@

where Fq is a Maxwell e lec t romagnet ic tensor given b y

Fij = Ail ] -- A~I i = Ai,] -- Aj. i , (8)

where A i is a four vee tor po ten t ia l and (,) s tands for par t i a l differentiat ion. The energy tensor , for a mass ive sealar field eoupled with e leet romagnet ie field will b e

Tij = ~ [ - - F i s Fsj 1 ]

-~- ~ g i j Fsp F~P §

I 1 ] @ 4 ~ 1 V, V j - - 2 g i : ( V s V s - m z V 2) . (9)

Then f rom (6) and (9), we get

_ g l t V,2 @ g4a V.2 @ m 2 V'z @ gtlgZ2(F12)z @ g~~ga3(F~a)2 _

__ g11944(F14)2 _ g22944 (F24 )2 _ g33g44~(F34)2 = 0 , ( 1 0 )

where I " = Ov/(9R and V" == ~V/0t etc. All the t e rms on l.h.s, of (10) are posit ive. Therefore to sat isfy the iden t i ty (10), each of its t e rms mus t vanish separa te ly , imply ing t h a t

V ' = V ' = 0, m = 0 ,

F12 = F13 = F14 F24 = Fa, = 0. (11)

I t is seen t h a t all the componen t s of Fi] vanish exeept F2a. But because of the axial sym.met ry imposed, the four vec tor po ten t i a l A i will not depend

upon q) and Z, i.e., A2, 3 : 0 : A3. 2 .

Hence b y (8) F23 --~ A2, 3 - - _/13, 2 ~--- 0 . ( 1 2 )

The Maxwell e lec t romagnet ic tensor being zero, its con t r ibu t ion to Tq is nil. I r is intercst ing to note t h a t ( from 11 and 12) the cont r ibu t ion of scalar mcson fields to Tq is also nil. In short Tq tu rns out to be zero and consequent ly gives an e m p t y space- t ime. Therefore we arr ive at a theorem:

" I n Ros~N's b imetr ic re la t iv i ty , the only possible solution of the a_~ally symmet r i c d is t r ibut ion-scalar meson, massive meson and Maxwell electro- magnet ie fields is a v a c u u m solut ion."**

I t has been observed t h a t the above theorem does not hold good for the axial s y m m e t r y revealed th rough MARDER'S metr ic [4] which involves three

paramete rs .

* In bimetrie theory ordinary differentiation of general relativity is replaced by y- differentiatioa.

** Axially symmetric vacuum solutions for biamtric relativity are developed and their significance etc. will appear in the next paper.

Acta Fhysica Academiae Scientiarum Hungt~rica'e 47, 1979

3 6 0 T.M. KARADE and Y, S. DHOBLE

In general relat ivi ty, the work of RoY and RAo [5] shows tha t the axially symmetr ic massive scalar field (also coupled with electromagnetic

fields) is non existent. Bu t it does admit scalar field and eleetromagnetie field.

Acknowledgement

The authors express their gratitude to the Referee for hiskind suggestions and eneoura- gement.

REFERENCES

1. N. ROS]~N, Bimetrie Theory of Gravitation, Topics in Theoretieal and Experimental Gravi- tation Physies, Edited by V. De Sabbata and J. Weber, Plenum Press, ~lew York, 273, 1977.

2. T. M. KAnA9E and Y. S. DHonr.E, eommunieated for publieation, 1979. 3. T. M. KAnAV~, Rey. Roum. Phys. (Rumania), 23, 425, 1978.

Reprinted: POST-RAAG Reports No. 79 (Japan) 1978. 4. L. MAnDER, Proe. Roy. Soe. Ser. A, 244, 524, 1958. 5. A. R. RoY and J. R. RAO, Commun. math. Phys., 27, 162, 1972.

Acta Physir Academiae Sdentiarum Hungaricae 47, 1979