NON-EXISTENCE OF HIGHER DIMENSIONAL AXIALLY SYMMETRIC FIELD IN ROSEN'S THEORY OF GRAVITATION

G. S. Khadekar, T. M. Karade

Department of Mathematics, Na#pur University, Nagpur, India

In Rosen's bimetric theory of gravitation the non-existence of higher dimensional axially symmetric massive scalar field and massive complex scalar field coupled with electromagnetic field is established.

1. I N T R O D U C T I O N

The non-availability of exact solutions of field equations of general relativity which represent spherical or plane gravitational waves prompted Einstein and Rosen [1] to search for empty space-time solutions representing cylindrical waves and obtain a cylindrically symmetric metric

(1.1) d~ 2 --- e2~-2a(dT 2 - dR2) - e-2#g 2 d~b 2 - e 2p dZ 2 ,

where o~ and fl are the functions of R and Tonly. The metric in (1.1) is widely known as the Einstein-Rosen (ER) metric and it plays a vital role in the study of gravitational radiation which provides a crucial link between general relativity and microscopic frontier of physics. In this paper the higher dimensional analogue of this metric is proposed. Consequently the non-existence of massive scalar field and massive complex scalar field coupled with electromagnetic field is achieved from the identities derived from the proposed metric.

2. BIMETRIC RELATIVITY A N D ER TYPE M E T R I C

The bimetric relativity is the theory of gravitation proposed by Rosen [2]. In this theory at each point of space-time we have two metrics:

(2.1) ds2 = ~ij dxi dxJ,

(2.2) dr72 = ~ij dxi dxJ.

The first metric tensor gij describes the four-dimensional curved space-time and thereby the gravitational field. The second metric tensor Plj refers to the flat space- time (four dimensional), whose curvature tensor (derived from Pij) vanishes and describes the inertial forces associated with the acceleration of the frame of reference.

The essential features of this theory are contained in the series of papers by Rosen [ 3 - 5 ] . The field equations of bimetric relativity from variational principle come out to be (2.3) /Cij = --8rck~/i = Ni/ - ~ N g i j ~ ~ " ,

and (2.4) Nj = � 8 9

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G. S. Khadekar et al.: Ax ia l l y symmetr ic f ie ld in Rosen's t h e o r y . . .

where the bar (/) stands for the covariant derivative with respect to ~j, ~ = (~/~),/2, -- det (~ij) and ~ = det (pij), and p*J = (cofactor of ~,~)/~. For its details one may refer to the work by Rosen cited above. One of the authors

(Karade) devoted papers [ 6 - 9 ] to this theory, n part of his work pertains to the non-existence of axially symmetric massive scalar field and massive complex scalar field.

In this paper we extend the study of non-existence of solutions to higher dimen- sional space-times. There is nothing unphysical in supposing higher dimensional space-times.

As a matter of fact the recent String theories could only be talked about if the space-time is 10-dimensional or 26-dimensional.

We propose the higher dimensional (say n-dimensional) ER type metric as n - - 4

(2.5) ds2 = -e2~-2P[ dR2 + E dR/2] -- e-2PR2 de 2 - e2t~ dZ2 + i = 1

+ e ( n - 3 ) ( 2 ~ - 2 # ) d T 2 .

Our convention is x 1 = R, x 2 = q~, x a = Z, x* = R~, x 5 = R2 . . . . . x "-1 = R,_ , , x" = T. The unknown variables ~ and fl are the functions of R and T only. The quantities bearing the cap (A) belong to the metric (1.1), e.g.

Nj. stands for Nj for four dimensional space-time (1.1), Nj stands for Nj for higher dimensional space-tim e (2.5).

The flat space-time corresponding to (2.5) is n - - 4

(2.6) daZ = - [ dRz + E O R ~ ] - - R 2d r 2 - dZ 2 + dT 2. i = 1

For this metric the non-vanishing Christoffel symbols of second kind F are

F22 = F21 = r22 = r22~ = 1 / R , F122 = r122 = - R .

Then the components of Nj satisfy the following identities

-~ (cd - if) + (e'" - fl") + sinh (2c~) = N I ,

1 N~ = if' + f f /R - R- ~ sinh (2c~) - B'" = ~r~,

N~ = - ( f l " + fl'/R - f l") = ~ 3 ,

N~ = - ( a " - f l " ) - 1 - + =

=

N 6 = ~t~,

n--1 A N n - 1 = N ~ ,

N~ = ( n - 3 ) ~ , Nj = 0 for i , j ,

Czech. J. Phys. B 39 (1989) 963

G. S. Khadekar et al.: Axially symmetric field in Rosen's theory.. .

where c~' = O~/OR and ~" = O~/OT etc. Now the straightforward calculations for the line element (2.5) yield

(2.7) K." = 0 .

Consequently (2.3) gives

( 2 . 8 ) r " = o .

3. H I G H E R D I M E N S I O N A L MA SSIV E M E S O N F I E L D C O U P L E D

W I T H E L E C T R O M A G N E T I C F I E L D

We assume the conventional forms of the electromagnetic field and energy tensor for the massive meson scalar field valid in our space-time (2.5) such that

1 [_F i sF jS + �88 ] ( 3 . 1 ) = ,

(3.2) M,j = 1 [ V iV J _ _}glj(V,sV, s _ rn2V2) ]

where Eij is the energy tensor for the electromagnetic field,

Fij is the Maxwell electromagnetic tensor satisfying

(3.3) Ffj = Ai , j - - A j , i ,

(here Af is a higher dimensional vector potential), (,) stands for partial differentiation, m = mass associated with the massive scalar field V, Mii = the energy tensor for massive scalar field, V,i = aV/t3x ~.

The energy tensor for a massive scalar field coupled with electromagnetic field will he

(3.4) T~.j = M i j + E i j .

From (2.8) and (3.4) we get

(3.5) {glag22(F12)2 + g ' l gaa (F ,3 )2 + gt 'g4a ' (F,4)2 + . . .

�9 .. + ga lg (n -1 ) (n - l ' ( r l ( n_ l ) )2 } --

-- {gHg'n(F~,32 + g22g'n(F2,)2 + g33g"(F3 , )2 + . . .

�9 .. + g ( " -~ ) ( ' - l ) g" (F( ,_~ ) ( , -1 ) , )2 } - g l~V '2 + g"nV'2 + m 2 V 2 = O.

Now as all the terms on the L. H. S. of (3.5) are positive, each of its terms vanishes separately to satisfy the identity (3.5). This implies that V' = V" = m = 0.

Therefore higher dimensional axially symmetric mass parameter of a massive scalar field coupled with an electromagnetic field vanishes for ER type metric in Rosen's bimetric theory of relativity. Moreover, it is seen that all the components

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G. S. Khadekar et al.: Axial ly symmetric f ield in Rosen's theory . . .

of Fii vanish except

F23 , F z 4 , . . . , F z ( n - 1 ), F34 , F35 , . . . , F 3 ( n - 1 ) , . . . , F ( n - 2 ) ( n - 1 ) �9

But because of axial symmetry A i are independent of O, Z, R1, R2 . . . . . Rn_ 4 and thus

F23 = . . . = F 2 ( , - 1 ) = F a 4 = . . . = F 3 ( n _ I ) . . . . = F ( n _ z ) ( n _ l ) = O .

Hence the contribution of Fij to Tij is nil. In consequence of which we arrive at a theorem: "In Rosen's bimetric relativity, the only possible higher dimensional solution of axially symmetric d i s t r i bu t ion - scalar meson, massive meson and Maxwell electromagnetic field - is a vacuum solution."

4. M A S S I V E M E S O N C O M P L E X S C A L A R F I E L D C O U P L E D W I T H E L E C T R O M A G N E T I C F I E L D

The energy momentum tensor for a complex scalar field V coupled with electro- magnetic field is written as,

(4.1) T,b = (Da v . DbV + DoV. D,V) - g,b(OeV. D p V - m2VV) -

__ FakFb k + 1 . l~ l~rs g~4ab~rs ~ 3

where Fab is defined in (3.3) and A~ satisfies gauge condition V~Aa -- 0.

The operator D. and its conjugate D. are defined by

(4.2) D, = V a + (4r01/2 leA,,

D a = V a -- (47Z) 1/2 ieAa,

where V indicates a covariant derivative related to ei~ and i = ( - 1) 1/2. The complex scalar field V satisfies the combined Einstein-Maxwell and Klein-Gordon equations yielding VV always positive (see Rao [10]). The mass and charged parameters of the field are denoted, respectively, by m and e, where e is related to the fine structure constant which in turn relates to the charge of an electron. From (2.8) and (4.1), we derive

(4.3) - { g l a D , V . D I V + g22D2V. D2V+ 033DaV.D3V+ ...

+ g ( n - 1 ) ( n - D D ( n - 1 ) V . D ( n _ I ) V -- g'" D .V .D .V}

+ �89 2 + 01~033(Fa3) 2 + ... + 9110("-a)("-')(F(,_l),) 2} - - �89 + . . . + . . . . . . + g 'n -1 ) (n -1 )gnn(F(n_ l )n )2 } + m 2 V V = O .

Again all the terms on the L. H. S. of (4.3) are positive because g11,822, g 3 3 , . . . g(n-1)(n-1) are all negative and g"", DIV. D~V, D2V. O z V . . . D n V . DnV are all positive and hence m = 0, etc.

Then follows the theorem:

Czech. J, Phys. B 39 (1989) 965

G. S . K h a d e k a r e t a l . : A x i a l l y s y m m e t r i c f i e l d in R o s e n ' s t h e o r y . . .

" I n Rosen 's bimetric relativity, the only possible higher dimensional solution of the axially symmetr ic field - complex massive scalar field, Maxwell electromagnetic field - is vacuum solut ion".

T h e a u t h o r s a r e t h a n k f u l t o t he re fe ree f o r h e l p f u l s u g g e s t i o n s .

A P P E N D I X

For the line element (2.5),

g l l = - - e Z ~ - Z P , g 2 2 = - - e - Z P R 2 , g 3 3 = - - e 2 #

g 4 4 = g 5 5 - . . = g ( n - a ) ( n - 1 ) = - - e 2 Z - a t ~

gnn = e ( n - 3 ) ( z a - 2 # ) , g lj = 0 , i :~ j

and for (2.6) the components of the metric tensor 7u are

711 - 1 , ])2Z - - R 2 "~ = , ])33 = - 1 ,

Y44 = 755 = -.. = 7(,-1)( .-1) = - 1 ,

7,,, = 1 , 7u = 0 , i # j .

2Uj. = y'p(gh'ghi/~)/fl ,

i.e.

N o w

Also

and

T h e n

= - ( g g ~ j r ~ , ) , p + - 2 N j ? ~p {(g hi ghj,~),a hi m --(Pj~),fli Fza,[gi h2 ghj,~r

F 2 r hi hi ~ m i - gh~gmjr~, -- F~,] - jakg ghz,~ - g gmzth , -- F~,] --

2 hi - - F ~ [ g g h j , 2 hi ~ m i - - g g m j l h ; . - - F j ; . ] ) .

N 1 + N 2 + N ~ = N 4 = N ~ .

N55 n - 1 . . . . . u . _ , = N," = s , ' .

N." = ( , - 3 ) N , " = ( , - 3 ) s , "

n--1 n = ~ N - = N 1 + N 2 + N ~ + N 4 + N ~ + . . . + N , _ , + N , ,

= ( n - 3) N4 4 + N , ~ = 2 ( n - - 3 )~r~ .

K~" = N , ~ - � 8 9 = ( n - 3) hr44- � 8 9 3)]_~4 . K," = 0 e t c .

Received 16 June 1988

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G. S. Khadekar et al.: Axial ly symmetric" f ield in Rosen's theory . . .

References

[1] Einstein A., Rosen N.: J. Frank. Inst. 223 (1937) 43. [2] Rosen N.: Bimetric Theory of Gravitation. Topics in Theoretical Physics (Ed. V. De Sabbata,

J. Weber). Plenum Press, New York, 1975, p. 273. [3] Rosen N.: Gen. Relativ. Gravit. 4 (1973) 435. [4] Rosen N.: Gen. Relativ. Gravit. 9 (1978) 339. [5] Rosen N.: Ann. Phys. 84 (1974) 455. [6] Karade T. M.: Rev. Roum. Phys. (Rumania) 23 (1978) 425. [7] Karade T. M.: Indian J. Pure Appl. Math. 11 (1980) 1202. [8] Karade T. M., Dhoble Y. S.: Curr. Sci. 48 (1979) 625. [9] Karade T. M., Dhoble Y. S.: Acta Phys. Hung. 47 (1979) 357.

[10] Rao J. R., Panda H. S., Nayak B. K.: Aust. J. Phys. 28 (1973) 353.

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