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Bimetric plane symmetric quantities in higher dimensional space-time.

1. INTRODUCTION

Various attempts have been made to introduce two metric at each point of space by Rosen [17], Eisenhart [4] and Logunov and Mestvirishvilli [14]. The geometry to that effect was applied by N. Rosen [17] in introducing a theory welknown as bimetric theory of relativity, which differs from the general theory of relativity of Einstein [1-3] and which was later developed by many scientists Falik and Rosen [6-7], De Liebscher [5], Yilmaz [26], Goldman and Rosen [8] Rosen and Rosen [25], Israelit and Rosen [16], Israelit [11,12], Rosen [18-24]. This theory has some advantages over the general relativity; the quantities such as christoffel symbols and others become tensors which otherwise in Riemannian geometry they are not.

Recently Nahatkar [16] have investigated the various bimetric quantities in plane symmetric space-times. Present work is the extension of the same in higher dimensional plane symmetric space-times. The behaviour of bimetric quantities [K.sub.nm], [A.sub.nm], [B.sub.nm] & [D.sub.nm] in plane symmetric higher dimensional space-time is studied. The space-time of various categories C (I), C(II), C(III) and C(IV) are also investigated.

We consider the plane symmetric metric

[ds.sup.2] = [e.sup.2A]([dt.sup.2] - [dx.sup.2]) - [e.sup.2B]([dy.sup.2] + [dz.sup.2] + [du.sup.2]), (1.1)

where A and B are functions of x and t with the convention

[x.sup.i] [equivalent to] (x, y, z, u, t), i= 1,2,3,4,5.

For our purpose, we adopt as the metric (1.1) and [a.sub.ij] the metric

[ds.sup.2] = ([dt.sup.2] - [dx.sup.2]) - [dy.sup.2] - [dz.sup.2] - [du.sup.2]). (1.2)

For the line elements (1.1) and (1.2), we have obtained following results.

2. BIMETRIC COVARIANT TENSORS OF ORDER TWO

(a) The tensor [K.sub.nm]

By definition, [K.sub.nm] = [[GAMMA].sup.l.sub.nm][[GAMMA].sup.p.sub.lp] - [[GAMMA].sup.p.sub.np] [[GAMMA].sup.p.sub.lm] or [K.sub.nm] = [K.sup.k.sub.nmk]

The bimetric plane symmetric covariant tensor [K.sub.nm] can be expressed in the form of four functions, [I.sub.1] ... [I.sub.4], of x and t as

[K.sub.1k] = ([I.sub.1, 0,0,0, [I.sub.2], [K.sub.2k] = (0,[I.sub.3],0,0,0), [K.sub.3k] = (0,0,0, [I.sub.3],0) [K.sub.5k] = ([I.sub.2],0,0,0, [I.sub.4])

where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where prime and dot represents differentiation with respect to x and t respectively.

(b) The tensor [A.sub.nm]

The non-vanishing components are [A.sub.1k] = ([H.sub.1], 0,0,0, [H.sub.2]), [A.sub.2k] = (0, [H.sub.3],0,0,0) [A.sub.3k] = (0,0[H.sub.3],0,0), [A.sub.4k] = (0,0,0,[H.sub.3],0), [A.sub.5k] = ([H.sub.2],0,0,0,[H.sub.4])

where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

(c) The tensor [B.sub.nm]

The total number of components are four and are given by

[B.sub.1k] = ([J.sub.1],0,0,0, [J.sub.2]), [B.sub.2k] = (0,[J.sub.3],0,0,0), [B.sub.3k] = (0,0,[J.sub.3],0,0), [B.sub.4k] = (0,0,0, [J.sub.3], 0), [B.sub.5k] = ([J.sub.2],0,0,0, [J.sub.4])

where [J.sub.1],.... [J.sub.4]are the functions of given as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

(d) The tensor [D.sub.nm]

There are five non-vanishing components of [D.sub.nm]

[D.sub.1k] = ([L.sub.1],0,0,0, [L.sub.2]), [D.sub.2k] = (0,[L.sub.3],0,0,0), [D.sub.3k] = (0,0,[L.sub.3],0,0), [D.sub.4k] = (0,0,0, [L.sub.3], 0), [D.sub.5k] = ([L.sub.4],0,0,0, [L.sub.5]),

where [L.sub.1],.... [L.sub.5] are the functions of x, t given by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

The investigation carried out results in the following theorem:

Theorem 2 : Each of the plane symmetric bimetric tensors [K.sub.nm], [A.sub.nm] and [B.sub.nm] are specified by four functions of x & t whereas the tensor [D.sub.nm] is specified by five functions of x & t.

3. STATIC SPACE-TIME OF CATEGORIES C(I), C(II), C(III) AND C(IV) Space-times C(I)

We know that the space-time C (I) corresponds to [A.sub.nm] = 0, i.e. [H.sub.i](x) = 0, i = 1,2,3,4

i.e. A" + 3B" = 0, -2A'B' + 2[B'.sup.2] + B" = 0, A" = 0 ....(i)

These equations leads to

A" = 0,B" = 0,

which on integration gives

A = ax + b, B = cx + d ... (ii)

where a, b, c, d are the constants of integration.

Using (i) in (ii), we get, a = c, or c = 0.

Then the space-time (1.1) reduces to

[ds.sup.2] = [e.sup.2(ax+b)]([dt.sup.2] - [dx.sup.2]) - [e.sup.2(ax+d)]([dy.sup.2] + [dz.sup.2] + [du.sup.2]),

by absorbing constants b and d in the differentials and c [not equal to] 0 leads to

[ds.sup.2] = [e.sup.2ax]([dt.sup.2] - [dx.sup.2] - [dy.sup.2] - [du.sup.2])

which is conformal to the flat. This confirms the following theorem.

Theorem 3: The plane symmetric static space-time of category C (I) is of the form

[ds.sup.2] = [e.sup.2ax]([dt.sup.2] - [dx.sup.2]) - [dy.sup.2] - [du.sup.2])

and is conformal to the flat space-time.

Space-time C (II)

By definition of C (II)

[K.sub.nm] = 0, i.e. [I.sub.i](x) = 0, i = 1,2,3,4.

i.e. A'B' - [B'.sup.2] = 0, -2A'B' + [B'.sup.2] = 0.

These equations leads to

(i) B' = 0 and A is undetermined. OR (ii) A = constant, B = constant The space-time (1.1) becomes

[ds.sup.2] = [e.sup.2A(x)]([dt.sup.2] - [dx.sup.2]) - [dy.sup.2] - [dx.sup.2] - [du.sup.2].

In the second case, it reduces to flat space-time of special relativity.

The above result can be expressed in the following theorem :

Theorem 4: The plane symmetric static space-time C (II) has the form

[ds.sup.2] = [e.sup.2A(x)]([dt.sup.2] - [dx.sup.2]) - [dy.sup.2] - [dx.sup.2] - [du.sup.2]).

Space-time C (III)

By definition of C (III), [B.sub.nm] = 0, i.e. [J.sub.i](x) = 0, i = 1, 2, 3, 4.

i.e. 2A" + 3B" + 2[A'.sup.2] = 0, [B'.sup.2] = 0, [A'.sup.2] = 0

which implies A = constant and B = constant.

Then the space-time (1.1) becomes

[ds.sup.2] = [e.sup.c]([dt.sup.2] - [dx.sup.2] - [dy.sup.2] - [dz.sup.2] - [du.sup.2]).

By absorbing constant in the differentials, we get

[ds.sup.2] = [dt.sup.2] - [dx.sup.2] - [dy.sup.2] - [dz.sup.2] - [du.sup.2],

which is the flat space-time.

The above investigation results in the following theorem

Theorem 5: The plane symmetric static space-time C(III) is the flat space-time of special relativity.

The above discussion results in the next theorem.

Theorem 6: The plane symmetric static space-time C(I) and C(III) are conformally related.

Space-time C(IV)

We know that the space-time C (IV) corresponds to [D.sub.nm] = 0 i.e. [L.sub.i](x) = 0, i = 1,.., 5 and hence we write

2A" + 3B" = 0, 2A'B' + 3[B'.sup.2] = 0.

Rejecting the non-trivial case A = constant and B = constant above equations are satisfied by

A = a + f (x), B = a - f(x).

Then follow the result in the form of following theorem

Theorem 7 : The plane symmetric static space-time C (IV) has the form

[ds.sup.2] = [e.sup.f(x)]([dt.sup.2] - [dx.sup.2]) - [e.sup.f(x)]([dy.sup.2] + [dz.sup.2] + [du.sup.2]).

CONCLUSION

The study of bimetric covariant tensors of order two is carried out with reference to plane symmetric five dimensional space-time (1.1). The forms of bimetric quantities [K.sub.nm], [A.sub.nm], [B.sub.nm] and [D.sub.nm] have been computed and are expressed in the form of theorems. It is established that the plane symmetric static space-time of category C(I) is conformal to the flat, the space-time of category C(III) is the flat space-time of special relativity. The plane symmetric static space-time C(I) and C(III) are conformally related. The present results obtained in higher dimensional space-time are resembles with that of obtained by Nahatkar [16].

ACKNOWLEDGEMENT

The authors are grateful to Dr. T.M. Karade (India) for his constant encouragement and fruitful discussion.

REFERENCES

[1.] Einstein, A.1912. "La theorie du rayonnement et les quanta", Edited by Langevin P and de Bronglie 450, Gauthier Villars, Paris, (1912).

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[3.] Einstein, A.1915b. "Preuss Akad Wiss Berlin", Sitzber, 799-801.

[4.] Eisenhart, L.P.1966. "Riemannian Geometry", Princeton University Press, (1966).

[5.] De, Liebscher,1975. GRG, 6: 277.

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[14.] Lagunov, A. and Mestvirishvili, M.1989. "The Relativistic Theory of Gravitation". Mir publishers, Moscow. (1989)

[15.] Landau, L.D. and Lifshitz, E.M. "The classical Theory of Fields", Course of Theoretical Physics, Vol.2, Fourth revised edition, USSR.

[16.] Nahatkar, M.2002. Ph. D. Thesis, Nagpur University, Nagpur) "Some Problems in relativity Theory", (2002).

[17.] Rosen, N.1940. Phy Rev, 57: 147..

[18.] Rosen, N.1973. GRG, 4: 435.

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S.D. Katore * and S.V. Thakare **

Department of Mathematics, R.A. Science College, Washim (M.S.), India
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Author:Katore, S.D.; Thakare, S.V.
Publication:Bulletin of Pure & Applied Sciences-Mathematics
Article Type:Report
Geographic Code:9INDI
Date:Jan 1, 2008
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