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Coupled Maxwell-Pseudoscalar Field from the Einstein-Mayer Theory.

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A N N A L E N D-ER PHYSIK
7. Folge. Band 44.1987. Heft 8, S. 553-632
Coupled Maxwell-Pseudoscalar Field from the Einstein- Mayer
Theory
By M. N. M~HANTA
and Y. K. GUPTA
University of Roorkee, India
A b s t r a c t . A coupled system of field equations representing interacting gravitational, electromagnetic and pseudoscalar fields is obtained using the five-dimensional formalism of Einstein and
Mayer (1931-1932). Solutions of the system for concrete cases are under investigation.
Die Kopplung yon Maxwellschem und pseudoskalarem Feld
in der Einstein-Mayer-Theorie
Inhaltsubersicht. Es werden Losungen der fiinf-dimensionalen Feldgleichungen von Einstein
und Mayer angegeben, die Wechselwirkungen zwischen dem Gravitationsfeld, dem elektromagnetischen Feld und einem pseudo-skalaren Feld darstellen.
The Einstein-Mayer Theory [l,21 was originally proposed for effecting a unification
of gravitation and electrodynamics on the lines of thgearlier Kaluza-Klein Theory [3,4].
I n the present paper we once again stick to the original version of the Einstein-Mayer
Theory and derive a consistent system of equations representing coupled gravitational,
electromagnetic and pseudoscalar fields. The system has the merit of resulting from
a purely geometrical procedure and thus the original aim of Einstein and Mayer of
representing gravitation, electromagnetism and matter by a purely geometric formalism
is partially realized if one thinks of the pseudoscalar field as a very special type of self.
gravitating fluid with pressure equal to the energy density. Detailed physical interpretation and solution of the field equations obtained, in concrete cases, is in progress.
I n this paper we first briefly remind about the basic mathematical apparatus and then
give the derivation of the field equations closely following [21 for coupled gravitational,
Maxwell and pseudoscalar fields.
Following [21 we take the field equations for the combined Einstein-Maxwell-Scalar
fields in the following “mixed” form,
Gop
1
Pop - - ; i - y o P = 0.l)
Multiplication by yz gives
G,,
Putting the skew symmetric part
G
4,p
+
+
= Y;Gop= Rqp+ Fq.8$’8p V*.8kVgkp
=0,
v;.p;t
Ann. Physik Leipzig 44 (1987)8
554
where Q1is a covariant vector field and
is easily seen that
- @t;L
@l;t
= @l,t - @t,l
ez'J1't is
the Levi Civita tensor. Then from (4)it
(6)
= 0,
i.e. G1G @ , l , where Q, is a pseudoscalar field.
Setting now the symmetric part
Qqp
- = 0,
(7)
we get the following eq.
RrlP
1
1
+4
qqpFX,Fat
) - ( Vq.Pk
(
-4 qqaR = - Fq.6Fxp
I.',k,
1
- 4qqJ)
9
(8)
writing
v zz vs"" v,,,.
(9)
Multiplication of eq. (1)by A" gives the first set of Maxwell's equations viz.
Pp;*
= Fk..v ; . P
(10)
*
Thus there is a source term representing the interaction of the Maxwell and the V field. The second set of Maxwell's equations will be taken in the usual form viz.
F[l.q;P]= F r g ; p
+
Fqp;,
+ FPr;q
(11)
= 0):
admitting a four potential A i (this A has a meaning different from the A introduced
earlier!). Using eq. (lo), (11)we have now the following identities:
(F
krFkp
1
- -o;F,,F"
4
EE
FkrFf; +
1
FLfik,r,Fkl'
= F kr F at V g t k .
(12)
From the system of eqs. ( 8 ) k (11)we now establish a consistent relativistic coupled
system which will represent the combined Einstein-Maxwell-Pseudoscalar fields by the
following procedure:
First we verify that eq. (10) satisfies the divergence relation identically. We write
this eq. as
1) The mathematical formalism and notation used are the same as in the Einstein-Afayer Theory
except that we assume the normalization condition for A" as A" A , = -1 instead of +l. This seems
t o be necessary t o get the sign of the e - m tensor of the electromagnetic field correct later on.
2) We adhere to the notation followed in 11, 21 even though the current practice involves a factor
1
-in this definition. The same will be true for the symbol { } when it is introduced pl:esently.
3
M. N. MAHANTAand Y. K. QUPTA, Maxwell-PseudoscalarFields in Einstein-Mayer Theory
556
Then
G$
FY:;@ - F r8 ;P Vr8" - Fr8V$p,
- -Fr8,,Vr6P using eq. (4),
because of skew symmetry of
_-- 1
Vr8pF[r8;@l
F and V tensors,
3
FE
= 0.
Now we consider eq. (8) written as
G,P- RP(14)
Taking covariant divergence we get
1
GPP;p= 7 R,,- FkgF,,VrtP
1
+ V*tpV8tg;PV- V,,,
4
(15)
using eq. (4) and (12).
Now
1
1
-Fn.,Frt V8tk= - - VkrtFIrtFklq
3 - - Vkr'(FrtP,,},
3
24
{ } has the usual meaning except that we drop the multiplier
(16)
i.e. it represents the
sum of all the possible permutations of the indices rtkq, taken with + for an even permutation and with - if it i s a n odd permutation.
To evaluate the third expression on the r.h.5. of (15) we consider the expression
;PI9
v8t,
$8 of the 24 terms of this in which plies on the left of the sign of covariant differentiation
gives the third term on the r.h.8. of (1.5).The other six terms where p lies on the right
1
of the sign of covariant differentiation give - - V,, each i.e.
2
1
v8tpV8tp;p
- v8""(V,t,;p}
= 18
f
1
yp.
Substituting in (15) from (16) and (17) we get,
'
Thus G;;,
= 0 if we add two more eqs. t o our system viz.
1
(R - 7v) = 0,
and
(17)
556
AM. Physik Leipzig 44 (1987)8
Both these equations are needed. Eq. (19) is needed to remove the indeterminancy in
the scalar curvature R in eq. (14) which now stands as
R:
-
1
:S
2
R =- F
(
Fps
+
1 -
‘
(
S;F,,Fat) - V,,,
V T d ’ P-
1
6
-qv),
(21)
and eq. (20) determines the scalar function @ in (5). In terms of components this eq.
reads
IV123;4
-
+
‘234;l
V341;2
-
‘412;3
= -(F23F14
f
’3IFZ4
+
’Ia’34).
(22)
Finally we explicitly introduce 0 in our system of field equations with the help
of
VU’P
= elsTd’9@,I ’
cbkq
= (-l/i-g)
eInl.9
= i 7 E l a k q9
where,
(23)
EbTd’g,
and
€hkq,
I
0 if any two indices are equal,
= &1 according as (Lkq) is an even or odd permutation
of (12 34).
Then
v,,, v a k p = 2(@,9J’ - S;@-u@,,),
and ’
V G V,,,V”P
Int rodncing the variables
= -G@juO.u.
M. N. MAHAXTAand Y.K. GFPTA, Maxwell-Pseudoscaler Fields in JCimtein-Mayer Theory
557
We finally get the following system of equations for coupled Einstein-Maxwell-Pseudoscalar fields :
-
The source term on the r.h.e. of (34) is the second invariant E H of the electromagnetic
field fik.
The kind of coupling between the Naxwell and pseudoscalar fields represented by
the eqs. (32), (34) seems t o be the novel feature introduced by the Einstein-Mayer approach. Usually the coupling of the fields R
i only through the gravitational field, direct
coupling rtmongst themselves being absent [8, 9, lo].
Acknowledgements. The authors would like to thank Prof. S. N. PANDEY
and
Dr. BANISingh of the Mathematics Department, University of Roorkee for helpfu1
discussions. Thanks are also due to the CSIR, India for support.
References
[l] EINSTEIN,
A.; (and W. MAYER):Sitzungsber. preuss. Akad. Wise., Phys. math. KI. (1931)
541.
[2] EIWSTEIPI,
A.; (and W. MAYER): Sitzungsber. preuss. Akad. Wiss., phys.-math. K1. (1932)
130.
131 KALUZA,TH.:Sitzungsber. preuss. Akad. Wiss., phys.-math. KI. (1921) 966.
[4] KLEIN,0.:2. Phys. 87 (1926) 896.
[6] -A,
M. N.; SVDEAKAB
RAO,A.: J. Math. P h p . (USA) 16 No. 2 (1976).
[GI 114arraNT~.M. N.: J. Math. Phys. (USA) 16 NO. 10 (1975).
[7] MAHAXTA,M. N.: Lett. Nuovo Cirnento 40 No. 1 (1984).
[8] RAO,J. R.; ROY,A. R.; TIWARI,
R. N.: Ann. Phys. (N.Y.) 69 (1972) 473.
[9] RAO,J. R.; ROY,
A. R.; TIWARI,
R. N.: Ann. Phys. (N.Y.) 79 (1973) 276.
[lo] TIWARI,R. N.: Ind. J. Pure and Appl. Math. 10 (3) (1979) 339.
[ll] MIE, G.: Ann. Phys. (Leipzig) 87 (1912) 511; 39 (1912) 1; 40 (1913) 1.
[12] EDDINGTON,
A. S.: Mathematic4 Theory of Relativity. Cambridge 1966, 194.
Bei der Redaktion eingegangen am 2. August 1988, revidiertes k u s k r i p t a m 30. September 1985.
Anschr. d. Verf.: Dr. M. N.MAH-4NT.k
Fellow, Indian Institute of Advanced Study
&strepati Nivas, Summer Hill
Shimls 17 1008,India
Dr. Y. K. GVFTA
Dept. of Mathematics, University of Roorkee
Roorkee 247677, India
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