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LlgTTER]g AL -NUOVO CIMENTO VOL. 27, ~. 17 26 Aprile 1980
On the Connection for the Brans-Dicke-Jordan Theory.
M. Gi]-RSES (*)
Max-Planck-I~stitut ],~r Physik und Astrophysik
8046
Garching bei Mu~che~i, Deutschla~id
M. ItAI.IL
Department o] Physics, Middle East Technical U~iversity - A~kara, Turkey
(ricevuto il 5 Febbraio 1980)
In an attempt to unify electromagnetic and gravitational fields, EISIgNHART (I) intro-
duced a method where the electromagnetic field acts as a part of a generalized asym-
metrical connection whose empty-space equations reproduce Einstein-Maxwell field equa-
tions. Although Eisenhart's choice for the generalized connection is not a suitable
one at least for the Reissner-Norsdrom case--his method is an attractive one to be
applied to the energy-momentum tensors other than the electromagnetic field. Inspired
from this work, we calculate the connection for Brans Dicke or Einstein-massless scalar
field theory (2).
The metric tensor gay of general relativity is known to couple with the energy-momen-
tum tensors of everything related with macroscopical quantities {like mass, electric charge
and angular momentum) except the gravitational field itself. Extension of general
relativity to nfieroscopical dimensions brought spin as a dynamical factor which could not
couple to anything until Riemannian geonletry was generalized to the Riemann-Cartan
geometry involving torsion suitably to couple with the energy-momentum of the spin (3).
This latter geometry is known also to arise naturally, provided we require local Poincar6
invariance in accordance with the Yang-Mills-Utiyaina technique of compensating
field (4-6).
In this note we shall consider a further extension of general relativity in the presence
of sources coupled with the gravitational field. This is achieved by releasing the metricity
condition
(1) V~g~ = 0,
(*) Alexander yon IIunboldt Fellow: on leave from M.E.T.U., Ankara, Turkey.
(1) L. P. EISENHART:
Proc. Nat. Acad. Sci. USA, 42,
249, 646, 878 (1956); 43, 333 (1957).
(2) A. JANIS, D. C. t/OBI:,~SON and J. ~'INICOL-R:
Phys. Rein,
186, 1729 (1969).
(3) E. ~,V. t{EHL, P. VAN DER HEYDE, G. D. KERLICK and. J. M. NESTER:
Rev. Mod. Phys.,
4B,
393 (1976).
0) C. N. YA~-~ and R. L. MILLS:
Phys. Rev.,
96, 191 (1954).
(5) I{. UTI]TAMA:
Phys. Rev.,
101, 1597 (1956).
(~) T. ~,V. I3. KIBBLE:
J. Math. Phys. (N. Y.),
2, 212 (1961).
562
ON THE CONNECTION FOR THE BRANS-DICKE-ffORDAN THEORY 5~
i.e.
covariant constancy of g~. Here V~ denotes the covariant differentiation with
respect to the symmetric (anti-symmetric) connection in Riemannian (Riemann-Cartan)
geometry. We shall therefore have geometries with
(2)
(Q~=Q~).
The nonmetricity tensor Q~ measures the nonmetric part of g~v and the covariant deriv-
ative is with respect to a general (asymmetrical) connection. From (2), the asym-
metrical connection is solved as
(3)
where
(4)
1
2 ~
,u fl - g (g"~'~ + g'~ '~ -- g"~'~) '
1
(the torsion tensor)
and the symmetric component will be denoted by
Defining further
(5)
we obtain the relations
(the contortion tensor),
SCnOUTEN (7) gave a classification of all possible geometries depending on the various
forms of the Q,3• and S~• tensors. The Riemann and Rieci tensors of the generalized
connection /~ become
(6)
R~.x ~ = K~ ~ + T~I~- T~I~[~ + T~Q ~ T~ Q- Tm ~ T~I q ,
(7)
Here K~ a and K~ denote Riemann and Ricci tensors, respectively, while (( bar >> rep-
resents covariant derivative, all with respect to the Riemannian connection (4). Let
us note that R~ # R~ in general.
(7) fro A, SCHOUTEN:
Ricci Calculus
(Berlin, 1954:).
564 M. Gih~SES and M. HALIL
We proceed now by imposing constraints on the contortion tensor T~ q which read
(8)
T~a ~
= 0,
(9) T.Z~]~ = 0
and therefore Eisenhart's field equations
(10) R~v(F) = O,
reduce to
(11)
K~ = T~q ~' T~,,~ e .
It is our belief that (10) unifies the gravitational field with classical matter fields
through the use of the nonmetric connection (3). Matter field equations are to be
identified with (8) and (9), while gravitational (Einstein) field equations are those given
by (11).
As an application we choose the contortion tensor as
(12) T~ =
(3g~v~o,vg m, + 6tJ,,~,, -- 6~9,~ ) ,
for which eq. (8) is identically satisfied, while (9) is equivalent to
(13) g~q~[~v = 0.
Equation (10), on the other hand, reads simply
(14) K m = nT,u~,v,
namely Einstein massless scalar field equations which are equivalent to Brans-Dicke-
Jordan theory of extended gravitation.
The extension of this work to electromagnetism (modification of Eisenhart), perfect
fluid and Yang-Mills fields coupled with fermion sources will be discussed elsewhere.
We would like to thank the Scientific and Technical Research Council of Turkey
(TBTAK) for financial support.
