In this situation as well validity of the above expression for all null vectors [l.sub.a], along with two times contracted Bianchi identity and covariant conservation of matter energy momentum tensor
, amounts to furnishing the ten components of Einstein's equations.
The known exact solutions may be classified into (at least) four classes , namely, the algebraic classification of conformal curvature, physical characterization of the energy momentum tensor
, existence, and structure of preferred vector fields and group of motions.
The energy momentum tensor
of the electromagnetic field [T.sup.[mu]v] not generally symmetric.
for all vector fields X, Y where S is the Ricci tensor of the type (0, 2), r is the scalar curvature, k is the gravitational constant and T is the energy momentum tensor
of type (0, 2).
So, for example, if one begins with a vacuum seed solution, then it is known that the Einstein tensor is zero and so only the conformal part must now be considered in conjunction with a perfect fluid energy momentum tensor
. Such solutions are referred to as conformally Ricci-flat spacetimes.
In the general theory of relativity, energy momentum tensor
plays an important role and the condition on energy momentum tensor
for a perfect fluid space time changes the nature of space time (5).
here [T.sub.ij] being the ordinary energy momentum tensor
associated to isotropic matter and radiation.
the new improved energy momentum tensor
is defined as
where k is the Einstein's gravitational constant, T is the energy momentum tensor
of type (0,2) given by
Although there are physical arguments for equating the Einstein tensor to the energy momentum tensor
([G.sub.[micro]v] = K[T.sub.[micro]v]), and thus into analogues for Newton's Law of Gravity, we note simply in this paper that Eq.
One way is to modify the dynamical field equations by taking negative pressure in the form of energy momentum tensor
Therefore, the nonzero components of the energy momentum tensor
from (11) using (13) are