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.
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).
where k is the Einstein's gravitational constant, T is the energy momentum tensor
of type (0,2) given by
The energy momentum tensor
of the electromagnetic field [T.sup.[mu]v] not generally symmetric.
Secondly, a dust-like energy momentum tensor
for a purely radial motion with account of an ultrarelativistic collapsing matter and thermally emitted radiation is obtained in subsec.
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.
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.
the new improved energy momentum tensor
is defined as
Therefore, the nonzero components of the energy momentum tensor
from (11) using (13) are
where R is the Ricci scalar, [R.sub.ij] is the Ricci tensor, [phi] is the Brans-Dicke scalar field, [bar.[omega]] is the dimensionless constant, and [T.sub.ij] is the energy momentum tensor
. The scalar fields satisfy the following equation: