Semiconformal curvature tensor and perfect fluid spacetimes in general relativity

The semiconformal curvature tensor and its divergence have been studied for perfect fluid spacetimes. It is seen, apart from other results, that the perfect fluid spacetimes with divergence-free semiconformal curvature tensor either satisfy the vacuum-like equation of state or represent FLRW cosmological model.


Introduction and preliminaries
Let the Einstein's field equations be Contraction of this equation by g ab leads to From Equation (2), Equation (1) becomes Ishii [1], introduced the set of conharmonic transformations as a subgroup of the conformal group of transformations. These conharmonic transformations are given byg bc = e 2β g bc , and satisfying the condition 1 where g bc andg bc are the metric tensors for Riemannian spaces V andṼ, respectively, and β is a real scalar function.
For n-dimensional Riemannian differentiable manifold (M n , g) (n ≥ 4), a rank four tensor L h bcd which is invariant under conharmonic transformation, is given by [2] where R h bcd , R bd are Riemann and Ricci curvature tensors respectively. The geometrical significance of conharmonic curvature tensor has been discussed by Shaikh and Kumar [3] while the physical significance of this tensor has been investigated by Abdussattar and Dwivedi [4] and Ahsan and Ali [5]. Kim [6,7] introduced a curvature like tensor such that it remains invariant under condition (5). He calls this new tensor as semiconformal curvature tensor and denotes it by P h bcd . For a Riemannian manifold (M n , g) this tensor is defined as provided the constants A and B are non-zero. Weyl conformal curvature tensor [8], is expressed as where R ab is Ricci tensor and R is the scalar curvature tensor. For A = 1 and B = −(1/(n − 2)), the semiconformal curvature tensor reduces to conformal curvature tensor, while for A = 1 and B = 0, it reduces to conharmonic curvature tensor. The Covariant form of the semiconformal curvature tensor can be written as It may be noted that the semiconformal curvature tensor P hbcd satisfies the following symmetry property P hbcd = −P bhcd = −P hbdc = P cdhb and P hbcd + P chbd + P bchd = 0.
Towards an attempt of obtaining the exact solution of Einstein field equations, the symmetry assumptions on spacetime help to achieve the target. These symmetries are defined through the vanishing of the Lie derivatives of certain tensor with respect to a spacelike, timelike or a null vector. The symmetries of spacetimes are also called as collineations. The notion of semiconformal curvature collineation, defined in terms of other curvature tensors, has been introduced by Pundeer et al. [9], who obtained the necessary and sufficient conditions under which a spacetime consisting electromagnetic fields may admit related collineations. Einstein's gravity is well connected with the Maxwell's electrodynamics, so the symmetries of spacetime related to electromagnetic fields can be coupled with the Maxwell's equations. Recently El-Salam [10] has used the technique of fractional calculas for the study of Maxwell's equations. This lucid technique may further be used for the symmetries of spacetime and the semiconformal curvature collineation may be analysed in this light. Abdussattar and Babita Dwivedi in their paper [4] have given a detailed study on the divergence of conharmonic curvature in perfect fluid spacetime. Also, the divergence of a W−curvature tensor in perfect fluid spacetime is studied by Ahsan and Ali [5], who have shown the representation of perfect fluid as Friedmann-Lemaitre-Robertson-Walker (FLRW) cosmological model under certain conditions. Motivated by these works on the divergence of curvature tensors we choose the semiconformal curvature tensor which is invariant under conharmonic transformation.
The present paper is yet another effort to study the semiconformal curvature tensor. In terms of the energy-momentum and Ricci tensors, the divergence of semiconformal curvature tensor has been expressed in Section 2. It may be noted that the divergence of semiconformal curvature tensor vanishes for Einstein spaces under certain conditions. Finally, in the last section a divergence-free semiconformal curvature tensor is considered for a perfect fluid spacetime and we found that if the semiconformal curvature tensor has zero divergence and the energy momentum tensor is of Codazzi type [11], then (μ + p) is constant. Furthermore, if the semiconformal curvature tensor is divergence-free then either μ + p = 0, i.e. the vacuum -like equation of state is satisfied bythe spacetime or the fluid is accelerationfree, vorticity-free, shear-free andrepresent a Friedmann-Lemaitre-Robertson-Walker (FLRW) cosmological model with μ − 3p = constant.

Spacetime with divergence of semiconformal curvature tensor
In four dimensional manifold, from Equation (7) the semiconformal curvature tensor is given by From Equations (6) and (8) for n = 4 the conharmonic and conformal curvature tensors may be written in the following form and Now we write Theorem 2.1: Semiconformal curvature tensor is proportional to conharmonic curvature tensor, if the scalar curvature is zero in the spacetime.

Proof:
The semiconformal curvature tensor in space of dimension n = 4, with the help of Equations (10) and (11), is given by while its contraction is given by which is also invariant under condition in Equation (5). Moreover, use of Equation (10) in (12), gives rise to which from Equation (13) leads to Thus if scalar curvature is zero then from Equation (14), semiconformal and conharmonic curvature tensors are proportional to each other.  (13) and (15), we write The second term of the right side of Equation (16) will vanish as the scalar curvature is zero. This completes the proof.
Further, from Equation (16) which on contracting over h and e leads to We know that the Bianchi identity is given by From Equations (17) and (18), we have which on using Equations (2) and (3), may be expressed as If the energy-momentum, T ab , is Codazzi type [11], that is, if it satisfies Yang's equation, then from Equations (21) and (20), the proof follows.
It is known that a n-dimensional Riemannian space is an Einstein space if its Ricci tensor R bc is proportional to the metric tensor, i.e. R bc = R/ng bc . In 4-dimensional spacetime of general relativity R bc = R/4g bc . Thus use of R bc = R/4g bc in Equation (22) completes the proof. Note that in this case, its scalar curvature R is constant [12], i.e. ∇ d R = 0 n is greater than or equal to 2. Therefore, from Equation (22), using the conditions of Einstein spacetime, we get divergence zero of the semiconformal curvature tensor.

Theorem 2.5: For the spaces of constant curvature, the divergence of semiconformal curvature tensor vanishes if the Ricci tensor is of Codazzi type and divergence-free.
Proof: The divergence of semiconformal curvature tensor, from Equation (22), may be expressed as We get immediate proof of the theorem, if the Ricci curvature tensor is of Codazzi type (see Equation (21)) and divergence-free.
3. Perfect fluid spacetime with divergence-free semiconformal curvature tensor Proof: For a perfect fluid spacetime, the energy momentum tensor is given by where p is the isotropic pressure, μ is the energy density and the fluid four-velocity vector is denoted by u a . Multiplying Equation (24) by g ab and using (timelike vector) u a u a = −1, we get Now, if the energy-momentum tensor is Codazzi type and the divergence of semiconformal curvature tensor vanishes, then Equation (22) implies Use of Equation (25) in Equation (26) leads to which on contraction by u d , leads to Here an overhead dot represents the covariant differential along the fluid flow vector and which on contracting with g bc leads tȯ In view of energy Equations (33), (38) yieldṡ Equation (37) with the help of force Equation (32) may be written as (μ + p)u b u c +ṗ(g bc + u b u c ) + (μ + p)∇ c u b = 0.