Abstract
We describe the use of conformal mappings as a mathematical mechanism to obtain exact solutions of the Einstein field equations in general relativity. The behaviour of the spacetime geometry quantities is given under a conformal transformation, and the Einstein field equations are exhibited for a perfect fluid distribution matter configuration. The field equations are simplified and then exact static and nonstatic solutions are found. We investigate the solutions as candidates to represent realistic distributions of matter. In particular, we consider the positive definiteness of the energy density and pressure and the causality criterion, as well as the existence of a vanishing pressure hypersurface to mark the boundary of the astrophysical fluid.
Highlights
The gravitational evolution of celestial bodies may be modeled by the Einstein field equations
X is a Killing vector X is a homothetic Killing vector X is a special conformal Killing vector X is a nonspecial conformal Killing vector approach has to do with the fact that the existence of conformal Killing vectors is known to simplify the field equations— in other words, they involve a geometric constraint
We have demonstrated that new static and nonstatic solutions of the Einstein field equations may be constructed from 46 f 45 x
Summary
The gravitational evolution of celestial bodies may be modeled by the Einstein field equations. X is a Killing vector X is a homothetic Killing vector X is a special conformal Killing vector X is a nonspecial conformal Killing vector approach has to do with the fact that the existence of conformal Killing vectors is known to simplify the field equations— in other words, they involve a geometric constraint This is in opposition to an algebraic constraint which may be imposed, for example, by demanding that the eigenvectors of the Weyl tensor have certain preferred alignments. For example, if one begins with a vacuum seed solution, it is known that the Einstein tensor is zero and so only the conformal part must be considered in conjunction with a perfect fluid energy momentum tensor Such solutions are referred to as conformally Ricci-flat spacetimes.
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