Generating elastic solutions of the Einstein field equations from the Schwarzschild vacuum solution

Abstract

The problem of generating solutions of the Einstein field equations with an elastic energy–momentum tensor from the Schwarzschild vacuum solution by means of conformal transformations is analyzed. Applying the formulation of relativistic elasticity, suitable conformal factors are obtained for static and non-static elastic spacetime configurations and particular solutions are presented. This work shows that the technique used here permits generating new elastic matter solutions from a vacuum spacetime.

Static conformal elastic solution of Einstein's field equations

“…From [21], we have found that the energy density ρ is positive for r > 3M , and the dominant energy condition (DEC) (ρ − |p t | ≥ 0, ρ − |p r | = ρ ≥ 0) is satisfied for particular ranges of r > 3M . Hence it is clear that both weak energy condition (WEC) and null energy condition (NEC) are validated for particular ranges of r > 3M .…”

“…One can see the detailed discussion for this case in [21], where the analytical solution of Eq. ( 54) is given.…”

“…The formulae (92) and ( 94) that we have used to evaluate the masses m + and m − , respectively, are compatible with the definitions to calculate the mass given in [8,21,28,29]. So, we have chosen the relevant definitions in this context under consideration.…”

“…One can conclude that the avenue adopted in the present work and Refs. [8,18,19,21], allows constructing the new exact solutions of the EFE with the elastic matter. We will carry out a study on trapped surfaces in constructed static conformal elastic solutions in the future.…”

“…The first part T ij E is material energy-momentum tensor and is considered as an elastic matter, for the purpose of the present work. The tensor T l i E is evaluated as follows [21]:…”