Generating Solutions of Einstein’s Equations

In this article the method of conformal transformation is used to generate perfect fluid solutions of Einstein’s equations from vacuum spacetimes. The conditions on the conformal factor in order to generate a perfect fluid spacetime are presented. With Minkowski as a seed spacetime, these conditions are applied and a particular solution is found. It belongs to the family of Stephani spacetimes and its physical properties are further investigated. Introduction Einstein’s equations governing general relativity as a theory of gravity form a system of 10 non-linear second order partial differential equations for 10 unknown elements of the metric tensor. This system is very complicated to solve in general, but there are many known solutions with symmetries, which allow us to carry out the calculations. We can then use the symmetric solution and try to transform it into a new one, perhaps with less symmetries. The generating method chooses a class of such transformations. Conformal transformation In this paper the generating method uses conformal transformation gμν = f −2gμν , (1) where gμν is the original (or ”seed”) metric and f is a scalar function. It is well known that the new metric gμν preserves null cones of the seed spacetime and therefore has the same causal structure. After choosing the seed metric, we have one degree of freedom in f to create the new spacetime. That is usually enough to break the symmetries of the original spacetime, but there are also some limitations. For example, two Einstein spaces that are properly conformally related (f is not constant) must be both vacuum pp-waves or one Minkowski and the other de Sitter, as was proved by Brinkmann [1925]. Daftardar-Gejji [1998] generalized this theorem to the case of two properly conformally related spacetimes with equal Einstein tensors and showed that both spacetimes are generalized (not necessarily vacuum) pp-waves. After a conformal transformation, the new Ricci tensor Rμν can be expressed in terms of f and quantities from the seed spacetime as follows Rμν = Rμν + 2 f f;μν + ( 1 f ¤f − 3 f2 ‖df‖2 ) gμν , (2) where the covariant derivative ’;’ is also taken in the seed spacetime, ¤f := gf;μν and ‖df‖2 := gf,μf,ν . In our case, the seed spacetime is chosen to be vacuum, i.e. Rμν = 0. After specifying the desired form of the new stress-energy tensor Tμν , we will obtain restrictions on the conformal factor f . Fluid from vacuum The new stress-energy tensor should not be vacuum because of the Brinkmann theorem. We choose to generate perfect fluid spacetimes, as did for example Loranger, Lake [2008], who started with an unphysical seed metric and obtained a family of static fluid spheres. The stress-energy tensor of a perfect fluid has the form Tμν = (ρ+ p)uμuν − pgμν , (3) 13 WDS'09 Proceedings of Contributed Papers, Part III, 13–19, 2009. ISBN 978-80-7378-103-3 © MATFYZPRESS