Generating solutions to the Einstein field equations

Abstract

Exact solutions to the Einstein field equations may be generated from already existing ones (seed solutions), that admit at least one Killing vector. In this framework, a space of potentials is introduced. By the use of symmetries in this space, the set of potentials associated to a known solution are transformed into a new set, either by continuous transformations or by discrete transformations. In view of this method, and upon consideration of continuous transformations, we arrive at some exact, stationary axisymmetric solutions to the Einstein field equations in vacuum, that may be of geometrical or/and physical interest.

On the properties of a deformed extension of the NUT space-time

“…(27) and a redefinition of the parameters like the ones given in Eqs. (28) and (29) may completely change the behaviour of the photon sphere.…”

“…This has led many authors in the past to consider 'rotating' generalization of static axially symmetric solutions (see for example [21][22][23][24][25][26]). In particular, stationary generalizations of the ZV metric have been studied in [27][28][29][30]. However, as it turns out, not all such generalizations describe rotating objects.…”

Degenerate metrics on a dual geometry of spherically symmetric space-time

“…The above equation contains a reduction in the number of terms, due to the simplification in the choice of parameters. This setting of values was motivated in a recent work [7], and we performed a comparision with a singularity structure of dual metrics in [8]. In a less restrictive situation, in the case where C 2 and C 3 are nonzero and arbitrary, the number of terms increases strongly so that we have to express a plethora of terms by a combination of factors such as the one given below:…”

“…Another extension of Schwarzschild space-time was considered in [7] where a new kind of Kretschmann scalar invariant was obtained, whose expression was compared recently with a dual version of a spherically symmetric space-time [8]. In the last case, was performed an analysis of duality from the geodesic perspective.…”

Generating solutions to the Einstein–Maxwell equations

“…Since ξ α ω α = 0 = L ξ ω a , it follows that ω α is a vector also on S. Now, given g αβ , we can determine ξ α , and through that, λ, h ab and ω α , as well. Reversely, upon consideration of λ, ξ α , h ab , and ω α , it is possible to reconstruct the metric g αβ of M (see, e.g., [15]). Notice that, since ω α is a vector also on S, indices can be raised either by h ab or by g αβ .…”