Spacetime Ehlers group: transformation law for the Weyl tensor

Abstract

The spacetime Ehlers group, which is a symmetry of the Einstein vacuum field equations for strictly stationary spacetimes, is defined and analyzed in a purely spacetime context (without invoking the projection formalism). In this setting, the Ehlers group finds its natural description within an infinite dimensional group of transformations that maps Lorentz metrics into Lorentz metrics and which may be of independent interest. The Ehlers group is shown to be well defined independently of the causal character of the Killing vector (which may become null on arbitrary regions). We analyze which global conditions are required on the spacetime for the existence of the Ehlers group. The transformation law for the Weyl tensor under Ehlers transformations is explicitly obtained. This allows us to study where, and under which circumstances, curvature singularities in the transformed spacetime will arise. The results of the paper are applied to obtain a local characterization of the Kerr-NUT metric.

Tópicos da correspondência fluido gravidade em espaços planos

“…Since the Ehlers group is developed within this formalism, it becomes necessary to adopt a different approach to this problem, or at least define the Ehlers group in such a way that they include the case of null Killing vectors. As stated on [39], there were examples suggesting that it would be possible to extend the Ehlers group to encompass null Killing vectors, and it is shown in [39] that this is indeed possible, if one no longer works in the projection formalism and instead works in a spacetime setting, i.e., using only spacetime objects. Moreover, it is shown that the Ehlers group can be included within an infinite-dimensional group of transformations mapping Lorentzian metrics into Lorentzian metrics.…”

“…We are soon going to describe eq. (4.1) in somewhat more detail, but the full treatment in which they are developed lies well outside the scope of our work, so we refer the reader to [39] for all details.…”

“…For convenience, we are going to adopt the nomenclature used in [39,40]: we shall refer to the general transformations described by eq. (4.1) as the generalized Ehlers group, while their subset which specifically maps vacuum solutions into vacuum solutions shall be called the spacetime Ehlers group.…”

“…Following [39], we will start with some definitions of objects that will be used throughout this text. As stated before, a full treatment lies outside the scope of this work, so we are only going to list the tools which are necessary for further development.…”

“…One such example is known as the Ehlers group [35][36][37][38][39], which has first been applied in the context of fluid/gravity correspondence in 2012 [40]. On the other hand, there is a known symmetry group of solutions of the Navier-Stokes equations [41], which might also be of interest whenever someone mentions a duality between gravitational and fluid dynamical systems.…”