The ultrarelativistic Kerr geometry and its energy-momentum tensor

Abstract

The ultrarelativistic limit of the Schwarzschild and the Kerr-geometry together with their respective energy-momentum tensors is derived. The approach is based on tensor-distributions making use of the underlying Kerr-Schild structure, which remains stable under the ultrarelativistic boost.

Impulsive waves in the Nariai universe

“…In particular, these belong to two distinct families in which the waves are either expanding or non-expanding, thus being understood as limiting cases of sandwich waves of the Robinson-Trautman or the Kundt classes, respectively. Specific non-expanding solutions were originally obtained by applying the Aichelburg-Sexl ultra-relativistic boost [4] to different elements of the Kerr-Newman and Weyl families (see, e.g., [5,6,7,8,9]). It has then been shown [10,11] that the whole class of non-expanding impulsive pure gravitational waves (with the only exception of plane waves) is generated by null particles with an arbitrary multipole structure, corresponding to singularities of the metric tensor.…”

Impulsive gravitational waves of massless particles in extended theories of gravity

“…This can be achieved by looking at the behavior of two pointlike objects in the gravitational field of the massless source. For models with minimal coupling, this can be seen by measuring the relative acceleration of two nearby test particles separated by n µ which is obtained from the geodesic deviation equation (25) and noting that for the spacetime under consideration, the components of the Riemann curvature tensor are proportional to the second derivatives of the wave profile K(u, x, y) with respect to the transverse coordinates. For models with nonminimal couplings where the particles do not move along geodesics as a result of extra forces coming from the coupling, the above equation should be modified by adding the relevant terms.…”

“…In fact, in the context of general relativity, the problem has been studied by boosting the Kerr metric in Refs. [24] and [25], by boosting the Kerr-Newman metric in Ref. [26], in the presence of a nonvanishing cosmological constant [27], for motion in Schwarzschild-Nordström and Schwarzschild-de Sitter spacetimes [28], for particles with arbitrary multipoles in Ref.…”

The use of generalized functions and distributions in general relativity

“…The ultrarelativistic limit of the latter in the (u, w, x, y) coordinate system is associated to δ(u)δ (2) (y, z)p a p b where p a = (1, 0, 0, 0), (33) which is just the energy-momentum tensor ofg ab . Indeed this observation was used by Balasin and Nachbagauer [6] to derive the ultrarelativistic limit of the Schwarzschild and Kerr geometries (see also [11]). It is also possible to calculate the ultrarelativistic limit of the Reissner-Nordstrøm solution.…”

“…Balasin and Nachbagauer considered rotating, charged, Kerr-Newman black-hole solutions in a number of papers ( [4,5,6,7,9,10]). The solutions considered have the feature that they are all examples of Kerr-Schild geometries.…”