Ultrarelativistic boost of a black hole in the magnetic universe of Levi-Civita--Bertotti--Robinson

Abstract

We consider an exact Einstein-Maxwell solution constructed by Alekseev and Garcia which describes a Schwarzschild black hole immersed in the magnetic universe of Levi-Civita, Bertotti and Robinson (LCBR). After reviewing the basic properties of this spacetime, we study the ultrarelativistic limit in which the black hole is boosted to the speed of light, while sending its mass to zero. This results in a non-expanding impulsive wave traveling in the LCBR universe. The wave front is a 2-sphere carrying two null point particles at its poles -- a remnant of the structure of the original static spacetime. It is also shown that the obtained line-element belongs to the Kundt class of spacetimes, and the relation with a known family of exact gravitational waves of finite duration propagating in the LCBR background is clarified. In the limit of a vanishing electromagnetic field, one point particle is pushed away to infinity and the single-particle Aichelburg-Sexl pp-wave propagating in Minkowski space is recovered.

Figures (2)

• Figure 1: The 2-hyperboloid visualizes the dS$_2$ factor AdS$_2$ factor of the LCBR universe (\ref{['BR']}) in the embedding coordinates \ref{['AdS2_emb']}. Each point corresponds to a 2-sphere of a constant radius $b$ in the four-dimensional spacetime. The parallel solid lines $Z_0+Z_1=0$ ($\Leftrightarrow Z_2=\pm b$) are the histories of two of these spheres, propagating at the speed of light from one side of the universe to the other. Cf. PodGri97 for a similar discussion in the case of the solutions of HotTan93.
• Figure 2: The conformal diagram of the LCBR universe (\ref{['BR']}) is that of its AdS$_2$ factor (cf. Carter73), with each point representing a two-dimensional sphere. The conformal spatial coordinate $R\in(0,\pi)$ is defined by $R=2\arctan(e^{{z}/b})$, and the timelike boundaries $R=0$ ($z=-\infty$) and $R=\pi$ ($z=+\infty$) correspond to null and spacelike infinity on opposite sides of the universe. The two solid lines represent the components $Z_2=\pm b$ of the null hypersurface $U=0$. Timelike geodesics emanating from $p$ reconverge at the image point $q$ (cf. HawEll73).