Boosted Schwarzschild Metrics from a Kerr-Schild Perspective

TL;DR

This paper investigates the differences between two Kerr-Schild representations of the Schwarzschild metric, revealing implications for gravitational memory effects and the consistency of boosts in nonlinear regimes.

Contribution

It demonstrates that the two Kerr-Schild versions have distinct Minkowski backgrounds and explores their implications for gravitational memory and boost symmetries.

Findings

The two Kerr-Schild backgrounds differ unexpectedly.

Nonlinear gravitational memory aligns with linear results only under specific boosts.

Boosts with respect to the ingoing background are consistent with no ingoing radiation.

Abstract

The Kerr-Schild version of the Schwarzschild metric contains a Minkowski background which provides a definition of a boosted black hole. There are two Kerr-Schild versions corresponding to ingoing or outgoing principle null directions. We show that the two corresponding Minkowski backgrounds and their associated boosts have an unexpected difference. We analyze this difference and discuss the implications in the nonlinear regime for the gravitational memory effect resulting from the ejection of massive particles from an isolated system. We show that the nonlinear effect agrees with the linearized result based upon the retarded Green function only if the velocity of the ejected particle corresponds to a boost symmetry of the ingoing Minkowski background. A boost with respect to the outgoing Minkowski background is inconsistent with the absence of ingoing radiation from past null infinity.