Penrose Limits and Spacetime Singularities

TL;DR

This paper provides a covariant method to characterize Penrose plane wave limits and demonstrates that many spacetime singularities, including black holes and cosmological models, lead to singular homogeneous plane waves with scale-invariant profiles.

Contribution

It introduces a covariant characterization of Penrose limits and analyzes their behavior near various spacetime singularities, revealing a universal scale-invariant profile for many cases.

Findings

Penrose limit profile matrix is the null geodesic deviation matrix restricted to the geodesic.

Singularities in black holes and cosmologies yield homogeneous plane waves with $A(u) \,\sim\ u^{-2}$.

Scale invariance of the plane wave profiles reflects the power-law nature of the original singularities.

Abstract

We give a covariant characterisation of the Penrose plane wave limit: the plane wave profile matrix $A(u)$ is the restriction of the null geodesic deviation matrix (curvature tensor) of the original spacetime metric to the null geodesic, evaluated in a comoving frame. We also consider the Penrose limits of spacetime singularities and show that for a large class of black hole, cosmological and null singularities (of Szekeres-Iyer ``power-law type''), including those of the FRW and Schwarzschild metrics, the result is a singular homogeneous plane wave with profile $A(u)\sim u^{-2}$, the scale invariance of the latter reflecting the power-law behaviour of the singularities.