Penrose limits and spacetime 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) ∼ u −2 , the scale invariance of the latter reflecting the power-law behaviour of the singularities. 1

Killing tensors in pp-wave spacetimes

“…For ǫ = 0 this value of κ corresponds to h = 3. The value κ = 3/16 can also arise in the regime ǫ = ±1, h ≥ 1, see [18] for details.…”

“…(ii) Physically, pp-wave spacetimes represent radiation moving at the speed of light, and so are of particular interest in the theory of gravitational radiation. Further, gravitational plane wave spacetimes have applications in String Theory [18] and arise naturally as the Penrose Limit [19] of any spacetime, see also [18].…”

Penrose limits and non-relativistic geometries

“…In this section, we use the covariant characterization of Penrose limit as given in [14]. In this approach, we directly arrive at the Penrose limit metric in Brinkmann coordinates and the plane wave is described by a wave profile A ij .…”

“…In this section, we review methods to obtain Penrose limits, especially the covariant characterization of Penrose limit as discussed in detail in [14]and [16]. This plane wave depends upon the metric we start with and the choice of the null geodesic along which the observer is moving.…”

Holography and Cosmological Singularities

“…In fact, it turns out that the pp-wave is singular if and only if the pre-Penrose limit original spacetime is singular [47]. The pp-wave space-time has space-like and null isometries.…”