The Universality of Penrose Limits near Space-Time Singularities

Abstract

We prove that Penrose limits of metrics with arbitrary singularities of power-law type show a universal leading u −2 -behaviour near the singularity provided that the dominant energy condition is satisfied and not saturated. For generic power-law singularities of this type the oscillator frequencies of the resulting homogeneous singular plane wave turn out to lie in a range which is known to allow for an analytic extension of string modes through the singularity. The discussion is phrased in terms of the recently obtained covariant characterisation of the Penrose limit; the relation with null geodesic deviation is explained in detail. 1

Scalar field probes of power-law space-time singularities

“…This result parallels that obtained in [1] for probes consisting of families of massless particles (null geodesic deviation, a.k.a. the Penrose Limit).…”

“…One minor difference between the results obtained here and those of [1] is that in the case of Penrose limits the strict DEC needed to be invoked only in the case of spacelike singularities, η = +1, timelike singularities always giving rise to plane waves with a u −2 -behaviour. This should be regarded as an indication (cf.…”

“…In particular, we will first show that the results obtained in [1], namely the scale invariant inverse square behaviour of the wave profile A ab (u), as well as a crucial [9,6] lower bound on the coefficients, have a precise and rather striking analogue in the case of a scalar field. Schematically, this analogy can be expressed as strict DEC ⇒ A ab (u) → c a δ ab u −2 scale invariant (c a ≥ −1/4) V eff (x) → cx −2 scale invariant (c ≥ −1/4) (1.3)…”

“…in [22,8] in related contexts. For decelerating cosmological FRW metrics, with cosmological scale factor (in comoving time) ∼ t h , 0 < h < 1, 25) with w the equation of state parameter, P = wρ, one finds [5,1] FRW : 26) as can be seen by going to conformal time. A routine calculation shows that the above result (2.18) for the purely x-dependent part of the effective potential (with x conformal time) is actually an exact result, and not an artefact of the near-singularity Szekeres-Iyer approximation.…”

“…in the vicinity of a generic singularity of this type. A first investigation along these lines was performed in [5,1] in the context of the Penrose Limit, i.e. of probing a space-time via the geodesic deviation of families of massless particles.…”