Scalar Field Probes of Power-Law Space-Time Singularities

TL;DR

This paper investigates the behavior of scalar waves near power-law space-time singularities, revealing a universal inverse square potential and demonstrating that certain singularities are quantum mechanically unresolved.

Contribution

It shows that the effective potential near these singularities universally exhibits an inverse square form under the DEC, extending previous null geodesic results to scalar fields.

Findings

Effective potential exhibits universal inverse square behavior near singularities.

Timelike singularities satisfying DEC are quantum mechanically singular.

Scalar field behavior near singularities depends on the x^{-2} coefficient.

Abstract

We analyse the effective potential of the scalar wave equation near generic space-time singularities of power-law type (Szekeres-Iyer metrics) and show that the effective potential exhibits a universal and scale invariant leading x^{-2} inverse square behaviour in the ``tortoise coordinate'' x provided that the metrics satisfy the strict Dominant Energy Condition (DEC). This result parallels that obtained in hep-th/0403252 for probes consisting of families of massless particles (null geodesic deviation, a.k.a. the Penrose Limit). The detailed properties of the scalar wave operator depend sensitively on the numerical coefficient of the x^{-2}-term, and as one application we show that timelike singularities satisfying the DEC are quantum mechanically singular in the sense of the Horowitz-Marolf (essential self-adjointness) criterion. We also comment on some related issues like the near-singularity behaviour of the scalar fields permitted by the Friedrichs extension.