Radiative falloff of a scalar field in a weakly curved spacetime without symmetries

TL;DR

This paper proves that a massless scalar field in a weakly curved, stationary, asymptotically flat spacetime decays over time following a universal inverse power-law, regardless of the spacetime's nonspherical features or angular momentum.

Contribution

It extends Price's falloff theorem to general weakly curved spacetimes without symmetry constraints, showing decay depends only on total mass.

Findings

Late-time decay follows inverse power-law determined by initial wave profile.

Decay behavior is independent of spacetime's angular momentum and multipole moments.

Contradictions with Kerr black hole studies are due to coordinate choices, not fundamental differences.

Abstract

We consider a massless scalar field propagating in a weakly curved spacetime whose metric is a solution to the linearized Einstein field equations. The spacetime is assumed to be stationary and asymptotically flat, but no other symmetries are imposed -- the spacetime can rotate and deviate strongly from spherical symmetry. We prove that the late-time behavior of the scalar field is identical to what it would be in a spherically-symmetric spacetime: it decays in time according to an inverse power-law, with a power determined by the angular profile of the initial wave packet (Price falloff theorem). The field's late-time dynamics is insensitive to the nonspherical aspects of the metric, and it is governed entirely by the spacetime's total gravitational mass; other multipole moments, and in particular the spacetime's total angular momentum, do not enter in the description of the field's late-time behavior. This extended formulation of Price's falloff theorem appears to be at odds with previous studies of radiative decay in the spacetime of a Kerr black hole. We show, however, that the contradiction is only apparent, and that it is largely an artifact of the Boyer-Lindquist coordinates adopted in these studies.