Late-time behaviors of scalar field modes for a collapsing null shell spacetime and for the Unruh state in Schwarzschild spacetime

TL;DR

This paper analyzes the late-time behavior of scalar field modes in collapsing null shell and Schwarzschild spacetimes, revealing power-law decay and oscillations linked to quasi-normal modes.

Contribution

It provides a detailed comparison of mode behaviors in collapsing null shell and Unruh states, highlighting the power-law decay and oscillatory features at late times.

Findings

Modes oscillate forever or decay as a power law

Power-law decay is preceded by quasi-normal mode oscillations

Differences between spacetimes diminish over time

Abstract

The behaviors of the modes for a massless minimally coupled scalar field are investigated for the Unruh state for Schwarzschild spacetime and the "in" vacuum state for a spacetime in which a spherically symmetric null shell collapses to form a nonrotating black hole. In both cases there are two different sets of solutions to the mode equation that make up the state. For both spacetimes, one set of modes oscillates forever with no damping of the oscillations and the other set approaches zero at late times. The difference between a mode that oscillates forever in the null-shell spacetime and the corresponding mode for the Unruh state vanishes as a power law in time. The modes that approach zero at late times also vanish at late times as a power law in time. In all cases the power-law damping is preceded by a period of oscillations that appear to be due to quasi-normal modes.