New solutions of the Ermakov-Pinney equation in curved space-time

TL;DR

This paper investigates solutions to the Ermakov-Pinney equation in various curved space-times, revealing behaviors relevant to black hole physics, cosmology, and gravitational waves, with implications for understanding wave evolution in these settings.

Contribution

It introduces new solutions of the Ermakov-Pinney equation in curved space-times and explores their physical implications in black hole, cosmological, and gravitational wave contexts.

Findings

Rapid blow-up of the Ermakov-Pinney field in cosmological and gravitational wave space-times

Spatially damped oscillations in Schwarzschild space-time

Phase function analysis across different curved space-time models

Abstract

An Ermakov-Pinney-like equation associated with the scalar wave equation in curved space-time is here studied. The example of Schwarzschild space-time considered in the present work shows that this equation can be viewed more as a model equation, with interesting applications in black hole physics. Other applications studied involve cosmological space-times (de Sitter) and pulse of plane gravitational waves. In all these cases the evolution of the Ermakov-Pinney field seems to be consistent with a rapid blow-up, unlike the Schwarzschild case where spatially damped oscillations are allowed. Eventually, the phase function is also evaluated in many of the above space-time models.