Hadamard Tail from Initial Data on the Light Cone

TL;DR

This paper implements a novel method to calculate the Hadamard tail of scalar fields in curved spacetimes with crossing null geodesics, specifically on Plebański-Hacyan spacetime, expanding understanding of wave propagation beyond flat geometries.

Contribution

It demonstrates the first implementation of the Characteristic Initial Data method for Hadamard tail calculation in a non-conformally flat spacetime with crossing null geodesics.

Findings

Calculated Hadamard tail in Plebański-Hacyan spacetime.

Analyzed tail variation with different coupling constants.

Proved the method's applicability in complex spacetime geometries.

Abstract

Field perturbations of a curved background spacetime generally propagate not only at the speed of light but also at all smaller velocities. This so-called $Hadamard\,tail$ contribution to wave propagation is relevant in various settings, from classical self-force calculations to communication between quantum particle detectors. One method for calculating this tail contribution is by integrating the homogeneous wave equation using Characteristic Initial Data on the light cone. However, to the best of our knowledge, this method has never been implemented before except in flat or conformally-flat spacetimes, where null geodesics emanating from a point do not cross. In this work, we implement this method on the black hole toy model Pleba\'nski-Hacyan spacetime, $\mathbb{M}_2\times\mathbb{S}^2$. We obtain new results in this spacetime by calculating the Hadamard tail of a scalar field everywhere where it is defined (namely, in the maximal normal neighbourhood of an arbitrary point) and investigate how it varies for various values of the coupling constant. This serves as a proof-of-concept for the Characteristic Initial Data method on spacetimes where null geodesics emanating from a point $do$ cross.