Non-local scalar fields in static spacetimes via heat kernels

TL;DR

This paper develops methods to solve non-local scalar field equations in static and ultrastatic spacetimes using heat kernel techniques, providing explicit Green's function estimates and examples.

Contribution

It introduces a heat kernel-based approach to analyze non-local scalar fields in static spacetimes, including explicit Green's function calculations and regularity discussions.

Findings

Derived static Green's function estimates using heat kernel bounds

Provided exact and approximate solutions for non-local equations in specific spacetimes

Discussed regularity properties of solutions in the static and ultrastatic cases

Abstract

We solve the non-local equation ${-e^{-\ell^2\square}\square{\phi}=J}$ for i) static scalar fields in static spacetimes and ii) time-dependent scalar fields in ultrastatic spacetimes. Corresponding equations are rewritten as non-local Poisson/inhomogeneous Helmholtz equations in compact and non-compact weighted/Riemannian manifolds using static/frequency-domain Green's functions, which can be computed from the heat kernels in the respective manifolds. With the help of the heat kernel estimates, we derive the static Green's function estimates and use them to discuss the regularity. We also present several examples of exact and estimated static/frequency-domain Green's functions.