# Green Functions and Radiation Reaction From a Spacetime Perspective

## TL;DR

This paper applies the matched expansions method to compute the self-force on a particle in curved spacetime, providing new insights into non-local effects and Green function calculations in black hole backgrounds.

## Contribution

It introduces a novel application of the Poisson-Wiseman-Anderson matched expansions method to compute the self-force in curved spacetime, including new transport equations for key bi-tensors.

## Key findings

- Successfully computed the Green function in Nariai spacetime
- Demonstrated the matched expansion method's effectiveness for self-force calculations
- Provided insights into the non-local properties of the self-force

## Abstract

The basis of this work is the first full application of the Poisson-Wiseman-Anderson method of `matched expansions' to compute the self-force acting on a point particle moving in a curved spacetime. The method employs two expansions for the Green function which are respectively valid in the `quasilocal' and `distant past' regimes, and which are matched together within the normal neighborhood. Building on a fundamental insight due to Avramidi, we provide a system of transport equations for determining key fundamental bi-tensors, including the tail-term, V(x,x'), appearing in the Hadamard form of the Green function. These bitensors are central to a broad range of problems from radiation reaction and the self-force to quantum field theory in curved spacetime and quantum gravity. Using their transport equations, we show how the quasilocal Green function may be computed throughout the normal neighborhood both numerically and as a covariant Taylor series expansion. These calculations are carried out for several black hole spacetimes. Finally, we present a complete application of the method of matched expansions. The calculation is performed in a static region of the spherically symmetric Nariai spacetime (dS_2 X S^2), where the matched expansion method is applied to compute the scalar self-force acting on a static particle. We find that the matched expansion method provides insight into the non-local properties of the self-force. The Green function in Schwarzschild spacetime is expected to share certain key features with Nariai. In this way, the Nariai spacetime provides a fertile testing ground for developing insight into the non-local part of the self-force on black hole spacetimes.