# Horizons as boundary conditions in spherical symmetry

## TL;DR

This paper develops a horizon-based formalism for describing near-horizon spacetime geometry in spherical symmetry, focusing on matter interactions and providing exact and approximate solutions.

## Contribution

It introduces a new horizon-based initial/final value formalism for spherically symmetric spacetimes with matter, including analytical and numerical methods.

## Key findings

- Exact solution for inward-falling null matter case.
- Analytical investigation of slowly evolving horizons with scalar fields.
- Proposed numerical approach for general scalar field interactions.

## Abstract

We initiate the development of a horizon-based initial (or rather final) value formalism to describe the geometry and physics of the near-horizon spacetime: data specified on the horizon and a future ingoing null boundary determine the near-horizon geometry. In this initial paper we restrict our attention to spherically symmetric spacetimes made dynamic by matter fields. We illustrate the formalism by considering a black hole interacting with a) inward-falling, null matter (with no outward flux) and b) a massless scalar field. The inward-falling case can be exactly solved from horizon data. For the more involved case of the scalar field we analytically investigate the near slowly evolving horizon regime and propose a numerical integration for the general case.