# Generalized no-hair theorems without horizons

## TL;DR

This paper extends no-hair theorems beyond black holes to include ultracompact objects with deep gravitational wells, showing that deviations from spherical symmetry diminish as the gravitational redshift increases, thus constraining exotic compact objects.

## Contribution

It generalizes Israel's no-hair theorem to static spacetimes without horizons, establishing constraints on deviations from spherical symmetry based on gravitational redshift.

## Key findings

- Deviations from Schwarzschild decrease with increasing redshift.
- No-hair theorems apply to ultracompact stars and wormholes.
- Israel's theorem is recovered at infinite redshift.

## Abstract

The simplicity of black holes, as characterized by no-hair theorems, is one of the most important mathematical results in the framework of general relativity. Are these theorems unique to black hole spacetimes, or do they also constrain the geometry around regions of spacetime with arbitrarily large (although finite) redshift? This paper presents a systematic study of this question and illustrates that no-hair theorems are not restricted to spacetimes with event horizons but are instead characteristic of spacetimes with deep enough gravitational wells, extending Israel's theorem to static spacetimes without event horizons that contain small deviations from spherical symmetry. Instead of a uniqueness result, we obtain a theorem that constrains the allowed deviations from the Schwarzschild metric and guarantees that these deviations decrease with the maximum redshift of the gravitational well in the external vacuum region. Israel's theorem is recovered continuously in the limit of infinite redshift. This result provides a first extension of no-hair theorems to ultracompact stars, wormholes, and other exotic objects, and paves the way for the construction of similar results for stationary spacetimes describing rotating objects.