  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.
