  We provide strong numerical evidence for a new no-scalar-hair theorem for
black holes in general relativity, which rules out spherical scalar hair of
static four dimensional black holes if the scalar field theory, when coupled to
gravity, satisfies the Positive Energy Theorem. This sheds light on the
no-scalar-hair conjecture for Calabi-Yau compactifications of string theory,
where the effective potential typically has negative regions but where
supersymmetry ensures the total energy is always positive. In theories where
the scalar tends to a negative local maximum of the potential at infinity, we
find the no-scalar-hair theorem holds provided the asymptotic conditions are
invariant under the full anti-de Sitter symmetry group.
