The Schwarzschild-anti-de Sitter metric is a static spherically symmetric metric tensor on a Lorentzian manifold solving
the vacuum Einstein equation with negative cosmological constant. For cosmological
constant 
its four-dimensional line element
is
 |
where 
and 
is the unit round metric on 
. For 
, the unique positive zero 
of 
bounds the static exterior and satisfies
 |
The constant-time spatial metric approaches hyperbolic space at infinity. This mass-radius relation is the model expression in the hyperbolic Penrose inequality.
