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Gravitational instanton
In
mathematical physics
and
differential geometry
, a
gravitational instanton
is a four-dimensional
complete
Riemannian manifold
satisfying the
vacuum
Einstein equations
. They are so named because they are analogues in
quantum theories of gravity
of
instantons
in Yang–Mills theory. In accordance with this analogy with self-dual Yang–Mills instantons, gravitational instantons are usually assumed to look like four dimensional
Euclidean space
at large distances, and to have a self-dual
Riemann tensor
. Mathematically, this means that they are asymptotically locally Euclidean (or perhaps asymptotically locally flat)
hyperkähler 4-manifolds
, and in this sense, they are special examples of
Einstein manifolds
. From a physical point of view, a gravitational instanton is a non-singular solution of the vacuum
Einstein equations
with
positive-definite
, as opposed to
Lorentzian
, metric.
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