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.
There are many possible generalizations of the original conception of a gravitational instanton: for example one can allow gravitational instantons to have a nonzero cosmological constant or a Riemann tensor which is not self-dual. One can also relax the boundary condition that the metric is asymptotically Euclidean.
There are many methods for constructing gravitational instantons, including the Gibbons–Hawking Ansatz, twistor theory, and the hyperkähler quotient construction.