Gravitational instantons are solutions of the four-dimensional Einstein or Einstein-Maxwell equations in Riemannian (Euclidean) signature that give complete metrics on non-compact four-manifolds with finite integrated squared Riemann curvature and that asymptotically approach flat space in a controlled way (ALE, ALF, and related asymptotics). Introduced in Hawking's Euclidean quantum gravity programme, they include analytic continuations of Lorentzian black holes -- Euclidean Schwarzschild and Kerr, obtained by making imaginary time periodic with period fixed by the surface gravity -- as well as examples with no Lorentzian analogue whose curvature is anti-self-dual, such as Eguchi-Hanson and anti-self-dual Taub-NUT. Their study links general relativity, Riemannian and hyperkähler geometry, self-duality, twistor theory, and the topology and classification of four-manifolds.
A review (based on the author's 2024 Cracow School of Theoretical Physics lectures) of gravitational instantons: solutions of the four-dimensional Riemannian (Euclidean) Einstein or Einstein-Maxwell …