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A gravitational instanton is a complete non-singular positive definite solution of the vacuum Einstein Equations . They have self-dual Riemann Tensor , which is equivalent to being Kähler -Einstein, and are analogous to Self-dual Yang-Mills Instantons .

Gravitational instantons fall into a few classes, such as asymptotically locally Euclidean spaces (ALE spaces), '''asymptotically locally flat spaces''' (ALF spaces). There also exist ALG spaces whose name is chosen by induction.


EXAMPLES


It will be convenient to write the instanton solutions below using right invariant (or left) one-forms on a Three-sphere S3. These can be defined in terms of Euler Angles by

:
\sigma_1^L = \sin \phi \, d heta - \cos \phi \sin heta \, d \psi

:
\sigma_2^L = \cos \phi d heta + \sin \phi \sin heta \, d \psi

:
\sigma_3^L = d \phi + \cos heta d \psi



Euclidean Taub-NUT



ds^2 = rac{1}{4} rac{r+n}{r-n} dr^2 + rac{r-n}{r+n} n^2 (\sigma_3^L)^2 + rac{1}{4}(r^2 - n^2) + (\sigma_2^L)^2



Eguchi-Hanson



ds^2 = \left( 1 - rac{a}{r^4} ight) ^{-1} dr^2 + rac{r^2}{4} \left( 1 - rac{a}{r^4} ight) (\sigma_3^L)^2 + rac{r^2}{4} + (\sigma_2^L)^2 .



Gibbons-Hawking multi-centre



ds^2 = rac{1}{V(\mathbf{x})} ( d au + \mathbf{\omega} \cdot d \mathbf{x})^2 + V(\mathbf{x}) d \mathbf{x} \cdot d \mathbf{x},


where



abla V = \pm