Gibbons–Hawking–York boundary term

In general relativity, the Gibbons–Hawking–York boundary term is a term that needs to be added to the Einstein–Hilbert action when the underlying spacetime manifold has a boundary.

The Einstein–Hilbert action is the basis for the most elementary variational principle from which the field equations of general relativity can be defined. However, the use of the Einstein–Hilbert action is appropriate only when the underlying spacetime manifold ${\displaystyle {\mathcal {M}}}$ is closed, i.e., a manifold which is both compact and without boundary. In the event that the manifold has a boundary ${\displaystyle \partial {\mathcal {M}}}$, the action should be supplemented by a boundary term so that the variational principle is well-defined.

The necessity of such a boundary term was first realised by York and later refined in a minor way by Gibbons and Hawking.

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