Gibbons–Hawking–York boundary term

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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 \mathcal{M} is closed, i.e., a manifold which is both compact and without boundary. In the event that the manifold has a boundary \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.

For a manifold that is not closed, the appropriate action is

\mathcal{S}_\mathrm{EH} + \mathcal{S}_\mathrm{GHY} = \frac{1}{16 \pi} \int_\mathcal{M} \mathrm{d}^4 x \, \sqrt{g} R + \frac{1}{8 \pi} \int_{\partial \mathcal{M}} \mathrm{d}^3 x \, \sqrt{h}K,

where \mathcal{S}_\mathrm{EH} is the Einstein–Hilbert action, \mathcal{S}_\mathrm{GHY} is the Gibbons–Hawking–York boundary term, h_{\alpha\beta} is the induced metric on the boundary and K is the trace of the second fundamental form. Varying the action with respect to the metric g_{\alpha\beta} gives the Einstein equations; the addition of the boundary term means that in performing the variation, the geometry of the boundary encoded in the induced metric h_{\alpha\beta} is fixed. There remains ambiguity in the action up to an arbitrary functional of the induced metric h_{\alpha\beta}.

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