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 closed manifold the appropriate action is

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

where $\mathcal{S}_{EH}$ is the Einstein-Hilbert action, $\mathcal{S}_{GHY}$ is the Gibbons-Hawking-York boundary term, $h αβ$ 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 $g αβ$ 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 $h αβ$ is fixed. There remains ambiguity in the action up to an arbitrary functional of the induced metric $h αβ$ .