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The Gibbons-Hawking-York (GHY) boundary term is given by $$S_{GH}=\frac{1}{8 \pi G}\int_{\partial M}\sqrt{|\gamma|}K,$$ where $\gamma_{ij}$ is the boundary induced metric, and $K$ is the trace of the extrinsic curvature.

Now, in some text, $K$ is given by $K=\gamma^{ij}\nabla_{i}n_{j}$, where $n_i$ is the outward-point unit normal vector. However, I'm not sure how to use the formula explicitly, i.e. given a gravitational theory with known metric and boundary, how to compute the extrinsic curvature and hence the GHY boundary term by using this formula. Can anyone help me with that? Specific examples will be especially appreciated.

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    $\begingroup$ Once you have the normal to the surface and the metric on the whole spacetime, there's a known formula for the induced metric on the hypersurface. Then, once you have that, you can compute $K$ explicitly and now you have everything you need. Then doing the integral is the usual matter of doing integrals in curved spacetime. Which part among these is the one which is causing problems? $\endgroup$ – John Donne Oct 17 at 17:17

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