Gibbons-Hawking-York boundary term

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The Einstein-Hilbert action should be supplemented by a boundary term so that variation of the action is well-defined. Such a boundary term was first discovered by York and popularized later by Gibbons and Hawking. If the manifold [\mathcal] has boundary [\partial \mathcal], then the full action including the boundary term is

[I_ + I_ := \frac \int_} d^4 x \sqrt R + \frac \int_} d^3 x \sqrtK],

where [h_] 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_] 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_] is fixed.. There remains ambiguity in the action up to an arbitrary functional of the induced metric [h_].

References

[J. W. York, "Role of conformal three-geometry in the dynamics of gravitation", Phys. Rev. Lett. 28, 1082 (1972).]
[G. W. Gibbons and S. W. Hawking, "Action integrals and partition functions in quantum gravity", Phys. Rev. D 15, 2752 (1977).]

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