Gibbons–Hawking–York boundary term
||This article may be too technical for most readers to understand. (December 2012)|
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 is closed, i.e., a manifold which is both compact and without boundary. In the event that the manifold has a boundary , the action should be supplemented by a boundary term so that the variational principle is well-defined.
For a manifold that isn't closed, the appropriate action is
where is the Einstein–Hilbert action, is the Gibbons–Hawking–York boundary term, is the induced metric on the boundary and is the trace of the second fundamental form. Varying the action with respect to the metric 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 is fixed. There remains ambiguity in the action up to an arbitrary functional of the induced metric .
- J. W. York (1972). "Role of conformal three-geometry in the dynamics of gravitation". Physical Review Letters 28 (16): 1082. Bibcode:1972PhRvL..28.1082Y. doi:10.1103/PhysRevLett.28.1082.
- G. W. Gibbons and S. W. Hawking (1977). "Action integrals and partition functions in quantum gravity". Physical Review D 15 (10): 2752. Bibcode:1977PhRvD..15.2752G. doi:10.1103/PhysRevD.15.2752.