Gibbons–Hawking–York boundary term
Term in general relativity / From Wikipedia, the free encyclopedia
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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 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.
The necessity of such a boundary term was first realised by James W. York and later refined in a minor way by Gary Gibbons and Stephen Hawking.
For a manifold that is not closed, the appropriate action is
where is the Einstein–Hilbert action, is the Gibbons–Hawking–York boundary term, is the induced metric (see section below on definitions) on the boundary, its determinant, is the trace of the second fundamental form, is equal to where the normal to is spacelike and where the normal to is timelike, and are the coordinates on the boundary. Varying the action with respect to the metric , subject to the condition
gives the Einstein equations; the addition of the boundary term means that in performing the variation, the geometry of the boundary encoded in the transverse metric is fixed (see section below). There remains ambiguity in the action up to an arbitrary functional of the induced metric .
That a boundary term is needed in the gravitational case is because , the gravitational Lagrangian density, contains second derivatives of the metric tensor. This is a non-typical feature of field theories, which are usually formulated in terms of Lagrangians that involve first derivatives of fields to be varied over only.
The GHY term is desirable, as it possesses a number of other key features. When passing to the Hamiltonian formalism, it is necessary to include the GHY term in order to reproduce the correct Arnowitt–Deser–Misner energy (ADM energy). The term is required to ensure the path integral (a la Hawking) for quantum gravity has the correct composition properties. When calculating black hole entropy using the Euclidean semiclassical approach, the entire contribution comes from the GHY term. This term has had more recent applications in loop quantum gravity in calculating transition amplitudes and background-independent scattering amplitudes.
In order to determine a finite value for the action, one may have to subtract off a surface term for flat spacetime:
where is the extrinsic curvature of the boundary imbedded flat spacetime. As is invariant under variations of , this addition term does not affect the field equations; as such, this is referred to as the non-dynamical term.