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{{Technical|date=December 2012}}
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 [[Einstein field equations|field equations of general relativity]] can be defined. However, the use of the Einstein–Hilbert action is appropriate only when the underlying spacetime manifold <math>\mathcal{M}</math> is [[Closed manifold|closed]], i.e., a manifold which is both [[compact manifold|compact]] and without boundary. In the event that the manifold has a boundary <math>\partial\mathcal{M}</math>, 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|York]] and later refined in a minor way by [[Gary Gibbons|Gibbons]] and [[Stephen Hawking|Hawking]].

For a manifold that isn't closed, the appropriate action is

:<math>\mathcal{S}_\mathrm{EH} + \mathcal{S}_\mathrm{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,</math>

where <math>\mathcal{S}_\mathrm{EH}</math> is the Einstein–Hilbert action, <math>\mathcal{S}_\mathrm{GHY}</math> is the Gibbons–Hawking–York boundary term, <math>h_{\alpha\beta}</math> is the [[induced metric]] on the boundary and <math>K</math> is the trace of the [[second fundamental form]]. Varying the action with respect to the metric <math>g_{\alpha\beta}</math> gives the [[Einstein field equations|Einstein equations]]; the addition of the boundary term means that in performing the variation, the geometry of the boundary encoded in the induced metric <math>h_{\alpha\beta}</math> is fixed. There remains ambiguity in the action up to an arbitrary functional of the induced metric <math>h_{\alpha\beta}</math>.

==References==
*{{cite journal |last=York |first=J. W. |authorlink=James W. York |year=1972|title=Role of conformal three-geometry in the dynamics of gravitation |journal=[[Physical Review Letters]] |volume=28 |issue=16 |doi=10.1103/PhysRevLett.28.1082 |bibcode=1972PhRvL..28.1082Y |page=1082}}
*{{cite journal |last1=Gibbons |first1=G. W. |authorlink1=Gary Gibbons |last2=Hawking |first2=S. W. |authorlink2=Stephen Hawking |title=Action integrals and partition functions in quantum gravity |year=1977 |journal=[[Physical Review D]] |volume=15 |issue=10 |doi=10.1103/PhysRevD.15.2752 |bibcode=1977PhRvD..15.2752G |page=2752}}

{{Stephen Hawking}}

{{DEFAULTSORT:Gibbons-Hawking-York boundary term}}
[[Category:Variational formalism of general relativity]]
[[Category:General relativity]]
[[Category:Lagrangian mechanics]]

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en>Sadads: removing orphan tags, removed orphan tag using AWB