Gibbons–Hawking–York term

Gibbons–Hawking–York term
Name	Gibbons–Hawking–York term
Field	Theoretical physics
Introduced	1977
Associated people	Stephen Hawking; Gary Gibbons; James York

Gibbons–Hawking–York term is a boundary term added to the Einstein–Hilbert action to make the variational principle for general relativity well posed on manifolds with boundary. It appears in formulations of classical General relativity and in path integral approaches to Quantum gravity, and it plays a central role in deriving correct boundary conditions for gravitational dynamics in spacetimes such as those studied by W. Israel, Arnowitt–Deser–Misner and in analyses related to Black hole thermodynamics and the Euclidean path integral approach championed by Stephen Hawking and James Hartle.

Introduction

The term supplements the Einstein–Hilbert action when the spacetime manifold has a nonempty boundary, ensuring that the action yields Einstein's equations under variations that hold the induced metric fixed on the boundary. In practice it is indispensable for deriving conserved quantities in the ADM formalism, for defining quasilocal energy in a manner compatible with work by Brown and York, and for constructing semiclassical approximations used by Gibbons and Hawking in studies of black hole entropy and instantons.

Derivation and mathematical formulation

The term is defined using the extrinsic curvature of the boundary embedded in the bulk manifold. Given a Lorentzian or Euclidean manifold with boundary as in treatments by Arnowitt, Deser, Misner, the bulk action S_EH = (1/16πG)∫R√g d^4x is supplemented by a boundary integral over the trace of the extrinsic curvature K. The addition mirrors constructions used in differential geometry by Gauss and Bonnet in the context of curvature integrals and aligns with hypersurface geometry developed by Cartan and Nash. For spacetime foliations akin to those in the ADM formalism, the boundary integrand is typically written as (1/8πG)∫K√h d^3y where h is the induced metric on the boundary hypersurface as in analyses by York and later refinements by Gibbons and Hawking.

Role in variational principle and boundary conditions

When varying the metric in the bulk while holding the induced metric on the boundary fixed — a prescription used in the work of Dirichlet boundary conditions in classical field theory and in semiclassical gravity calculations by Hartle and Hawking — the bare Einstein–Hilbert action yields unwanted boundary terms containing normal derivatives of the metric. Inclusion of the boundary term cancels these contributions, making the variational problem well posed. This cancellation is analogous to adding surface terms in the action principles used by Noether and in canonical treatments by Dirac. It permits consistent imposition of boundary data in problems studied by Israel and in derivations of the first law of black hole mechanics pursued by Bardeen, Carter, and Hawking.

Applications in general relativity and quantum gravity

The boundary term is essential in calculating gravitational entropy in contexts explored by Bekenstein and Hawking, and in defining semiclassical partition functions for gravitational instantons examined by Gibbons and Hawking. It appears in computations of quasilocal energy and momentum in the framework developed by Brown and York, in conserved charges associated with asymptotic symmetries such as those of Arnowitt–Deser–Misner and Bondi, and in holographic renormalization techniques used in studies linking Anti-de Sitter space with Conformal field theory dualities advocated by Maldacena. In canonical quantization programs influenced by Wheeler and DeWitt, proper boundary terms are necessary for defining the Hamiltonian and for ensuring that generators of diffeomorphisms are functionally differentiable as in work by Regge and Teitelboim.

Extensions and generalizations

Generalizations include boundary terms appropriate for higher-derivative gravity theories such as Gauss–Bonnet gravity and Lovelock theories studied by Lovelock, where analogous surface contributions must be added to the action to obtain well-posed variational principles. In asymptotically Anti-de Sitter space contexts, additional counterterms inspired by Henningson and Skenderis are introduced to render the action finite and to define conserved quantities consistent with the AdS/CFT correspondence postulated by Maldacena. For timelike, spacelike, or null boundaries the structure of the boundary contribution and the required boundary data differ; treatments of null surfaces by Parattu and collaborators build on earlier analyses by Hayward and Booth.

Examples and computations

Canonical computations of black hole entropy via Euclidean methods by Gibbons and Hawking use the boundary term to obtain the correct semiclassical action for the Schwarzschild instanton studied originally in works on gravitational thermodynamics by Bekenstein and Hawking. In asymptotically flat spacetimes, evaluations of the ADM mass by Arnowitt, Deser, and Misner implicitly rely on boundary contributions equivalent to the term. For cosmological instantons considered by Hartle and Hawking the inclusion of the surface term ensures consistent saddle-point evaluations of the gravitational path integral and the resulting predictions for semiclassical tunneling amplitudes similar to those examined by Vilenkin and Coleman.

Historical context and attribution

The importance of boundary terms in gravitational actions traces through the development of differential geometry and variational calculus in the 19th and 20th centuries, including contributions by Gauss, Riemann, and Hilbert. The specific formulation widely used in contemporary general relativity and quantum gravity literature was articulated in papers by Gibbons and Hawking and was clarified in canonical language by York, with complementary perspectives provided by Arnowitt, Deser, and Misner. Subsequent elaborations and applications were advanced by researchers including Brown, York, Regge, Teitelboim, Maldacena, and many others.

Category:General relativity