Gibbons–Hawking–York boundary term

Gibbons–Hawking–York boundary term
Name	Gibbons–Hawking–York boundary term
Field	Theoretical physics
Discovered	1977
Discoverer	G. W. Gibbons; Stephen Hawking; James W. York

Gibbons–Hawking–York boundary term The Gibbons–Hawking–York boundary term is a surface contribution added to the Einstein–Hilbert action to produce a well-posed variational principle for spacetime manifolds with boundary. It was introduced to resolve boundary-value issues in classical general relativity and plays a central role in semiclassical approaches to Hawking radiation, Hawking's path integral formulation, and canonical treatments associated with Arnowitt–Deser–Misner decomposition. The term influences computations in Black hole thermodynamics, ADM mass evaluations, and modern treatments of holography such as the AdS/CFT correspondence.

Introduction

The boundary term was formulated to supplement the Einstein–Hilbert action used in Albert Einstein's theory of gravitation so that variations holding the induced metric fixed on a timelike or spacelike boundary yield Einstein's field equations without additional boundary contributions. Its introduction is intimately connected to work by G. W. Gibbons, Stephen Hawking, and James W. York Jr. and complements foundational formalisms like the Hamiltonian formulation of general relativity and the ADM formalism. The need for the boundary term arises from surface terms produced by second derivatives of the metric in the bulk action, a problem encountered in analyses by researchers using techniques related to Noether's theorem, Variational principles, and boundary conditions in classical field theory.

Mathematical definition

Mathematically, the boundary term adds an integral over the hypersurface ∂M of the trace K of the extrinsic curvature associated with the embedding of ∂M into the bulk manifold M. In formulae, the full action includes the Einstein–Hilbert action plus 2 times the integral of K times the induced metric determinant on ∂M; this is constructed using the unit normal vector field n^a and the induced metric h_{ab} on the boundary. The extrinsic curvature K_{ab} is defined by projecting the Levi-Civita connection along tangential directions and taking the Lie derivative of h_{ab} along n^a, concepts that are standard in treatments by authors working on the Gauss–Codazzi equations, Riemann curvature tensor, and junction conditions like those of Israel.

Role in variational principle and action principle

Including the boundary term is required to ensure a well-posed variational principle when the induced metric on ∂M is held fixed, paralleling approaches used in classical mechanics and field theory by practitioners influenced by Lagrangian mechanics and the Hamiltonian methods pioneered by figures such as Pierre-Simon Laplace in conceptual lineage. Without the term, variations produce normal-derivative contributions that spoil stationarity of the action under metric variations that keep only the induced metric fixed. The boundary term thus ensures that solutions extremize the total action, enabling linkage to conserved quantities through constructions akin to ADM energy, Komar mass, and surface integrals used in studies by researchers of asymptotically flat spacetimes and asymptotically Anti-de Sitter spacetimes.

Applications in general relativity and quantum gravity

The boundary term is pivotal in deriving black hole entropy formulas in semiclassical gravity frameworks developed by Stephen Hawking, Bekenstein, and later workers in black hole thermodynamics. It appears in path-integral computations used in Euclidean quantum gravity approaches advanced by G. W. Gibbons and Stephen Hawking, and underlies derivations of partition functions for gravitational ensembles studied by researchers in statistical mechanics of gravitating systems. The term is also essential in holographic renormalization used in the AdS/CFT correspondence introduced by Maldacena and applied in research by groups at Princeton University, Institute for Advanced Study, and elsewhere to define finite boundary stress tensors and conserved charges.

Derivation and examples

Derivations typically start from the variation of the Einstein–Hilbert action and isolate boundary contributions arising from integrating by parts terms involving second derivatives of the metric. The added boundary integral cancels those undesirable terms when the induced metric is fixed. Classic examples include computations for the Schwarzschild solution studied by Schwarzschild, Kerr solution derived by Kerr, and cosmological models related to Friedmann solutions, where the Gibbons–Hawking–York term contributes to proper evaluation of action differences between competing saddle points. In Euclidean approaches to black hole thermodynamics as used by G. W. Gibbons and Stephen Hawking, the term is used to compute regularized on-shell actions that yield free energies and entropies.

Extensions and generalizations

Generalizations include boundary terms adapted to higher-derivative theories such as Gauss–Bonnet gravity studied by researchers following work by Lovelock, and modifications appropriate for non-Riemannian geometries considered by investigators of Teleparallel gravity and f(R) gravity frameworks explored by scholars at institutions like Cambridge University and Harvard University. For asymptotically noncompact boundaries, additional counterterms and background subtraction procedures—developed in the context of holographic renormalization—are introduced following methodologies by groups led by Skenderis and Balasubramanian. Junction conditions for thin shells and braneworld scenarios use variants related to the Israel junction conditions applied in models inspired by Randall–Sundrum model research.

Computation and regularization techniques

Practical computations employ background subtraction, counterterm subtraction, and holographic renormalization to render on-shell actions finite in asymptotically flat or asymptotically Anti-de Sitter settings, techniques refined by researchers in the AdS/CFT correspondence community and quantum gravity groups at Stanford University and University of California, Berkeley. Numerical relativity applications compute boundary contributions when imposing outer boundary conditions in evolutions pioneered by teams at Caltech and Max Planck Institute. Regularization often parallels methods in quantum field theory used by workers referencing Dimensional regularization, Pauli–Villars regularization, and heat kernel techniques employed in semiclassical gravity computations associated with Stephen Hawking and others.

Category:General relativity