Israel junction conditions

Israel junction conditions
Name	Israel junction conditions
Field	General relativity
Introduced by	Werner Israel
Year	1966
Related	Einstein field equations; Darmois junction conditions; thin shell formalism

Israel junction conditions are matching conditions in general relativity that relate discontinuities of the extrinsic curvature on a non-null hypersurface to the surface stress–energy tensor of a thin shell. They provide a rigorous prescription for gluing two distinct spacetime manifolds across a hypersurface, connecting solutions such as the Schwarzschild metric, Kerr metric, Reissner–Nordström metric, and cosmological spacetimes like Friedmann–Lemaître–Robertson–Walker metric.

Introduction

The Israel junction conditions were introduced by Werner Israel in 1966 to extend earlier results by Georges Darmois and to systematize the treatment of singular hypersurfaces in Einstein field equations. They serve in analyses ranging from classical models by Roy Kerr and Karl Schwarzschild to modern uses in brane cosmology and the ADM formalism. The formalism links intrinsic geometry of the hypersurface, influenced by Bernard Schutz and Stephen Hawking, to extrinsic curvature jumps governed by the Einstein tensor and conserved by identities related to Noether's theorem.

Mathematical formulation

The mathematical statement involves two oriented Lorentzian manifolds M+ and M− with metrics g+_{ab} and g−_{ab} joined across a hypersurface Σ whose induced metric h_{ij} is required to be continuous; this continuity echoes conditions used by Darmois and by works on gravitational collapse by Oppenheimer–Snyder. Denote the extrinsic curvature as K^{±}_{ij} constructed using unit normal n^a (choice analogous to conventions in the Arnowitt–Deser–Misner formalism). The first junction condition: h^{+}_{ij} = h^{-}_{ij} (continuity of induced metric) appears in studies by Misner Thorne Wheeler and John Wheeler. The second junction condition relates the jump [K_{ij}] = K^{+}_{ij} − K^{-}_{ij} to the surface stress–energy S_{ij} via S_{ij} = −(1/8π) ( [K_{ij}] − h_{ij} [K] ), a relation used in analyses by Israel and applied in papers by Tom Banks and Lisa Randall. The conservation equation D_j S^{ij} = −[T_{ab} n^a h^{b i}] follows from the contracted Bianchi identities and has parallels with flux conditions studied by Steven Weinberg.

Physical interpretation and applications

Physically, the conditions describe thin shells of matter or energy localized on Σ that generate discontinuities in extrinsic curvature; such shells appear in models of domain walls in field theory studied by Alex Vilenkin and Tom Kibble, in bubble nucleation scenarios analyzed by Sidney Coleman, and in astrophysical thin-shell wormholes following the work of Morris and Thorne. Applications include constructing Schwarzschild-de Sitter shells relevant to studies by Gibbons and Hawking, modeling the Israel–Wilson–Perjés solutions, and building junctions in cosmic string spacetimes investigated by A. Vilenkin and B. Gott. In brane world models influenced by Lisa Randall and Raman Sundrum the junction conditions take a central role in relating bulk dynamics to brane stress tensors, linking to concepts from Kaluza–Klein theory and AdS/CFT correspondence explored by Juan Maldacena.

Examples and special cases

Common examples include matching an interior Friedmann–Lemaître–Robertson–Walker metric (used in Oppenheimer–Snyder collapse) to an exterior Schwarzschild metric across a collapsing star surface; thin shell wormholes constructed from two copies of Reissner–Nordström or Schwarzschild spacetimes; and domain walls in de Sitter and anti-de Sitter backgrounds relevant to Coleman–De Luccia transitions. Special cases treat null hypersurfaces, where the null formulation by Barrabès and Israel modifies the extrinsic curvature concept, and vacuum shells where S_{ij}=0 reduces to the Darmois conditions. Another special scenario is cylindrical symmetry invoking solutions by Einstein–Rosen and Levi-Civita; rotating shells relate to matching with the Kerr metric as studied by Bardeen.

Derivations and proofs

Derivations proceed by integrating the Einstein field equations across Σ, employing Gaussian normal coordinates adapted to Σ as in the approach used by Arnowitt, Deser, and Misner. One constructs distributional curvature tensors, showing that the Riemann tensor contains delta-function singularities proportional to [K_{ij}], and then equates coefficients to obtain the surface stress–energy expression, following steps akin to those in texts by Wald and Hawking and Ellis. Alternative derivations use Hamiltonian techniques from the ADM formalism and variational principles with Gibbons–Hawking–York boundary terms introduced by York and Gibbons to handle surface contributions consistently. Rigorous coordinate-independent proofs employ tubular neighborhood theorems and matching lemmas developed in differential geometry by John M. Lee and in work by Yvonne Choquet-Bruhat.

Generalizations and extensions

Extensions include junction conditions in higher-curvature theories like Gauss–Bonnet gravity and Lovelock gravity where surface terms generalize Israel’s relation and have been studied by David Lovelock and Ted Jacobson. In string theory, junctions on D-branes and in M-theory compactifications involve modified matching conditions tied to localized sources such as Dp-branes and NS5-branes. Quantum-corrected semiclassical junctions incorporate expectation values of the stress–energy tensor and anomalies studied by Stephen Fulling and Paul Davies. Numerical relativity implementations adapt the junction formalism for interface conditions in simulations by groups around Pretorius and Lindblom, while recent work explores junctions in non-Riemannian geometry and teleparallel formulations examined by Aldrovandi and Pereira.

Category:General relativity