Singular hypersurfaces and thin shells in general relativity

We present extensions to the <TT> GRJunction</TT> computer algebra program, which allow the study of null boundary surfaces and thin shells in general relativity. We summarize the null formalism due to Barrabès and Israel and highlight those steps which differ from the timelike/spacelike cases. <TT> GRJunction</TT> has been used to verify a number of results from the literature. We then present two new results calculated with the aid of <TT> GRJunction</TT>. These are the junction of two Kerr - Newman solutions at a non-horizon straddling null shell in the slow rotation limit and the exact junction of two Kerr - Newman solutions at a horizon straddling shell.

We classify all fundamental electrically charged thin shells in general relativity, i.e., static spherically symmetric perfect fluid thin shells with a Minkowski spacetime interior and a Reissner-Nordstr\"om spacetime exterior, characterized by the spacetime mass and electric charge. The fundamental shell can exist in three states, nonextremal, extremal, and overcharged. The nonextremal state allows the shell to be located such that its radius can be outside its own gravitational radius, or can be inside its own Cauchy radius. The extremal state allows the shell to be located such that its radius can be outside its own gravitational radius, or can be inside it. The overcharged state allows the shell to be located anywhere. There is a further division, one has to specify the orientation of the shell, i.e., whether the normal out of the shell points toward increasing or decreasing radii. There is still a subdivision in the extremal state when the shell is at the gravitational radius, in that the shell can approach it from above or from below. The shell is assumed to be composed of an electrically charged perfect fluid, and the energy conditions are tested. Carter-Penrose diagrams are drawn for the shell spacetimes. There are fourteen cases in the classification of the fundamental shells, namely, nonextremal star shells, nonextremal tension shell black holes, nonextremal tension shell regular and nonregular black holes, nonextremal compact shell naked singularities, Majumdar-Papapetrou star shells, extremal tension shell singularities, extremal tension shell regular and nonregular black holes, Majumdar-Papapetrou compact shell naked singularities, Majumdar-Papapetrou shell quasiblack holes, extremal null shell quasinonblack holes, extremal null shell singularities, Majumdar-Papapetrou null shell singularities, overcharged star shells, and overcharged compact shell naked singularities.

Singular hypersurfaces and thin shells in general relativity

Following Israel's study of singular hypersurfaces and thin shells in general relativity, the complete set of Einstein's field equations in the presence of a bubble boundary SIGMA is reviewed for all spherically symmetric embedding four-geometries ${M}^{\ifmmode\pm\else\textpm\fi{}}$. The mapping that identifies points between the boundaries ${\ensuremath{\Sigma}}^{+}$ and ${\ensuremath{\Sigma}}^{\mathrm{\ensuremath{-}}}$ is obtained explicitly when the regions ${M}^{+}$ and ${M}^{\mathrm{\ensuremath{-}}}$ are described by a de Sitter and a Minkowski metric, respectively. In addition, the evolution of a bubble with vanishing surface energy density is studied in a spatially flat Robertson-Walker space-time, for region ${M}^{\mathrm{\ensuremath{-}}}$ radiation dominated with a vanishing cosmological constant, and an energy equation in ${M}^{+}$ determined by the matching. It is found that this type of bubble leads to a ``worm-hole'' matching; that is, an infinite extent exterior of a sphere is joined across the wall to another infinite extent exterior of a sphere. Interior-interior matches are also possible. Under this model, solutions for a bubble following a Hubble law are analyzed. Numerical solutions for bubbles with constant tension are also obtained.

We present the <TT> GRjunction</TT> computer algebra program which allows the study of non-null boundary surfaces and thin shells in general relativity. Implementing the Darmois - Israel thin-shell formalism requires a careful selection of definitions and algorithms to ensure that results are generated in a straightforward way. We have used the package to correctly reproduce a wide variety of examples from the literature. In this paper <TT> GRjunction</TT> is used to perform two new calculations: joining two Kerr solutions with differing masses and angular momenta along a thin shell in the slow rotation limit, and the calculation of the stress - energy of a Curzon wormhole. The Curzon wormhole has the interesting property that shells located at radius R < 2m have regions which satisfy the weak energy condition.

Thin shells in general relativity

Quantized shells as a tool for studying semiclassical effects in general relativity

This paper shows how the structure and dynamics of a thin shell traveling at the speed of light can be obtained from a simple and convenient prescription that is a straightforward extension and continuous limit of the familiar extrinsic-curvature algorithm for subluminal shells. It allows the space-time coordinates to be chosen freely and independently on the two sides of the shell. The prescription is applied to several examples of interest in general relativity and cosmology.

A comment on thin shells in general relativity

Thin shells in general relativity have been used in the past as keystones to obtain realistic models of cosmological and astrophysical situations. A crucial role for these developments was played by the compact description of their dynamics in terms of Israel’s junction conditions. Starting from this geometrical formulation we present a problem related to the WKB regime of shell dynamics and suggest a possible solution. General relativistic shells are an interesting system in general relativity and because of the simple geometrical description of their dynamics provided by Israel’s junction conditions [1] they became preferred models for many crucial aspects of astrophysical and cosmological situations (see [2] for a more complete bibliography on the subject). Many of these models have been developed under the assumption of spherical symmetry, but (as it happens for instance in the case of gravitational collapse [3]) this does not seem a severe restriction and it is likely that the obtained results can be extended to more general situations. On the other hand, the reduction in the number of degrees of freedom that it is possible to obtain in the spherically symmetric case makes simpler the development of effective models and more transparent the discussion of the interesting subtleties � To appear in the Proceedings of the 6th International Symposium on Frontiers in Funda

The Israel equations for thin shells in General Relativity are derived directly from the least action principle. The method is elaborated for obtaining the equations for double layers in quadratic gravity from the least action principle.

We give a comparative description of monopole and electrically charged spherically symmetric dust thin shells. Herewith we consider two of the most interesting configurations: the hollow shell and shell, surrounding a body with opposite charge. The classification of shells in accordance with the types of black holes and traversable wormholes is constructed. The theorems for the parameters of turning points are proved. Also for atomlike configurations the effects of screening (electrical case) and amplification (monopole case) of the internal mass by shell charge are studied. Finally, one considers the quantum aspects; herewith, exact solutions of wave equations and bound states spectra are found.

A comment of thin shells in general relativity

A distributional method to solve the Einstein’s field equations for thin shells is formulated. The familiar field equations and jump conditions of Darmois-Israel formalism are derived. A careful analysis of the Bianchi identities shows that, for cases under consideration, they make sense as distributions and lead to jump conditions of Darmois-Israel formalism.

The variational principle for a thin dust shell in General Relativity is constructed. The principle is compatible with the boundary-value problem of the corresponding Euler–Lagrange equations, and leads to “natural boundary conditions” on the shell. These conditions and the gravitational field equations which follow from an initial variational principle, are used for elimination of the gravitational degrees of freedom. The transformation of the variational formula for spherically-symmetric systems leads to two natural variants of the effective action. One of these variants describes the shell from a stationary interior observer’s point of view, another from the exterior one. The conditions of isometry of the exterior and interior faces of the shell lead to the momentum and Hamiltonian constraints. The canonical equivalence of the mentioned systems is shown in the extended phase space. Some particular cases are considered.