Shells In General Relativity Research Articles

Dynamics of Rotating Cylindrical Shells in General Relativity

Cylindrical spacetimes with rotation are studied using the Newmann-Penrose formulas. By studying null geodesic deviations the physical meaning of each component of the Riemann tensor is given. These spacetimes are further extended to include rotating dynamic shells, and the general expression of the surface energy-momentum tensor of the shells is given in terms of the discontinuation of the first derivatives of the metric coefficients. As an application of the developed formulas, a stationary shell that generates the Lewis solutions, which represent the most general vacuum cylindrical solutions of the Einstein field equations with rotation, is studied by assuming that the spacetime inside the shell is flat. It is shown that the shell can satisfy all the energy conditions by properly choosing the parameters appearing in the model, provided that $ 0 \le \sigma \le 1$, where $\sigma$ is related to the mass per unit length of the shell.

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

Thin shells in general relativity can be used both as models of collapsing objects and as probes in the space-time outside compact sources. Therefore they provide a useful tool for the analysis of the final fate of collapsing matter and of the effects induced in the matter by strong gravitational fields. We describe the radiating shell as a (second quantized) many-body system with one collective degree of freedom, the (average) radius, by means of an effective action which also entails a thermodynamic description. Then we study some of the quantum effects that occur in the matter when the shell evolves from an (essentially classical) large initial radius towards the singularity and compute the corresponding backreaction on its trajectory.

Effective action and thermodynamics of radiating shells in general relativity

An effective action is obtained for the area and mass aspect of a thin shell of radiating self-gravitating matter. On following a mini-superspace approach, the geometry of the embedding space-time is not dynamical but fixed to be either Minkowski or Schwarzschild inside the shell and Vaidya in the external space filled with radiation. The Euler-Lagrange equations of motion are discussed and shown to entail the expected invariance of the effective Lagrangian under time reparametrization. They are equivalent to the usual junction equations and suggest a macroscopic quasi-static thermodynamic description.

On the Motion of Shells in General Relativity

We use Lanczos equations to analyze the motionof shells in the gravitational field of a sphericallysymmetric central body. A first integral of Israel'smatching conditions is used to show that the motion of the shell depends on the equation of stateof the shell matter. In particular, the case of abarotropic equation of state is analyzed. The questionof how the type of motion determines the equation of state for the matter of the shell is alsoinvestigated.

Nonradial linear oscillations of shells in general relativity

Section V discusses quasinormal modes. It is shown how the transition conditions at the shell which are derived in Sec. III can be used to calculate numerically the quasinormal mode frequencies of the shell. Section VI deals with the Newtonian limit. The ordinary differential equations describing linearized self-gravitating Newtonian shells are derived as limits from the relativistic equations. This limit process is rather subtle, because the Gaussian gauge, which is natural in the relativistic picture, is not so natural for the usual formulation of Newton’s theory in an inertial frame. Nevertheless, the correct Newtonian equations can be found as a limit from Einstein’s equations and the transition conditions at the shell.

Monopole and electrically charged dust thin shells in general relativity: Classical and quantum comparison of hollow and atomlike configurations

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.

Junctions and thin shells in general relativity using computer algebra: II. The null formalism

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.

Some physical consequences of abrupt changes in the multipole moments of a gravitating body

The Barrab\`es-Israel theory of light-like shells in General Relativity is used to show explicitly that in general a light-like shell is accompanied by an impulsive gravitational wave. The gravitational wave is identified by its Petrov Type N contribution to a Dirac delta-function term in the Weyl conformal curvature tensor (with the delta-function singular on the null hypersurface history of the wave and shell). An example is described in which an asymptotically flat static vacuum Weyl space-time experiences a sudden change across a null hypersurface in the multipole moments of its isolated axially symmetric source. A light-like shell and an impulsive gravitational wave are identified, both having the null hypersurface as history. The stress-energy in the shell is dominated (at large distance from the source) by the jump in the monopole moment (the mass) of the source with the jump in the quadrupole moment mainly responsible for the stress being anisotropic. The gravitational wave owes its existence principally to the jump in the quadrupole moment of the source confirming what would be expected.

The equivalence of Darmois-Israel and distributional method for 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.

Junctions and thin shells in general relativity using computer algebra: I. The Darmois - Israel formalism

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.

Erratum: "Self-energy of a thin charged shell in general relativity"

Erratum: "Self-energy of a thin charged shell in general relativity"

Thin shells in general relativity and cosmology: The lightlike limit.

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.

Self-energy of a thin charged shell in general relativity.

The self-energy of a thin charged shell is calculated within general relativity with the help of the Gauss-Codazzi and Lanczos equations. For the special case of a dust shell, our result agrees with that of Arnowitt, Deser, and Misner, but we find a different value for the lower bound on the radius of the shell and thus also for the total mass.

Inflation and bubbles 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.

Evolution of Spherical Shells in General Relativity

We discuss the evolution of thin spherical shells within General Relativity. We start directly from the Einstein equations coupled to an arbitrary spherical matter distribution, and show that the resulting dynamics can be solved exactly in the limit when this distribution approaches an infinitesimally thin shell. This reproduces a previously well known result by W. Israel, but contrary to his method our procedure can be generalized to systematically take into account finite thickness corrections.

Thin cylindrical shells in general relativity

The sources of static cylindrically symmetric fields in the form of thin cylindrical shells are studied. The results are compared (in the case of the thin shell consisting of orbiting particles with the result of Raychaudhuri and Som). A special solution is also found, represented by a collapsing shell which is the source of the static field.

A comment of thin shells in general relativity

The special type of space-time metrix is related to a plane source.

A comment on thin shells in general relativity

The special type of space-time metrix is related to a plane matter source.

Charged shells in general relativity and their gravitational collapse

The boundary conditions for the gravitational and electromagnetic fields on charged shells are formulated. The shells are characterized in geometrical terms and the boundary conditions are manifestly covariant. The formalism is applied to the study of the gravitational collapse of homogeneous charged spherical shells. The law of conservation of energy, obtained by integrating the equations of motion of the shell and interpreted in full analogy with the special theory of relativity, is the starting point in analyzing the equilibrium states and the motion of the shells. It is concluded that no charge, however high it may be, can stop the gravitational collapse of the shell below the upper Nordstrom radius.

Singular hypersurfaces and thin shells in general relativity

Singular hypersurfaces and thin shells in general relativity