We investigate the variational principle for the gravitational field in the presence of thin shells of completely unconstrained signature (generic shells). Such variational formulations have been given before for shells of timelike and null signatures separately, but so far no unified treatment exists. We identify the shell equation as the natural boundary condition associated with a broken extremal problem along a hypersurface where the metric tensor is allowed to be nondifferentiable. Since the second order nature of the Einstein-Hilbert action makes the boundary value problem associated with the variational formulation ill-defined, regularization schemes need to be introduced. We investigate several such regularization schemes and prove their equivalence. We show that the unified shell equations derived from this variational procedure reproduce past results obtained via distribution theory by Barrabes and Israel for hypersurfaces of fixed causal type and by Mars and Senovilla for generic shells. These results are expected to provide a useful guide to formulating thin shell equations and junction conditions along generic hypersurfaces in modified theories of gravity.
Recent studies relating the approximations for the equations of state for thin shells and their consequent perturbative evolution are extended to thin-shell wormholes in theories beyond general relativity and more than four spacetime dimensions. The assumption of equations of state of the same form for static and slowly evolving shells appears as a strong restriction excluding the possibility of oscillatory evolutions. Then the new results considerably differ from previous ones obtained within the usual linearized approach.
A simple and direct, based on the equations of motion, derivation of the variational principle and effective actions for a spherical charged dust shell in general relativity is offered. This principle is based on the relativistic version of the DÆAlembert principle of virtual displacements and leads to the effective actions for the shell, which describe the shell from the point of view of the exterior or interior stationary observers. Herewith, sides of the shell are considered independently, in the coordinates of the interior or exterior region of the shell. Canonical variables for a charged …
A recently proposed variational approach for general relativity where, in addition to the metric tensor, two independent affine connections enter the action as dynamical variables, is revised. Field equations always reduce to the Einstein field equations for any dependence of the matter action upon an independent connection.
We study spherically symmetric timelike thin-shells in $3+1-$dimensional bulk spacetime with a variable equation of state for the fluid presented on the shell. In such a fluid the angular pressure $p$ is a function of both surface energy density $σ$ and the radius $R$ of the thin-shell. Explicit cases of the thin shells connecting two non-identical cloud of strings spacetimes and a flat Minkowski spacetime to the Schwarzschild metric are investigated.
A consistent variational derivation of the Majorana 4-spinor field equations coupled to Einstein's theory of gravitation is given. The equivalence of the first and the second order variational field equations is explicitly demonstrated. The Lagrange multiplier 2-forms we use turn out to be precisely the Belinfante-Rosenfeld 2-forms that are needed to symmetrize the canonical energy-momentum tensor of the Majorana spinor.
In this paper we consider singular timelike spherical hypersurfaces embedded in a $D$-dimensional spherically symmetric bulk spacetime which is an electrovacuum solution of Einstein equations with cosmological constant. We analyse the different possibilities regarding the orientation of the gradient of the standard $r$ coordinate in relation to the shell. Then we study the dynamics according to Einstein equations for arbitrary matter satisfying the dominant energy condition. In particular, we thoroughly analyse the asymptotic dynamics for both the small and large-shell-radius limits. We also s…
It is shown that exact spherically symmetric solutions to Einstein's Field Equations exist such that, over an open region of the spacetime, they are singularity free, satisfy the dominant energy condition, represent elastic matter with a well defined constitutive function, and are such that elastic perturbations propagate causally. Two toy-models are then built up in which a thick elastic, spherically symmetric shell with the above properties, separates two Robertson-Walker regions corresponding to different values of the curvature $k$. The junction conditions (continuity of the first and seco…
The problem of formulating synchronous variational principles in the context of General Relativity is discussed. Based on the analogy with classical relativistic particle dynamics, the existence of variational principles is pointed out in relativistic classical field theory which are either asynchronous or synchronous. The historical Einstein-Hilbert and Palatini variational formulations are found to belong to the first category. Nevertheless, it is shown that an alternative route exists which permits one to cast these principles in terms of equivalent synchronous Lagrangian variational formul…
By a spherical gravitating condenser we mean two concentric charged shells made of perfect fluids restricted by the condition that the electric field is nonvanishing only between the shells. Flat space is assumed inside the inner shell. By using Israel's formalism we first analyze the general system of N shells and then concentrate on the two-shell condensers. Energy conditions are taken into account; physically interesting cases are summarized in two tables, but also more exotic situations in which, for example, the inner shell may occur below the inner horizon of the corresponding Reissner-N…
In this note several formulae that follow from W. Israel's thin shell equation of motion are derived. We review the main results of Israel's seminal papers. The Israel equation of motion is solved for the three regimes of total energy E<0, E=0, and E>0. This family of solutions is analogous to the three classes of solutions of the Lemaitre-Tolman-Bondi equations. We also derive the constraint that the gravitational mass is always positive. The main result of this note is the expression for the interior and exterior metrics in terms of proper time on the shell. In these coordinates the time dil…
In this work, we present the Einstein field equations (EFE) in the framework of a modified matter source and thus find out the solutions for the Wormhole. To obtain characteristic solutions set, we employ two kinds of equations of states (EOS), i.e., the van der Waals and polytropic EOS to the EFE. We adopt embedding class as a general technique for the system and discuss several physical attributes of the stellar system under embedded wormhole solutions. Detailed discussion about the matter contents of the thin-shell developed around the obtained wormhole solutions in the background of Schwar…
The spherically symmetric thin shells are the nearest generalizations of the point-like particles. Moreover, they serve as the simple sources of the gravitational fields both in General Relativity and much more complex quadratic gravity theories. We are interested in the special and physically important case when all the quadratic in curvature tensor (Riemann tensor) and its contractions (Ricci tensor and scalar curvature) terms are present in the form of the square of Weyl tensor. By definition, the energy-momentum tensor of the thin shell is proportional to Dirac delta-function. We construct…
We construct thin-shell electrically charged wormholes in d-dimensional general relativity with a cosmological constant. The wormholes constructed can have different throat geometries, namely, spherical, planar and hyperbolic. Unlike the spherical geometry, the planar and hyperbolic geometries allow for different topologies and in addition can be interpreted as higher-dimensional domain walls or branes connecting two universes. In the construction we use the cut-and-paste procedure by joining together two identical vacuum spacetime solutions. Properties such as the null energy condition and ge…
We derive the expressions for the local, on-shell closed co-dimension 2 forms in the Palatini formulation of general relativity and explicitly show their on-shell equivalence to those of the metric formulation. When compared to other first order formulations, two subtleties have to be addressed during the construction: off-shell non-metricity and the fact that the transformation of the connection under infinitesimal diffeomorphisms involves second order derivatives of the associated vector fields.
We consider the construction of generic spherically symmetric thin-shell traversable wormhole spacetimes in standard general relativity. By using the cut-and-paste procedure, we comprehensively analyze the stability of arbitrary spherically symmetric thin-shell wormholes to linearized spherically symmetric perturbations around static solutions. While a number of special cases have previously been dealt with in scattered parts of the literature, herein we take considerable effort to make the analysis as general and unified as practicable. We demonstrate in full generality that stability of the …
A new variational principle for General Relativity, based on an action functional $I\/(Φ,\nabla)\/$ involving both the metric $Φ\/$ and the connection $\nabla\/$ as independent, \emph{unconstrained\/} degrees of freedom is presented. The extremals of $I\/$ are seen to be pairs $\/(Φ,\nabla)\/$ in which $Φ\/$ is a Ricci flat metric, and $\nabla\/$ is the associated Riemannian connection. An application to Kaluza's theory of interacting gravitational and electromagnetic fields is discussed.
Einstein-Gauss-Bonnet gravity is a generalization of the general relativity to higher dimensions in which the first and second-order terms correspond to general relativity and Einstein-Gauss-Bonnet gravity respectively. We construct a new class of five-dimensional (5D) thin-shell wormholes by the `Cut-Paste' technique from black holes in Einstein-Gauss-Bonnet gravity inspired by non-commutative geometry starting with a static spherically symmetric, Gaussian mass distribution as a source and for this structural form of the thin shell wormhole we have explored several salient features of the sol…
Using thin shell formalism we construct two solutions of intra-universe wormholes. The first model is a cosmological analog of the Aichelburg-Schein timehole, while another one is an intra-universe form of the Bronnikov-Ellis solution.
We report off-shell Noether currents obtained from off-shell Noether potentials for first-order general relativity described by $n$-dimensional Palatini and Holst Lagrangians including the cosmological constant. These off-shell currents and potentials are achieved by using the corresponding Lagrangian and the off-shell Noether identities satisfied by diffeomorphisms generated by arbitrary vector fields, local $SO(n)$ or $SO(n-1,1)$ transformations, `improved diffeomorphisms', and the `generalization of local translations' of the orthonormal frame and the connection. A remarkable aspect of our …
We consider thin spherical shells of matter in both Newtonian gravity and general relativity, and examine their equilibrium configurations and dynamical stability. Thin-shell models are admittedly a poor substitute for realistic stellar models. But the simplicity of the equations that govern their dynamics, compared with the much more complicated mechanics of a self-gravitating fluid, allows us to deliver, in a very direct and easy manner, powerful insights regarding their equilibria and stability. We explore, in particular, the link between the existence of a maximum mass along a sequence of …
We study some properties of static spherically symmetric elastic bodies in general relativity using both analytical and numerical tools. The materials considered belong to the class of John elastic materials and reduce to perfect fluids when the rigidity parameter is set to zero. We find numerical support that such elastic bodies exist with different possible shapes (balls, single shells and multiple shells) and that their gravitational redshift can be very large ($z\approx 2.8$) without violating the dominant energy condition. Moreover we show that the elastic body has finite radius even in t…
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-Nordström 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 radiu…
The bimetric variational principle is a subtle reinterpretation of general relativity that assumes the spacetime connection to be generated by an independent metric. Unlike the so called Palatini formalism that promotes the connection into a fundamental field, the new variational principle results in a physically distinct theory since the potential for the connection carries new degrees of freedom. The connection-generating metric naturally allows also an antisymmetric component. This sets torsion propagating! It is also shown here that while in the most straightforward generalization of the E…
Interaction / collision of two concentric spherical thin-shells of linear fluid resulting in collapse has been considered recently. We show that addition of finely tuned electric charges on the shells apart from the cosmological constant serves to delay the collapse indefinitely, yielding an ever colliding system of two concentric fluid shells. Given the finely tuned charges this provides an example of a perpetual two-body motion in general relativity.
We study a black hole of mass $M$, enclosed within a spherical box, in equilibrium with its Hawking radiation. We show that the spacetime geometry inside the box is described by the Oppenheimer-Volkoff equations for radiation, except for a {\em thin shell} around the horizon. We use the maximum entropy principle to show that the invariant width of the shell is of order $\sqrt{M}$, its entropy is of order $M$ and its temperature of order $1/\sqrt{M}$ (in Planck units). Thus, the width of the shell is {\em much larger} than the Planck length. Our approach is to insist on thermodynamic consistenc…
Based on the recently introduced black-bounce spacetimes, we shall consider the construction of the related spherically symmetric thin-shell traversable wormholes within the context of standard general relativity. All of the really unusual physics is encoded in one simple parameter $a$ which characterizes the scale of the bounce. Keeping the discussion as close as possible to standard general relativity is the theorist's version of only adjusting one feature of the model at a time. We shall modify the standard thin-shell traversable wormhole construction, each bulk region now being a black-bou…
I present the first analytical study of gravitational collapse in a compact CMC foliation with $S^3$ spatial topology. The solutions I find, in this context, will be both solutions of Shape Dynamics and General Relativity. The aim is to describe a system undergoing gravitational collapse in Shape Dynamics, so a well-justified and useful simplification is to assume spherical symmetry. This kills all the local gravitational degrees of freedom, but some nontrivial degrees of freedom are recovered by introducing matter. The simplest form of matter is infinitely thin spherical shells of dust, of wh…
A mathematical complication due to an unnecessary formal assumption concerning the variational principle of general relativity theory, which apparently bothered Einstein and Hilbert, is shown and cleared up. Some historical confusion seems caused by the impossibility to use the conventional Euler-Lagrange formalism directly there, which even otherwise is nothing but one of various possible procedures to apply the superior principle of least action. Correspondingly to the absence of any direct calculation in the literature so far, only a numerical modification in parts - explicitly taken into a…
We investigate a new thin-shell wormhole constructed by surgically grafting two regular charged black holes arising from the action using nonlinear electrodynamics coupled to general relativity. The stress-energy components within the shell violate the null and weak energy conditions but obey the strong energy condition. We study the stability in two ways: (i) taking a specific equation of state at the throat and (ii) analyzing the stability to linearized spherically symmetric perturbations about a static equilibrium solution. Various other aspects of this thin-shell wormhole are also analyzed…
We discuss a relativistic model for heat conduction, building on a convective variational approach to multi-fluid systems where the entropy is treated as a distinct dynamical entity. We demonstrate how this approach leads to a relativistic version of the Cattaneo equation, encoding the finite thermal relaxation time that is required to satisfy causality. We also show that the model naturally includes the non-equilibrium Gibbs relation that is a key ingredient in most approaches to extended thermodynamics. Focussing on the pure heat conduction problem, we compare the variational results to the …
We derive a general set of acceptable junction conditions for $F(T)$ gravity via the variational principle. The analysis is valid for both the traditional form of $F(T)$ gravity theory as well as the more recently introduced Lorentz covariant theory of Krššák and Saridakis. We find that the general junction conditions derived, when applied to simple cases such as highly symmetric static or time dependent geometries (such as spherical symmetry) imply both the Synge junction conditions as well as the Israel-Sen-Lanczos-Darmois junction conditions of General Relativity. In more complicated scenar…
