Electrodynamics and spacetime geometry I: Foundations

We explore the intimate connection between spacetime geometry and electrodynamics. This link is already implicit in the constitutive relations between the field strengths and excitations, which are an essential part of the axiomatic structure of electromagnetism, clearly formulated via integration theory and differential forms. We briefly review the foundations of electromagnetism based on charge and magnetic flux conservation, the Lorentz force and the constitutive relations which introduce the spacetime metric. We then proceed with the tensor formulation by assuming local, linear, homogeneous and isotropic constitutive relations, and explore the physical, observable consequences of Maxwell's equations in curved spacetime. The field equations, charge conservation and the Lorentz force are explicitly expressed in general (pseudo) Riemanian manifolds. The generalized Gauss and Maxwell-Amp\`{e}re laws, as well as the wave equations, reveal potentially interesting astrophysical applications. In all cases new electromagnetic couplings and related phenomena are induced by spacetime curvature. The implications and possible applications for gravity waves detection are briefly addressed. At the foundational level, we discuss the possibility of generalizing the vacuum constitutive relations, by relaxing the fixed conditions of homogeneity and isotropy, and by assuming that the symmetry properties of the electro-vacuum follow the spacetime isometries. The implications of this extension are briefly discussed in the context of the intimate connection between electromagnetism and the geometry (and causal structure) of spacetime.Comment: 21 page

Electrodynamics and spacetime geometry: Astrophysical applications

After a brief review of the foundations of (pre-metric) electromagnetism, we explore some physical consequences of electrodynamics in curved spacetime. In general, new electromagnetic couplings and related phenomena are induced by the spacetime curvature. The applications of astrophysical interest considered here correspond essentially to the following geometries: the Schwarzschild spacetime and the spacetime around a rotating spherical mass in the weak field and slow rotation regime. In the latter, we use the Parameterised Post-Newtonian (PPN) formalism. We also explore the hypothesis that the electric and magnetic properties of vacuum reflect the spacetime isometries. Therefore, the permittivity and permeability tensors should not be considered homogeneous and isotropic a priori. For spherical geometries we consider the effect of relaxing the homogeneity assumption in the constitutive relations between the fields and excitations. This affects the generalized Gauss and Maxwell-Amp\`{e}re laws where the electric permittivity and magnetic permeability in vacuum depend on the radial coordinate in accordance with the local isometries of space. For the axially symmetric geometries we relax both the assumptions of homogeneity and isotropy. We explore simple solutions and discuss the physical implications related to different phenomena such as: the decay of electromagnetic fields in the presence of gravity, magnetic terms in Gauss law due to the gravitomagnetism of the spacetime around rotating objects, a frame-dragging effect on electric fields and the possibility of a spatial (radial) variability of the velocity of light in vacuum around spherical astrophysical objects for strong gravitational fields.Comment: 20 pages. Applications of the general formalism developed in arXiv:1602.01492. V2: typos corrected and references added. V3: 15 pages; revised version to appear in The European Physical Journal Plu

Closed timelike curves and causality violation

The conceptual definition and understanding of time, both quantitatively and qualitatively is of the utmost difficulty and importance. As time is incorporated into the proper structure of the fabric of spacetime, it is interesting to note that General Relativity is contaminated with non-trivial geometries which generate closed timelike curves. A closed timelike curve (CTC) allows time travel, in the sense that an observer that travels on a trajectory in spacetime along this curve, may return to an event before his departure. This fact apparently violates causality, therefore time travel and it's associated paradoxes have to be treated with great caution. The paradoxes fall into two broad groups, namely the consistency paradoxes and the causal loops. A great variety of solutions to the Einstein field equations containing CTCs exist and it seems that two particularly notorious features stand out. Solutions with a tipping over of the light cones due to a rotation about a cylindrically symmetric axis and solutions that violate the energy conditions. All these aspects are analyzed in this review paper.Comment: 19 pages, 6 figures. Invited chapter to appear in an edited collection 'Classical and Quantum Gravity: Theory, Analysis and Applications

Stability of phantom wormholes

It has recently been shown that traversable wormholes may be supported by phantom energy. In this work phantom wormhole geometries are modelled by matching an interior traversable wormhole solution, governed by the equation of state $p=\omega \rho$ with $\omega<-1$, to an exterior vacuum spacetime at a finite junction interface. The stability analysis of these phantom wormholes to linearized spherically symmetric perturbations about static equilibrium solutions is carried out. A master equation dictating the stability regions is deduced, and by separating the cases of a positive and a negative surface energy density, it is found that the respective stable equilibrium configurations may be increased by strategically varying the wormhole throat radius. The first model considered, in the absence of a thin shell, is that of an asymptotically flat phantom wormhole spacetime. The second model constructed is that of an isotropic pressure phantom wormhole, which is of particular interest, as the notion of phantom energy is that of a spatially homogeneous cosmic fluid, although it may be extended to inhomogeneous spherically symmetric spacetimes.Comment: 9 pages, 9 figures, Revtex4. V2: five references adde

Traversable wormholes supported by cosmic accelerated expanding equations of state

We explore the possibility that traversable wormholes be supported by specific equations of state responsible for the present accelerated expansion of the Universe, namely, phantom energy, the generalized Chaplygin gas, and the van der Waals quintessence equation of state.Comment: 3 pages, contribution to the proceedings of MG11, Berlin, 23-29 July, 2006; based on an invited talk in the parallel session GT5, Wormholes, Energy Conditions and Time Machine

Exotic solutions in General Relativity: Traversable wormholes and 'warp drive' spacetimes

The General Theory of Relativity has been an extremely successful theory, with a well established experimental footing, at least for weak gravitational fields. Its predictions range from the existence of black holes, gravitational radiation to the cosmological models, predicting a primordial beginning, namely the big-bang. All these solutions have been obtained by first considering a plausible distribution of matter, and through the Einstein field equation, the spacetime metric of the geometry is determined. However, one may solve the Einstein field equation in the reverse direction, namely, one first considers an interesting and exotic spacetime metric, then finds the matter source responsible for the respective geometry. In this manner, it was found that some of these solutions possess a peculiar property, namely 'exotic matter,' involving a stress-energy tensor that violates the null energy condition. These geometries also allow closed timelike curves, with the respective causality violations. These solutions are primarily useful as 'gedanken-experiments' and as a theoretician's probe of the foundations of general relativity, and include traversable wormholes and superluminal 'warp drive' spacetimes. Thus, one may be tempted to denote these geometries as 'exotic' solutions of the Einstein field equation, as they violate the energy conditions and generate closed timelike curves. In this article, in addition to extensively exploring interesting features, in particular, the physical properties and characteristics of these 'exotic spacetimes,' we also analyze other non-trivial general relativistic geometries which generate closed timelike curves.Comment: 52 pages, 20 figures, RevTex4. Invited chapter to appear in an edited collection 'Classical and Quantum Gravity Research Progress', Nova Science Publisher

Stable dark energy stars

The gravastar picture is an alternative model to the concept of a black hole, where there is an effective phase transition at or near where the event horizon is expected to form, and the interior is replaced by a de Sitter condensate. In this work, a generalization of the gravastar picture is explored, by considering a matching of an interior solution governed by the dark energy equation of state, $\omega\equiv p/ \rho<-1/3$, to an exterior Schwarzschild vacuum solution at a junction interface. The motivation for implementing this generalization arises from the fact that recent observations have confirmed an accelerated cosmic expansion, for which dark energy is a possible candidate. Several relativistic dark energy stellar configurations are analyzed by imposing specific choices for the mass function. The first case considered is that of a constant energy density, and the second choice, that of a monotonic decreasing energy density in the star's interior. The dynamical stability of the transition layer of these dark energy stars to linearized spherically symmetric radial perturbations about static equilibrium solutions is also explored. It is found that large stability regions exist that are sufficiently close to where the event horizon is expected to form, so that it would be difficult to distinguish the exterior geometry of the dark energy stars, analyzed in this work, from an astrophysical black hole.Comment: 10 pages, 6 figures, Revtex4. V2: comments and references added, 11 pages. V3: Significant additions and clarifications, 12 page

From the Flamm-Einstein-Rosen bridge to the modern renaissance of traversable wormholes

We consider the possibility of multiply-connected spacetimes, ranging from the Flamm-Einstein-Rosen bridge, geons, and the modern renaissance of traversable wormholes. A fundamental property in wormhole physics is the flaring-out condition of the throat, which through the Einstein field equation entails the violation of the null energy condition. In the context of modified theories of gravity, it has also been shown that the normal matter can be imposed to satisfy the energy conditions, and it is the higher order curvature terms, interpreted as a gravitational fluid, that sustain these non-standard wormhole geometries, fundamentally different from their counterparts in general relativity. We explore interesting features of these geometries, in particular, the physical properties and characteristics of these `exotic spacetimes'.Comment: 20 pages. MG14 rapporteur article based on the AT3 parallel session. Includes a brief review of wormhole physics and of the contributions to the AT3 sessio

Stable dark energy stars: An alternative to black holes?

In this work, a generalization of the Mazur-Mottola gravastar model is explored, by considering a matching of an interior solution governed by the dark energy equation of state, $\omega\equiv p/ \rho<-1/3$, to an exterior Schwarzschild vacuum solution at a junction interface, situated near to where the event horizon is expected to form. The motivation for implementing this generalization arises from the fact that recent observations have confirmed an accelerated cosmic expansion, for which dark energy is a possible candidate.Comment: 3 pages, contribution to the proceedings of MG11, Berlin, 23-29 July, 2006; based on an invited talk in the parallel session BHT5, Alternative Black Hole Model

Traversable wormholes supported by dark gravity

A fundamental property in wormhole physics is the flaring-out condition of the throat, which through the Einstein field equation entails the violation of the null energy condition. In the context of modified theories of gravity, it has also been shown that the normal matter can be imposed to satisfy the energy conditions, and it is the higher order curvature terms, interpreted as a gravitational fluid, that sustain these non-standard wormhole geometries, fundamentally different from their counterparts in general relativity. We review recent work in wormhole physics in the context of modified theories of gravity.Comment: 3 pages; contribution to the proceedings of the Thirteenth Marcel Grossmann Meeting, Stockholm University, Sweden, 1-7 July, 2012; based on a talk in the AT3 "Gravitational Fields with Sources, Regular Black Holes, Quasiblack Holes, and Analog Black Holes" parallel sessio