Narrow Results

Among all electromagnetic theories which (a) are derivable from a Lagrangian, (b) satisfy the dominant energy condition, and (c) in the weak field limit coincide with classical linear electromagnetics, we identify a certain subclass with the property that the corresponding spherically symmetric, asymptotically flat, electrostatic spacetime metric has the mildest possible singularity at its center, namely, a conical singularity on the time axis. The electric field moreover has a point defect on the time axis, its total energy is finite, and is equal to the ADM mass of the spacetime. By an appropriate scaling of the Lagrangian, one can arrange the total mass and total charge of these spacetimes to have any chosen values. For small enough mass-to-charge ratio, these spacetimes have no horizons and no trapped null geodesics. We also prove the uniqueness of these solutions in the spherically symmetric class, and we conclude by performing a qualitative study of the geodesics and test-charge trajectories of these spacetimes.

After formulating the notion of integrability for axially symmetric harmonic maps from ${ℝ}^3$ into symmetric spaces, we give a complete and rigorous proof that, subject to some mild restrictions on the target, all such maps are integrable. Furthermore, we prove that a variant of the inverse scattering method, called vesture (dressing) can always be used to generate new solutions for the harmonic map equations starting from any given solution. In particular, we show that the problem of finding $N$-solitonic harmonic maps into a non-compact Grassmann manifold $SU(p,q)/S(U(p)\times U(q))$ is completely reducible via the vesture (dressing) method to a problem in linear algebra which we prove is solvable in general. We illustrate this method, and establish its agreement with previously known special cases, by explicitly computing a 1-solitonic harmonic map for the two cases $(p=1,q=1)$ and $(p=2,q=1)$ and showing that the family of solutions obtained in each case contains respectively the Kerr family of solutions to the Einstein vacuum equations, and the Kerr–Newman family of solutions to the Einstein–Maxwell equations in the hyperextreme sector of the corresponding parameters.

Introducing the Lie algebroid generalized tangent bundle of a Kaluza–Klein bundle, we develop the theory of general distinguished linear connections for this space. In particular, using the Lie algebroid generalized tangent bundle of the Kaluza–Klein vector bundle, we present the (g, h)-lift of a curve on the base M and we characterize the horizontal and vertical parallelism of the (g, h)-lift of accelerations with respect to a distinguished linear (ρ, η)-connection. Moreover, we study the torsion, curvature and Ricci tensor field associated to a distinguished linear (ρ, η)-connection and we obtain the identities of Cartan and Bianchi type in the general framework of the Lie algebroid generalized tangent bundle of a Kaluza–Klein bundle. Finally, we introduce the theory of (pseudo) generalized Kaluza–Klein G-spaces and we develop the Einstein equations in this general framework.

We obtain some basic estimates for a Monge–Ampère type equation introduced by Moncrief in the study of the Relativistic Teichmüller Theory. We then give another proof of the parametrization of the Teichmüller space obtained by Moncrief. Our approach provides yet another proof of the classical Teichmüller theorem that the Teichmüller space of a compact oriented surface of genus $g(\mathrm{\Sigma })>1$ is diffeomorphic to the disk of dimension $6g(\mathrm{\Sigma })-6$. We also give another proof of properness of a certain energy function on the Teichmüller space.

We study the Einstein equations in the static cylindrically symmetric case with the stress–energy tensor of the form $T_{\nu }^{\mu }=\mathrm{\text{diag}}\{\mu ,-\alpha \mu ,-\beta \mu ,-\gamma \mu \}$, where $\mu$ is an unknown function and $\alpha$, $\beta$, $\gamma$ are arbitrary real constants ($\alpha$ is assumed to be nonzero). The stress–energy tensor of this form includes as special cases several well-known solutions, such as the perfect fluid solution with the barotropic equation of state, the solution with the static electric field and the solution with the massless scalar field. We solve the Einstein equations with this stress–energy tensor and study some properties of the obtained metric.

Rastall theory propounded some five decades ago belongs to a class of modified theories of gravity. Such theories are motivated by the need to modify general relativity suitably in order to address some problems not explained by the standard theory. Amongst such issues are the observed accelerated expansion of the universe, motion in extremely high gravity regimes and explanations for the discrepancy in the value of the cosmological constant between quantum gravity and experimentation. Recently, it has been claimed that the Rastall theory is trivially equivalent to the standard Einstein theory. We investigate this claim in the context of stellar structure and elementary requirements for physical plausibility. We consider the analogue of the Saslaw et al. [Astrophys. J.471, 571 (1996)] isothermal model of general relativity and show that the Rastall version satisfies the basic requirements unlike its counterpart. Then, we examine in turn the consequences of suppressing one of the inverse square law fall off of the energy density or the linear equation of state. Imposing a linear barotropic equation of state, we find a generalized de Sitter spacetime as an exact solution of the Rastall equations. In addition, the case of a constant spatial gravitational potential is studied. In each case, we note that the physics of the Rastall model differs from that of the Einstein version.

Following recent advances in the local theory of current-algebraic orbifolds, we study various geometric properties of the general WZW orbifold, the general coset orbifold and a large class of (nonlinear) sigma model orbifolds. Phase-space geometry is emphasized for the WZW orbifolds — while for the sigma model orbifolds we construct the corresponding sigma model orbifold action, which includes the previously-known general WZW orbifold action and general coset orbifold action as special cases. We focus throughout on the twisted Einstein tensors with diagonal monodromy, including the twisted Einstein metric, the twisted B field and the twisted torsion field of each orbifold sector. Finally, we present strong evidence for a conjectured set of twisted Einstein equations which should describe those sigma model orbifolds in this class which are also one-loop conformal.

A value of the cosmological constant in a toy model of the five-dimensional universe is calculated in such a manner that it remains in agreement with both astronomical observations and the quantum field theory concerning the zero-point fluctuations of the vacuum. The (negative) cosmological constant is equal to the inverse of the Planck length squared, which means that in the toy model the vanishing of the observed value of the cosmological constant is a consequence of the existence of an energy cutoff exactly at the Planck level. In turn, a model for both a virtual and a real particle–antiparticle pair is proposed which describes properly some energetic properties of both the vacuum fluctuations and created particles, as well as it allows one to calculate the discrete "bare" values of an elementary particle's mass, electric charge and intrinsic angular momentum (spin) at the energy cutoff. The relationships between the discussed model and some phenomena such as the Zitterbewegung and the Unruh–Davies effect are briefly analyzed, too. The proposed model also allows one to derive the Lorentz transformation and the Maxwell equations while considering the properties of the vacuum filled with the sea of virtual particles and their antiparticles. Finally, the existence of a finite value of the vacuum-energy density resulting from the toy model leads us to the formulation of dimensionless Einstein equations which may be derived from the Lagrangian with a dimensionless (naively renormalized) coupling constant.

In this paper, we present a three-parameter model for the distributions of “dark matter” around local spherically symmetric and static distributions of normal matter, usually referred to as “dark matter halos”. The model is based on the assumption that the spontaneous symmetry breaking mechanism involving the Higgs field in the standard model of elementary particle physics leads to the existence of a constant distribution of vacuum energy throughout space–time. This is to be interpreted as a constant distribution of proper energy, as measured by local observers in their proper reference frames. Due to the presence of this universal distribution of vacuum energy, the model requires the introduction of the cosmological term in the Einstein field equations, in order to regulate the behavior of the resulting geometry at cosmologically large distances. By implementing the model on galactic scales, we are able to establish the gravitational consequences of such a homogeneous distribution of proper energy density at these scales. This results in equivalent halo masses and orbital velocity curves that, at least qualitatively, match the observations for galactic “dark matter” halos. Although the detailed realization of the model presented and calculated here cannot be construed as a precise representation of galactic structure, given that most of the actual structure of the galaxy, such as the galactic disk itself, is ignored, and replaced by a single spherically symmetric bulge, in this paper we do introduce the conceptual framework that may be used to construct a more complete galactic model in the future.

The most popular theories of everything are various versions of the superstring theory. The theories require existence of additional space dimensions, vibrations of which create the material particles in $3\mathrm{\text{D}}$ space. The additional space dimensions are understood as being currently smaller than the Planck Length and due to this not directly observable. We search for multidimensional models of the Universe (one time dimension; three isotropic, flat external dimensions, and $n$-internal dimensions), which satisfy the multidimensional Einstein equations and which started from the same radius of all of the internal and external dimensions, with an anisotropic energy–momentum tensor. Analytical solution of $(N+1)$-dimensional Einstein equation in a reparameterized time is reminded and discussed. The energy–momentum tensor is solely responsible for expansion of the external dimensions and shrinking of the internal ones; and to obtain this behavior of the space the tensor needs to fulfill some conditions i.e. the energy–momentum tensor cannot include only radiation, vacuum and baryonic matter. For the behavior of the physical space consistent with the one observed in our Universe, the dark energy and/or dark matter have to exist.

Class of axially symmetric solutions of vacuum Einstein field equations including the Papapetrou solution as particular case has been found. It has been shown that the derived solution describes the external axial symmetric gravitational field of the source with nonvanishing mass. The general solution is obtained for this class of functions. As an example of physical application, the spacetime metric outside a line gravitomagnetic monopole has been obtained from Papapetrou solution of vacuum equations of gravitational field.

We discuss simple vacuum solutions to the Einstein Equations in five-dimensional space–times compactified in two different ways. In such spaces, one black hole phase and more than one black string phase may exist. Several old metrics are adapted to new background topologies to yield new solutions to the Einstein Equations. We then briefly talk about the angular momentum they may carry, the horizon topology and phase transitions that may occur.

We introduce a simple and straightforward averaging procedure, which is a generalization of one which is commonly used in electrodynamics, and show that it possesses all the characteristics we require for linearized averaging in general relativity and cosmology — for weak-field and perturbed FLRW situations. In particular, we demonstrate that it yields quantities which are approximately tensorial in these situations, and that its application to an exact FLRW metric yields another FLRW metric, to first-order in integrals over the local coordinates. Finally, we indicate some important limits of any linearized averaging procedure with respect to cosmological perturbations which are the result of averages over large amplitude small and intermediate scale inhomogeneities, and show our averaging procedure can be approximately implemented by that of Zotov and Stoeger in these cases.

Although it ranks amongst the oldest of problems in classical general relativity, the challenge of finding new exact solutions for spherically symmetric perfect fluid spacetimes is still ongoing because of a paucity of solutions which exhibit the necessary qualitative features compatible with observational evidence. The problem amounts to solving a system of three partial differential equations in four variables, which means that any one of four geometric or dynamical quantities must be specified at the outset and the others should follow by integration. The condition of pressure isotropy yields a differential equation that may be interpreted as second-order in one of the space variables or also as first-order Ricatti type in the other space variable. This second option has been fruitful in allowing us to construct an algorithm to generate a complete solution to the Einstein field equations once a geometric variable is specified ab initio. We then demonstrate the construction of previously unreported solutions and examine these for physical plausibility as candidates to represent real matter. In particular we demand positive definiteness of pressure, density as well as a subluminal sound speed. Additionally, we require the existence of a hypersurface of vanishing pressure to identify a radius for the closed distribution of fluid. Finally, we examine the energy conditions. We exhibit models which display all of these elementary physical requirements.

Exact solutions to the Einstein field equations may be generated from already existing ones (seed solutions), that admit at least one Killing vector. In this framework, a space of potentials is introduced. By the use of symmetries in this space, the set of potentials associated to a known solution is transformed into a new set, either by continuous transformations or by discrete transformations. In view of this method, and upon consideration of continuous transformations, we arrive at some exact, stationary axisymmetric solutions to the Einstein field equations in vacuum, that may be of geometrical or/and physical interest.

In this paper, we review and analyze four specific general-relativistic problems in which gravitomagnetism plays an important role: the dragging of magnetic fields around rotating black holes, dragging inside a collapsing slowly rotating spherical shell of dust, compared with the dragging by rotating gravitational waves. We demonstrate how the quantum detection of inertial frame dragging can be accomplished by using the Unruh–DeWitt detectors. Finally, we shall briefly show how “instantaneous Machian gauges” can be useful in the cosmological perturbation theory.

There is much talk of a “mysterious form of energy”, called “dark energy” that forms the bulk of the energy content of the universe. Perhaps the most mysterious aspect of it is why should it be regarded as mysterious in the first place. This question is discussed in the context of the development of relativity and relativistic cosmology. It will be argued that there is no good reason to treat it as other than Einstein’s cosmological constant.

When the vacuum Einstein equations are cast in the form of Hamiltonian evolution equations, the initial data lie in the cotangent bundle of the manifold of Riemannian metrics on a Cauchy hypersurface Σ. As in every Lagrangian field theory with symmetries, the initial data must satisfy constraints. But, unlike those of gauge theories, the constraints of general relativity do not arise as momenta of any Hamiltonian group action. In this paper, we show that the bracket relations among the constraints of general relativity are identical to the bracket relations in the Lie algebroid of a groupoid consisting of diffeomorphisms between space-like hypersurfaces in spacetimes. A direct connection is still missing between the constraints themselves, whose definition is closely related to the Einstein equations, and our groupoid, in which the Einstein equations play no role at all. We discuss some of the difficulties involved in making such a connection. In an appendix, we develop some aspects of diffeology, the basic framework for our treatment of function spaces.

Stochastic Einstein equations are considered when three-dimensional space metric γ_ij are stochastic functions. The probability density for the stochastic quantities is connected with Perelman's entropy functional. As an example, the Friedman Universe is considered. It is shown that for the Friedman Universe the dynamical evolution is not changed. The connection between general relativity and Ricci flow is discussed.

In the present paper we obtain, in a covariant form, and in their full generality, the field equations in a relativistic general Kaluza–Klein space. This is done by using the Riemannian horizontal connection defined in [3], and some 4D horizontal tensor fields, as for instance: horizontal Ricci tensor, horizontal Einstein gravitational tensor field, horizontal electromagnetic energy–momentum tensor field, etc. Also, we present some inter-relations between STM theory and brane-world theory. This enables us to introduce in brane theory some electromagnetic potentials constructed by means of the warp function.