Vacuum Field Equations Research Articles

Critical behavior of the black hole/black string transition

We numerically construct static localized black holes in five and six spacetime dimensions which are solutions to Einstein’s vacuum field equations with one compact periodic dimension. In particular, we investigate the critical regime in which the poles of the localized black hole are about to merge. A well adapted multi-domain pseudo-spectral scheme provides us with accurate results and enables us to investigate the phase diagram of those localized solutions within the critical regime, which goes far beyond previous results. We find that in this regime the phase diagram possesses a spiral structure adapting to the one recently found for non-uniform black strings. When approaching the common endpoint of both phases, the behavior of physical quantities is described by complex critical exponents giving rise to a discrete scaling symmetry. The numerically obtained values of the critical exponents agree remarkably well with those derived from the double-cone metric.

Characteristic formulation for metric f(R) gravity

In recent years, the Characteristic formulation of numerical relativity has found increasing use in the extraction of gravitational radiation from numerically generated spacetimes. In this paper, we formulate the Characteristic initial value problem for $f(R)$ gravity. We consider, in particular, the vacuum field equations of Metric $f(R)$ gravity in the Jordan frame, without utilising the dynamical equivalence with scalar-tensor theories. We present the full hierarchy of non-linear hypersurface and evolution equations necessary for numerical implementation in both tensorial and eth forms. Furthermore, we specialise the resulting equations to situations where the spacetime is almost Minkowski and almost Schwarszchild using standard linearization techniques. We obtain analytic solutions for the dominant $\ell=2$ mode and show that they satisfy the concomitant constraints. These results are ideally suited as testbed solutions for numerical codes. Finally, we point out that the Characteristic formulation can be used as a complementary analytic tool to the $1+1+2$ semi-tetrad formulation.

Definition of the relativistic geoid in terms of isochronometric surfaces

We present a definition of the geoid that is based on the formalism of general relativity without approximations; i.e. it allows for arbitrarily strong gravitational fields. For this reason, it applies not only to the Earth and other planets but also to compact objects such as neutron stars. We define the geoid as a level surface of a time-independent redshift potential. Such a redshift potential exists in any stationary spacetime. Therefore, our geoid is well defined for any rigidly rotating object with constant angular velocity and a fixed rotation axis that is not subject to external forces. Our definition is operational because the level surfaces of a redshift potential can be realized with the help of standard clocks, which may be connected by optical fibers. Therefore, these surfaces are also called isochronometric surfaces. We deliberately base our definition of a relativistic geoid on the use of clocks since we believe that clock geodesy offers the best methods for probing gravitational fields with highest precision in the future. However, we also point out that our definition of the geoid is mathematically equivalent to a definition in terms of an acceleration potential, i.e. that our geoid may also be viewed as a level surface orthogonal to plumb lines. Moreover, we demonstrate that our definition reduces to the known Newtonian and post-Newtonian notions in the appropriate limits. As an illustration, we determine the isochronometric surfaces for rotating observers in axisymmetric static and axisymmetric stationary solutions to Einstein's vacuum field equation, with the Schwarzschild metric, the Erez-Rosen metric, the q-metric and the Kerr metric as particular examples.

A boosted Kerr black hole solution and the structure of a general astrophysical black hole

A solution of Einstein's vacuum field equations that describes a boosted Kerr black hole relative to an asymptotic Lorentz frame at the future null infinity is derived. The solution has three parameters (mass, rotation and boost) and corresponds to the most general configuration that an astrophysical black hole must have; it reduces to the Kerr solution when the boost parameter is zero. In this solution the ergosphere is north-south asymmetric, with dominant lobes in the direction opposite to the boost. However the event horizon, the Cauchy horizon and the ring singularity, which are the core of the black hole structure, do not alter, being independent of the boost parameter. Possible consequences for astrophysical processes connected with Penrose processes in the asymmetric ergosphere are discussed.

On the dark matter as a geometric effect in $$ f (\mathcal {R})$$ f ( R ) gravity

A mysterious type of matter is supposed to exist, because the observed rotational velocity curves of particle moving around the galactic center and the expected rotational velocity curves do not match. There are also a number of proposals in the modified gravity for this discrepancy. In this contrast, in $2008$, B$\ddot{\text{o}}$hmer et al. presented an interesting idea in (Astropart Phys 29(6):386-392, 2008) where they showed that a $f(\mathcal{R})$ gravity model could actually explain dark matter to be a geometric effect only. They solved the gravitational field equations in vacuum using generic $f(\mathcal{R})$ gravity model for constant velocity regions and found that the resulting modifications in the Einstein-Hilbert Lagrangian is of the form $\mathcal{R}^{1+m}$, where $m=V_{tg}^2/c^2$; $V_{tg}$ being the tangential velocity of the test particle moving around galactic dark matter region and, $c$, the speed of light. From observations it is known that $m\approx\mathcal{O}(10^{-6})$. In this article, we perform two things (1) We show that the form of $f(\mathcal{R})$ they claimed is not correct. In doing the calculations, we found that when the radial component of the metric for constant velocity regions is a constant then the exact solutions for $f(\mathcal{R})$ obtained is of the form of $\mathcal{R}^{1-\alpha}$ which corresponds to a negative correction rather than positive, $\alpha$ is a function of $m$. (2) We also show that we can not have an analytic solution of $f(\mathcal{R})$ for all values of tangential velocity including the observed value of tangential velocity $200-300$Km/s if the radial coefficient of the metric which describes the dark matter regions is \emph{not a constant}. Thus, we have to rely on the numerical solutions to get an approximate model for dark matter in $f(\mathcal{R})$ gravity.

On the equivalence of approximate stationary axially symmetric solutions of the Einstein field equations

We study stationary axially symmetric solutions of the Einstein vacuum field equations that can be used to describe the gravitational field of astrophysical compact objects in the limiting case of slow rotation and slight deformation. We derive explicitly the exterior Sedrakyan-Chubaryan approximate solution, and express it in analytical form, which makes it practical in the context of astrophysical applications. In the limiting case of vanishing angular momentum, the solution reduces to the well-known Schwarzschild solution in vacuum. We demonstrate that the new solution is equivalent to the exterior Hartle-Thorne solution. We establish the mathematical equivalence between the Sedrakyan-Chubaryan, Fock-Abdildin and Hartle-Thorne formalisms.

Cosmological models in Weyl geometrical scalar-tensor theory

We investigate cosmological models in a recently proposed geometrical theory of gravity, in which the scalar field appears as part of the space-time geometry. We extend the previous theory to include a scalar potential in the action. We solve the vacuum field equations for different choices of the scalar potential and give a detailed analysis of the solutions. We show that in some cases a cosmological scenario is found that seems to suggest the appearance of a geometric phase transition. We build a toy model, in which the accelerated expansion of the early universe is driven by pure geometry.

Wormholes with asymptotic Lifshitz scaling in Hořava gravity

We study static spherically symmetric solutions of the nonprojectable Hořava theory with and without cosmological constant. The solutions we find are two-side wormholes and (single-side) naked singularities. Interestingly, in the case of negative cosmological constant we find that in the exterior side the wormhole acquires an asymptotic scaling between space and time equal to the scaling of the Lifshitz solution, which was previously found to be a vacuum solution of the same theory. This result leads us to pose the question whether in the case of negative cosmological constant the asymptotic anisotropic Lifshitz scaling is a generic feature of the vacuum field equations rather than the asymptotic AdS-like scaling.

Characterization of (asymptotically) Kerr–de Sitter-like spacetimes at null infinity**Preprint UWThPh-2016-5.

We investigate solutions to Einstein's vacuum field equations with positive cosmological constant Λ which admit a smooth past null infinity à la Penrose and a Killing vector field whose associated Mars–Simon tensor (MST) vanishes. The main purpose of this work is to provide a characterization of these spacetimes in terms of their Cauchy data on . Along the way, we also study spacetimes for which the MST does not vanish. In that case there is an ambiguity in its definition which is captured by a scalar function Q. We analyze properties of the MST for different choices of Q. In doing so, we are led to a definition of ‘asymptotically Kerr–de Sitter-like spacetimes’, which we also characterize in terms of their asymptotic data on .

Generating solutions to the Einstein field equations

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.

Type II universal spacetimes

We study type II universal metrics of the Lorentzian signature. These metrics simultaneously solve vacuum field equations of all theories of gravitation with the Lagrangian being a polynomial curvature invariant constructed from the metric, the Riemann tensor and its covariant derivatives of an arbitrary order. We provide examples of type II universal metrics for all composite number dimensions. On the other hand, we have no examples for prime number dimensions and we prove the non-existence of type II universal spacetimes in five dimensions. We also present type II vacuum solutions of selected classes of gravitational theories, such as Lovelock, quadratic and gravities.

Cartan's soldered spaces and conservation laws in physics

In this paper, we will introduce a generalized soldering p-forms geometry, which can be the right framework to describe many new approaches and concepts in modern physics. Here we will treat some aspects of the theory of local cohomology in fields theory or more precisely the theory of soldering-form conservation laws in physics. We provide some illustrative applications, primarily taken from the Einstein equations of general theory of relativity and Yang–Mills theory. This theory can be considered to be a generalization of Noether's theory of conserved current to differential forms of any degree. An essential result of this, is that the conservation of the energy–momentum in general relativity, is linked to the fact that the vacuum field equations are equivalent to the integrability conditions of a first-order system of differential equations. We also apply the idea of the soldered space and the integrability conditions to the case of Yang–Mills theory. The mathematical framework, where these intuitive considerations would fit naturally, can be used to describe also the dynamics of changing manifolds.

Novel Remarks on Point Mass Sources, Firewalls, Null Singularities and Gravitational Entropy

A continuous family of static spherically symmetric solutions of Einstein’s vacuum field equations with a spatial singularity at the origin \( r = 0 \) is found. These solutions are parametrized by a real valued parameter \( \lambda \) (ranging from 0 to 1) and such that the radial horizon’s location is displaced continuously towards the singularity (\( r = 0 \)) as \( \lambda \) increases. In the extreme limit \( \lambda = 1\), the location of the singularity and horizon merges leading to a null singularity. In this extreme case, any infalling observer hits the null singularity at the very moment he/she crosses the horizon. This fact may have important consequences for the resolution of the fire wall problem and the complementarity controversy in black holes. An heuristic argument is provided how one might avoid the Hawking particle emission process in this extreme case when the singularity and horizon merges. The field equations due to a delta-function point-mass source at \( r = 0 \) are solved and the Euclidean gravitational action corresponding to those solutions is evaluated explicitly. It is found that the Euclidean action is precisely equal to the black hole entropy (in Planck area units). This result holds in any dimensions \( D \ge 3 \).

Exact cosmological solutions for MOG

We find some new exact cosmological solutions for the covariant scalar-tensor-vector gravity theory, the so-called MOdified Gravity (MOG). The exact solution of the vacuum field equations has been derived. Also, for non vacuum cases we have found some exact solutions with the aid of the Noether symmetry approach. More specifically, the symmetry vector and also the Noether conserved quantity associated to the point-like Lagrangian of the theory have been found. Also we find the exact form of the generic vector field potential of this theory by considering the behavior of the relevant point-like Lagrangian under the infinitesimal generator of the Noether symmetry. Finally, we discuss the cosmological implications of the solutions.

A new solution of Einstein’s vacuum field equations

A new solution of Einstein's vacuum field equations is discovered which appears as a generalization of the well-known Ozsvath-Schucking solution and explains its source of curvature which has otherwise remained hidden. Curiously, the new solution has a vanishing Kretschmann scalar and is singularity-free despite being curved. The discovery of the new solution is facilitated by a new insight which reveals that it is always possible to define the source of curvature in a vacuum solution in terms of some dimensional parameters. As the parameters vanish, so does the curvature. The new insight also helps to make the vacuum solutions Machian.

Schwarzschild and Kerr solutions of Einstein's field equation: An Introduction

Starting from Newton's gravitational theory, we give a general introduction into the spherically symmetric solution of Einstein's vacuum field equation, the Schwarzschild(–Droste) solution, and into one specific stationary axially symmetric solution, the Kerr solution. The Schwarzschild solution is unique and its metric can be interpreted as the exterior gravitational field of a spherically symmetric mass. The Kerr solution is only unique if the multipole moments of its mass and its angular momentum take on prescribed values. Its metric can be interpreted as the exterior gravitational field of a suitably rotating mass distribution. Both solutions describe objects exhibiting an event horizon, a frontier of no return. The corresponding notion of a black hole is explained to some extent. Eventually, we present some generalizations of the Kerr solution.

Gravitational self-force in nonvacuum spacetimes

The gravitational self-force has thus far been formulated in background spacetimes for which the metric is a solution to the Einstein field equations in vacuum. While this formulation is sufficient to describe the motion of a small object around a black hole, other applications require a more general formulation that allows for a nonvacuum background spacetime. We provide a foundation for such extensions, and carry out a concrete formulation of the gravitational self-force in two specific cases. In the first we consider a particle of mass $m$ and scalar charge $q$ moving in a background spacetime that contains a background scalar field. In the second we consider a particle of mass $m$ and electric charge $e$ moving in an electrovac spacetime. The self-force incorporates all couplings between the gravitational perturbations and those of the scalar or electromagnetic fields. It is expressed as a sum of local terms involving tensors defined in the background spacetime and evaluated at the current position of the particle, as well as tail integrals that depend on the past history of the particle. Because such an expression may not be a useful starting point for an explicit evaluation of the self-force, we also provide covariant expressions for the singular potentials, expressed as local expansions near the world line; these can be involved in the construction of effective extended sources for the regular potentials, or in the computation of regularization parameters when the self-force is computed as a sum over spherical-harmonic modes.

Exact solution of vacuum field equation in Finsler spacetime

We suggest that the vacuum field equation in Finsler spacetime is equivalent to vanishing of Ricci scalar. Schwarzschild metric can be deduced from a solution of our field equation if the spacetime preserve spherical symmetry. Supposing spacetime to preserve the symmetry of "Finslerian sphere", we find a non-Riemannian exact solution of the Finslerian vacuum field equation. The solution is similar to the Schwarzschild metric. It reduces to Schwarzschild metric while the Finslerian parameter $\epsilon$ vanishes. It is proved that the Finslerian covariant derivative of the geometrical part of the gravitational field equation is conserved. The interior solution is also given. We get solutions of geodesic equation in such a Schwarzschild-like spacetime, and show that the geodesic equation returns to the counterpart in Newton's gravity at weak field approximation. The celestial observations give constraint on the Finslerian parameter $\epsilon<10^{-4}$. And the recent Michelson-Morley experiment requires $\epsilon<10^{-16}$. The counterpart of Birkhoff's theorem exist in Finslerian vacuum. It shows that the Finslerian gravitational field with the symmetry of "Finslerian sphere" in vacuum must be static.

Conformally Covariant Systems of Wave Equations and their Equivalence to Einstein’s Field Equations

We derive, in 3 + 1 spacetime dimensions, two alternative systems of quasi-linear wave equations, based on Friedrich’s conformal field equations. We analyse their equivalence to Einstein’s vacuum field equations when appropriate constraint equations are satisfied by the initial data. As an application, the characteristic initial value problem for the Einstein equations with data on past null infinity is reduced to a characteristic initial value problem for wave equations with data on an ordinary light-cone.

Weyl gravity as general relativity

When the full connection of Weyl conformal gravity is varied instead of just the metric, the resulting vacuum field equations reduce to the vacuum Einstein equation, up to the choice of local units, if and only if the torsion vanishes. This result differs strongly from the usual fourth-order formulation of Weyl gravity.