Dimensional General Relativity Research Articles

Wormholes in higher dimensional space-time: Exact solutions and their linear stability analysis

We derive the simplest traversable wormhole solutions in $n$-dimensional general relativity, assuming static and spherically symmetric spacetime with a ghost scalar field. This is the generalization of the Ellis solution (or the so-called Morris-Thorne's traversable wormhole) into a higher-dimension. We also study their stability using linear perturbation analysis. We obtain the master equation for the perturbed gauge-invariant variable and search their eigenvalues. Our analysis shows that all higher-dimensional wormholes have an unstable mode against the perturbations with which the throat radius is changed. The instability is consistent with the earlier numerical analysis in four-dimensional solution.

On the implementation of the canonical quantum simplicity constraint

In this paper, we discuss several approaches to solve the quadratic and linear simplicity constraints in the context of the canonical formulations of higher dimensional general relativity and supergravity developed in our companion papers. Since the canonical quadratic simplicity constraint operators have been shown to be anomalous in any dimension D ⩾ 3 in Class. Quantum Grav. 30 045003, non-standard methods have to be employed to avoid inconsistencies in the quantum theory. We show that one can choose a subset of quadratic simplicity constraint operators which are non-anomalous among themselves and allow for a natural unitary map of the spin networks in the kernel of these simplicity constraint operators to the SU(2)-based Ashtekar–Lewandowski Hilbert space in D = 3. The linear constraint operators on the other hand are non-anomalous by themselves; however, their solution space is shown to differ in D = 3 from the expected Ashtekar–Lewandowski Hilbert space. We comment on possible strategies to make a connection to the quadratic theory. Also, we comment on the relation of our proposals to the existing work in the spin foam literature and how these works could be used in the canonical theory. We emphasize that many ideas developed in this paper are certainly incomplete and should be considered as suggestions for possible starting points for more satisfactory treatments in the future.

New variables for classical and quantum gravity in all dimensions: III. Quantum theory

We quantize the new connection formulation of (D + 1)-dimensional general relativity developed in our companion papers by loop quantum gravity (LQG) methods. It turns out that all the tools prepared for LQG straightforwardly generalize to the new connection formulation in higher dimensions. The only new challenge is the simplicity constraint. While its ‘diagonal’ components acting at edges of spin-network functions are easily solved, its ‘off-diagonal’ components acting at vertices are non-trivial and require a more elaborate treatment.

Chapter 3. Topology and Uniqueness of Higher Dimensional Black Holes

We review recent results concerning general properties of higher dimensional black holes. The topics selected with particular focus are those concerning topology, symmetry, and uniqueness properties of asymptotically flat vacuum black holes in higher dimensional general relativity.

Weak-field approximation of effective gravitational theory with local Galilean invariance

We examine the weak-field approximation of locally Galilean invariant gravitational theories with general covariance in a (4+1)-dimensional Galilean framework. The additional degrees of freedom allow us to obtain Poisson, diffusion, and Schrödinger equations for the fluctuation field. An advantage of this approach over the usual (3+1)-dimensional General Relativity is that it allows us to choose an ansatz for the fluctuation field that can accommodate the field equations of the Lagrangian approach to MOdified Newtonian Dynamics (MOND) known as AQUAdratic Lagrangian (AQUAL). We investigate a wave solution for the Schrödinger equations.

Non-existence of asymptotically flat geons in (2 + 1) gravity

Geons, small topological structures that exhibit particle properties such as charge and angular momentum without the presence of matter sources, have been extensively discussed in (3 + 1)-dimensional general relativity. Given the recent renewal of interest in (2 + 1) gravity, it is natural to ask whether or not the notion of geons extends to three dimensions. We prove here that, in contrast to the (3 + 1)-dimensional case, there are no (2 + 1)-dimensional asymptotically flat solutions of the vacuum Einstein or Einstein–Maxwell equations containing geons. In contrast, (2 + 1)-dimensional asymptotically anti-de Sitter spacetimes can indeed contain geons; however, the geons are always hidden behind a single black hole horizon. We also prove sufficient conditions for the non-existence of (2 + 1)-dimensional asymptotically flat geon-containing solutions.

Dimensional Analysis and General Relativity

Newton's law of gravitation is a central topic in the first-year physics curriculum. A lecturer can go beyond the physical details and use the history of gravitation to discuss the development of scientific ideas; unfortunately, the most recent chapter in this history, general relativity, is not covered in first-year courses. This paper discusses some topics that can be introduced with the judicious use of the dimensionless quantity GM/Rc2: deflection of light by a massive object, perihelion precession, gravitational redshifts, and the Global Positioning System (GPS).

Macrostates Thermodynamics and Its Stable Classical Limit in Global One-Dimensional Quantum General Relativity

Global One Dimensional Quantum General Relativity is the toy model with nontrivial eld theoretical content, describing classical one-dimensional massive bosonic elds related to any 3 + 1 metric, where the dimension is a volume of threedimensional embedding. In fact it constitutes the midisuperspatial Quantum Gravity model. We use one-particle density operator method in order to building macrostates thermodynamics related with any 3 + 1 metric. Taking the Boltzmann gas limit, which is given by the energy equipartition law for the Bose Einstein gas of space quantum states generated from the Bogoliubov vacuum, we receive consistent with General Relativity thermodynamical degrees of freedom number. Concepts of Physics, Vol. VI, No. 1(2009) DOI: 10.2478/v10005-009-0002-5 19 It con rm that the proposed Quantum Gravity toy model has well-de ned classical limit in accordance with classical gravity theory. 20 Concepts of Physics, Vol. VI, No. 1 (2009) Macrostates thermodynamics ...

A double Myers-Perry black hole in five dimensions

Using the inverse scattering method we construct a six-parameter family of exact, stationary, asymptotically flat solutions of the 4+1 dimensional vacuum Einstein equations, with U(1)^2 rotational symmetry. It describes the superposition of two Myers-Perry black holes, each with a single angular momentum parameter, both in the same plane. The black holes live in a background geometry which is the Euclidean C-metric with an extra flat time direction. This background possesses conical singularities in two adjacent compact regions, each corresponding to a set of fixed points of one of the U(1) actions in the Cartan sub-algebra of SO(4). We discuss several aspects of the black holes geometry, including the conical singularities arising from force imbalance, and the torsion singularity arising from torque imbalance. The double Myers-Perry solution presented herein is considerably simpler than the four dimensional double Kerr solution and might be of interest in studying spin-spin interactions in five dimensional general relativity.

Symmetry Properties of Black Holes in Higher Dimensional General Relativity

Symmetry Properties of Black Holes in Higher Dimensional General Relativity

Collision of high-energy closed strings: Formation of a ringlike apparent horizon

We study collisions of two high-energy closed strings in the framework of $D$-dimensional general relativity. The model of a high-energy closed string is introduced as a $pp$-wave generated by a ring-shaped source with the radius $R$. At the instant of the collision, the positions of two strings are assumed to coincide precisely. In this setup, we study the formation of two kinds of apparent horizons (AHs): the AH of topology ${S}^{D\ensuremath{-}2}$ (the black hole AH) and the AH of topology ${S}^{1}\ifmmode\times\else\texttimes\fi{}{S}^{D\ensuremath{-}3}$ (the black ring AH). These two AHs are solved numerically and the conditions for the formation of the two AHs are clarified in terms of the ring radius $R$. Specifically, we demonstrate that the black ring AH forms for sufficiently large $R$. The effects of an impact parameter and the relative orientation of incoming strings in more general cases are briefly discussed.

Five-dimensional rotating black hole in a uniform magnetic field: The gyromagnetic ratio

In four dimensional general relativity, the fact that a Killing vector in a vacuum spacetime serves as a vector potential for a test Maxwell field provides one with an elegant way of describing the behaviour of electromagnetic fields near a rotating Kerr black hole immersed in a uniform magnetic field. We use a similar approach to examine the case of a five dimensional rotating black hole placed in a uniform magnetic field of configuration with bi-azimuthal symmetry, that is aligned with the angular momenta of the Myers-Perry spacetime. Assuming that the black hole may also possess a small electric charge we construct the 5-vector potential of the electromagnetic field in the Myers-Perry metric using its three commuting Killing vector fields. We show that, like its four dimensional counterparts, the five dimensional Myers-Perry black hole rotating in a uniform magnetic field produces an inductive potential difference between the event horizon and an infinitely distant surface. This potential difference is determined by a superposition of two independent Coulomb fields consistent with the two angular momenta of the black hole and two nonvanishing components of the magnetic field. We also show that a weakly charged rotating black hole in five dimensions possesses two independent magnetic dipole moments specified in terms of its electric charge, mass, and angular momentum parameters. We prove that a five dimensional weakly charged Myers-Perry black hole must have the value of the gyromagnetic ratio g=3.

Anisotropic inflation and the origin of four large dimensions

In the context of (4 + d)-dimensional general relativity, we propose an inflationary scenario wherein three spatial dimensions grow large, while d extra dimensions remain small. Our model requires that a self-interacting d-form acquires a vacuum expectation value along the extra dimensions. This causes three spatial dimensions to inflate, whilst keeping the size of the extra dimensions nearly constant. We do not require an additional stabilization mechanism for the radion, as stable solutions exist for flat, and for negatively curved compact extra dimensions. From a four-dimensional perspective, the radion does not couple to the inflaton; and, the small amplitude of the CMB temperature anisotropies arises from an exponential suppression of fluctuations, due to the higher-dimensional origin of the inflaton. The mechanism triggering the end of inflation is responsible both for heating the universe and for avoiding violations of the equivalence principle due to coupling between the radion and matter.

Weak gravity in the Dvali-Gabadadze-Porrati braneworld model

We analyze the weak gravity in the braneworld model proposed by Dvali-Gabadadze-Porrati, in which the unperturbed background spacetime is given by five dimensional Minkowski bulk with a brane which has the induced Einstein Hilbert term. This model has a critical length scale $r_c$. Naively, we expect that the four dimensional general relativity (4D GR) is approximately recovered at the scale below $r_c$. However, the simple linear perturbation does not work in this regime. Only recently the mechanism to recover 4D GR was clarified under the restriction to spherically symmetric configurations, and the leading correction to 4D GR was derived. Here, we develop an alternative formulation which can handle more general perturbations. We also generalize the model by adding bulk cosmological constant and the brane tension.

Static and rotating electrically charged black holes in three-dimensional Brans-Dicke gravity theories

We obtain static and rotating electrically charged black holes of a Einstein-Maxwell-Dilaton theory of the Brans-Dicke type in (2+1)-dimensions. The theory is specified by three fields, the dilaton, the graviton and the electromagnetic field, and two parameters, the cosmological constant and the Brans-Dicke parameter. It contains eight different cases, of which one distinguishes as special cases, string theory, general relativity and a theory equivalent to four dimensional general relativity with one Killing vector. We find the ADM mass, angular momentum, electric charge and dilaton charge and compute the Hawking temperature of the solutions. Causal structure and geodesic motion of null and timelike particles in the black hole geometries are studied in detail.

Perfect fluid of -branes, 2D dilaton gravity and the big-bang

Abstract This paper starts by building the energy–momentum tensor of a perfect fluid of p -branes coupled to ( p+4 )-dimensional general relativity. Having three homogeneous and isotropic macroscopical spatial dimensions, the system gravity/fluid can be reduced to an effective theory over the branes. For the string fluid ( p=1 ) the effective theory is nothing but the 2D dilaton gravity where the potential for the scalar field, which is the scale factor of the macroscopical space, is fixed by the state equation and the three-dimensional geometry. This theory can be solved allowing us to compare some relevant aspects in our homogeneous and isotropic string cosmologies with those of the Robertson–Walker ones. In particular, unlike the point-particle models, the existence of an initial singularity is strongly sensitive to the state equation, and it is remarkable that this model picks out the radiation state equation as the canonical case where the big-bang is kinematically forbidden. Moreover, we cannot reduce the Robertson–Walker cosmologies to the limit when the string size approaches to zero, because the existence of an upper bound on the string size is not compatible with the big-bang. Some examples are presented.

Remarks on the reduced phase space of -dimensional gravity on a torus in the Ashtekar formulation

We examine the reduced phase space of the Barbero-Varadarajan solutions of the Ashtekar formulation of (2 + 1)-dimensional general relativity on a torus. We show that it is a finite-dimensional space due to the existence of an infinite-dimensional residual gauge invariance which reduces the infinite-dimensional space of solutions to a finite-dimensional space of gauge-inequivalent solutions. This is in agreement with general arguments which imply that the number of physical degrees of freedom for (2 + 1)-dimensional Ashtekar gravity on a torus is finite.

Asymptotic structure of symmetry-reduced general relativity

Gravitational waves with a space-translation Killing field are considered. In this case, the 4-dimensional Einstein vacuum equations are equivalent to the 3-dimensional Einstein equations with certain matter sources. This interplay between 4- and 3- dimensional general relativity can be exploited effectively to analyze issues pertaining to 4 dimensions in terms of the 3-dimensional structures. An example is provided by the asymptotic structure at null infinity: While these space-times fail to be asymptotically flat in 4 dimensions, they can admit a regular completion at null infinity in 3 dimensions. This completion is used to analyze the asymptotic symmetries, introduce the analog of the 4-dimensional Bondi energy-momentum and write down a flux formula. The analysis is also of interest from a purely 3-dimensional perspective because it pertains to a diffeomorphism invariant 3-dimensional field theory with {\it local} degrees of freedom, i.e., to a midi-superspace. Furthermore, due to certain peculiarities of 3 dimensions, the description of null infinity does have a number of features that are quite surprising because they do not arise in the Bondi-Penrose description in 4 dimensions.

Spinning black holes in (2 + 1)-dimensional string and dilaton gravity

We present a new class of spinning black hole solution in (2 + 1)-dimensional general relativity minimally coupled to a dilaton with potential e bƒΛ . When b = 4, the corresponding spinning black hole is a solution of low energy (2 + 1)-dimensional string gravity. Apart from the limiting case of the BTZ black hole, these spinning black holes have no inner horizon and a curvature singularity only at the origin. We compute the mass and angular momentum parameters of the solutions at spatial infinity, as well as their temperature and entropy.

Erratum: Static charged black holes in (2+1)-dimensional dilaton gravity

A one parameter family of static charged black hole solutions in $(2+1)$-dimensional general relativity minimally coupled to a dilaton $\phi\propto ln({r\over\beta})$ with a potential term $e^{b\phi}\Lambda$ is obtained. Their causal strutures are investigated, and thermodynamical temperature and entropy are computed. One particular black hole in the family has the same thermodynamical properties as the Schwarzschild black hole in $3+1$ dimensions. Solutions with cosmological horizons are also discussed. Finally, a class of black holes arising from the dilaton with a negative kinetic term (tachyon) is briefly discussed.