Five-dimensional Vacuum Einstein Equations Research Articles

Stationary and biaxisymmetric four-soliton solution in five dimensions

Using the inverse scattering method to solve the five-dimensional vacuum Einstein equations, we construct an asymptotically flat four-soliton solution as a stationary and bi-axisymmetric solution. We impose certain boundary conditions on this solution so that it includes a rotating black hole whose horizon-cross section is topologically a lens space of L(2,1). The solution has nine parameters but three only is physically independent due to the constraint equations. The remaining degrees of freedom correspond to the mass and two independent angular momenta of the black hole. We analyze a few simple cases in detail, in particular, the static case with two zero-angular momenta and the stationary case with a single non-zero angular momentum.

Twisting algebraically special solutions in five dimensions

We determine the general form of the solutions of the five-dimensional vacuum Einstein equations with cosmological constant for which (i) the Weyl tensor is everywhere type II or more special in the null alignment classification of Coley et al, and (ii) the 3 × 3 matrix encoding the expansion, shear and twist of the aligned null direction has rank 2. The dependence of the solution on two coordinates is determined explicitly, so the Einstein equation reduces to PDEs in the three remaining coordinates, just as for four-dimensional (4d) algebraically special solutions. The solutions fall into several families. One of these consists of warped products of 4d algebraically special solutions. The others are new.

Boundary value problem for black rings

We study the boundary value problem for asymptotically flat stationary black ring solutions to the five-dimensional vacuum Einstein equations. Assuming the existence of two additional commuting axial Killing vector fields and the horizon topology of ${\mathrm{S}}^{1}\ifmmode\times\else\texttimes\fi{}{\mathrm{S}}^{2}$, we show that the only asymptotically flat black ring solution with a regular horizon is the Pomeransky-Sen'kov black ring solution.

INDUCING CHARGES AND CURRENTS FROM EXTRA DIMENSIONS

In a particular variant of Kaluza–Klein theory, the so-called induced-matter theory (IMT), it is shown that any configuration of matter may be geometrically induced from a five-dimensional vacuum space. By using a similar approach we show that any distribution of charges and currents may also be induced from a five-dimensional space. Although in the case of IMT the geometry is Riemannian and the fundamental equations are the five-dimensional Einstein equations in vacuum, here we consider a Minkowskian geometry and five-dimensional Maxwell equations in vacuum.

A VACUUM-LIKE CONFIGURATION IN GENERAL RELATIVITY AS A MANIFESTATION OF A LORENTZ-INVARIANT MODE OF FIVE-DIMENSIONAL GRAVITY

A Lorentz-invariant cosmological model is constructed within the framework of five-dimensional gravity. The five-dimensional theorem which is analogical to the generalized Birkhoff theorem is proved, that corresponds to the Kaluza's "cylinder condition." The five-dimensional vacuum Einstein equations have an integral of motion corresponding to this symmetry, the integral of motion is similar to the mass function in general relativity (GR). Space closure with respect to the extra dimensionality follows from the requirement of the absence of a conical singularity. Thus, the Kaluza–Klein (KK) model is realized dynamically as a Lorentz-invariant mode of five-dimensional general relativity. After the dimensional reduction and conformal mapping the model is reduced to the GR configuration. It contains a scalar field with a vanishing conformally invariant energy–momentum tensor on the flat space–time background. This zero mode can be interpreted as a vacuum configuration in GR. As a result the vacuum-like configuration in GR can be considered as a manifestation of the Lorentz-invariant empty five-dimensional space.

New axisymmetric stationary solutions of five-dimensional vacuum Einstein equations with asymptotic flatness

New axisymmetric stationary solutions of the vacuum Einstein equations in five-dimensional asymptotically flat spacetimes are obtained by using solitonic solution-generating techniques. The new solutions are shown to be equivalent to the four-dimensional multi-solitonic solutions derived from particular class of four-dimensional Weyl solutions and to include different black rings from those obtained by Emparan and Reall.

Energy-dependent phenomenological metrics and five-dimensional Einstein equations

We propose a new Kaluza-Klein-like scheme based on a five-dimensional Riemannian space in which energy plays the role of the fifth dimension and spacetime is deformed. The solutions of the five-dimensional Einstein equations in vacuum allow us to recover, as special cases, the energy-dependent phenomenological metrics, describing the four fundamental interactions, recently derived from the analysis of some experimental data.

Energy as a Fifth Dimension

A recent analysis of the experimental data on some physical phenomena ruled by the four fundamental interactions (electromagnetic, weak, strong and gravitational) seems to show the possibility of describing such interactions in terms of a deformation of the usual Minkowski spacetime, with a metric whose coefficients do depend on the energy of the process considered. In this paper, we show that such results can be accounted for in terms of a Kaluza-Klein-like scheme, based on a five-dimensional Riemannian space in which energy plays the role of the fifth dimension. The corresponding five-dimensional Einstein equations in vacuum are solved in some cases of physical relevance and it is shown that all the phenomenological metrics describing the four fundamental forces are recovered as special cases of the classes of solutions found. Possible developments of the formalism are also briefly outlined.

Energy and Relativistic Clock Rates in Five Dimensions

In the framework of a Kaluza-Klein-like scheme, based on a five-dimensional Riemannian space in which energy plays the role of the fifth dimension, we discuss a class of solutions of the five-dimensional Einstein equations in vacuum, which allows us to recover the energy-dependent phenomenological metric for gravitation, recently derived from the analysis of some experimental data concerning the slowing down of clock rates in the gravitational field of Earth.

Solution of the five-dimensional vacuum Einstein equations in Kaluza-Klein theory

In a recent paper Ross obtained the five-dimensional vacuum Einstein equations in Kaluza-Klein theory with energy-momentum tensor equal to zero and solved the equations for a particular case. Here we obtain the complete set of solutions of these equations.

SL(3, [formula omitted]) representation for invariance transformations in five-dimensional gravity

The invariance transformations of the axisymmetric five-dimensional vacuum Einstein equations in a represenation of the group SL(3, R ) in the potential space (analogous to the Ernst potential space). Using this formulation, an exact class of stationary axisymmetric solutions is generated, which contains among others Belinsky-Ruffini, Kramer and the Kerr-NUT solutions, as particular cases.

Rotating Kaluza-Klein monopoles and dyons

Conformastationary solutions of the five-dimensional vacuum Einstein equations, depending on one or two harmonic potentials, are constructed. We thus obtain solutions describing systems of rotating electric or magnetic monopoles, as well as rotating dyon solutions, satisfying the principle of equivalence.

Solutions of five-dimensional general relativity without spatial symmetry

Conformastationary solutions of the five-dimensional vacuum Einstein equations, depending on one or two harmonic potentials, are constructed. The solutions depending on one potential fall in three distinct classes. Solutions of two of these classes may be combined to yield a class of solutions depending on two potentials, which correspond to the Israel-Wilson-Perjes solutions of the Einstein-Maxwell theory. The asymptotically flat solutions of this class describe systems of rotating electric or magnetic monopoles.

Comments on windows to extra dimensions

We comment upon a static, three-dimensional spherically symmetric solution of the five-dimensional vacuum Einstein equations discussed recently by Davidson and Owen. We also present a generalisation of this solution that includes the effects of a static electric charge.

Axisymmetric regular multiwormhole solutions in five-dimensional general relativity

A class of regular, asymptotically flat solutions to the five-dimensional vacuum Einstein equations with a two-parameter Abelian isometry group is constructed, under the additional assumption of axial symmetry in three-dimensional space. The possibility of interpreting these multiwormhole solutions as multiparticle systems is discussed.

Massive from massless regular solutions in five-dimensional general relativity

The regular multiwormhole solutions to the five-dimensional vacuum Einstein equations, previously obtained, are generalized to massive solutions, interpreted as systems of extended particles.

Kaluza-Klein Monopole

By adding the trivial term $\ensuremath{-}d{t}^{2}$ one can convert the four-dimensional (positive-definite) Newman-Unti-Tamburino line element into a static solution of the five-dimensional vacuum Einstein equations. Interpreted $\stackrel{`}{a}\mathrm{la}$ Kaluza and Klein, the solution describes a monopole with a charge-to-mass ratio of $\sqrt{2}$ in rationalized Planck units. Although it is a perfectly regular five-geometry it appears singular from a four-dimensional perspective.

Spherically symmetric solutions in five-dimensional general relativity

The most general time-independent spherically symmetric (in the usual three space dimensions) solution to the five-dimensional vacuum Einstein equations is found, subject to the existence of a Killing vector in the fifth direction. The significance of these solutions is discussed within the context of a previously proposed extension of the Kaluza-Klein model in which the universe, although (4+1)-dimensional, has evolved over cosmic times into an effectively (3+l)-dimensional one.