Spacelike Killing Vector Research Articles

Revisiting Buchdahl transformations: new static and rotating black holes in vacuum, double copy, and hairy extensions

This paper investigates Buchdahl transformations within the framework of Einstein and Einstein-Scalar theories. Specifically, we establish that the recently proposed Schwarzschild–Levi-Civita spacetime can be obtained by means of a Buchdahl transformation of the Schwarschild metric along the spacelike Killing vector. The study extends Buchdahl’s original theorem by combining it with the Kerr–Schild representation. In doing so, we construct new vacuum-rotating black holes in higher dimensions which can be viewed as the Levi-Civita extensions of the Myers–Perry geometries. Furthermore, it demonstrates that the double copy scheme within these new generated geometries still holds, providing an example of an algebraically general double copy framework. In the context of the Einstein-Scalar system, the paper extends the corresponding Buchdahl theorem to scenarios where a static vacuum seed configuration, transformed with respect to a spacelike Killing vector, generates a hairy black hole spacetime. We analyze the geometrical features of these spacetimes and investigate how a change of frame, via conformal transformations, leads to a new family of black hole spacetimes within the Einstein-Conformal-Scalar system.

Exploring the shear viscosity in four-dimensional planar black holes beyond general relativity

The well known shear viscosity to entropy density ratio ($\eta /s$) cannot be computed when the black hole space-time has zero thermodynamic entropy. This is the case, for example, when General Relativity in four dimensions is complemented with Critical Gravity, or in particular scenarios within the Four-dimensional-scalar-Gauss-Bonnet theories. Recently, it has been shown that the zero entropy situation can be overcome in these examples by introducing suitable matter fields. With this at hand, in this paper we analyze each case and the impact of these extra sources in the ratio, in terms of their new parameters. We find that, while the $\eta /s$ ratio remains constant and insensitive in the former, this is not the case for the latter. To perform the calculations, we construct a Noether charge using a space-like Killing vector. The accuracy of the aforementioned findings is supported by the Kubo formalism.

Shear viscosity from black holes in generalized scalar-tensor theories in arbitrary dimensions

In higher dimensions, we study Degenerate-Higher-Order-Scalar-Tensor theories and we derive solutions that resemble the Schwarzschild Anti-de Sitter black holes. We compute their thermodynamic quantities following the Wald formalism, satisfying the First Law of Thermodynamics and a higher dimensional Smarr relation. Constructing a Noether charge with a suitable choice of a space-like Killing vector, we obtain the shear viscosity of the non-gravitational dual field theory, where for a suitable choice of the couplings functions, the Kovtun-Son-Starinets bound is violated. These results are corroborated by the calculation of the Green's functions following the Kubo formalism.

Electromagnetism from 5D gravity: beyond the Maxwell equations

Ever with the work of Kaluza, it has been known that 4D Einstein- and Maxwell-type equations emerge from the equations for 5D gravity, in Ricci-flat space-times having a space-like Killing vector. We revisit these equations and compare them with the Maxwell equations and the Ohm’s law. Although 5D gravity and traditional electromagnetic theory are mathematically related, a paradigm shift in traditional electromagnetic concepts is required for unification. First, both the Maxwell equations and the Ohm’s law are found among the field equations for 5D gravity, which turns Ohm’s law into a field equation. Second, the concept of 4-current density, bringing together electrical charge and current density, is no longer fundamental for the electromagnetic theory resulting from 5D gravity. Third, the 5D gravity equations suggest a generalization for the model of the perfect conductor, to describe a broader range of electromagnetic phenomena and bypass the concept of 4-current density. We thus propose an amendment to the Maxwell equations, based on the field equations for 5D gravity, and discuss a few applications. Namely, we discuss the propagation of electromagnetic waves through an imperfect conductor and topics in electrostatics and magnetostatics, touching ground with the Hall effect and the phenomenon of the persistent current.

Five-dimensional Brans–Dicke compactified universe dominated by a varying speed of light

We extend the model of a 5D Brans–Dicke gravity theory reduced to 4D through the presence of a hypersurface-orthogonal space-like killing vector field in the underlying 5D spacetime by including a varying speed of light. The resulting model is characterized by the presence of two scalar fields. We focus on late-time power law solutions which emerge in general when scalar fields couple to spacetime curvature and do not contradict the SNIa astrophysical data. Analytic solutions in 4-dimensions are derived and late-time accelerated expansion was found. The universe is dominated by dark energy, free from phantom field and is characterized by a decaying energy matter density, decaying scalar fields, and a decreasing celerity of light. The model is confronted with astrophysical observations and is found to fit these data.

Deforming black holes with odd multipolar differential rotation boundary

Motivated by the novel asymptotically global AdS$_4$ solutions with deforming horizon in [JHEP {\bf 1802}, 060 (2018)], we analyze the boundary metric with odd multipolar differential rotation and numerically construct a family of deforming solutions with tripolar differential rotation boundary, including two classes of solutions: solitons and black holes. We find that the maximal values of the rotation parameter $\varepsilon$, below which the stable large black hole solutions could exist, are not a constant for $T> T_{schw}=\sqrt{3}/2\pi\simeq0.2757$. When temperature is much higher than $ T_{schw}$, even though the norm of Killing vector $\partial_{t}$ keeps timelike for some regions of $\varepsilon<2$, solitons and black holes with tripolar differential rotation could be unstable and develop hair due to superradiance. As the temperature $T$ drops toward $T_{schw}$, we find that though there exists the spacelike Killing vector $\partial_{t}$ for some regions of $\varepsilon>2$, solitons and black holes still exist and do not develop hair due to superradiance. Moreover, for $T\leqslant T_{schw}$, the curves of entropy firstly combine into one curve and then separate into two curves again, in the case of each curve there are two solutions at a fixed value of $\varepsilon$. In addition, we study the deformations of horizon for black holes by using an isometric embedding in the hyperbolic three-dimensional space. Furthermore, we also study the quasinormal modes of the solitons and black holes, which have analogous behaviours to that of dipolar rotation and quadrupolar rotation.

Geodesic structure of naked singularities in AdS3 spacetime

We present a complete study of the geodesics around naked singularities in AdS$_3$, the three-dimensional anti-de Sitter spacetime. These stationary spacetimes, characterized by two conserved charges --mass and angular momentum--, are obtained through identifications along spacelike Killing vectors with a fixed point. They are interpreted as massive spinning point particles, and can be viewed as three-dimensional analogues of cosmic strings in four spacetime dimensions. The geodesic equations are completely integrated and the solutions are expressed in terms of elementary functions. We classify different geodesics in terms of their radial bounds, which depend on the constants of motion. Null and spacelike geodesics approach the naked singularity from infinity and either fall into the singularity or wind around and go back to infinity, depending on the values of these constants, except for the extremal and massless cases for which a null geodesic could have a circular orbit. Timelike geodesics never escape to infinity and do not always fall into the singularity, namely, they can be permanently bounded between two radii. The spatial projections of the geodesics (orbits) exhibit self-intersections, whose number is particularly simple for null geodesics. As a particular application, we also compute the lengths of fixed-time spacelike geodesics of the static naked singularity using two different regularizations.

Note on the Noether charge and holographic transports

We clarify the relation between the Noether charge associated to an arbitrary vector field and the equations of motions by revisiting Wald formalism. For a time-like Killing vector, aspects of the Noether charge suggest that it is dual to the heat current in the boundary for general holographic theories. For a space-like Killing vector, we interpret the Noether charge (at the transverse direction) as shear stress of the dual fluid so we can compute the ratio of shear viscosity to entropy density by simply using the infrared data on the black hole event horizon. We test the new method for Einstein gravity and Gauss-Bonnet gravity and find that it produces correct results for both cases even in the presence of additional matter fields.

Phantom metrics with Killing spinors

We study metric solutions of Einstein–anti-Maxwell theory admitting Killing spinors. The analogue of the IWP metric which admits a space-like Killing vector is found and is expressed in terms of a complex function satisfying the wave equation in flat (2+1)-dimensional space–time. As examples, electric and magnetic Kasner spaces are constructed by allowing the solution to depend only on the time coordinate. Euclidean solutions are also presented.

Numerical evolution of plane gravitational waves in the Friedrich-Nagy gauge

The first proof of well-posedness of an initial boundary value problem for the Einstein equations was given in 1999 by Friedrich and Nagy. They used a frame formalism with a particular gauge for formulating the equations. This `Friedrich-Nagy' (FN) gauge has never been implemented for use in numerical simulations before because it was deemed too complicated. In this paper we present an implementation of the FN gauge for systems with two commuting space-like Killing vectors. We investigate the numerical performance of this formulation for plane wave space-times, reproducing the well-known Khan-Penrose solution for colliding impulsive plane waves and exhibiting a gravitational wave `ping-pong'.

Five-dimensional generalized $$f(R)$$ f ( R ) gravity with curvature–matter coupling

The generalized \(f(R)\) gravity with curvature–matter coupling in five-dimensional (5D) spacetime can be established by assuming a hypersurface-orthogonal space-like Killing vector field of 5D spacetime, and it can be reduced to the 4D formalism of FRW universe. This theory is quite general and can give the corresponding results for Einstein gravity, and \(f(R)\) gravity with both no-coupling and non-minimal coupling in 5D spacetime as special cases, that is, we would give some new results besides previous ones given by Huang et al. in Phys Rev D 81:064003, 2010. Furthermore, in order to get some insight into the effects of this theory on the 4D spacetime, by considering a specific type of models with \(f_{1}(R)=f_{2}(R)=\alpha R^{m}\) and \(B(L_{m})=L_{m}=-\rho \), we not only discuss the constraints on the model parameters \(m,n\), but also illustrate the evolutionary trajectories of the scale factor \(a(t)\), the deceleration parameter \(q(t)\), and the scalar field \(\epsilon (t),\phi (t)\) in the reduced 4D spacetime. The research results show that this type of \(f(R)\) gravity models given by us could explain the current accelerated expansion of our universe without introducing dark energy.

The Generalized f(R) Model with Coupling in 5D Spacetime

The generalized f(R) gravity with coupling in five-dimensional (5D) spacetime is studied on the basis of both the 4D generalized f(R) gravity with coupling and the pure f(R) gravity without coupling in 5D spacetime. Specifically, by assuming a hypersurface-orthogonal space-like Killing vector field in 5D spacetime, the generalization of the 4D generalized f(R) gravity to 5D can be realized, which can give the reduced 4-metric coupled with two scalar fields. In particular, we discuss a special class of models, i.e., f1(R) = f2(R) = αRm (m ≠ 1), and choose B(Lm) = Lm = −ρ in the homogeneous and isotropic cosmology with the 4D Friedmann—Robertson—Walker metric. The numerical analysis shows that the parameter m can be constrained by means of the current observations for the deceleration parameter, which implies that this generalized f(R) model with coupling in 5D spacetime can account for the present accelerated expansion of the universe.

SPACETIME SINGULARITIES: RECENT DEVELOPMENTS

Recent developments concerning oscillatory spacelike singularities in general relativity are taking place on two fronts. The first treats generic singularities in spatially homogeneous cosmology, most notably Bianchi types VIII and IX. The second deals with generic oscillatory singularities in inhomogeneous cosmologies, especially those with two commuting spacelike Killing vectors. This paper describes recent progress in these two areas: in the spatially homogeneous case, focus is on mathematically rigorous results, while analytical and numerical results concerning generic behavior and so-called recurring spike formation are the main topics in the inhomogeneous case. Unifying themes are connections between asymptotic behavior, hierarchical structures and solution generating techniques, which provide hints for a link between the nature of generic singularities and a hierarchy of hidden asymptotic symmetries.

Bianchi class B spacetimes with electromagnetic fields

We carry out a thorough analysis on a class of cosmological space-times which admit three spacelike Killing vectors of Bianchi class B and contain electromagnetic fields. Using dynamical system analysis, we show that a family of electro-vacuum plane-wave solutions of the Einstein-Maxwell equations is the stable attractor for expanding universes. Phase dynamics are investigated in detail for particular symmetric models. We integrate the system exactly for some special cases to confirm the qualitative features. Some of the obtained solutions have not been presented previously to the best of our knowledge. Finally, based on those analyses, we discuss the relation between those homogeneous models and perturbations of open Friedmann-Lemaitre-Robertson-Walker universes. We argue that the electro-vacuum plane-wave modes correspond to a certain long-wavelength limit of electromagnetic perturbations.

Hybrid quantization: From Bianchi I to the Gowdy model

The Gowdy cosmologies are vacuum solutions to the Einstein equations which possess two space-like Killing vectors and whose spatial sections are compact. We consider the simplest of these cosmological models: the case where the spatial topology is that of a three-torus and the gravitational waves are linearly polarized. The subset of homogeneous solutions to this Gowdy model are vacuum Bianchi I spacetimes with a three-torus topology. We deepen the analysis of the loop quantization of these Bianchi I universes adopting the improved dynamics scheme put forward recently by Ashtekar and Wilson-Ewing. Then, we revisit the hybrid quantization of the Gowdy $T^3$ cosmologies by combining this loop quantum cosmology description with a Fock quantization of the inhomogeneities over the homogeneous Bianchi I background. We show that, in vacuo, the Hamiltonian constraint of both the Bianchi I and the Gowdy models can be regarded as an evolution equation with respect to the volume of the Bianchi I universe. This evolution variable turns out to be discrete, with a strictly positive minimum. Furthermore, we argue that this evolution is well-defined inasmuch as the associated initial value problem is well posed: physical solutions are completely determined by the data on an initial section of constant Bianchi I volume. This fact allows us to carry out to completion the quantization of these two cosmological models.

Killing vectors and anisotropy

We consider an action that can generate fluids with three unequal stresses for metrics with a spacelike Killing vector. The parameters in the action are directly related to the stress anisotropies. The field equations following from the action are applied to an anisotropic cosmological expansion and an extension of the Gott-Hiscock cosmic string.

N-dimensional plane symmetric solution with perfect fluid source

A new class of plane symmetric solution sourced by a perfect fluid is found in our recent work. An n-dimensional ( n ⩾ 4 ) global plane symmetric solution of Einstein field equation generated by a perfect fluid source is investigated, which is the direct generalization of our previous 4-dimensional solution. One time-like Killing vector and ( n − 2 ) ( n − 1 ) / 2 space-like Killing vectors, which span a Euclidean group G ( n − 2 ) ( n − 1 ) / 2 , are permitted in this solution. The regions of the parameters constrained by weak, strong and dominant energy conditions for the source are studied. The boundary condition to match to n-dimensional Taub metric and Minkowski metric are analyzed respectively.

Towards canonical quantum gravity for G1 geometries in 2+1 dimensions with a Λ-term

The canonical analysis and subsequent quantization of the (2+1)-dimensional action of pure gravity plus a cosmological constant term is considered, under the assumption of the existence of one spacelike Killing vector field. The proper imposition of the quantum analogues of two linear (momentum) constraints reduces an initial collection of state vectors, consisting of all smooth functionals of the components (and/or their derivatives) of the spatial metric, to particular scalar smooth functionals. The demand that the midi-superspace metric (inferred from the kinetic part of the quadratic (Hamiltonian) constraint) must define on the space of these states an induced metric whose components are given in terms of the same states, which is made possible through an appropriate re-normalization assumption, severely reduces the possible state vectors to three unique (up to general coordinate transformations) smooth scalar functionals. The quantum analogue of the Hamiltonian constraint produces a Wheeler–DeWitt equation based on this reduced manifold of states, which is completely integrated.

The conformal Penrose limit: back to square one

We show that the conformal Penrose limit is an ordinary plane wave limit in a higher dimensional framework which resolves the spacetime singularity. The higher dimensional framework is provided by Ricci-flat manifolds which are of the form MD = Md × B, where Md is an Einstein spacetime that has a negative cosmological constant and admits a spacelike conformal Killing vector, and B is a complete Sasaki–Einstein space with constant sectional curvature. We define the Kaluza–Klein metric of MD through the conformal Killing potential of Md and prove that Md has a conformal Penrose limit if and only if MD has an ordinary plane wave limit. Further properties of the limit are discussed.

Cosmology in three dimensions: steps towards the general solution

We use covariant and first-order formalism techniques to study the properties of general relativistic cosmology in three dimensions. The covariant approach provides an irreducible decomposition of the relativistic equations, which allows for a mathematically compact and physically transparent description of the three-dimensional spacetimes. Using this information we review the features of homogeneous and isotropic 3D cosmologies, provide a number of new solutions and study gauge invariant perturbations around them. The first-order formalism is then used to provide a detailed study of the most general 3D spacetimes containing perfect-fluid matter. Assuming the material content to be dust with comoving spatial 2-velocities, we find the general solution of the Einstein equations with a non-zero (and zero) cosmological constant and generalize known solutions of Kriele and the 3D counterparts of the Szekeres solutions. In the case of a non-comoving dust fluid we find the general solution in the case of one non-zero fluid velocity component. We consider the asymptotic behaviour of the families of 3D cosmologies with rotation and shear and analyse their singular structure. We also provide the general solution for cosmologies with one spacelike Killing vector, find solutions for cosmologies containing scalar fields and identify all the PP-wave 2 + 1 spacetimes.