Hypersurface-orthogonal Killing Vector Research Articles

Hawking-Ellis type of matter on Killing horizons in symmetric spacetimes

Spherically, plane, or hyperbolically symmetric spacetimes with an additional hypersurface orthogonal Killing vector are often called ``static'' spacetimes even if they contain regions where the Killing vector is nontimelike. It seems to be widely believed that an energy-momentum tenor for a matter field compatible with these spacetimes in general relativity is of the Hawking-Ellis type I everywhere. We show in arbitrary $n(\ensuremath{\ge}3)$ dimensions that, contrary to popular belief, a matter field on a Killing horizon is not necessarily of type I but can be of type II. Such a type-II matter field on a Killing horizon is realized in the Gibbons-Maeda-Garfinkle-Horowitz-Strominger black hole in the Einstein-Maxwell-dilaton system and may be interpreted as a mixture of a particular anisotropic fluid and a null dust fluid.

Nonstaticity with type II, III, or IV matter field in $$f(R_{\mu \nu \rho \sigma },g^{\mu \nu })$$ gravity

In all $n(\ge 3)$-dimensional gravitation theories whose Lagrangians are functions of the Riemann tensor and metric, we show that static solutions are absent unless the total energy-momentum tensor for matter fields is of type I in the Hawking-Ellis classification. In other words, there is no hypersurface-orthogonal timelike Killing vector in a spacetime region with an energy-momentum tensor of type II, III, or IV. This asserts that, if back-reaction is taken into account to give a self-consistent solution, ultra-dense regions with a semiclassical type-IV matter field cannot be static even with higher-curvature correction terms. As a consequence, a static Planck-mass relic is possible as a final state of an evaporating black hole only if the semiclassical total energy-momentum tensor is of type I.

Dynamical equilibrium states of a class of irrotational non-orthogonally transitive G 2 cosmologies: II. Models with one hypersurface-orthogonal Killing vector field

We consider a class of inhomogeneous self-similar cosmological models in which the perfect fluid flow is tangential to the orbits of a three-parameter similarity group. It is assumed throughout that the source is a perfect fluid with linear equation of state. We restrict the similarity group to possess both an Abelian G 2, and a single hypersurface orthogonal Killing vector field, and we restrict the fluid flow to be orthogonal to the orbits of the Abelian G 2. The temporal evolution of the models is forced to be power law, due to the similarity group, and the Einstein field equations reduce to a three-dimensional autonomous system of ordinary differential equations which is qualitatively analysed in order to determine the spatial structure of the models. The existence of two classes of well-behaved models is demonstrated. The first of these is asymptotically spatially homogeneous and matter dominated, and the second is vacuum dominated and either asymptotically spatially homogeneous or acceleration dominated, at large spatial distances.

Doubly torqued vectors and a classification of doubly twisted and Kundt spacetimes

The simple structure of doubly torqued vectors allows for a natural characterization of doubly twisted down to warped spacetimes, as well as Kundt spacetimes down to PP waves. For the first ones the vectors are timelike, for the others they are null. We also discuss some properties, and their connection to hypersurface orthogonal conformal Killing vectors, and null Killing vectors.

On uniqueness of static asymptotically anti-de Sitter black hole

In this paper, the asymptotically anti-de Sitter spacetime with a hypersurface orthogonal Killing vector field is discussed near the conformal boundary by using Newman–Penrose formalism. We first investigate the multipole structure of a static, asymptotically anti-de Sitter spacetime. We note that, different to asymptotically flat case in which the multipoles are contained in , the multipoles of a static, asymptotically anti-de Sitter spacetime are all contained in . We then show that the Schwarzschild anti-de Sitter spacetime is the only asymptotically anti-de Sitter, vacuum, static, axisymmetric, type-D analytic solution of vacuum Einstein equation with a negative cosmological constant in a Bondi coordinate neighborhood of the conformal boundary if the spherical harmonic expansion of is finite.

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.

Embedding Galilean and Carrollian geometries. I. Gravitational waves

The aim of this series of papers is to generalize the ambient approach of Duval et al. regarding the embedding of Galilean and Carrollian geometries inside gravitational waves with parallel rays. In this paper (Paper I), we propose a generalization of the embedding of torsionfree Galilean and Carrollian manifolds inside larger classes of gravitational waves. On the Galilean side, the quotient procedure of Duval et al. is extended to gravitational waves endowed with a lightlike hypersurface-orthogonal Killing vector field. This extension is shown to provide the natural geometric framework underlying the generalization by Lichnerowicz of the Eisenhart lift. On the Carrollian side, a new class of gravitational waves – dubbed Dodgson waves – is introduced and geometrically characterized. Dodgson waves are shown to admit a lightlike foliation by Carrollian manifolds and furthermore to be the largest subclass of gravitational waves satisfying this property. This extended class allows us to generalize the embedding procedure to a larger class of Carrollian manifolds that we explicitly identify. As an application of the general formalism, (Anti) de Sitter spacetime is shown to admit a lightlike foliation by codimension one (A)dS Carroll manifolds.

Exotic massive gravity: Causality and a Birkhoff-like theorem

We study the local causality issue via the Shapiro time-delay computations in the on-shell consistent exotic massive gravity in three dimensions. The theory shows time-delay as opposed to time-advance despite having a ghost at the linearized level both for asymptotically flat and anti-de Sitter spacetimes. We also prove a Birkhoff-like theorem: any solution with a hypersurface orthogonal non-null Killing vector field is conformally flat; and find some exact solutions.

Multiple Killing horizons and near horizon geometries

Near horizon geometries with multiply degenerate Killing horizons are considered, and their degenerate Killing vector fields identified. We prove that they all arise from hypersurface-orthogonal Killing vectors of any cut of with the inherited metric—cuts are spacelike co-dimension two submanifolds contained in . For each of these Killing vectors on a given cut, there are three different possibilities for the near horizon metric which are presented explicitly. The structure of the metric for near horizon geometries with multiple Killing horizons of order is thereby completely determined, and in particular we prove that the cuts on must be warped products with maximally symmetric fibers (ergo of constant curvature). The question whether multiple degenerate Killing horizons may lead to inequivalent near horizon geometries by using different degenerate Killings is addressed, and answered on the negative: all near horizon geometries built from a given multiple degenerate Killing horizon (using different degenerate Killings) are locally isometric.

Entropy of a box of gas in an external gravitational field revisited

Earlier it was shown that the entropy of an ideal gas, contained in a box and moving in a gravitational field, develops an area dependence when it approaches the horizon of a static, spherically symmetric spacetime. Here we extend the above result in two directions; viz., to (a) the stationary axisymmteric spacetimes and (b) time dependent cosmological spacetimes evolving asymptotically to the de Sitter or the Schwarzschild de Sitter spacetimes. While our calculations are exact for the stationary axisymmetric spacetimes, for the cosmological case we present an analytical expression of the entropy when the spacetime is close to the de Sitter or the Schwarzschild de Sitter spacetime. Unlike the static spacetimes, there is no hypersurface orthogonal timelike Killing vector field in these cases. Nevertheless, the results hold and the entropy develops an area dependence in the appropriate limit.

Coordinate independent expression for transverse trace-free tensors

The transverse and trace-free (TT) part of the extrinsic curvature represents half of the dynamical degrees of freedom of the gravitational field in the 3 + 1 formalism. As such, it is part of the freely specifiable initial data for numerical relativity. Though TT tensors in three-space possess only two component degrees of freedom, they cannot ordinarily be given solely by two scalar potentials. Such expressions have been derived, however, in coordinate form, for all TT tensors in flat space which are also translationally or axially symmetric (Conboye and Murchadha 2014 Class. Quantum Grav. 31 085019). Since TT tensors are conformally covariant, these also give TT tensors in conformally flat space. In this article, the work above has been extended by giving a coordinate-independent expression for these TT tensors. The translational and axial symmetry conditions have also been generalized to invariance along any hypersurface orthogonal Killing vector.

Static axisymmetric Einstein equations in vacuum: Symmetry, new solutions, and Ricci solitons

An explicit one-parameter Lie point symmetry of the four-dimensional vacuum Einstein equations with two commuting hypersurface-orthogonal Killing vector fields is presented. The parameter takes values over all of the real line and the action of the group can be effected algebraically on any solution of the system. This enables one to construct particular one-parameter extended families of axisymmetric static solutions and cylindrical gravitational wave solutions from old ones, in a simpler way than most solution-generation techniques, including the prescription given by Ernst for this system. As examples, we obtain the families that generalize the Schwarzschild solution and the $C$-metric. These in effect superpose a Levi-Civita cylindrical solution on the seeds. Exploiting a correspondence between static solutions of Einstein's equations and Ricci solitons (self-similar solutions of the Ricci flow), this also enables us to construct new steady Ricci solitons.

Self-Dual Conformal Gravity

We find necessary and sufficient conditions for a Riemannian four-dimensional manifold (M, g) with anti-self-dual Weyl tensor to be locally conformal to a Ricci-flat manifold. These conditions are expressed as the vanishing of scalar and tensor conformal invariants. The invariants obstruct the existence of parallel sections of a certain connection on a complex rank-four vector bundle over M. They provide a natural generalisation of the Bach tensor which vanishes identically for anti-self-dual conformal structures. We use the obstructions to demonstrate that LeBrun’s anti-self-dual metrics on connected sums of $${\mathbb{CP}^2}$$ s are not conformally Ricci-flat on any open set. We analyze both Riemannian and neutral signature metrics. In the latter case we find all anti-self-dual metrics with a parallel real spinor which are locally conformal to Einstein metrics with non-zero cosmological constant. These metrics admit a hyper-surface orthogonal null Killing vector and thus give rise to projective structures on the space of β-surfaces.

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.

Killing-Yano tensors in spaces admitting a hypersurface orthogonal Killing vector

Methods are presented for finding Killing-Yano tensors, conformal Killing-Yano tensors, and conformal Killing vectors in spacetimes with a hypersurface orthogonal Killing vector. These methods are similar to a method developed by the authors for finding Killing tensors. In all cases one decomposes both the tensor and the equation it satisfies into pieces along the Killing vector and pieces orthogonal to the Killing vector. Solving the separate equations that result from this decomposition requires less computing than integrating the original equation. In each case, examples are given to illustrate the method.

Rigid supersymmetric backgrounds of minimal off-shell supergravity

We discuss various aspects of rigid supersymmetry within minimal N=1 off-shell supergravity using the old and new minimal formulations both in Lorentzian and Euclidean signatures. In particular, we construct all rigid supersymmetry backgrounds with a hypersurface orthogonal Killing vector. In the Lorentzian signature we show that AdS_4 provides a rigid supersymmetric background in both formulations albeit with different amounts of preserved supersymmetry. In the Euclidean signature we find new backgrounds of the old-minimal supergravity, including squashed four-spheres and a half-BPS version of flat space.

Optical metrics and projective equivalence

Trajectories of light rays in a static spacetime are described by unparametrized geodesics of the Riemannian optical metric associated with the Lorentzian spacetime metric. We investigate the uniqueness of this structure and demonstrate that two different observers, moving relative to one another, who both see the Universe as static may determine the geometry of the light rays differently. More specifically, we classify Lorentzian metrics admitting more than one hyper-surface orthogonal timelike Killing vector and analyze the projective equivalence of the resulting optical metrics. These metrics are shown to be projectively equivalent up to diffeomorphism if the static Killing vectors generate a group $SL(2,\mathbb{R})$, but not projectively equivalent in general. We also consider the cosmological $C$ metrics in Einstein-Maxwell theory and demonstrate that optical metrics corresponding to different values of the cosmological constant are projectively equivalent.

Decoupling and reduction in Chern–Simons modified gravity

We show that for four-dimensional spacetimes with a non-null hypersurface orthogonal Killing vector and for a Chern–Simons (CS) background (non-dynamical) scalar field, which is constant along the Killing vector, the source-free equations of CS modified gravity decouple into their Einstein and Cotton constituents. Thus, the model supports only general relativity solutions. We also show that, when the cosmological constant vanishes and the gradient of the CS scalar field is parallel to the non-null hypersurface orthogonal Killing vector of constant length, CS modified gravity reduces to topologically massive gravity in three dimensions. Meanwhile, with the cosmological constant such a reduction requires an appropriate source term for CS modified gravity.

Conformal Symmetries of Spherical Spacetimes

We investigate the conformal geometry of spherically symmetric spacetimes in general without specifying the form of the matter distribution. The general conformal Killing symmetry is obtained subject to a number of integrability conditions. Previous results relating to static spacetimes are shown to be a special case of our solution. The general inheriting conformal symmetry vector, which maps fluid flow lines conformally onto fluid flow lines, is generated and the integrability conditions are shown to be satisfied. We show that there exists a hypersurface orthogonal conformal Killing vector in an exact solution of Einstein’s equations for a relativistic fluid which is expanding, accelerating and shearing.