Killing vectors in empty space algebraically special metrics. I

Empty space algebraically special metrics possessing an expanding degenerate principal null vector and a Killing vector are investigated. It is shown that the Killing vector falls into one of two classes. The class containing all asymptotically timelike Killing vectors is investigated in detail and the associated metrics are identified. Several theorems concerning these metrics are given, among which is a proof that if the metric is regular and possesses an asymptotically timelike Killing vector, then it must be typeD. In addition some relations between Killing vectors in general spaces are developed along with a set of tetrad symmetry equations stronger than those of Killing.     

The Carter constant and Petrov classification of the Vaidya–Einstein–Kerr spacetime

The existence of the Carter constant in the Vaidya–Einstein–Kerr (VEK) spacetime and its relation to the Petrov type is investigated. This spacetime is an example of a black hole in an asymptotically non-flat background. We construct the Carter constant and obtain the Killing tensor in the VEK spacetime. The Newman–Penrose formalism is employed to obtain the spin coefficients. We present a complete (Petrov) classification of the VEK spacetime and the special case of the non-rotating Vaidya–Einstein–Schwarzschild spacetime. We demonstrate explicitly that both spacetimes are of type-D.     

Noether定理与黑洞的质量公式及黑洞熵

运用并发展了协变相空间的Noether荷方法,对于真空广义相对论稳态轴对称黑洞得到:黑洞质量公式是关于Killing向量场和完整Cauchy面上的零Noether荷以及黑洞力学第一定律.对于一大类向量场,利用零标架方法证明在视界附近的约化代数的中心项为零.这表明,Carlip用纯粹对称性分析的方法来解释黑洞熵的微观起源值得商榷.     

On ellipsoidal solutions of Liouville's equation in general relativity

A relativistic, collisionless gas of gravitating particles all having the same proper mass (possibly equal to zero) is studied under the assumption that the oneparticle distribution function is locally ellipsoidal in momentum space with respect to some timelike vector field (observer). Liouville's equation implies that the distribution function depends only on a quadratic form in the 4- momenta, whose coefficients are a Killing tensor in the case of non- vanishing proper mass, and a conformal Killing tensor in the case of vanishing rest mass of the particles. It is suggested that cosmological models of Bianchi-type I can be described in terms of ellipsoidal momentum distribution functions whose ellipsoidal tensor is built out of the Killing vectors associated with the spatial homogeneity.     

Timelike Killing vectors and ergo surfaces in non-asymptotically flat spacetimes

Ergo surfaces are investigated in spacetimes with a cosmological constant. We find the existence of multiple timelike Killing vectors, each corresponding to a distinct ergo surface, with no one being preferred. Using a kinematic invariant, which provides a measure of hypersurface orthogonality, we explore its potential role in selecting a preferred timelike Killing vector and consequently a unique ergo surface.     

Cold plasma dispersion relations in the vicinity of a Schwarzschild black hole horizon

We apply the ADM 3 + 1 formalism to derive the general relativistic magnetohydrodynamic equations for cold plasma in spatially flat Schwarzschild metric. Respective perturbed equations are linearized for non-magnetized and magnetized plasmas both in non-rotating and rotating backgrounds. These are then Fourier analyzed and the corresponding dispersion relations are obtained. These relations are discussed for the existence of waves with positive angular frequency in the region near the horizon. Our results support the fact that no information can be extracted from the Schwarzschild black hole. It is concluded that negative phase velocity propagates in the rotating background whether the black hole is rotating or non-rotating.     

Black hole solutions in heterotic string theory on a torus

We construct the general electrically charged, rotating black hole solution in the heterotic string theory compactified on a six-dimensional torus and study its classical properties. This black hole is characterized by its mass, angular momentum, and a 28-dimensional electric charge vector. We recover the axion-dilaton black holes and Kaluza-Klein black holes for special values of the charge vector. For a generic black hole of this kind, the 28-dimensional magnetic dipole moment vector is not proportional to the electric charge vector, and we need two different gyromagnetic ratios for specifying the relation between these two vectors. We also give an algorithm for constructing a 58 parameter rotating dyonic black hole solution in this theory, characterized by its mass, angular momentum, a 28-dimensional electric charge vector and a 28-dimensional magnetic charge vector. This is the most general asymptotically flat black hole solution in this theory consistent with the no-hair theorem.     

Rigidity in vacuum under conformal symmetry

Motivated in part by Eardley et al. (Commun Math Phys 106(1):137–158, 1986), in this note we obtain a rigidity result for globally hyperbolic vacuum spacetimes in arbitrary dimension that admit a timelike conformal Killing vector field. Specifically, we show that if M is a Ricci flat, timelike geodesically complete spacetime with compact Cauchy surfaces that admits a timelike conformal Killing field X, then M must split as a metric product, and X must be Killing. This gives a partial proof of the Bartnik splitting conjecture in the vacuum setting.     

Existence and stability of circular orbits in static and axisymmetric spacetimes

The existence and stability of timelike and null circular orbits (COs) in the equatorial plane of general static and axisymmetric (SAS) spacetime are investigated in this work. Using the fixed point approach, we first obtained a necessary and sufficient condition for the non-existence of timelike COs. It is then proven that there will always exist timelike COs at large \(\rho \) in an asymptotically flat SAS spacetime with a positive ADM mass and moreover, these timelike COs are stable. Some other sufficient conditions on the stability of timelike COs are also solved. We then found the necessary and sufficient condition on the existence of null COs. It is generally shown that the existence of timelike COs in SAS spacetime does not imply the existence of null COs, and vice-versa, regardless whether the spacetime is asymptotically flat or the ADM mass is positive or not. These results are then used to show the existence of timelike COs and their stability in an SAS Einstein-Yang-Mills-Dilaton spacetimes whose metric is not completely known. We also used the theorems to deduce the existence of timelike and null COs in some known SAS spacetimes.     

Supersymmetry of the extreme rotating toroidal black hole

We study the supersymmetry of the charged rotating toroidal black hole solutions found by Lemos and Zanchin, and show that the only configurations that are supersymmetric are: (i) the non-rotating electrically charged naked singularities already studied by Caldarelli and Klemm, and (ii) an extreme rotating toroidal black hole with zero magnetic and electric charges. For this latter case, the extreme uncharged black hole, we calculate the Killing spinors and show that the configuration preserves the same supersymmetries as the background spacetime.     

A Note on Classification of Spatially Homogeneous Rotating Space-Times According to Their Teleparallel Killing Vector Fields in Teleparallel Theory of Gravitation

In this paper we classify spatially homogeneous rotating space-timesaccording to their teleparallel Killing vector fields using direct integration technique. It turns out that the dimension of the teleparallel Killing vector fields is 5 or 10. In the case of 10 teleparallel Killing vector fields the space-time becomes Minkowski and all the torsion components are zero. Teleparallel Killing vector fields in this case are exactly the same as in general relativity. In the cases of 5 teleparallel Killing vector fields we get two more conservation laws in the teleparallel theory of gravitation. Here we also discuss some well-known examples of spatially homogeneous rotating space-times according to their teleparallel Killing vector fields.     

(Conformal) Killing Vectors in the Newman-Penrose Formalism

Finding (conformal) Killing vectors of a given metric can be a difficult task. This paper presents an efficient technique for finding Killing, homothetic, or even proper conformal Killing vectors in the Newman-Penrose (NP) formalism. Leaning on, and extending, results previously derived in the GHP formalism we show that the (conformal) Killing equations can be replaced by a set of equations involving the commutators of the Lie derivative with the four NP differential operators, applied to the four coordinates.It is crucial that these operators refer to a preferred tetrad relative to the (conformal) Killing vectors, a notion to be defined. The equations can then be readily solved for the Lie derivative of the coordinates, i.e. for the components of the (conformal) Killing vectors. Some of these equations become trivial if some coordinates are chosen intrinsically (where possible), i.e. if they are somehow tied to the Riemann tensor and its covariant derivatives.If part of the tetrad, i.e. part of null directions and gauge, can be defined intrinsically then that part is generally preferred relative to any Killing vector. This is also true relative to a homothetic vector or a proper conformal Killing vector provided we make a further restriction on that intrinsic part of the tetrad. If because of null isotropy or gauge isotropy, where part of the tetrad cannot even in principle be defined intrinsically, the tetrad is defined only up to (usually) one null rotation parameter and/or a gauge factor, then the NP-Lie equations become slightly more involved and must be solved for the Lie derivative of the null rotation parameter and/or of the gauge factor as well. However, the general method remains the same and is still much more efficient than conventional methods.Several explicit examples are given to illustrate the method.     

The analog of electric and magnetic fields in stationary gravitational systems

Newtonian and Machian aspects of the stationary gravitational field are brought into formal analogy with a stationary electromagnetic field. The electromagnetic vector potential equals (up to a factor) the timelike Killing vector field. The current density is given by the contraction of the Killing vector with the Ricci tensor. A coordinate-dependent split in electric and magnetic field vectors is given, and some results of classical electrodynamics are used to illustrate the analogy. In the linearized theory, the usual Maxwell equations are obtained. The analogy also holds from the point of view of particle motion. The geodesic equation is brought into a special form that exhibits an analog to the Lorentz force. Two examples (which have played an important role in the theoretical discovery of Machian effects) are considered.     

General nonextremal rotating black holes in minimal five-dimensional gauged supergravity

We construct the general solution for nonextremal charged rotating black holes in five-dimensional minimal gauged supergravity. They are characterized by four nontrivial parameters: namely, the mass, the charge, and the two independent rotation parameters. The metrics in general describe regular rotating black holes, providing the parameters lie in appropriate ranges so that naked singularities and closed timelike curves (CTCs) are avoided. We calculate the conserved energy, angular momenta, and charge for the solutions, and show how supersymmetric solutions arise in a Bogomol'nyi-Prasad-Sommerfield limit. These have naked CTCs in general, but for special choices of the parameters we obtain new regular supersymmetric black holes or smooth topological solitons.     

Relation between asymmetrical orbital angular momentum and irradiance-profile rotation for partially coherent beams

On the basis of the irradiance-moment formalism for describing general partially coherent beams, we investigate the relation between the spatial orientation of the transverse beam profile upon propagation and the asymmetrical part of its orbital angular momentum (OAM). More specifically, a necessary and sufficient condition (one-to-one correspondence) is shown between freely propagating non-rotating beams and vanishing asymmetrical OAM. As a corollary, it is obtained that any beam emerging from optical systems that transform a rotating field into a non-rotating beam exhibits vortex OAM only.     

Does Charge Contribute to the Frame Dragging of?Spacetime?

Electrically charged systems bound by a strong gravitational force can sustain a huge amount of electric charge (up to 10²⁰ C) against Coulomb repulsion. General relativistically such systems form a stable hydrostatic configuration both in the non-rotating and rotating cases. Here we study the effects of electric charge (electric energy density) on the spacetime outside a rotating electrically charged system bound by a strong gravitational force. In particular we investigate the effect of charge density on frame-dragging of spacetime in the exterior region. Using the coupled Einstein-Maxwell equations it is found that in the slow rotation approximation charge accumulation not only acts like an additional mass, thus modifying the spherically symmetric part of the spacetime, the electric charge also contributes directly to the dragging of spacetime. A modified Lense-Thirring formula for the spacetime frame dragging frequency is obtained and its implication for rotating charged compact stars is discussed.     

Arnowitt-Deser-Misner energy and g00

For a stationary, asymptotically flat space-time the “Komar energy”, associated with the time-like Killing vector and the ADM energy are equal when the latter is evaluated on a Cauchy surface which is asymptotically at rest relative to the Killing vector. The implicationd of this result on the positivity-of-energy problem in General Relativity are discussed.     

Solution Generating with Perfect Fluids

We apply a technique, due to Stephani, for generating solutions of the Einstein-perfect-fluid equations. This technique is similar to the vacuum solution generating techniques of Ehlers, Harrison, Geroch and others. We start with a seed solution of the Einstein-perfect-fluid equations with a Killing vector. The seed solution must either have (i) a spacelike Killing vector and equation of state P = or (ii) a timelike Killing vector and equation of state + 3P = 0. The new solution generated by this technique then has the same Killing vector and the same equation of state. We choose several simple seed solutions with these equations of state and where the Killing vector has no twist. The new solutions are twisting versions of the seed solutions.     

Grand Canonical Ensembles in General Relativity

We develop a formalism for general relativistic, grand canonical ensembles in space-times with timelike Killing fields. Using that, we derive ideal gas laws, and show how they depend on the geometry of the particular space-times. A systematic method for calculating Newtonian limits is given for a class of these space-times, which is illustrated for Kerr space-time. In addition, we prove uniqueness of the infinite volume Gibbs measure, and absence of phase transitions for a class of interaction potentials in anti-de Sitter space.