Spinor fields and symmetries of the spacetime

We show that in the background of a stationary and axisymmetric black hole, there is a particular spinor field whose “conserved current” interpolates between the null Killing vector on the horizon and the time Killing vector at the spatial infinity. The spinor field only needs to satisfy a very general and simple constraint.     

Stationary spinning strings and symmetries of classical spacetimes

We explore the symmetries of classical stationary spacetimes in terms of the dynamics of a spinning string described by a worldsheet supersymmetric action. We show that for stationary configurations of the string, the action reduces to that for a pseudo-classical spinning point particle in an effective space, which is a conformally scaled quotient space of the original spacetime. As an example, we consider the stationary spinning string in the Kerr–Newman spacetime, whose motion is equivalent to that of the spinning point particle in the three-dimensional effective space. We present the Killing tensor as well as the spin-valued Killing vector of this space. However, the nongeneric supersymmetry corresponding to the Killing–Yano tensor of the Kerr–Newman spacetime is lost in the effective space.     

The commutation property of a stationary,axisymmetric system

It is shown that in studies of space-time systems which are both stationary and axisymmetric, no generality is lost by considering only cases where the stationary and axisymmetric actions (or equivalently the two corresponding Killing vector fields) commute.     

Noether versus Killing symmetry of conformally flat Friedmann metric

In a recent study Noether symmetries of some static spacetime metrics in comparison with Killing vectors of corresponding spacetimes were studied. It was shown that Noether symmetries provide additional conservation laws that are not given by Killing vectors. In an attempt to understand how Noether symmetries compare with conformal Killing vectors, we find the Noether symmetries of the flat Friedmann cosmological model. We show that the conformally transformed flat Friedman model admits additional conservation laws not given by the Killing or conformal Killing vectors. Inter alia, these additional conserved quantities provide a mechanism to twice reduce the geodesic equations via the associated Noether symmetries.     

A Higher Dimensional Stationary Rotating Black Hole Must be Axisymmetric

A key result in the proof of black hole uniqueness in 4-dimensions is that a stationary black hole that is “rotating”—i.e., is such that the stationary Killing field is not everywhere normal to the horizon—must be axisymmetric. The proof of this result in 4-dimensions relies on the fact that the orbits of the stationary Killing field on the horizon have the property that they must return to the same null geodesic generator of the horizon after a certain period, P. This latter property follows, in turn, from the fact that the cross-sections of the horizon are two-dimensional spheres. However, in spacetimes of dimension greater than 4, it is no longer true that the orbits of the stationary Killing field on the horizon must return to the same null geodesic generator. In this paper, we prove that, nevertheless, a higher dimensional stationary black hole that is rotating must be axisymmetric. No assumptions are made concerning the topology of the horizon cross-sections other than that they are compact. However, we assume that the horizon is non-degenerate and, as in the 4-dimensional proof, that the spacetime is analytic.     

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.     

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.     

On finding hypersurface orthogonal Killing fields

The conditions for a space-time to admit a hypersurface-orthogonal Killing vector field are studied. Tests are derived first for space-times that admit two commuting Killing vectors, both for the timelike-spacelike (TS) case and the spacelike-spacelike case (SS); these give the condition for linear combinations of the two known Killing vectors to be hypersurface orthogonal. Illustrations are given in some stationary axisymmetric (TS) metrics and cylindrically symmetric (SS) metrics. In the general case, conditions for static symmetry are already known. We give the similar conditions for a spacelike hypersurface-orthogonal Killing field.     

On the ‘Stationary Implies Axisymmetric’ Theorem for Extremal Black Holes in Higher Dimensions

All known stationary black hole solutions in higher dimensions possess additional rotational symmetries in addition to the stationary Killing field. Also, for all known stationary solutions, the event horizon is a Killing horizon, and the surface gravity is constant. In the case of non-degenerate horizons (non-extremal black holes), a general theorem was previously established [24] proving that these statements are in fact generally true under the assumption that the spacetime is analytic, and that the metric satisfies Einstein’s equation. Here, we extend the analysis to the case of degenerate (extremal) black holes. It is shown that the theorem still holds true if the vector of angular velocities of the horizon satisfies a certain “diophantine condition,” which holds except for a set of measure zero.     

Projective differential geometry and geodesic conservation laws in general relativity,II: Conservation laws

We examine the geodesic conservation laws associated with the projective actions discussed in our earlier paper with the same overall title. Using the Cartan formalism, a one-to-one correspondence between a class of these actions and all geodesic conservation laws is possible. In particular there is a natural geometric interpretation of Killing tensors. Homothetic motions are shown to correspond to conserved quantities on all geodesies (not just null ones). The same approach identifies homothetic Killing tensors and a universal quadratic first integral which reduces to the conformai Killing tensor case on null geodesics.     

An exact conformal symmetry Ansatz on Kaluza-Klein reduced TMG

Using a Kaluza-Klein dimensional reduction, and further imposing a conformal Killing symmetry on the reduced metric generated by the dilaton, we show an Ansatz that yields many of the known stationary axisymmetric solutions to TMG.     

Conditions for non-existence of static or stationary,Einstein–Maxwell,non-inheriting black-holes

We consider asymptotically-flat, static and stationary solutions of the Einstein equations representing Einstein–Maxwell space–times in which the Maxwell field is not constant along the Killing vector defining stationarity, so that the symmetry of the space-time is not inherited by the electromagnetic field. We find that static degenerate black hole solutions are not possible and, subject to stronger assumptions, nor are static, non-degenerate or stationary black holes. We describe the possibilities if the stronger assumptions are relaxed.     

Solving Bäcklund transformations with harmonic mappings via Killing vectors

On the basis of Killing vectors, a systematic method for solving Bäcklund transformations of the Euler equation in the harmonic mapping theory is presented. As an application, the Ernst equation for stationary axisymmetrical gravitation is discussed, and the Ehler transformation of that equation is obtained.     

State Independence for Tunnelling Processes Through Black Hole Horizons and Hawking Radiation

Tunnelling processes through black hole horizons have recently been investigated in the framework of WKB theory, discovering an interesting interplay with Hawking radiation. In this paper, we instead adopt the point of view proper of QFT in curved spacetime, namely, we use a suitable scaling limit towards a Killing horizon to obtain the leading order of the correlation function relevant for the tunnelling. The computation is done for a certain large class of reference quantum states for scalar fields, including Hadamard states. In the limit of sharp localization either on the external side or on opposite sides of the horizon, the quantum correlation functions appear to have thermal nature. In both cases the characteristic temperature is referred to the surface gravity associated with the Killing field and thus connected with the Hawking one. Our approach is valid for every stationary charged rotating non-extremal black hole. However, since the computation is completely local, it covers the case of a Killing horizon which just temporarily exists in some finite region, too. These results provide strong support to the idea that the Hawking radiation, which is detected at future null infinity and needs some global structures to be defined, is actually related to a local phenomenon taking place even for local geometric structures (local Killing horizons), existing just for a while.     

Algebraic structure of general electromagnetic fields and energy flow

The algebraic structures of a general electromagnetic field and its energy–momentum tensor in a stationary space–time are analyzed. The explicit form of the reference frame in which the energy of the field appears at rest is obtained in terms of the eigenvectors of the electromagnetic tensor and the existing Killing vector. The case of a stationary electromagnetic field is also studied and a comparison is made with the standard short-wave approximation. The results can be applied to the general case of a structured light beams, in flat or curved spaces. Bessel beams are worked out as example.     

Global and Uniqueness Properties of Stationary and Static Spacetimes with Outer Trapped Surfaces

Global properties of maximal future Cauchy developments of stationary, m-dimensional asymptotically flat initial data with an outer trapped boundary are analyzed. We prove that, whenever the matter model is well posed and satisfies the null energy condition, the future Cauchy development of the data is a black hole spacetime. More specifically, we show that the future Killing development of the exterior of a sufficiently large sphere in the initial data set can be isometrically embedded in the maximal Cauchy development of the data. In the static setting we prove, by working directly on the initial data set, that all Killing prehorizons are embedded whenever the initial data set has an outer trapped boundary and satisfies the null energy condition. By combining both results we prove a uniqueness theorem for static initial data sets with outer trapped boundary.     

Birkhoff系统的一类新型守恒量

给出了Birkhoff系统的一类新型守恒量。首先，建立了Birkhoff系统的运动方程及其Mei对称性的定义和判据；其次，给出了系统的一类新型守恒量的存在定理，并导出了用于确定无限小生成元的广义Killing方程；最后，建立了守恒定理的逆定理 关键词： Birkhoff系统 Mei对称性 守恒量 Killing方程     

Motions in Taub-NUT–de Sitter spinning spacetime

We investigate the geodesic motion of pseudo-classical spinning particles in the Taub-NUT–de Sitter spacetime. We obtain the conserved quantities from the solutions of the generalized Killing equations for spinning spaces. Applying the formalism the motion of a pseudo-classical Dirac fermion is analyzed on a cone and plane.     

A comment on the uniqueness theorems for stationary black holes

A simple local geometric condition is given that is sufficient to restrict the possible variety of exterior fields of all stationary axisymmetric black hole spaces to depend only on a finite number of parameters. It is discussed how this condition could be used to gain insight into the nature of the Carter-Robinson uniqueness theorem. Also a new coordinate system is constructed for all stationary axially symmetric space-times possessing a bifurcate Killing horizon. It covers a whole neighborhood of the horizons and of the bifurcation axis and possesses special geometric properties that are easy to visualize. The Kerr metric together with its spin coefficients and Weyl tensor components are described in the new coordinates.     

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.