Stationary Axisymmetric Spacetimes Research Articles

Flat deformation of a spacetime admitting two commuting Killing fields

It is shown that, given an analytic Lorentzian metric on a 4-manifold, gab, which admits two Killing vector fields, there exists a local deformation law ηab = a gab + b Hab, where Hab is a two-dimensional projector, such that ηab is flat and admits the same Killing vectors. We also characterize the particular case when the projector Hab coincides with the quotient metric. We apply some of our results to general stationary axisymmetric spacetimes.

Spacetime encodings. IV. The relationship between Weyl curvature and Killing tensors in stationary axisymmetric vacuum spacetimes

The problem of obtaining an explicit representation for the fourth invariant of geodesic motion (generalized Carter constant) of an arbitrary stationary axisymmetric vacuum spacetime generated from an Ernst Potential is considered. The coupling between the non-local curvature content of the spacetime as encoded in the Weyl tensor, and the existence of a Killing tensor is explored and a constructive, algebraic test for a fourth order Killing tensor suggested. The approach used exploits the variables defined for the B\"{a}ckland transformations to clarify the relationship between Weyl curvature, constants of geodesic motion, expressed as Killing tensors, and the solution generation techniques. A new symmetric non-covariant formulation of the Killing equations is given. This formulation transforms the problem of looking for fourth-order Killing tensors in 4D into one of looking for four interlocking two-manifolds admitting fourth-order Killing tensors in 2D.

Rediscussion on Black Hole Angular Momentum

An exact formula calculating a stationary axisymmetric black hole’s angular momentum is given. As a check on the formula, the Kerr black hole’s angular momentum is calculated. To obtain a correct result, we do the integration computating the angular momentum over both the event horizon’s 2-dimensional section ∂ B + and the inner horizon’s 2-dimensional section ∂ B −, which is different from the definition of the original formula proposed by Bardeen and others. The angular momentums of two other categories of stationary axisymmetric non-Kerr black holes are calculated too. Because of the existing of the naked singularities, their angular momentums either go to infinity or can not be well-defined. These results indicate that the black hole thermodynamic first law can not be set up in these stationary axisymmetric non-Kerr black hole spacetimes. Without the Cosmic Censor Conjecture, the thermodynamic laws exclude these black holes.

Thin accretion disks in stationary axisymmetric wormhole spacetimes

In this paper, we study the physical properties and the equilibrium thermal radiation emission characteristics of matter forming thin accretion disks in stationary axially symmetric wormhole spacetimes. The thin disk models are constructed by taking different values of the wormhole’s angular velocity, and the time averaged energy flux, the disk temperature, and the emission spectra of the accretion disks are obtained. Comparing the mass accretion in a rotating wormhole geometry with the one of a Kerr black hole, we verify that the intensity of the flux emerging from the disk surface is greater for wormholes than for rotating black holes with the same geometrical mass and accretion rate. We also present the conversion efficiency of the accreting mass into radiation, and show that the rotating wormholes provide a much more efficient engine for the transformation of the accreting mass into radiation than the Kerr black holes. Therefore specific signatures appear in the electromagnetic spectrum of thin disks around rotating wormholes, thus leading to the possibility of distinguishing wormhole geometries by using astrophysical observations of the emission spectra from accretion disks.

Physical frames along circular orbits in stationary axisymmetric spacetimes

Three natural classes of orthonormal frames, namely Frenet-Serret, Fermi-Walker and parallel transported frames, exist along any timelike world line in spacetime. Their relationships are investigated for timelike circular orbits in stationary axisymmetric spacetimes, and illustrated for black hole spacetimes.

A new derivation of the conformally flat stationary cyclic non-circular spacetimes

We present an alternative way to derive the conformally flat stationary cyclic non-circular spacetimes. We show that there is no room for stationary axisymmetric non-circular axisymmetric spacetimes. We reproduce the well know results for this sort of spacetimes recently reported in [1].

Equatorial symmetry/antisymmetry of stationary axisymmetric electrovac spacetimes: II

In this second paper devoted to the equatorial symmetry/antisymmetry of stationary axisymmetric electrovac spacetimes, we show how two theorems proved in our previous paper (Ernst et al 2006 Class. Quantum Grav. 23 4945) can be utilized to construct exact solutions that are equatorially symmetric or antisymmetric.

TELEPARALLEL ENERGY–MOMENTUM DISTRIBUTION OF LEWIS–PAPAPETROU SPACETIMES

In this paper, we find the energy–momentum distribution of stationary axisymmetric spacetimes in the context of teleparallel theory by using Möller prescription. The metric under consideration is the generalization of the Weyl metrics called the Lewis–Papapetrou metric. The class of stationary axisymmetric solutions of the Einstein field equations has been studied by Galtsov to include the gravitational effect of an external source. Such spacetimes are also astrophysically important as they describe the exterior of a body in equilibrium. The energy density turns out to be nonvanishing and well-defined and the momentum becomes constant except along θ-direction. It is interesting to mention that the results reduce to the already available results for the Weyl metrics when we take ω = 0.

Lyapunov indices with two nearby trajectories in a curved spacetime

We compare three methods for computing invariant Lyapunov exponents (LEs) in general relativity. They involve the geodesic deviation vector technique (M1), the two-nearby-orbits method with projection operations and with coordinate time as an independent variable (M2), and the two-nearby-orbits method without projection operations and with proper time as an independent variable (M3). An analysis indicates that M1 and M3 do not need any projection operation. In general, the values of LEs from the three methods are almost the same. As an advantage, M3 is simpler to use than M2. In addition, we propose to construct the invariant fast Lyapunov indictor (FLI) with two-nearby-trajectories and give its algorithm in order to quickly distinguish chaos from order. Taking a static axisymmetric spacetime as a physical model, we apply the invariant FLIs to explore the global dynamics of phase space of the system where regions of chaos and order are clearlyidentified.

Equatorial symmetry/antisymmetry of stationary axisymmetric electrovac spacetimes

Two theorems are proved concerning how stationary axisymmetric electrovac spacetimes that are equatorially symmetric or equatorially antisymmetric can be characterized correctly in terms of the Ernst potentials and Φ or in terms of axis-data.

Energy of general four-dimensional stationary axisymmetric spacetime in the teleparallel geometry

A field equation with a cosmological constant term is derived and the energy of a general four-dimensional stationary axisymmetric spacetime is studied in the context of the Hamiltonian formulation of the teleparallel equivalent of general relativity (TEGR). We find that, by means of the integral form of the constraint equations of the formalism naturally without any restriction on the metric parameters, the energy for the asymptotically flat/de Sitter/anti-de Sitter stationary spacetimes in the Boyer–Lindquist coordinate can be expressed as . It is surprising to learn that the energy expression is only relevant to the metric components grr, gθθ and gϕϕ. As examples, by using this formula we calculate the energies of the Kerr–Newman (KN), Kerr–Newman anti-de Sitter (KN–AdS), Kaluza–Klein and Cvetič–Youm spacetimes.

Frenet–Serret formalism for null world lines

The Frenet–Serret curve analysis is extended from non-null to null trajectories in a generic spacetime using the Newman–Penrose formalism, recovering old results which are not well known and clarifying the associated Fermi–Walker transport which has been left largely unexplored in the literature. This machinery is then used to discuss null circular orbits in stationary axisymmetric spacetimes using the Kerr spacetime as a concrete example, and to integrate the equations of parallel transport along null geodesics in any spacetime.

Note on reflection symmetry in stationary axisymmetric electrovacuum spacetimes

Recently, Kordas (1995 Class. Quantum Grav. 12 2037) and Meinel and Neugebauer (1995 Class. Quantum Grav. 12 2045) studied the conditions for reflection symmetry in stationary axisymmetric spacetimes in vacuum. They found that a solution to the Einstein field equations is reflectionally symmetric if their Ernst potential (ρ = 0, z) = e(z) on a portion of the positive z-axis extending to infinity satisfies the condition e+(z)e*+(−z) = 1. In this note, we formulate analogous conditions for two complex Ernst potentials in electrovacuum. We also present the special case of rational axis potentials.

Supersymmetry, exact Foldy-Wouthuysen transformation, and gravity

Gravitational effects in relativistic quantum mechanics are investigated. The Dirac particle in general space-time metrics is considered and the Dirac Hamiltonian is constructed in terms of the Newman-Penrose formalism. To discuss the physical meaning of the Dirac Hamiltonian, it is necessary to perform the Foldy-Wouthuysen transformation. In most cases this transformation exists only in an approximate form. In this paper we show that for supersymmetric Dirac Hamiltonians not depending explicitly on time the exact Foldy-Wouthuysen transformation can always be constructed. Further, we derive criteria for spin coefficients for which the accompanying Dirac Hamiltonian is supersymmetric. These criteria are fulfilled by the class of static axisymmetric space-time metrics. For the subclass of stationary metrics, the exact Foldy-Wouthuysen transformation is calculated and the transformed Dirac Hamiltonian is derived. Recently, Obukhov constructed a different exact Foldy-Wouthuysen transformation for that class of space-time metrics and calculated the Dirac Hamiltonian in the Foldy-Wouthuysen representation. We show that the expansion series in orders of $1/m{c}^{2}$ of our and Obukhov's Dirac Hamiltonians coincide.

HAWKING RADIATION VIA TUNNELING FROM KERR BLACK HOLES

In this paper, Parikh's recent work, which treats Hawking radiation as a tunneling process in a spherically symmetric spacetime, is extended to the stationary axisymmetric Kerr spacetime. A generalized Painleve coordinate system (Painleve–Kerr coordinates) is introduced. The new coordinates are well-behaved at the event horizon. Constant-time slices are just flat Euclidean space in radial and satisfy Landau's condition of clock synchronization. The tunneling process across the Kerr black hole event horizon is investigated and the correctd radiation spectrum is obtained. The result is consistent with an underlying unitary theory.

Static axisymmetric spacetimes with prescribed multipole moments

In this paper we develop a method of finding the static axisymmetric spacetime corresponding to any given set of multipole moments. In addition to an implicit algebraic form for the general solution, we also give a power series expression for all finite sets of multipole moments. As conjectured by Geroch we prove in the special case of axisymmetry, that there is a static spacetime for any given set of multipole moments subject to a (specified) convergence criterion. We also use this method to confirm a conjecture of Hernández-Pastora and Martín concerning the monopole–quadrupole solution.

Corrections and comments on the multipole moments of axisymmetric electrovacuum spacetimes

Following the method of Hoenselaers and Perjés, we present a new corrected and dimensionally consistent set of multipole gravitational and electromagnetic moments for stationary axisymmetric spacetimes. Furthermore, we use our results to compute the multipole moments, both gravitational and electromagnetic, of a Kerr–Newman black hole.

GEOMETRIC TRANSPORT ALONG CIRCULAR ORBITS IN STATIONARY AXISYMMETRIC SPACETIMES

Parallel transport along circular orbits in orthogonally transitive stationary axisymmetric spacetimes is described explicitly relative to Lie transport in terms of the electric and magnetic parts of the induced connection. The influence of both the gravito-electromagnetic fields associated with the zero angular momentum observers and of the Frenet–Serret parameters of these orbits as a function of their angular velocity is seen on the behavior of parallel transport through its representation as a parameter-dependent Lorentz transformation between these two inner-product preserving transports which is generated by the induced connection. This extends the analysis of parallel transport in the equatorial plane of the Kerr spacetime to the entire spacetime outside the black hole horizon, and helps give an intuitive picture of how competing "central attraction forces" and centripetal accelerations contribute with gravitomagnetic effects to explain the behavior of the 4-acceleration of circular orbits in that spacetime.

A novel derivation for Kerr metric in Papapetrou gauge

We present a simple novel derivation, ab initio, of the equations appropriate for stationary axisymmetric spacetimes using the Papapetrou form of the metric (Papapetrou gauge). It is shown that using coordinates which preserve the Papapetrou gauge three separated solutions of the Ernst equations appear in the case of the Kerr metric. In this context a parameter arises which represents topological defects induced by an infinite static string along the z axis. Finally, we discuss a simple solution that may be derived from the Kerr ansatz.

Erratum: Conformally flat stationary axisymmetric spacetimes [Phys. Rev. D66, 124018 (2002)

It is shown that within conformally flat stationary axisymmetric spacetimes, besides of the static family, there exists a new class of metrics, which is always stationary and axisymmetric. All these spacetimes, the static and the stationary ones, are endowed with an arbitrary function depending on the two non--Killingian coordinates. The explicit form of this function can be determined once the coupled matter, i.e., the energy--momentum tensor is given. One might hope possible extensions of this result to black holes on two--branes in four dimensions.