Stationary Axisymmetric Spacetimes Research Articles

Boyer–Lindquist Space-Times and Beyond: Metamaterial Analogues for Arbitrary Space-Times

Analogue space-times (and in particular metamaterial analogue space-times) have a long varied and rather complex history. Much of the previous related work to this field has focused on spherically symmetric models; however, axial symmetry is much more relevant for mimicking astrophysically interesting systems that are typically subject to rotation. Now it is well known that physically reasonable stationary axisymmetric space-times can, under very mild technical conditions, be put into Boyer–Lindquist form. Unfortunately, a metric presented in Boyer–Lindquist form is not well adapted to the “quasi-Cartesian” metamaterial analysis that we developed in our previous articles on “bespoke analogue space-times”. In the current article, we shall first focus specifically on various space-time metrics presented in Boyer–Lindquist form, and subsequently determine a suitable set of equivalent metamaterial susceptibility tensors in a laboratory setting. We shall then turn to analyzing generic space-times, not even necessarily stationary, again determining a suitable set of equivalent metamaterial susceptibility tensors. Perhaps surprisingly, we find that the well-known ADM formalism proves to be not particularly useful, and that it is instead the dual “threaded” (Kaluza–Klein–inspired) formalism that provides much more tractable results. While the background laboratory metric is (for mathematical simplicity and physical plausibility) always taken to be Riemann flat, we will allow for arbitrary curvilinear coordinate systems on the flat background space-time. Finally, for completeness, we shall reconsider spherically symmetric space-times, but now in general spherical polar coordinates rather than quasi-Cartesian coordinates. In summary, this article provides a set of general-purpose calculational tools that can readily be adapted for mimicking various interesting (curved) space-times by using nontrivial susceptibility tensors in general (background-flat) laboratory settings.

Rotating regular black holes and other compact objects with a Tolman-type potential as a regular interior for the Kerr metric

In this paper, we obtain a new class of stationary axisymmetric spacetimes by using the Gürses–Gürsey metric with an appropriate mass function in order to generate a rotating core of matter that may be smoothly matched to the exterior Kerr metric. The same stationary spacetimes may be obtained by applying a slightly modified version of the Newman–Janis algorithm to a nonrotating spherically symmetric seed metric. The starting spherically symmetric configuration represents a nonisotropic de Sitter-type fluid whose radial pressure [Formula: see text] satisfies an state equation of the form [Formula: see text], where the energy density [Formula: see text] is chosen to be the Tolman-type-VII energy density [R. C. Tolman, Phys. Rev. 55, 364 (1939)]. The resulting rotating metric is then smoothly matched to the exterior Kerr metric, and the main properties of the obtained geometries are investigated. All the solutions considered in this study are regular in the sense they are free of curvature singularities. Depending on the relative values of the total mass m and rotation parameter a, the resulting stationary spacetimes represent different kinds of rotating compact objects such as regular black holes, extremal regular black holes, and regular starlike configurations.

Stable photon orbits in stationary axisymmetric spacetimes with an electromagnetic field and a cosmological constant

Stable photon orbits in stationary axisymmetric spacetimes with an electromagnetic field and a cosmological constant

Explicit formulae for surface gravities in stationary circular axi-symmetric spacetimes

Using minimalist assumptions we develop a natural functional decomposition for the spacetime metric, and explicit tractable formulae for the surface gravities, in arbitrary stationary circular (PT symmetric) axisymmetric spacetimes. We relate rigidity results, (the existence of a Killing horizon), and the zeroth law to the absence of curvature singularities at the would-be horizons. These observations are of interest to both observational astrophysicists (modelling the cold, dark, heavy objects at the center of most spiral galaxies), and to the analogue spacetime community, (wherein the presence of naked singularities is not necessarily deprecated, and the occurrence of non-Killing horizons is relatively common).

Vertical Equilibrium Model for Accretion Flow: An Exact General Relativistic Self-consistent Analysis

We provide a complete general relativistic and fluid dynamical treatment to derive an expression for the relativistic accretion rate in curved spacetime. We exploit the Killing symmetry of stationary axisymmetric spacetime to obtain the conserved current, using which we find an integral relation between mass flux and height of the accretion disk. In comparison with the existing different models of thin- and slim-disk approaches our analysis is self-consistent and mathematically rigorous. We reviewed the previous work and surveyed the recent status in this field. Our formula is different from the existing expression in the literature. The difference arises solely due to the rotation parameter of the compact star. The effect becomes important for slim/thick disks and for rapidly spinning compact stars. Thus one needs to alter the formula for Newtonian or spherically symmetric spacetime while studying the accretion flows around a Kerr black hole. Our formula can also be used to connect the variability of the vertical structure of the accretion disk through the luminosity variation of X-ray binaries, and can thereby explain the origin of the hysteresis loop of the spectral states.

Metric for two unequal extreme Kerr-Newman black holes

In the present paper, within the framework of stationary axisymmetric spacetimes, binary systems composed of two unequal co- and counter-rotating extreme Kerr-Newman black holes separated by a massless strut are reported. The metric describing both configurations is introduced in a closed analytical form in terms of five arbitrary parameters: the masses Mi, electric charges Qi, and a coordinate distance R. We obtain novel results from these configurations; in particular, those related to the merging process.

Repulsive Casimir force in stationary axisymmetric spacetimes

We study the influence of stationary axisymmetric spacetimes on Casimir energy. We consider a massive scalar field and analyze its dependence on the apparatus orientation with respect to the dragging direction associated with such spaces. We show that, for an apparatus orientation not considered before in the literature, the Casimir energy can change its sign, producing a repulsive force. As applications, we analyze two specific metrics: one associated with a linear motion of a cylinder and a circular equatorial motion around a gravitational source described by Kerr geometry.

Physical interpretation of Newman-Janis rotating systems. II. General systems

Drake and Szekeres have extended the Newman-Janis algorithm to produce stationary axisymmetric spacetimes from general static spherically symmetric solutions of the Einstein equations. The algorithm mathematically generates an energy-momentum tensor for the rotating solution, but the rotating and nonrotating system may or may not represent the same physical system, in the sense of both being a perfect fluid, or an electromagnetic field, or a $\Lambda$-term, and so on. In Part I (arxiv:2104.02255), we compared the structure of the eigenvalues and eigenvectors of the rotating and nonrotating energy-momentum tensors (their Segre types) and looked for the existence of equations of state relating the rotating energy density and principal pressures for Kerr-Schild systems. Here we extend our analysis to general static spherically symmetric systems obtained according to the Drake-Szekeres generalization of the Newman-Janis algorithm. We find that these rotating systems can have almost all Segre types except [31] and [(31)]. Moreover, the Segre type of the spacetime can change severely in passing from the nonrotating to the rotating configurations, for example to $[11Z\bar{Z}]$ from seed systems which were initially [(111,1)]. We also find conditions dictating how many equations of state may exist in a Drake-Szekeres system.

The effect of stationary axisymmetric spacetimes in interferometric visibility

In this article, we consider a scenario in which a spin-1/2 quanton goes through a superposition of co-rotating and counter-rotating circular paths, which play the role of the paths of a Mach–Zehnder interferometer in a stationary and axisymmetric spacetime. Since the spin of the particle plays the role of a quantum clock, as the quanton moves in a superposed path it gets entangled with the momentum (or the path), and this will cause the interferometric visibility (or the internal quantum coherence) to drop, since, in stationary axisymmetric spacetimes there is a difference in proper time elapsed along the two trajectories. However, as we show here, the proper time of each path will couple to the corresponding local Wigner rotation, and the effect in the spin of the superposed particle will be a combination of both. Besides, we discuss a general framework to study the local Wigner rotations of spin-1/2 particles in general stationary axisymmetric spacetimes for circular orbits.

Universal properties of light rings for stationary axisymmetric spacetimes

Light rings (LRs) play an important role in gravitational wave observations and black hole photographs. In this paper, we investigate general features of LRs in stationary, axisymmetric, asymptotically flat spacetimes with or without horizons. For a nonextremal black hole, we show explicitly that there always exist at least two LRs propagating in opposite directions, where the outermost one is radially unstable. For an extremal black hole, we show that there exists at least one retrograde LR. We find that there is at least one LR which is angularly stable. The stability analysis does not involve any energy condition. Our method also applies to horizonless spacetimes and we prove that LRs always appear in pairs. Only some natural and generic assumptions are used in our proof. The results are applicable to general relativity as well as most modified theories of gravity. In contrast to previous works on this issue, we obtain much stronger results with a much more straightforward approach.

On homothetic Killing vectors in stationary axisymmetric vacuum spacetimes

In this paper, we consider homothetic Killing vectors in the class of stationary axisymmetric vacuum (SAV) spacetimes, where the components of the vectors are functions of the time and radial coordinates. In this case, the component of any homothetic Killing vector along the [Formula: see text] direction must be constant. First, it is shown that either the component along the radial direction is constant or we have the proportionality [Formula: see text], where [Formula: see text]. In both cases, complete analyses are carried out and the general forms of the homothetic Killing vectors are determined. The associated conformal factors are also obtained. The case of vanishing twist in the metric, i.e. [Formula: see text] is considered and the complete forms of the homothetic Killing vectors are determined, as well as the associated conformal factors.

Constraining the spacetime spin using time delay in stationary axisymmetric spacetimes

Total travel time t and time delay Delta t between images of gravitational lensing (GL) in the equatorial plane of stationary axisymmetric (SAS) spacetimes for null and timelike signals with arbitrary velocity are studied. Using a perturbative method in the weak field limit, t in general SAS spacetimes is expressed as a quasi-series of the impact parameter b with coefficients involving the source-lens distance r_s and lens-detector distancesr_d, signal velocity v, and asymptotic expansion coefficients of the metric functions. The time delay Delta t to the leading order(s) were shown to be determined by the spacetime mass M, spin angular momentum a and post-Newtonian parameter gamma , and kinematic variables r_s,~r_d,~v and source angular position beta . When beta ll sqrt{aM}/r_{s,d}, Delta t is dominated by the contribution linear to spin a. Modeling the Sgr A* supermassive black hole as a Kerr–Newman black hole, we show that as long as beta lesssim 1.5times 10^{-5} [^{prime prime }], then Delta t will be able to reach the mathcal {O}(1) second level, which is well within the time resolution of current GRB, gravitational wave and neutrino observatories. Therefore measuring Delta t in GL of these signals will allow us to constrain the spin of the Sgr A*.

Perturbative deflection angle for signal with finite distance and general velocities

We propose a perturbative method to compute the deflection angle of both null and massive particles for source and detector at finite distance. This method applies universally to the motion of particles with general velocity in the equatorial plane of stationary axisymmetric spacetimes or static spherical symmetric spacetimes that are asymptotically flat. The resultant deflection angle automatically arranges into a quasi-inverse series form of the impact parameter, with coefficients depending on the metric functions, the signal velocity and the source and detector locations through the apparent angles. In the large impact parameter limit, the series coefficients are reduced to rational functions of sine/cosine functions of the zeroth order apparent angle.

Photon regions and umbilic conditions in stationary axisymmetric spacetimes

Photon region (PR) in the strong gravitational field is defined as a compact region where photons can travel endlessly without going to infinity or disappearing at the event horizon. In Schwarzschild metric PR degenerates to the two-dimensional photon sphere r=3r_g/2 where closed circular photon orbits are located. The photon sphere as a three-dimensional hypersurface in spacetime is umbilic (its second quadratic form is pure trace). In Kerr metric the equatorial circular orbits have different radii for prograde, r_p, and retrograde, r_r, motion (where r is Boyer–Lindquist radial variable), while for r_p<r<r_r the spherical orbits with constant r exist which are no more planar, but filling some spheres. These spheres, however, do not correspond to umbilic hypersurfaces. In more general stationary axisymmetric spacetimes not allowing for complete integration of geodesic equations, the numerical integration show the existence of PR as well, but the underlying geometric structure was not fully identified so far. Here we suggest geometric description of PR in generic stationary axisymmetric spacetimes, showing that PR can be foliated by partially umbilic hypersurfaces, such that the umbilic condition holds for classes of orbits defined by the foliation parameter. New formalism opens a way of analytic description of PR in stationary axisymmetric spacetimes with non-separable geodesic equations.

Dirac spin and angular momentum of stationary axisymmetric space–time in the teleparallel gravity

In the framework of a teleparallel equivalent to general relativity, stationary axisymmetric space–time in the teleparallel gravity for two different sets of tetrad fields have been investigated. For these sets, we have obtained the expressions for the torsion vector, torsion axial-vector and the angular momentum of the solution. We found that the obtained expressions of the torsion axial-vector and the angular momentum are, in general, quite different in both sets of tetrad fields, while the expressions for the torsion vector have the same value. Moreover, the vector part connected with Dirac spin has been evaluated as well. Finally, special cases of the stationary axisymmetric space–time are discussed.

Dirac fermion, cosmological event horizons, and quantum entanglement

We discuss the field quantisation of a free massive Dirac fermion in the two causally disconnected static patches of the de Sitter spacetime, by using mode functions that are normalisable on the cosmological event horizon. Using this, we compute the entanglement entropy of the vacuum state corresponding to these two regions, for a given fermionic mode. Further extensions of this result to more general static spherically symmetric and stationary axisymmetric spacetimes are discussed. For the stationary axisymmetric Kerr-de Sitter spacetime in particular, the variations of the entanglement entropy with respect to various eigenvalues and spacetime parameters are depicted numerically. We also comment on such variations when instead we consider the non-extremal black hole event horizon of the same spacetime.

Completing characterization of photon orbits in Kerr and Kerr-Newman metrics

Recently, several new characteristics have been introduced to describe null geodesic structure of stationary spacetimes, such as photon regions (PR) and transversely trapping surfaces (TTS). The former are three-dimensional domains confining the spherical photon orbits, while the latter are closed two-surface of spherical topology which fill other regions called TTR. It is argued that in generic stationary axisymmetric spacetime it is natural to consider also the non-closed TTSs of the geometry of spherical cups, satisfying the same conditions ("partial" TTS or PTTS), which fill the three-dimensional regions, PTTR. We then show that PR, TTR and PTTR together with the corresponding anti-trapping regions constitute the complete set of regions filling the entire three-space (where timelike surfaces are defined) of Kerr-like spacetimes. This construction provides a novel optical description of such spacetimes without recurring to explicit solution of the geodesic equations. Applying this analysis to Kerr-Newman metrics (including the overspinning ones) we reveal four different optical types for different sets of the rotation and charge parameters. To illustrate their properties we extend Synge analysis of photon escape in the Schwarzschild metric to stationary spacetimes and construct density graphs describing escape of photons from all the above regions.

Limitations of the pseudo-Newtonian approach in studying the accretion flow around a Kerr black hole

We study the relativistic accretion flow around a generic stationary axisymmetric space-time and obtain an effective potential ($\Phi^{\rm eff}$) that accurately mimics the general relativistic features of Kerr black hole having spin $0\le a_k <1$. Considering the accretion disc confined around the equatorial plane of a rotating black hole and using the relativistic equation of state, we examine the properties of the relativistic accretion flow and compare it with the same obtained form semi-relativistic as well as non-relativistic accretion flows. Towards this, we first investigate the transonic properties of the accretion flow around the rotating black hole where good agreement is observed for relativistic and semi-relativistic flows. Further, we study the non-linearities such as shock waves in accretion flow. Here also we find that the shock properties are in agreement for both relativistic and semi-relativistic flows irrespective of the black hole spin ($a_k$), although it deviates significantly for non-relativistic flow. In fact, when the particular shocked solutions are compared for flows with identical outer boundary conditions, the positions of shock transition in relativistic and semi-relativistic flows agree well with deviation of $6-12\%$ for $0 \le a_k \le 0.99$, but vast disagreement is observed for non-relativistic flow. In addition, we compare the parameter space for shock to establish the fact that relativistic as well as semi-relativistic accretion flow dynamics do show close agreement irrespective of $a_k$ values, whereas non-relativistic flow fails to do so. With these findings, we point out that semi-relativistic flow including $\Phi^{\rm eff}$ satisfactorily mimics the relativistic accretion flows around Kerr black hole. Finally, we discuss the possible implications of this work in the context of dissipative advective accretion flow around Kerr black holes.

Axisymmetric Pure Radiation Space–Time with Causality-Violating Geodesics

We present a stationary axisymmetric space–time admitting circular closed timelike geodesics everywhere within a finite region of space. The space–time is free from curvature divergence and is locally isometric to a nonvacuum pp-wave space–time. The matter–energy content is a pure radiation field and satisfies the null energy condition (NEC), and the metric is of type N in the Petrov classification scheme. Finally, we demonstrate the existence of timelike and null circular geodesic paths for this metric.

Carter's constant and superintegrability

Carter's constant is a nontrivial conserved quantity of motion of a particle moving in stationary axisymmetric spacetime. In the version of the theorem originally given by Carter, due to the presence of two Killing vectors, the system effectively has two degrees of freedom. We propose an extension to the first version of Carter's theorem to a system having three degrees of freedom to find two functionally independent Carter-like integrals of motion. We further generalize the theorem to a dynamical system with [Formula: see text] degrees of freedom. We further study the implications of Carter's constant to superintegrability and present a different approach to probe a superintegrable system. Our formalism gives another viewpoint to a superintegrable system using the simple observation of separable Hamiltonian according to Carter's criteria. We then give some examples by constructing some two-dimensional superintegrable systems based on this idea and also show that all three-dimensional simple classical superintegrable potentials are also Carter separable.