An exact solution of Einstein's field equations in empty space first found in 1985 by Quevedo and Mashhoon is analyzed in detail. This solution generalizes Kerr spacetime to include the case of matter with arbitrary mass quadrupole moment and is specified by three parameters, the mass $M$, the angular momentum per unit mass $a$ and the quadrupole parameter $q$. It reduces to the Kerr spacetime in the limiting case $q=0$ and to the Erez-Rosen spacetime when the specific angular momentum $a$ vanishes. The geometrical properties of such a solution are investigated. Causality violations, directional singularities and repulsive effects occur in the region close to the source. Geodesic motion and accelerated motion are studied on the equatorial plane which, due to the reflection symmetry property of the solution, turns out to be also a geodesic plane.
The Kerr metric is known to present issues when trying to find an interior solution. In this work we continue in our efforts to construct a more realistic exterior metric for astrophysical objects. A new approximate metric representing the spacetime of a charged, rotating and slightly-deformed body is obtained by perturbing the Kerr-Newman metric to include the mass-quadrupole and quadrupole-quadrupole orders. It has a simple form, because is Kerr-Newman-like. Its post-linear form without charge coincides with post-linear quadrupole-quadrupole metrics already found.
A new approximate metric representing the spacetime of a rotating deformed body is obtained by perturbing the Kerr metric to include til the second order of the quadrupole moment. It has a simple form, because is Kerr-like. Its Taylor expansion form coincides with second order quadrupole metrics with slow rotation already found. Moreover, it can be transformed to an improved Hartle-Thorne metric, this guarantees its validity to be useful in studying compact object, and that it is possible to find an inner solution.
The features of equatorial motion of an extended body in Kerr spacetime are investigated in the framework of the Mathisson-Papapetrou-Dixon model. The body is assumed to stay at quasi-equilibrium and respond instantly to external perturbations. Besides the mass, it is completely determined by its spin, the multipolar expansion being truncated at the quadrupole order, with a spin-induced quadrupole tensor. The study of the radial effective potential allows to analytically determine the ISCO shift due to spin and the associated frequency of the last circular orbit.
We discuss motions of extended bodies in Kerr spacetime by using Mathisson-Papapetrou-Dixon equations. We firstly solve the conditions for circular orbits, and calculate the orbital frequency shift due to the mass quadrupoles. The results show that we need not consider the spin-induced quadrupoles in extreme-mass-ratio inspirals for spatial gravitational wave detectors. We quantitatively investigate the temporal variation of rotational velocity of the extended body due to the coupling of quadrupole and background gravitational field. For generic orbits, we numerically integrate the Mathisson-P…
The equatorial motion of extended bodies in a Kerr spacetime is investigated in the framework of the Mathisson-Papapetrou-Dixon model, including the full set of effective components of the quadrupole tensor. The numerical integration of the associated equations shows the specific role of the mass and current quadrupole moment components. While most of the literature on this topic is limited to spin-induced (purely electric) quadrupole tensor, the present analysis highlights the effect of a completely general quadrupole tensor on the dynamics. The contribution of the magnetic-type components is…
Extracting Energy from a Non-Kerr Rotating Spacetime with an Anomalous Quadrupole Moment via Magnetic Reconnection
School of Material Science and Engineering, Chongqing Jiaotong University（重庆交通大学材料科学与工程学院）；College of Physics and Optoelectronic Engineering, Chongqing Normal University（重庆师范大学物理与光电工程学院）
The innermost stable circular orbit equation of a test particle is obtained for an approximate Kerr-like spacetime with quadrupole moment. We derived the effective potential for the radial coordinate by the Euler-Lagrange method. This equation can be employed to measure the mass quadrupole by observational means, because from this equation a quadratic polynomial for the quadrupole moment can be found. As expected, the limiting cases of this equation are found to be the known cases of Kerr and Schwarzschild.
The motion of a body endowed with dipolar as well as quadrupolar structure is investigated in the Kerr background according to the Dixon's model, extending a previous analysis done in the Schwarzschild background. The full set of evolution equations is solved under the simplifying assumptions of constant frame components for both the spin and the quadrupole tensors and that the center of mass moves along an equatorial circular orbit, the total four-momentum of the body being aligned with it. We find that the motion deviates from the geodesic one due to the internal structure of the body, leadi…
Measurability of Quadrupole Deviations from Kerr in Binary black hole Mergers
Shanghai Astronomical Observatory（上海天文台）；School of Astronomy and Space Science, University of Chinese Academy of Sciences（中国科学院大学天文与空间科学学院）；School of Fundamental Physics and Mathematical Sciences, Hangzhou Institute for Advanced Study, UCAS（杭州高等研究院基础物理与数学学院）；International Centre for Theoretical Physics Asia-Pacific（亚太理论物理中心）
We present a new solution in Einstein-Maxwell theory which can be considered as the magnetized version of Kerr-Taub-NUT solution. Some properties of the spacetime are discussed. We also compute the entropy of extremal black hole in the spacetime by using the Kerr/CFT correspondence approach.
We describe a new numerical algorithm for ray tracing in the external spacetimes of spinning compact objects characterized by arbitrary quadrupole moments. Such spacetimes describe non-Kerr vacuum solutions that can be used to test the no-hair theorem in conjunction with observations of accreting black holes. They are also appropriate for neutron stars with spin frequencies in the 300-600 Hz range, which are typical of the bursting sources in low-mass X-ray binaries. We use our algorithm to show that allowing for the quadrupole moment of the spacetime to take arbitrary values leads to observab…
An exact expression derived in the literature for the rate of dragging of inertial frames (Lense-Thirring (LT) precession) in a general stationary spacetime, is reviewed. The exact LT precession frequencies for Kerr, Kerr-Taub-NUT and Taub-NUT spacetimes are explicitly derived. Remarkably, in the case of the zero angular momentum Taub-NUT spacetime, the frame-dragging effect is shown {\it not} to vanish, when considered for spinning test gyroscopes. The result becomes sharper for the case of vanishing ADM mass of that spacetime. We clarify how our results are consistent with claims in the rece…
It is not currently known how to put the Kerr spacetime metric into the so-called Gordon form, although the closely related Kerr-Schild form of the Kerr metric is well known. A Gordon form for the Kerr geometry, if it could be found, would be particularly useful in developing analogue models for the Kerr spacetime, since the Gordon form is explicitly given in terms of the 4-velocity and "refractive index" of an effective medium. In the current article we report progress toward this goal. First we present the Gordon form for an approximation to Kerr spacetime in the slow-rotation limit, obtaine…
The deviation from geodesic motion of the world line of an extended body endowed with multipolar structure up to the mass quadrupole moment is studied in the Kerr background according to the Mathisson-Papapetrou-Dixon model. The properties of the quadrupole tensor are clarified by identifying the relevant components which enter the equations of motion, leading to the definition of an effective quadrupole tensor sharing its own algebraic symmetries, but also obeying those implied by the Mathisson-Papapetrou-Dixon model itself. The equations of motion are then solved analytically in the limit of…
Testing the Kerr hypothesis beyond the quadrupole with GW241011
Indian Institute of Technology Madras（印度马德拉斯理工学院）；University of Birmingham（伯明翰大学）；The University of Texas at Austin（得克萨斯大学奥斯汀分校）；Chennai Mathematical Institute（金奈数学研究所）；Max Planck Institute for Gravitational Physics (Albert Einstein Institute)（马克斯·普朗克引力物理研究所（爱因斯坦研究所））；Leibniz Universität Hannover（汉诺威莱布尼茨大学）
Equatorial motion of test particles in the Kerr-de Sitter spacetimes is considered. Circular orbits are determined, their properties are discussed for both the black-hole and naked-singularity spacetimes, and their relevance for thin accretion discs is established.
The static Kottler metric is the Schwarzschild vacuum metric extended to include a cosmological constant. Angular momentum is added to the Kottler metric by using Newman and Janis' complexifying algorithm. The new metric is the Lambda generalization of the Kerr spacetime. It is stationary, axially symmetric, Petrov type II, and has Kerr-Schild form.
The astrophysical importance of the Kerr spacetime cannot be overstated. Of the currently known exact solutions to the Einstein field equations, the Kerr spacetime stands out in terms of its direct applicability to describing astronomical black hole candidates. In counterpoint, purely mathematically, there is an old classical result of differential geometry, due to Darboux, that all 3-manifolds can have their metrics recast into diagonal form. In the case of the Kerr spacetime the Boyer-Lindquist coordinates provide an explicit example of a diagonal spatial 3-metric. Unfortunately, as we demon…
Using the energy momentum complex given by Møller in 1978 based on the absolute parallelism, the energy distribution in Kerr spacetime is evaluated. The energy with this spacetime is found to be the same as it was earlier evaluated using different definitions mainly based on the metric tensor.
We provide a prescription to solve the metric completion problem in gravitational self-force calculations on a Kerr spacetime by fixing the remaining gauge freedom. We discuss the explicit example of eccentric equatorial orbits, recovering all limiting cases already studied in the literature of eccentric orbits in Schwarzschild as well as circular orbits in both Schwarzschild and Kerr spacetimes.
We prove, outside the influence region of a ball of radius $R_0$ centered in the origin of the initial data hypersurface, the existence of global solutions near to Kerr spacetime, provided that the initial data are sufficiently near to those of Kerr; This external region is the "far" part of the outer region of the perturbed Kerr spacetime. Moreover if we assume that the corrections to the Kerr metric decay sufficiently fast, $o(r^{-3})$, we prove that the various null components of the Riemann tensor decay in agreement with the "Peeling theorem".
We will show that the Nariai metric, i.e. the static spherically symmetric vacuum spacetime with a cosmological constant, admits a conformally Kerr-Schild spacetime representation. We find the vacuum solutions of the Einstein-Maxwell equations for the Nariai metric using the Horsky-Mitskievich generating conjecture.
We have studied numerically the shadows of a non-Kerr rotating compact object with quadrupole mass moment, which belongs to Manko-Novikov family. The non-integrable photon motion caused by quadrupole mass moment affects sharply the shadow of the compact object. As the deviation parameter related to quadrupole mass moment is negative, the shadow of compact object is prolate and there are two disconnected main shadows with eyebrows located symmetrically on both sides of the equatorial plane. As the deviation parameter is positive, the shadow becomes oblate and the main shadow is joined together …
The Kerr spacetime is perhaps the most astrophysically important of the currently known exact solutions to the Einstein field equations. Whenever spacetimes can be put in unit-lapse form it becomes possible to identify some very straightforward timelike geodesics, (the "rain" geodesics), making the physical interpretation of these spacetimes particularly clean and elegant. The most well-known of these unit-lapse formulations is the Painleve-Gullstrand form of the Schwarzschild spacetime, though there is also a Painleve-Gullstrand form of the Lense-Thirring (slow rotation) spacetime. More radic…
We consider the Maxwell field in the exterior of a Kerr black hole. For this system, we propose a geometric construction of generalized Klein-Gordon equation called Fackerell-Ipser equation. Our model is based on conformal Yano-Kiling tensor (CYK tensor). We present non-standard properties of CYK tensors in the Kerr spacetime which are useful in electrodynamics.
We obtain a characterization of the Kerr metric among stationary, asymptotically flat, vacuum spacetimes, which extends the characterization in terms of the Simon tensor (defined only in the manifold of trajectories) to the whole spacetime. More precisely, we define a three index tensor on any spacetime with a Killing field, which vanishes identically for Kerr and which coincides in the strictly stationary region with the Simon tensor when projected down into the manifold of trajectories. We prove that a stationary asymptotically flat vacuum spacetime with vanishing spacetime Simon tensor is l…
In this work we study the quasilocal energy as in [11] for a constant radius surface in Kerr spacetime in Boyer-Lindquist coordinates. We show that under suitable conditions for isometric embedding, for a stationary observer the quasilocal energy defined in [11] for constant radius in a Kerr like spacetime is exactly equal to the Brown-York quasilocal energy [2]. By some careful estimations, we show that for a constant radius surface in the Kerr spacetime which is outside the ergosphere the embedding conditions for the previous result are satisfied. We discuss extremal solutions as described i…
In this article we study the innermost stable circular orbit (ISCO) of electrically charged particles in the electrically charged Reissner-Nordström spacetime, the Kerr-Newman spacetime and the Kerr-Sen spacetime. We find that the radius of the ISCO increases with an increasing particle-black hole charge product $|qQ|$ in the case of attractive Coulomb interaction $qQ<0$. For repulsive Coulomb interaction, the ISCO radius first decreases to a minimum and then increases again, until it diverges as the charge product approaches one. If the charge $Q$ of the black hole is very small, the minimum …
We determine all spacetime flows which lead to a Hamiltonian dynamics in the space of general relativistic initial data sets with asymptotically Kerr-de Sitter ends. The corresponding Hamiltonians are calculated. Some implications for black-hole thermodynamics are pointed out.
We show that the~braneworld rotating Kerr--Newman black hole and naked singularity spacetimes with both positive and negative braneworld tidal charge parameter can be separated into fourteen classes according to properties of circular geodesics governing the~Keplerian accretion. We determine efficiency of the~Keplerian accretion disks for all braneworld Kerr--Newman spacetimes. We demonstrate occurrence of an~infinitely deep gravitational potential in Kerr--Newman naked singularity spacetimes having the~braneworld dimensionless tidal charge $b \in (1/4,1)$ and the~dimensionless spin $a \in (2\…
In this paper, we investigate Hawking radiation of massive spin-1 particles from 5-dimensional Kerr-Gödel spacetime. By applying the WKB approximation and the Hamilton-Jacobi ansatz to the relativistic Proca equation, we obtain the quantum tunneling rate of the massive vector particles. Using the obtained tunneling rate, we show how one impeccably computes the Hawking temperature of the 5-dimensional Kerr-Gödel spacetime.
Gaussian curvature of the two-surface r=0, t=const is calculated for the Kerr-de Sitter and Kerr-Newman-de Sitter solutions, yielding non-zero analytical expressions for both the cases. The results obtained, on the one hand, exclude the possibility for that surface to be a disk and, on the other hand, permit one to establish a correct geometrical interpretation of that surface for each of the two solutions.
We find a new class of exact solutions in the Einstein-Maxwell theory by employing the Ernst magnetization process to the Kerr-Newman-Taub-NUT spacetimes. We study the solutions and find that they are regular everywhere. We also find the quasilocal conserved quantities for the spacetimes, the corresponding Smarr formula and the first law of thermodynamics.
One of the most important sources for future space-borne gravitational wave detectors such as TianQin and LISA is EMRI. It happens when a stellar orgin compact object orbiting around a massive black hole(MBH) in the center of galaxies and has many benefits in the study of astrophysics and fundamental theories. One of the most important objectives is to test the no-hair theorem by measuring the quadrupole moment of the MBH. This requires us to estimate the parameters of an EMRI system accurately enough, which means we also need an accurate waveform templet for this process. Based on the fast an…
The causal properties of the family of Kerr-de Sitter spacetimes are analyzed and compared to those of the Kerr family. At first we show that a Kerr-de Sitter spacetime can be viewed as an assembly of Carter's blocks i.e. four dimensional spacetime regions contained within Killing horizons or a Killing horizon and the asymptotic de Sitter region. From this perspective and leaving aside topological identifications, the causal properties of a Kerr de Sitter spacetime are determined by the causal properties of the individual Carter's blocks viewed as spacetimes in their own right. We show that an…
We analyze the stability of circular geodesics for timelike as well as null geodesics of the Kerr BH spacetime with rotation parameter on the equatorial plane by Lyapunov stability analysis. Also, we verify the results of stability by presenting the phase portrait for both timelike and null geodesics. Further, by reviewing the Kosambi-Cartan-Chern (KCC) theory, we analyze the Jacobi stability for Kerr spacetime and present a comparative study of the methods used for stability analysis of geodesics.
The second-order post-Newtonian solution for the light propagation in the field of Kerr-Newman black hole is achieved via an iterative method. Based on this result, we further obtain the second-order post-Newtonian light deflection in Kerr-Newman spacetime, which is formulated in an united form for any arbitrarily incident directions. All results are exhibited in the coordinate system constituted by the initial light-direction vector, the impact vector, and their cross-product.
We study the absorption probability and Hawking radiation spectra of a phantom scalar field in the Kerr black hole spacetime. We find that the presence of the negative kinetic energy terms modifies the standard results in the greybody factor, super-radiance and Hawking radiation. Comparing with the usual scalar particle, the phantom scalar emission is enhanced in the black hole spacetime.
It was recently discovered that Killing horizons in the generic Kerr-NUT-(anti) de Sitter spacetimes are projectively singular, i.e. their spaces of the null generators have singular geometry. Only if the cosmological constant takes the special value determined by the Kerr and NUT parameters, and the radius of the horizon, then the corresponding horizon does not suffer that problem. In the current paper, the projectively non-singular horizons are investigated. They are found to be cosmological and non-extremal. Every projectively non-singular horizon can be used to define a global completion o…
We generalize to Kerr spacetime previous gravitational self-force results on gyroscope precession along circular orbits in the Schwarzschild spacetime. In particular we present high order post- Newtonian expansions for the gauge invariant precession function along circular geodesics valid for arbitrary Kerr spin parameter and show agreement between these results and those derived from the full post-Newtonian conservative dynamics. Finally we present strong field numerical data for a range of the Kerr spin parameter, showing agreement with the GSF-PN results, and the expected lightring divergen…
We describe the hyperboloidal compactification for Teukolsky equations in Kerr spacetime. We include null infinity on the numerical grid by attaching a hyperboloidal layer to a compact domain surrounding the rotating black hole and the orbit of an inspiralling point particle. This technique allows us to study, for the first time, gravitational waveforms from large- and extreme-mass-ratio inspirals in Kerr spacetime extracted at null infinity. Tests and comparisons of our results with previous calculations establish the accuracy and efficiency of the hyperboloidal layer method.
A radial deformation of the Kerr metric is studied which leads to the appearance of wormholes in certain cases. {\bf These wormholes can be thought of as being formed by two distinct but identical asymptotically non-flat spacetimes joined at the throat}. The status of energy conditions are discussed in details. The components of the energy-momentum tensor of the spacetime and its eigenvalues are explicitly given. It is shown that the resulting spacetime, represents a family of exact anisotropic fluid solutions.
We have investigated the acceleration of the unbound high energy particles moving along the rotation axis in the Kerr-Taub-NUT spacetime, and then study the dependence of the repulsive effects on the NUT charge for the particles in the spacetime. Whether the repulsive effects with the NUT charge become stronger depends on the Carter constant, the position and velocity of the particles themselves. We also present numerically the changes of the observable velocity and acceleration with the NUT charge for the unbound particles in the Kerr-Taub-NUT spacetime.
We study causal geodesics in the equatorial plane of the extremal Kerr-Taub-NUT spacetime, focusing on the Innermost Stable Circular Orbit (ISCO),and compare its behaviour with extant results for the ISCO in the extremal Kerr spacetime. Calculation of the radii of the direct ISCO, its Kepler frequency, and rotational velocity show that the ISCO coincides with the horizon in the exactly extremal situation. We also study geodesics in the strong {\it non}-extremal limit, i.e., in the limit of vanishing Kerr parameter (i.e., for Taub-NUT and massless Taub-NUT spacetimes as special cases of this sp…
We study the strong gravitational lensing on the equatorial plane of a quasi-Kerr compact object with arbitrary quadrupole moments which can be used to model the super-massive central object of the galaxy. We find that, when the quadrupolar correction parameter $ξ$ takes the positive (negative) value, the photon-sphere radius $r_{ps}$, the minimum impact parameter $u_{ps}$, the coefficient $\bar{b}$, the relative magnitudes $r_m$ and the angular position of the relativistic images $θ_{\infty}$ are larger (smaller) than the results obtained in the Kerr black hole, but the coefficient $\bar{a}$,…
We discuss the tension between the possible existence of Painleve-Gullstrand coordinate systems versus the explicit geometrical features of the Kerr spacetime; a subject of interest to Professor Thanu Padmanabhan in the weeks immediately preceding his unexpected death. We shall carefully distinguish strong and weak Painleve-Gullstrand coordinate systems, and conformal variants thereof, cataloguing what we know can and cannot be done -- sometimes we can make explicit global statements, sometimes we must resort to implicit local statements. For the Kerr spacetime the best that seems to be achiev…
We analyze the behavior of causal geodesics on a Kerr-de Sitter spacetime with particular emphasis on their completeness property. We set up an initial value problem (IVP) whose solutions lead to a global understanding of causal geodesics on these spacetime. Causal geodesics that avoid the rotation axis are complete except the ones that hit the ring-like curvature singularity and those that encounter the ring singularity are necessary equatorial ones. We also show the existence of geodesics that cross or lie on the rotation axis. The equations governing the latter family show the repulsive nat…
General relativistic tidal equations are formulated with respect to the rest frame of a central gravitational source described by the Kerr gravitational field. Specifically, observers that are spatially at rest in the exterior Kerr spacetime are considered in detail; in effect, these fiducial observers define the rest frame of the Kerr source. The generalized Jacobi equation for the free motion of a test particle is worked out within the Fermi coordinate system established along the world line of a fiducial observer at rest on the symmetry axis of the exterior Kerr background. The nature of th…
