Holes In General Relativity Research Articles

General spherical harmonic bra-ket overlap integrals of trigonometric functions

Closed formulas in terms of double sums of Clebsch–Gordan coefficients are computed for the evaluation of bra-ket spherical harmonic overlap integrals of a wide class of trigonometric functions. These analytical expressions can find useful application in problems involving non-separable wave equations, e.g. general-relativistic perturbation theory, electromagnetism, quantum theory, etc, wherein the overlap integrals arise from the coupling among different angular modes. We provide some examples related to linear perturbations of spinning black holes in general relativity and modified gravity, in which the analytical formulas for the overlap integrals are particularly useful to compute the black-hole spectrum.

Constraining a modified gravity theory in strong gravitational lensing and black hole shadow observations

We study the strong gravitational lensing effect around rotating black holes in different gravity theories. By calculating the deflection angle of strong gravitational lensing, we evaluate the lensing observables including the image position, separation, magnification, and the time delays between the relativistic images of different rotating black holes. We argue that the differences in image positions, separations between the rotating black hole in modified gravity (MOG) theory and the Kerr black hole in general relativity (GR) are more significant in SgrA* than those in M87*; however the differences in time delays between rotating black holes in MOG and GR are shorter in SgrA* than that in M87*. Our evaluations on lensing observables in the strong gravity regime can help to distinguish the MOG from GR. Furthermore, we investigate the shadow observables of different rotating black holes. Employing the Event Horizon Telescope observations on the angular shadow radius for supermassive M87* and SgrA* black holes, respectively, we estimate the ranges of MOG parameter and obtain its upper limit constraint $0.350\ensuremath{\lesssim}{\ensuremath{\alpha}}_{\mathrm{up}}\ensuremath{\lesssim}0.485$ and $0.162\ensuremath{\lesssim}{\ensuremath{\alpha}}_{\mathrm{up}}\ensuremath{\lesssim}0.285$ correspondingly, relating to black hole spins. This is the first constraint on the MOG parameter for rotating supermassive black holes from Event Horizon Telescope observations on the angular shadow radius. Our constraint on the MOG parameter is much tighter compared with the result obtained from the orbital precession of the S2 star.

Inspiraling compact objects with generic deformations

Self-gravitating bodies can have an arbitrarily complex shape, which implies a much richer multipolar structure than that of a black hole in General Relativity. With this motivation, we study the corrections to the dynamics of a binary system due to generic, nonaxisymmetric mass quadrupole moments to leading post-Newtonian (PN) order. Utilizing the method of osculating orbits and a multiple scale analysis, we find analytic solutions to the precession and orbital dynamics of a (generically eccentric) binary in terms of the dimensionless modulus parameters $\epsilon_{m}$, corresponding to axial $m=1$ and polar $m=2$ corrections from oblateness/prolateness. The solutions to the precession dynamics are exact for $0 \le \epsilon_{2} < 1$, and perturbative in $\epsilon_{1} \ll 1$. We further compute the leading order corrections to the gravitational wave amplitude and phase for a quasi-circular binary due to mass quadrupole effects. Making use of the stationary phase approximation and shifted uniform asymptotics (SUA), the corrections to the phase enter at relative 2PN order, while the amplitude modulations enter at -0.5PN order with a SUA amplitude correction at 3.25PN order, relative 2PN order to the leading order SUA correction. By investigating the dephasing due to generic quadrupole moments, we find that a phase difference $\gtrsim 0.1$~radians is achievable for $\epsilon_{m} \gtrsim 10^{-3}$, which suggests that constraints with current and future ground-based gravitational wave detectors are possible. Our results can be implemented in parameter estimation studies to constrain generic multipolar deformations of the Kerr geometry and of neutron stars.

Deciphering signatures of Bardeen black holes from the observed quasi-periodic oscillations

Quasi-periodic oscillations (QPOs) observed in the power spectrum of black holes are unique observational probes to the background spacetime since they can be directly related to the timescales associated with the motion of matter orbiting in the vicinity of the black hole horizon. In this regard, the high frequency QPOs (HFQPOs) are particularly interesting as they occur in commensurable pairs, the most common ratio being the 3:2 twin peak QPOs. The theoretical models which aim to explain these QPOs express the observed frequencies in terms of the epicyclic motion of test particles in a given background spacetime. In this work, we study the signatures of Bardeen spacetime from the observed QPOs in the black hole power spectrum. Bardeen black holes are rotating, regular black holes with a magnetic monopole charge. Such regular backgrounds are theoretically interesting as they can potentially evade the curvature singularity, otherwise unavoidable in general relativistic black holes. We perform a χ2 analysis by comparing the available observations of the quasi-periodic oscillations from black hole sources with the relevant theoretical models and note that the Kerr black holes in general relativity are observationally more favored compared to black holes with a monopole charge. Our analysis reveals that black holes with very high monopole charges are disfavored from QPO related observations.

Energy contents of stringy charged black hole in general relativity and teleparallel gravity

In the framework of general relativity (GR) and teleparallel equivalent to GR (TEGR), we investigate the energy–momentum localization of a stringy charged black hole by using Bergman–Thompson’s prescription and the Papapetrou’s prescription. Moreover, we investigate the gravitational energy–momentum (GEM) in TEGR. Our results are consistent with the results of previous studies. Finally, we calculate the GEM flux and gravitational center of mass in TEGR.

How not to extract information from black holes: Cosmic censorship as a guiding principle

Black holes in general relativity are commonly believed to evolve towards a Schwarzschild state as they gradually lose angular momentum and electrical charge under Hawking evaporation. However, when Kim and Wen applied quantum information theory to Hawking evaporation and argued that Hawking particles with maximum mutual information could dominate the emission process, they found that charged black holes tend towards extremality. In view of some evidence pointing towards extremal black holes being effectively singular, this would violate the cosmic censorship conjecture. Nevertheless, since the Kim-Wen model is too simplistic (e.g. it assumes a continuous spectrum of particles with arbitrary charge-to-mass ratio), one might hope that a more realistic model could avoid this problem. In this work, we show that having only a finite species of charged particles would actually worsen the situation, with some end states becoming a naked singularity. With this model as an example, we emphasize the need to study whether charged black holes can violate cosmic censorship under a given model of Hawking evaporation.

Investigating the relation between gravitational wave tests of general relativity

Gravitational wave observations of compact binary coalescences provide precision probes of strong-field gravity. There is thus now a standard set of null tests of general relativity (GR) applied to LIGO-Virgo detections and many more such tests proposed. However, the relation between all these tests is not yet well understood. We start to investigate this by applying a set of standard tests to simulated observations of binary black holes in GR and with phenomenological deviations from GR. We consider four types of tests: residuals, inspiral-merger-ringdown consistency, parameterized phasing (two varieties), and modified dispersion relation. We also check the consistency of the unmodeled reconstruction of the waveforms with the waveform recovered using GR templates. These tests are applied to simulated observations similar to GW150914 with both large and small deviations from GR and similar to GW170608 just with small deviations from GR. We find that while very large deviations from GR are picked up with high significance by almost all tests, more moderate deviations are picked up by only a few tests, and some deviations are not recognized as GR violations by any test at the moderate signal-to-noise ratios we consider. Moreover, the tests that identify various deviations with high significance are not necessarily the expected ones. In particular, the parameterized tests recover PN test parameters much closer to zero than their true values in some cases. Additionally, we find that of the GR deviations we consider, the residuals test is only able to detect extreme deviations from GR that no longer look like binary black hole coalescences in GR. The reconstruction comparison shows more promise for detecting relatively small GR deviations in an unmodeled framework, at least for high-mass systems.

Gravitational lensing for stationary axisymmetric black holes in Eddington-inspired Born-Infeld gravity

Recent years have witnessed a surge of interest of the lensing of the black holes arising from general as well as other modified theories of gravity due to the experimental data available from the Event Horizon Telescope (EHT) results. The EHT may open a new door indicating the possible existence of the rotating black hole solutions in modified theories of gravity in the strong field regime. With this motivation, we investigate in the present paper the equatorial lensing $(\ensuremath{\theta}=\ensuremath{\pi}/2)$ by a recently obtained exact rotating black holes solution in Eddington-inspired Born-Infeld (EiBI) theory in both the strong- and weak-field limits. Such black holes are the modification of Kerr-Newman black holes in general relativity, characterized by their mass $M$, the charge $Q$, and the rotation parameter $a$. and an additional term $\ensuremath{\epsilon}$ accounting for the correction to the Kerr-Newman solutions. We show numerically the variations of the impact parameter ${u}_{m}$, the light deflection coefficients $p$ and $q$, and the total azimuthal bending angle ${\ensuremath{\alpha}}_{D}$ and find a close dependence of these quantities on the charge parameter ${r}_{q}$, the correction term $\ensuremath{\epsilon}$, and the spin $a$. We also calculate the angular position ${\ensuremath{\theta}}_{\ensuremath{\infty}}$, the angular separation $s$, and the magnification of the relativistic images. In addition, we also discuss the weak lensing of the black holes in Eddington-inspired Born-Infeld (EiBI) theory using the Gauss-Bonnet theorem. We calculate the weak lensing parameter and find its variation with different values of the parameters ${r}_{q}$ and $\ensuremath{\epsilon}$.

Ladder symmetries of black holes. Implications for love numbers and no-hair theorems

It is well known that asymptotically flat black holes in generalrelativity have a vanishing static, conservative tidal response. We show that this is a result of linearly realized symmetries governingstatic (spin 0,1,2)perturbations around black holes. The symmetries have a geometric origin: in the scalar case, they arise from the (E)AdS isometries of a dimensionally reduced black hole spacetime. Underlying the symmetries is a ladder structure which can be used to construct the full tower of solutions,and derive their general properties: (1) solutions that decay withradius spontaneously break the symmetries, and mustdiverge at the horizon;(2) solutions regular at the horizon respect the symmetries, andtake the form of a finite polynomial that grows with radius.Taken together, these two properties imply that static response coefficients — and in particular Love numbers — vanish. Moreover, property (1) is consistent with the absence of black holes with linear (perturbative) hair. We also discuss the manifestation of these symmetries in the effective point particle description of a black hole, showing explicitly that for scalar probesthe worldline couplings associated with a non-trivial tidal response and scalar hair must vanish in order for the symmetries to be preserved.

Asymptotics of linear differential systems and application to quasinormal modes of nonrotating black holes

The traditional approach to perturbations of nonrotating black holes in General Relativity uses the reformulation of the equations of motion into a radial second-order Schr\"odinger-like equation, whose asymptotic solutions are elementary. Imposing specific boundary conditions at spatial infinity and near the horizon defines, in particular, the quasi-normal modes of black holes. For more complicated equations of motion, as encountered for instance in modified gravity models with different background solutions and/or additional degrees of freedom, we present a new approach that analyses directly the first-order differential system in its original form and extracts the asymptotic behaviour of perturbations, without resorting to a second-order reformulation. As a pedagogical illustration, we apply this treatment to the perturbations of Schwarzschild black holes and then show that the standard quasi-normal modes can be obtained numerically by solving this first-order system with a spectral method. This new approach paves the way for a generic treatment of the asymptotic behaviour of black hole perturbations and the identification of quasi-normal modes in theories of modified gravity.

Gravitational-wave constraints on the GWTC-2 events by measuring the tidal deformability and the spin-induced quadrupole moment

Gravitational waves from compact binary coalescences provide a unique laboratory to test properties of compact objects. As alternatives to the ordinary black holes in general relativity, various exotic compact objects have been proposed. Some of them have largely different values of the tidal deformability and spin-induced quadrupole moment from those of black holes, and their binaries could be distinguished from the binary black hole by using gravitational waves emitted during their inspiral regime, excluding the highly model-dependent merger and ringdown regimes. We reanalyze gravitational waves from low-mass merger events in the GWTC-2, detected by the Advanced LIGO and Advanced Virgo. Focusing on the influence of tidal deformability and spin-induced quadrupole moment in the inspiral waveform, we provide model-independent constraints on deviations from the standard binary black hole case. We find that all events that we have analyzed are consistent with the waveform of the binary black hole in general relativity. Bayesian model selection shows that the hypothesis that the binary is composed of exotic compact objects is disfavored by all events.

Radiative classical gravitational observables at mathcal{O} (G3) from scattering amplitudes

We compute classical gravitational observables for the scattering of two spinless black holes in general relativity and mathcal{N} =8 supergravity in the formalism of Kosower, Maybee, and O’Connell (KMOC). We focus on the gravitational impulse with radiation reaction and the radiated momentum in black hole scattering at mathcal{O} (G3) to all orders in the velocity. These classical observables require the construction and evaluation of certain loop-level quantities which are greatly simplified by harnessing recent advances from scattering amplitudes and collider physics. In particular, we make use of generalized unitarity to construct the relevant loop integrands, employ reverse unitarity, the method of regions, integration-by-parts (IBP), and (canonical) differential equations to simplify and evaluate all loop and phase-space integrals to obtain the classical gravitational observables of interest to two-loop order. The KMOC formalism naturally incorporates radiation effects which enables us to explore these classical quantities beyond the conservative two-body dynamics. From the impulse and the radiated momentum, we extract the scattering angle and the radiated energy. Finally, we discuss universality of the impulse in the high-energy limit and the relation to the eikonal phase.

Rotating black holes in general relativity coupled to nonlinear electrodynamics

We find an exact spherically symmetric magnetically charged black hole solution to general relativity (GR) coupled to nonlinear electrodynamics (NED) with an appropriate Lagrangian density. In turn, starting with this spherical black hole as a seed metric, we construct a rotating spacetime, a modification of Kerr black hole, using the revised Newman–Janis algorithm that depends on mass, spin, and a NED parameter g. We find an exact expression for thermodynamic quantities of the black holes like the mass, Hawking temperature, entropy, heat capacity, and free energy expressed in terms of horizon radius, and they show significant deviations from the Kerr case owing to NED. We also calculate analytical expressions for effective Komar mass and angular momentum for the rotating black hole and demonstrate that the Komar conserved quantity corresponding to the null Killing vector at horizon obeys Kχ=2S+T+. The radiating counterpart renders a generalization of Carmeli’s spacetime as well as Vaidya’s spacetime in the appropriate limits.

Quantum corrections to tidal Love number for Schwarzschild black holes

A sum rule for tidal Love number is derived from quantum field theory computations, which relates tidal susceptibility of a spinless body to transition rates of its single graviton emission processes. An analogous sum rule for electromagnetism is given as an example, which is substantiated by comparing to the solved problem of the hydrogen atom. Based on the semiclassical Hawking radiation spectrum, a finite nonvanishing value for quantum corrections to the Love number of Schwarzschild black holes in general relativity is computed using the sum rule, which is known to classically vanish.

Modeling and detecting resonant tides of exotic compact objects

The event horizon of a black hole in general relativity absorbs all infalling radiation. Any observation of the contrary would immediately challenge the expectation that astrophysical black holes are described by the vacuum Kerr geometry. If a putative black hole does reflect part of the ingoing radiation, its quasinormal mode structure is drastically altered. Low frequency modes can be introduced that are resonantly excited during the inspiral of a binary system. We study the resulting phase shift of the gravitational wave signal. Building on neutron star results, we obtain a model-independent expression for the phase shift that depends only on quasinormal modes and Love numbers of the compact object. We find that the phase shift might be detectable with Einstein Telescope for asymmetric binaries in high signal-to-noise events ($\ensuremath{\sim}{10}^{3}$), but by far cannot explore the Planck scale.

Numerical black hole solutions in modified gravity theories: Axial symmetry case

We extend a recently developed numerical code to obtain stationary, axisymmetric solutions that describe rotating black hole spacetimes in a wide class of modified theories of gravity. The code utilizes a relaxed Newton-Raphson method to solve the full nonlinear modified Einstein's Equations on a two-dimensional grid with a Newton polynomial finite difference scheme. We validate this code by considering static and axisymmetric black holes in General Relativity. We obtain rotating black hole solutions in scalar-Gauss-Bonnet gravity with a linear (linear scalar-Gauss-Bonnet) and an exponential (Einstein-dilaton-Gauss-Bonnet) coupling and compare them to analytical and numerical perturbative solutions. From these numerical solutions, we construct a fitted analytical model and study observable properties calculated from the numerical results.

Testing gravity of a disformal Kerr black hole in quadratic degenerate higher-order scalar-tensor theories by quasi-periodic oscillations

By using the relativistic precession model, we have studied frequencies of quasi-periodic oscillations in the spacetime of a disformal Kerr black hole. This black hole owns an extra disformal parameter and belongs to a class of non-stealth solutions in quadratic degenerate higher-order scalar-tensor (DHOST) theories. Our result shows that only the periastron precession frequency is related to the disformal parameter, while the azimuthal frequency and the nodal precession frequency are identical with those in the usual Kerr black hole in general relativity. Combing with the observation data of GRO J1655-40, we fit parameters of the disformal Kerr black hole, and find that the disformal parameter α is almost negative in the range of 1 σ, which implies the negative disformal parameter α is favored by the observation data of GRO J1655-40.

Parameter estimation of hairy Kerr black holes from its shadow and constraints from M87*

ABSTRACT The recently obtained hairy Kerr black holes, due to additional sources or surrounding fluid, like dark matter, with conserved energy-momentum tensor, have a deviation α and primary hair l0, apart from rotation parameter a and mass M. In the wake of the Event Horizon Telescope (EHT) observations of the supermassive black hole M87*, a recent surge in interest in black hole shadows suggests comparing the black holes in general relativity and modified theories of gravity to assess these models’ differences. Motivated by this, we take on an extensive study of the rotating hairy Kerr black holes, which encompasses, in particular cases, the Kerr black hole (α = 0). We investigate ergosphere and shadows of the black holes to infer that their size and shape are affected due to the l0 and are found to harbour a richer chaotic structure. In particular, the hairy Kerr black holes possess smaller size but more distorted shadows when compared with Kerr black holes. We also estimate the parameters l0 and a associated with hairy Kerr black holes using the shadow observables. The inferred circularity deviation ΔC ≤ 0.1 for the M87* black hole is satisfied, whereas shadow angular diameter θd = 42 ± 3μas, within 1σ region, for a given choice of α, places bounds on the parameters a and l0. Interestingly, the shadow axial ratio obeying 1 &amp;lt; Dx ≲ 4/3 is in agreement with the EHT results and thus eventuates in the hairy Kerr black holes being suitable candidates for astrophysical black holes.

A nonsingular, anisotropic universe in a black hole with torsion and particle production

We consider a universe formed in a black hole in general relativity with spin and torsion. The interior of a Schwarzschild black hole can be represented by the Kantowski–Sachs metric that describes a closed anisotropic universe. We use this metric to derive the Einstein–Cartan field equations with a relativistic spin fluid as a source. We show that torsion may prevent a singularity and replace it with a nonsingular bounce if particle production dominates over shear. Particle production after the last bounce can generate a finite period of inflation, during which the universe expands and isotropizes to the current state. This scenario is only approximate: the Kantowski–Sachs metric is never reached and should be replaced with a more general metric that tends to that of a 3-sphere.

New spinorial approach to mass inequalities for black holes in general relativity

A new spinorial strategy for the construction of geometric inequalities involving the Arnowitt-Deser-Misner mass of black hole systems in general relativity is presented. This approach is based on a second order elliptic equation (the approximate twistor equation) for a valence 1 Weyl spinor. This has the advantage over other spinorial approaches to the construction of geometric inequalities based on the Sen-Witten-Dirac equation that it allows us to specify boundary conditions for the two components of the spinor. This greater control on the boundary data has the potential of giving rise to new geometric inequalities involving the mass. In particular, it is shown that the mass is bounded from below by an integral functional over a marginally outer trapped surface (MOTS) which depends on a freely specifiable valence 1 spinor. From this main inequality, by choosing the free data in an appropriate way, one obtains a new nontrivial bounds of the mass in terms of the inner expansion of the MOTS. The analysis makes use of a new formalism for the $1+1+2$ decomposition of spinorial equations.