We study the geometry of the event horizon of a spacetime in which a small compact object plunges into a large Schwarzschild black hole. We first use the Regge-Wheeler and Zerilli formalisms to calculate the metric perturbations induced by this small compact object, then find the new event horizon by propagating null geodesics near the unperturbed horizon. A caustic is shown to exist before the merger. Focusing on the geometry near the caustic, we show that it is determined predominantly by large-l perturbations, which in turn have simple asymptotic forms near the point at which the particle plunges into the horizon. It is therefore possible to obtain an analytic characterization of the geometry that is independent of the details of the plunge. We compute the invariant length of the caustic. We further show that among the leading-order horizon area increase, half arises from generators that enter the horizon through the caustic, and the rest arises from area increase near the caustic, induced by the gravitational field of the compact object.
In this second part of a two-part paper, we discuss numerical simulations of a head-on merger of two non-spinning black holes. We resolve the fate of the original two apparent horizons by showing that after intersecting, their world tubes "turn around" and continue backwards in time. Using the method presented in the first paper to locate these surfaces, we resolve several such world tubes evolving and connecting through various bifurcations and annihilations. This also draws a consistent picture of the full merger in terms of apparent horizons, or more generally, marginally outer trapped surf…
We model an extreme mass ratio merger (EMR) as a point particle radially plunging into a large Schwarzschild black hole. We assume that the mass of the point particle, $μ$, is much smaller than the black hole mass M. Under this assumption we can employ the Zerilli formalism modified to include a source term which arises from the energy-momentum tensor of the small object. We solve the Zerilli equation by numerically evolving initial data. Then, we ray trace the null geodesics of the event horizon from after the merger backward in time to extract the geometry of the perturbed event horizon. Fur…
In a binary black hole merger, it is known that the inspiral portion of the waveform corresponds to two distinct horizons orbiting each other, and the merger and ringdown signals correspond to the final horizon being formed and settling down to equilibrium. However, we still lack a detailed understanding of the relation between the horizon geometry in these three regimes and the observed waveform. Here we show that the well known inspiral chirp waveform has a clear counterpart on black hole horizons, namely, the shear of the outgoing null rays at the horizon. We demonstrate that the shear beha…
In this work, we uncover new features in the evolution of the deformations of the dynamical horizon geometry in a binary black hole merger scenario using numerical relativity. First, in the inspiral phase, owing to the deformations, the dynamical horizons of the two black holes are found to steadily acquire multipole moments that vanish when the horizons are isolated. Out of these, the dominant moment is found to be the quadrupole moment. Second, we show that they encode detailed information about the dynamics of the binary black hole system. The dominant quadrupole multipole moment is particu…
We resolve the fate of the two original apparent horizons during the head-on merger of two non-spinning black holes. We show that following the appearance of the outer common horizon and subsequent inter-penetration of the original horizons, they continue to exist for a finite period of time before they are individually annihilated by unstable MOTSs. The inner common horizon vanishes in a similar, though independent, way. This completes the understanding of the analogue of the event horizon's "pair of pants" diagram for the apparent horizon. Our result is facilitated by a new method for locati…
The merger of a binary black hole gives birth to a highly distorted final black hole. The gravitational radiation emitted as this black hole relaxes presents us with the unique opportunity to probe extreme gravity and its connection with the dynamics of the black hole horizon. Using numerical relativity simulations, we demonstrate a connection between a concrete observable feature in the gravitational waves and geometrical features on the dynamical apparent horizon of the final black hole. Specifically, we show how the line-of-sight passage of a "cusp"-like defect on the horizon of the final b…
In classical numerical relativity, marginally outer trapped surfaces (MOTSs) are the main tool to locate and characterize black holes. For five decades it has been known that during a binary merger, a new outer horizon forms around the initial apparent horizons of the individual holes once they are sufficiently close together. However the ultimate fate of those initial horizons has remained a subject of speculation. Recent axisymmetric studies have shed new light on this process and this pair of papers essentially completes that line of research: we resolve the key features of the post-swallow…
We numerically study the algebraic properties of the Weyl tensor through the merger of two non-spinning black holes (BHs). We are particularly interested in the conjecture that for such a vacuum spacetime, which is zeroth-order algebraically general, a geometric horizon (GH), on which the spacetime is algebraically special and which is identified by the vanishing of a complex scalar invariant (${\mathcal{D}}$), characterizes a smooth foliation independent surface (horizon) associated with the BH. In the first simulation we investigate the level-$0$ sets of $\text{Re}({\mathcal{D}})$ (since $\t…
Methods are presented to define and compute source multipoles of dynamical horizons in numerical relativity codes, extending previous work from the isolated and dynamical horizon formalisms in a manner that allows for the consideration of horizons that are not axisymmetric. These methods are then applied to a binary black hole merger simulation, providing evidence that the final remnant is a Kerr black hole, both through the (spatially) gauge-invariant recovery of the geometry of the apparent horizon, and through a detailed extraction of quasinormal ringing modes directly from the strong-field…
We study in detail the dynamics and stability of marginally trapped surfaces during a binary black hole merger. This is the second in a two-part study. The first part studied the basic geometric aspects of the world tubes traced out by the marginal surfaces and the status of the area increase law. Here we continue and study the dynamics of the horizons during the merger, again for the head-on collision of two non-spinning black holes. In particular we follow the spectrum of the stability operator during the course of the merger for all the horizons present in the problem and implement systemat…
The behavior of quasi-local black hole horizons in a binary black hole merger is studied numerically. We compute the horizon multipole moments, fluxes and other quantities on black hole horizons throughout the merger. These lead to a better qualitative and quantitative understanding of the coalescence of two black holes; how the final black hole is formed, initially grows and then settles down to a Kerr black hole. We calculate the rate at which the final black hole approaches equilibrium in a fully non-perturbative situation and identify a time at which the linear ringdown phase begins. Final…
We study curvature invariants in a binary black hole merger. It has been conjectured that one could define a quasi-local and foliation independent black hole horizon by finding the level--$0$ set of a suitable curvature invariant of the Riemann tensor. The conjecture is the geometric horizon conjecture and the associated horizon is the geometric horizon. We study this conjecture by tracing the level--$0$ set of the complex scalar polynomial invariant, $\mathcal{D}$, through a quasi-circular binary black hole merger. We approximate these level--$0$ sets of $\mathcal{D}$ with level--$\varepsilon…
We find the first binary black hole event horizon with a toroidal topology. It had been predicted that generically the event horizons of merging black holes should briefly have a toroidal topology, but such a phase has never been seen prior to this work. In all previous binary black hole simulations, in the coordinate slicing used to evolve the black holes, the topology of the event horizon transitions directly from two spheres during the inspiral to a single sphere as the black holes merge. We present a coordinate transformation to a foliation of spacelike hypersurfaces that "cut a hole" thro…
We discuss the transition from quasi-circular inspiral to plunge of a system of two nonrotating black holes of masses $m_1$ and $m_2$ in the extreme mass ratio limit $m_1m_2\ll (m_1+m_2)^2$. In this limit, we compare the merger waveforms obtained by two different methods: a {\it numerical} (Regge-Wheeler-Zerilli) one, and an {\it analytical} (Effective One Body) one. This is viewed as a contribution to the matching between analytical and numerical methods.
Event horizons are the defining physical features of black hole spacetimes, and are of considerable interest in studying black hole dynamics. Here, we reconsider three techniques to localise event horizons in numerical spacetimes: integrating geodesics, integrating a surface, and integrating a level-set of surfaces over a volume. We implement the first two techniques and find that straightforward integration of geodesics backward in time to be most robust. We find that the exponential rate of approach of a null surface towards the event horizon of a spinning black hole equals the surface gravi…
We argue that the event horizon of a binary black hole merger, in the extreme-mass-ratio limit where one of the black holes is much smaller than the other, can be described in an exact analytic way. This is done by tracing in the Schwarzschild geometry a congruence of null geodesics that approaches a null plane at infinity. Its form can be given explicitly in terms of elliptic functions, and we use it to analyze and illustrate the time-evolution of the horizon along the merger. We identify features such as the line of caustics at which light rays enter the horizon, and the critical point at wh…
We present results obtained by scattering a scalar field off the curved background of a coalescing binary black hole system. A massless scalar field is evolved on a set of fixed backgrounds, each provided by a spatial hypersurface generated numerically during a binary black hole merger. We show that the scalar field scattered from the merger region exhibits quasinormal ringing once a common apparent horizon surrounds the two black holes. This occurs earlier than the onset of the perturbative regime as measured by the start of the quasinormal ringing in the gravitational waveforms. We also use …
The understanding of strong-field dynamics near black-hole horizons is a long-standing and challenging prob- lem in general relativity. Recent advances in numerical relativity and in the geometric characterization of black- hole horizons open new avenues into the problem. In this first paper in a series of two, we focus on the analysis of the recoil occurring in the merger of binary black holes, extending the analysis initiated in [1] with Robinson- Trautman spacetimes. More specifically, we probe spacetime dynamics through the correlation of quantities defined at the black-hole horizon and at…
An important physical phenomenon that manifests itself during the inspiral of two orbiting compact objects is the tidal deformation of each under the gravitational influence of its companion. In the case of binary neutron star mergers, this tidal deformation and the associated Love numbers have been used to probe properties of dense matter and the nuclear equation of state. Non-spinning black holes on the other hand have a vanishing (field) tidal Love number in General Relativity. This pertains to the deformation of the asymptotic gravitational field. In certain cases, especially in the late s…
We study binary black hole mergers in the extreme mass-ratio limit. We determine the energy, angular momentum, and linear momentum of the post-merger, remnant black hole. Unlike previous works, we perform our analysis directly in the test-particle limit by solving the Regge-Wheeler-Zerilli wave equation with a source that moves along a geodesic. We rely on the fact that toward the merger, small mass-ratio binary systems follow a quasiuniversal geodesic trajectory. This formalism captures the final premerger stages of small mass-ratio binaries and thus provides a straightforward universal descr…
Recent advances in numerical relativity have revealed how marginally trapped surfaces behave when black holes merge. It is now known that interesting topological features emerge during the merger, and marginally trapped surfaces can have self-intersections. This paper presents the most detailed study yet of the physical and geometric aspects of this scenario. For the case of a head-on collision of non-spinning black holes, we study in detail the world tube formed by the evolution of marginally trapped surfaces. In the first of this two-part study, we focus on geometrical properties of the dyna…
We study the open and closed axisymmetric marginally outer trapped surfaces contained in leaves of constant Painlevé-Gullstrand time for Schwarzschild spacetimes. We identify a family of closed MOTS in the black hole interior characterized by an arbitrary number of self-intersections. This suggests that the self-intersecting behaviour reported in [Phys.Rev.D 100, 084044 (2019)] may be a far more generic phenomenon than expected. We also consider open surfaces, finding that their behaviour is highly constrained but includes surfaces with multiple self-intersections inside the horizon. We argue …
We analyse the formation and evolution of the event horizon of a black hole binary merger when the black holes are charged. We find that the presence of charge influences the properties of the merger and can be useful for investigating the validity of various theories of modified gravity and several proposals for dark matter candidates. It can moreover give insights into various aspects of astrophysical phenomena involving black holes, such as degeneracies in the gravitational wave parameter determination. We perform our analysis both analytically and numerically, in $D=4$ dimensions, in the E…
The correspondence between black holes and colliding waves extends to cover the near horizon geometry of rotating black holes and colliding waves with cross polarization. Extreme Kerr and Kerr-Newman geometries are given as examples.
We study event horizons of non-axisymmetric black holes and show how features found in axisymmetric studies of colliding black holes and of toroidal black holes are non-generic and how new features emerge. Most of the details of black hole formation and black hole merger are known only in the axisymmetric case, in which numerical evolution has successfully produced dynamical space-times. The work that is presented here uses a new approach to construct the geometry of the event horizon, not by locating it in a given spacetime, but by direct construction. In the axisymmetric case, our method pro…
The strong version of the nonviolent nonlocality proposal of Giddings predicts "strong but soft" quantum metric fluctuations near black hole horizons in an attempt to resolve the information paradox. To study observable signatures of this proposal, we numerically solve Einstein's equations modified by these fluctuations and analyze the gravitational wave signal from the inspiral and merger of two black holes. In a model of evolution for such fluctuations, we show that they lead to significant deviations in the observed waveform, even when the black holes are still well separated, and could pot…
Inter-University Centre for Astronomy and Astrophysics（大学间天文学与天体物理中心）；International Centre for Theoretical Sciences, Tata Institute of Fundamental Research（塔塔基础研究所国际理论科学中心）；Gran Sasso Science Institute (GSSI)（格兰萨索科学研究所）；INFN, Laboratori Nazionali del Gran Sasso（意大利国家核物理研究所格兰萨索国家实验室）；Department of Physics and Astronomy, Washington State University（华盛顿州立大学物理与天文系）；School of Physics and Astronomy and Institute for Gravitational Wave Astronomy, University of Birmingham（伯明翰大学物理与天文学院及引力波天文研究所）
Is the graviton massless? This problem was addressed in the literature at a phenomenological level, using modified dispersion relations for gravitational waves, in linearized calculations around flat space. Here, we perform a detailed analysis of the gravitational waveform produced when a small particle plunges or inspirals into a large non-spinning black hole. Our results should presumably also describe the gravitational collapse to black holes and explosive events such as supernovae. In the context of a theory with massive gravitons and screening, merging objects up to $1\,{\rm Gpc}$ away or…
Research on extracting science from binary-black-hole (BBH) simulations has often adopted a "scattering matrix" perspective: given the binary's initial parameters, what are the final hole's parameters and the emitted gravitational waveform? In contrast, we are using BBH simulations to explore the nonlinear dynamics of curved spacetime. Focusing on the head-on plunge, merger, and ringdown of a BBH with transverse, antiparallel spins, we explore numerically the momentum flow between the holes and the surrounding spacetime. We use the Landau-Lifshitz field-theory-in-flat-spacetime formulation of …
Building up on previous work, we present a new calculation of the gravitational wave (GW) emission generated during the transition from quasi-circular inspiral to plunge, merger and ringdown by a binary system of nonspinning black holes, of masses $m_1$ and $m_2$, in the extreme mass ratio limit, $m_1 m_2\ll(m_1+m_2)^2$. The relative dynamics of the system is computed {\it without making any adiabatic approximation} by using an effective one body (EOB) description, namely by representing the binary by an effective particle of mass $μ=m_1 m_2/(m_1+m_2)$ moving in a (quasi-)Schwarzschild backgro…
We discuss the transition from quasi-circular inspiral to plunge of a system of two nonrotating black holes of masses $m_1$ and $m_2$ in the extreme mass ratio limit $m_1m_2\ll (m_1+m_2)^2$. In the spirit of the Effective One Body (EOB) approach to the general relativistic dynamics of binary systems, the dynamics of the two black hole system is represented in terms of an effective particle of mass $μ\equiv m_1m_2/(m_1+m_2)$ moving in a (quasi-)Schwarzschild background of mass $M\equiv m_1+m_2$ and submitted to an ${\cal O}(μ)$ radiation reaction force defined by Padé resumming high-order Post-…
It is expected that all astrophysical black holes in equilibrium are well described by the Kerr solution. Moreover, any black hole far away from equilibrium, such as one initially formed in a compact binary merger or by the collapse of a massive star, will eventually reach a final equilibrium Kerr state. At sufficiently late times in this process of reaching equilibrium, we expect that the black hole is modeled as a perturbation around the final state. The emitted gravitational waves will then be damped sinusoids with frequencies and damping times given by the quasinormal mode spectrum of the …
Static solutions of the Einstein-Yang-Mills-Higgs system containing extreme black holes are studied. The field equations imply strong restrictions on boundary values of all fields at the horizon. If the Yang-Mills radial electric field $E$ is non-zero there, then all fields at the horizon take values in the centralizer of $E$. For the particular case of SU(3), there are two different kinds of centralizers: two-dimensional abelian (Cartan subalgebra) and four-dimensional (su(2)$\times$u(1)) ones. The two-dimensional centralizer admits only constant fields: even the geometry of the horizon is th…
This work addresses and solves the problem of generically tracking black hole event horizons in computational simulation of black hole interactions. Solutions of the hyperbolic eikonal equation, solved on a curved spacetime manifold containing black hole sources, are employed in development of a robust tracking method capable of continuously monitoring arbitrary changes of topology in the event horizon, as well as arbitrary numbers of gravitational sources. The method makes use of continuous families of level set viscosity solutions of the eikonal equation with identification of the black hole…
We investigate the properties of the event horizon in the merger between a large black hole and a smaller neutron star. We find that, if the star is compact enough, then, in its rest frame a horizon begins to grow inside the star before it merges with the black hole, in a manner analogous to the growth of the event horizon in stellar collapse. We may say that, ahead of its fall into the larger black hole, the neutron star begins to become a black hole itself. We discuss how the phenomenon, even if not directly observable, can be invariantly characterized. We demonstrate it quantitatively by ex…
We construct the covariantly defined multipole moments on the common horizon of an equal-mass, non-spinning, quasicircular binary-black-hole system. We see a strong correlation between these multipole moments and the gravitational waveform. We find that the multipole moments are well described by the fundamental quasinormal modes at sufficiently late times. For each multipole moment, at least two fundamental modes of different $\ell$ are detectable in the best model. These models provide faithful estimates of the true mass and spin of the remnant black hole. We also show that by including over…
The inspiral and merger of a binary black hole system generally leads to an asymmetric distribution of emitted radiation, and hence a recoil of the remnant black hole directed opposite to the net linear momentum radiated. The recoil velocity is generally largest for comparable mass black holes and particular spin configurations, and approaches zero in the extreme mass ratio limit. It is generally believed that for extreme mass ratios eta<<1, the scaling of the recoil velocity is V {\propto} eta^2, where the proportionality coefficient depends on the spin of the larger hole and the geometry of …
This work establishes critical phenomena in the topological transition of black hole coalescence. We describe and validate a computational front tracking event horizon solver, developed for generic studies of the black hole coalescence problem. We then apply this to the Kastor - Traschen axisymmetric analytic solution of the extremal Maxwell - Einstein black hole merger with cosmological constant. The surprising result of this computational analysis is a power law scaling of the minimal throat proportional to time. The minimal throat connecting the two holes obeys this power law during a short…
We discuss the merger process of binary black holes with Hawking radiation taken into account. Besides the redshifted radiation to infinity, binary black holes can exchange radiation between themselves, which is first redshifted and then blueshifted when it propagates from one hole to the other. The exchange rate should be large when the temperature-divergent horizons are penetrating each other to form a single horizon with unique temperature. This will cause non-negligible mass and angular momentum transfer between the black holes during the merging process of the horizons. We further argue i…
In numerical simulations of black hole binaries, Pretorius and Khurana [Class. Quant. Grav. {\bf 24}, S83 (2007)] have observed critical behaviour at the threshold between scattering and immediate merger. The number of orbits scales as $n\simeq -γ\ln|p-p_*|$ along any one-parameter family of initial data such that the threshold is at $p=p_*$. Hence they conjecture that in ultrarelavistic collisions almost all the kinetic energy can be converted into gravitational waves if the impact parameter is fine-tuned to the threshold. As a toy model for the binary, they consider the geodesic motion of a …
Dirac equation is solved in the near horizon limit geometry of an extreme Kerr black hole. We decouple equations first as usual, into an axial and angular part. The axial equation turns out to be independent of the mass and is solved exactly. The angular equation reduces, in the massless case, to a confluent Heun equation. In general for nonzero mass, the angular equation is expressible at best, as a set of coupled first order differential equations apt for numerical investigation. The axial potentials corresponding to the associated Schrodinger-type equations and their conserved currents are …
We present a formula that relates the variations of the area of extreme throat initial data with the variation of an appropriate defined mass functional. From this expression we deduce that the first variation, with fixed angular momentum, of the area is zero and the second variation is positive definite evaluated at the extreme Kerr throat initial data. This indicates that the area of the extreme Kerr throat initial data is a minimum among this class of data. And hence the area of generic throat initial data is bounded from below by the angular momentum. Also, this result strongly suggests th…
We study gravitational-wave emission from the quasi-circular, extreme mass ratio inspiral of compact objects of mass m0 into massive objects of mass M>>m0 whose external metric is identical to the Schwarzschild metric, except for the absence of an event horizon. To be specific we consider one of the simplest realizations of such an object: a nonrotating thin-shell gravastar. The power radiated in gravitational waves during the inspiral shows distinctive peaks corresponding to the excitation of the polar oscillation modes of the gravastar. For ultra-compact gravastars the frequency of these pea…
We determine all infinitesimal transverse deformations of extreme horizons in Einstein-Maxwell theory that preserve axisymmetry. In particular, we show that the general static transverse deformation of the AdS(2) X S(2) near-horizon geometry is a two-parameter family, which contains the known extreme charged, accelerating, static black hole solution held in equilibrium by an external electric or magnetic field (Ernst solution) and a special case of the extreme Kerr-Newman-Melvin solution. More generally, we find a three-parameter family of deformations of the extreme Kerr-Newman horizon, which…
A massless scalar field exhibits an instability at the event horizon of an extreme black hole. We study numerically the nonlinear evolution of this instability for spherically symmetric perturbations of an extreme Reissner-Nordstrom (RN) black hole. We find that generically the endpoint of the instability is a non-extreme RN solution. However, there exist fine-tuned initial perturbations for which the instability never decays. In this case, the perturbed spacetime describes a time-dependent extreme black hole. Such solutions settle down to extreme RN outside, but not on, the event horizon. The…
We present a proof-of-concept study, based on numerical-relativity simulations, of how gravitational waves (GWs) from neutron star merger remnants can probe the nature of matter at extreme densities. Phase transitions and extra degrees of freedom can emerge at densities beyond those reached during the inspiral, and typically result in a softening of the equation of state (EOS). We show that such physical effects change the qualitative dynamics of the remnant evolution, but they are not identifiable as a signature in the GW frequency, with the exception of possible black-hole formation effects.…
It is shown that the near horizon geometry of a generic extreme regular black hole solution of Einstein gravity coupled to nonlinear electrodynamics is described by the $AdS_2 \times S^2$ spacetime.
