Session E12: Spacetime Structure of Binary Black Hole Simulations

Visualizing the Weyl Curvature Tensor: Frame-Drag Vortex Lines and Tidal Tendex Lines

When one slices spacetime into space plus time, the Weyl curvature tensor gets split into two symmetric, trace-free tensors: its ``electric'' part, which describes tidal forces, and its ``magnetic'' part, which describes differential frame dragging. The electric part is completely characterized by {\it tidal tendex lines} (integral curves of its eigenvectors) and their {\it tendicities} (eigenvalues); and the magnetic part, by corresponding {\it frame-drag vortex lines} and their {\it vorticities}. We will discuss the physical meanings of these quantities and their use to visualize spacetime curvature, and we will illustrate them for stationary situations: a spinning body in linearized theory, and a Kerr black hole.   

Vortex and Tendex Lines in Post-Newtonian and Black-Hole Perturbation Spacetimes

We explore the behavior of frame-drag vortex lines and their vorticities, and tidal tendex lines and their tendicities (see above abstract by Thorne et al.) in analytic, dynamical situations --- specifically, for a plane gravitational wave, for a binary system in the Post-Newtonian approximation, and for quadrupolar quasinormal modes of a Schwarzschild black hole. Using pictures of the vortex and tendex lines, and the Maxwell-like evolution equations for the electric and magnetic parts of the Weyl tensor, we show how dynamical near-zone vortex and tendex lines give rise to gravitational waves as one moves outward into the wave zone.   

Further interpretation and application of frame-drag vorticity and tidal tendicity

Previous talks in this session presented concepts of frame-drag vorticity and tidal tendicity in the bulk spatial slices in curved spacetimes. Analogous concepts can be defined directly on two-surfaces, such as slices of black hole horizons, defining ``horizon vortexes'' and ``horizon tendexes.'' I will discuss these concepts, their relationship to the piercing of the two-surface by the vortex lines and tendex lines, to the geometry (intrinsic and extrinsic) of the two-surface, and to existing constructions defining black hole spin and other source multipoles. I will also present simulations of equal-mass black hole binaries in the so-called ``extreme kick'' configuration carried out recently with the SpEC code, and interpret the maximization of the kick in terms of these concepts of vorticity and tendicity.   

Computation of Vortex and Tendex Lines in Numerical Simulations, and Their Behaviors in a Head-on Collision of Spinning Black Holes

In this talk, I will describe the frame-drag vortex lines and tidal tendex lines (see previous abstracts) in the context of numerical relativity. Specifically, I will discuss how these lines can be computed in numerical-relativity simulations in general, and I will then explore their behaviors in a specific simulation (performed using the SpEC code): the head-on merger of two identical black holes with oppositely directed, transverse spins. As the holes merge, their merged horizon acquires four vortexes, two with each sign of vorticity. The closest vortex lines, those which connect to the horizon, simply wrap around the hole connecting horizon vortexes of like sign. As the hole settles down to Schwarzschild, the horizon vortexes slosh (their vorticities oscillate in sign). With each oscillation, they eject a self-contained packet of vortex lines that becomes a gravitational wave as it travels into the wave zone.   

The Meanings of Frame-Drag Vortex Lines and Tidal Tendex Lines

Frame-drag vortex lines and tidal tendex lines provide a tensorial visualization of gravitational waves and their generation that contains more information than traditional scalar quantities such as the Newman-Penrose $\Psi_4$. By examining the example of a head-on merger of identical black holes with transverse spins, I will present some relationships between the vortex and tendex viewpoint and other formalisms in wave zone. I will also discuss analogies to electromagnetism, with the aim of demonstrating how one might build up intuition into the information embodied in vortex and tendex lines, and their vorticities and tendicities.   

Binary-black-hole mergers with spins beyond the Bowen-York limit

Some accretion models predict black holes with spins that are nearly extremal. When (possibly rapidly spinning) black holes spiral together and merge, the emitted gravitational waves, the properties (mass, spin, velocity) of the remnant, and the behavior of the strongly curved spacetime near the holes' horizons can only be explored using numerical simulations. Most binary-black-hole (BBH) simulations use initial data based on the Bowen-York extrinsic curvature and consequently cannot represent merging holes with spins larger than 0.93 (the ``Bowen-York limit''). In this talk, I will describe results (waveform, remnant properties, and vorticity and tendicity of the strongly curved spacetime) from simulations [using the SpEC code (black-holes.org/SpEC.html)] of BBH inspirals, mergers, and ringdowns that use suitable initial data to achieve spins beyond the Bowen-York limit.   

Intermediate-mass-ratio black hole binaries: numerical relativity meets perturbation theory

We perform a series of full numerical simulations of nonspinning black holes with mass ratios $q=1/10$ and $q=1/15$. We compare gravitational radiation waveforms as computed by full numerical and perturbative evolutions. In the source terms of the Regge-Wheller-Zerilli Schwarzschild perturbations equations we use the full numerical tracks transformed into the Schwarzschild gauge. We then extend this perturbative formalism to take into account small intrinsic spins of the large black hole (valid for spins $|a/M|<0.3$). Including the final spins improves the overlap functions when comparing full numerical and perturbative waveforms, reaching 99.5\% for the leading $(\ell,m)=(2,2)$ and (3,3) modes, and 98.3\% for the nonleading (2,1) mode in the $q=1/10$ case, which includes 8 orbits before merger. For the $q=1/15$ case, we obtain overlaps near 99.7\% for all three modes. We discuss the modeling of the full inspiral and merger based on a combined matching of post-Newtonian, full numerical, and geodesic trajectories.   

Hybrid black-hole binary initial data

Traditional black-hole binary puncture initial data is conformally flat. This unphysical assumption is coupled with a lack of radiation signature from the binary's past life. As a result, waveforms extracted from evolutions of this data display an abrupt jump. In Kelly et al. [Class.Quant.Grav.27:114005,2010], a new binary black-hole initial data with radiation contents derived in the post-Newtonian (PN) calculation was adapted to puncture evolutions in numerical relativity. This data satisfies the constraint equations to the 2.5PN order, and contains a transverse-traceless ``wavy'' metric contribution, violating the standard assumption of conformal flatness. Although the evolution contained less spurious radiation, there were undesired features; the unphysical horizon mass loss and the large initial orbital eccentricity. Introducing a hybrid approach to the initial data evaluation, we significantly reduce these undesired features.   

Simulations of Binary Black Hole Mergers in Gaseous Disks

Binary black hole mergers in the presence of gaseous accretion flows are prime candidates for simultaneous observations of both gravitational waves and electromagnetic signals. We study such systems using our fully general relativistic hydrodynamics code, focusing on potentially observable electromagnetic signatures. We outline recent developments in our study, which explores the final stages of binary black hole mergers inside an adiabatic (except for shocks), disk-like accretion flow. We discuss the dynamical response of the disk to the inspiral and merger of the black holes, treating the optically thin electromagnetic radiation as a perturbation. We identify characteristic, observable changes in the electromagnetic luminosity during the merger.