Session L12: Strong-field gravity: black holes, event horizons, and cosmology

The beginnings of black hole horizons

The beginning of a black hole horizon is the set of points where generators enter the horizon. Several properties of this ``entry set" and the early horizon near it are shown: It is the locus of the horizon's self-intersections, and it is spacelike of dimension zero, one or two, where this is defined. It is connected but can bifurcate in possibly complicated ways. On spacelike surfaces the entry of generators manifests itself in a kink in the horizon. The kinks propagate at superluminal speed until they ``run out of steam," slow down to light speed and disappear. Kinks generally run from the main collapse region to secondary collapse events until no more new generators enter the horizon. This is illustrated by collapse of two mass concentrations, and by the case of a large number of particles.   

Event Horizons from black-hole rings

We construct and evolve non-rotating vacuum initial data with a ring singularity, based on a simple extension of the standard Brill-Lindquist multiple black-hole initial data, and search for event horizon with spatial slices that are topologically tori. We find, that it is not possible to produce a finite-sized toroidal horizon, which only occurs in a singular limit where the horizon width has zero size, indicating the presence of a naked singularity.   

Dynamical Horizons of Distorted Rotating Black Holes

We present numerical simulations of a rotating black hole distorted by a pulse of ingoing gravitational radiation. For strong pulses, we find up to five concentric marginally outer trapped surfaces. These trapped surfaces appear and disappear in pairs, so that the total number of such surfaces at any given time is odd. During the highly dynamical regime, we analyze the structure of marginally trapped tubes in the context of the dynamical horizon formalism, and evaluate the energy flux across them via the dynamical horizon flux law. We also evaluate the angular momentum flux across them via the generalized Damour-Navier-Stokes equation. Finally, we contrast the behavior of the marginally trapped tubes with the event horizon.   

Generic Ringdown Frequencies at the Birth of a Kerr Black Hole

The final phase of the gravitational radiation emitted at the birth of a Kerr black hole is made up of decaying sinusoids, oscillating at the complex quasi-normal mode (QNM) frequencies, and weak power law tails. In particular, the QNMs arise as eigenmodes of the homogeneous linearized Einstein's equations, plus appropriate boundary conditions. We present a new set of generic ``ringdown'' frequencies, which have been found by analytically evolving generic, regular initial data for the vacuum Teukolsky equation in a near horizon expansion, following the strategy of Mino and Brink (2008). This set of modes has frequencies proportional to the horizon frequency of the black hole, and decay rates proportional to the surface gravity. We comment on our results in the context of this previous study. A full understanding of the ringdown frequency modes is essential for creation of accurate gravitational waveforms for use in gravitational wave detection. We explore the utility of these new modes in fitting a ringdown gravitational waveform produced by the numerical evolution of a binary inspiral.   

A physically motivated framework to describe black hole perturbations

Black hole perturbation theory is typically studied on time surfaces that extend between the bifurcation sphere and spatial infinity. I will argue that---from a physical point of view---it is favorable to employ time surfaces that extend between the future event horizon and future null infinity. I will demonstrate how this new approach resolves problems related to the representation of quasinormal mode eigenfunctions, the construction of short-ranged potentials for curvature perturbations, and the numerical calculation of gravitational waveforms as measured by an idealized observer.   

Late--time Kerr tails: ``up" and ``down" excitations

We revisit the question of Kerr spacetime late--time scalar--field tails numerically, specifically the excitation of ``up" and ``down" modes. Specifically, an ``up" mode is an excited $(\ell,m)$ mode because of an initial $(\ell',m)$ mode for $\ell>\ell'$. The definition of a ``down" mode is commensurate. We propose to generalize the Barack--Ori formula for the decay rate of any tail multipole given a generic initial data set, to the contribution of any initial multipole mode. Our proposal leads to a much simpler expression for the late--time power law index. Specifically, we propose that the late--time decay rate of a kinematically allowed $Y_{\ell m}$ spherical harmonic multipole moment because of an initial $Y_{\ell' m}$ multipole is independent of the azimuthal number $m$, and is given by $t^{-n}$, where $n=\ell'+\ell+1$ for $\ell<\ell'$ and $n=\ell'+\ell+3$ for $\ell\ge\ell'$. The independence of $m$ may be surprising because of the explicit dependence of the Green's function on $m$. The much greater complexity of the usual Hod formula is an artifact of the additional requirement of describing the slowest damped mode.   

Finding Mixmaster Dynamics in Collapsing Cosmological Spacetimes

Numerical and analytic studies indicate that collapsing, generic cosmological spacetimes exhibit local Mixmaster dynamics, at least in part. At a heuristic level, the method of consistent potentials---seeing if local Kasner behavior is consistent---can make plausible the Mixmaster behavior. However, spacetimes with one, two, and three spatial symmetries that are known to exhibit local Mixmaster dynamics also appear to exhibit qualitatively different interactions when described by potentials in a natural set of variables. If one assumes, for example, that a Bianchi IX spatially homogeneous model is a special case of a $U(1)$-symmetric spatially inhomogeneous model, transformations between the usual variables in each case yield predictions for new $U(1)$ dynamics that might be found in numerical simulations. However, generic $T^2$-symmetric models do not appear to allow similar constructions within the standard Belinskii-Khalatnikov-Lifshitz time variable. While this has been described previously (see B.K. Berger, Class. Quant. Grav. {\bf 21} (2004) S81-S96 and references therein), some open questions remain. These will be explored and, perhaps, resolved.   

A new framework for analyzing the effects of small scale inhomogeneities in cosmology

While the universe is spatially homogeneous and isotropic on large scales, it is extremely inhomogeneous on small scales, where fractional density perturbations can be much larger than one. With this in mind, several authors have proposed that, when properly averaged, non-linear terms in the Einstein equation can mimic Dark Energy. We present a non-vacuum generalization of the ``short-wave approximation'', which takes such large density fluctuations on short scales into account in a mathematically rigorous manner. We prove that if the stress energy tensor of matter satisfies the weak energy condition, then the zeroth order ``effective stress energy tensor'' associated with the small scale density and metric fluctuations must be trace-free and satisfy the weak energy condition as well. We conclude from this that small scale density inhomogeneities cannot account for the acceleration of the scale factor. However, at first order, we find that linearized cosmological perturbation theory can be corrected to include small but non-zero backreaction effects of small scale density inhomogeneities.   

Violation of the equivalence principle via the Unruh effect

We show that in certain situations the equivalence principle - stated as the local equality of acceleration with a gravitational field - is violated by the Unruh effect. For a black hole space-time with a horizon the EP holds, while for a gravitating body without a horizon (e.g. the Earth, a white dwarf, a neutron star) the EP is violated.