Over the past three decades, black holes have played an important role in quantum gravity, mathematical physics, numerical relativity and gravitational wave phenomenology. However, conceptual settings and mathematical models used to discuss them have varied considerably from one area to another. Over the last five years a new, quasi-local framework was introduced to analyze diverse facets of black holes in a unified manner. In this framework, evolving black holes are modelled by dynamical horizons and black holes in equilibrium by isolated horizons. We review basic properties of these horizons and summarize applications to mathematical physics, numerical relativity, and quantum gravity. This paradigm has led to significant generalizations of several results in black hole physics. Specifically, it has introduced a more physical setting for black hole thermodynamics and for black hole entropy calculations in quantum gravity, suggested a phenomenological model for hairy black holes, provided novel techniques to extract physics from numerical simulations, and led to new laws governing the dynamics of black holes in exact general relativity.

/pdf/isolated-and-dynamical-horizons-and-their-applications-2ojdjee2p1.pdf

Isolated and dynamical horizons and their applications

https://www.luth.obspm.fr/~luthier/gourgoulhon/pdf/GourgJ06a.pdf

A 3+1 perspective on null hypersurfaces and isolated horizons

This paper presents a quasilocal method of studying the physics of dynamical black holes in numerical simulations. This is done within the dynamical horizon framework, which extends the earlier work on isolated horizons to time-dependent situations. In particular: (i) We locate various kinds of marginal surfaces and study their time evolution. An important ingredient is the calculation of the signature of the horizon, which can be either spacelike, timelike, or null. (ii) We generalize the calculation of the black hole mass and angular momentum, which were previously defined for axisymmetric isolated horizons to dynamical situations. (iii) We calculate the source multipole moments of the black hole which can be used to verify that the black hole settles down to a Kerr solution. (iv) We also study the fluxes of energy crossing the horizon, which describes how a black hole grows as it accretes matter and/or radiation. We describe our numerical implementation of these concepts and apply them to three specific test cases, namely, the axisymmetric head-on collision of two black holes, the axisymmetric collapse of a neutron star, and a nonaxisymmetric black hole collision with nonzero initial orbital angular momentum.

/pdf/introduction-to-dynamical-horizons-in-numerical-relativity-344ed91ptm.pdf

Introduction to dynamical horizons in numerical relativity

Using ideas employed in higher dimensional gravity, non-expanding, weakly isolated and isolated horizons are introduced and analyzed in 2+1 dimensions. While the basic definitions can be taken over directly from higher dimensions, their consequences are somewhat different because of the peculiarities associated with 2+1 dimensions. Nonetheless, as in higher dimensions, we are able to: i) analyze the horizon geometry in detail; ii) introduce the notions of mass, charge and angular momentum of isolated horizons using geometric methods; and, iii) generalize the zeroth and the first laws of black hole mechanics. The Hamiltonian methods also provide, for the first time, expressions of total angular momentum and mass of charged, rotating black holes and their relation to the analogous quantities defined at the horizon. We also construct the analog of the Newman-Penrose framework in 2+1 dimensions which should be useful in a wide variety of problems in 2+1 dimensional gravity.

https://www.intlpress.com/site/pub/files/_fulltext/journals/atmp/2002/0006/0003/ATMP-2002-0006-0003-a003.pdf

Isolated horizons in $2 + 1$ gravity

Over the past three decades, black holes have played an important role in quantum gravity, mathematical physics, numerical relativity and gravitational wave phenomenology. However, conceptual settings and mathematical models used to discuss them have varied considerably from one area to another. Over the last five years a new, quasi-local framework was introduced to analyze diverse facets of black holes in a unified manner. In this framework, evolving black holes are modeled by dynamical horizons and black holes in equilibrium by isolated horizons. We review basic properties of these horizons and summarize applications to mathematical physics, numerical relativity and quantum gravity. This paradigm has led to significant generalizations of several results in black hole physics. Specifically, it has introduced a more physical setting for black hole thermodynamics and for black hole entropy calculations in quantum gravity; suggested a phenomenological model for hairy black holes; provided novel techniques to extract physics from numerical simulations; and led to new laws governing the dynamics of black holes in exact general relativity.

/pdf/isolated-and-dynamical-horizons-and-their-applications-31osa7qy4z.pdf

The isolated horizon framework is applied to numerical relativity.
This framework describes black holes which are in equilibrium in a
spacetime which is otherwise fully dynamical. We introduce
isolated horizons in a way which is directly applicable in
numerical evolutions. In particular, we present and numerically
implement a coordinate invariant method for calculating the mass
and angular momentum of isolated horizons. We also show how
isolated horizons can be used to study the local geometry of a
black hole and extract invariant gravitational waveforms. We
apply some results from the isolated horizon framework to
calculate the binding energy between two black holes. This will be
useful in comparing different initial data sets which represent
similar physical situations. Finally, we also present a method
for studying the physics of dynamical black holes. In particular,
we derive balance laws relating the flux of matter
fields/radiation with the change in the physical parameters of the
black hole such as angular momentum and mass.

/pdf/isolated-horizons-in-numerical-relativity-1zyu30jsw5.pdf

Badri Krishnan

Papers

Isolated horizons in numerical relativity