166 research outputs found
Simple Analytic Models of Gravitational Collapse
Most general relativity textbooks devote considerable space to the simplest
example of a black hole containing a singularity, the Schwarzschild geometry.
However only a few discuss the dynamical process of gravitational collapse, by
which black holes and singularities form. We present here two types of analytic
models for this process, which we believe are the simplest available; the first
involves collapsing spherical shells of light, analyzed mainly in
Eddington-Finkelstein coordinates; the second involves collapsing spheres
filled with a perfect fluid, analyzed mainly in Painleve-Gullstrand
coordinates. Our main goal is pedagogical simplicity and algebraic
completeness, but we also present some results that we believe are new, such as
the collapse of a light shell in Kruskal-Szekeres coordinates.Comment: Submitted to American Journal of Physic
Painleve-Gullstrand Coordinates for the Kerr Solution
We construct a coordinate system for the Kerr solution, based on the zero
angular momentum observers dropped from infinity, which generalizes the
Painleve-Gullstrand coordinate system for the Schwarzschild solution. The Kerr
metric can then be interpreted as describing space flowing on a (curved)
Riemannian 3-manifold. The stationary limit arises as the set of points on this
manifold where the speed of the flow equals the speed of light, and the
horizons as the set of points where the radial speed equals the speed of light.
A deeper analysis of what is meant by the flow of space reveals that the
acceleration of free-falling objects is generally not in the direction of this
flow. Finally, we compare the new coordinate system with the closely related
Doran coordinate system.Comment: 6 pages; v2: new section, matches final published version; v3: sign
error in the expression of the function delta correcte
Fermion absorption cross section of a Schwarzschild black hole
We study the absorption of massive spin-half particles by a small
Schwarzschild black hole by numerically solving the single-particle Dirac
equation in Painleve-Gullstrand coordinates. We calculate the absorption cross
section for a range of gravitational couplings Mm/m_P^2 and incident particle
energies E. At high couplings, where the Schwarzschild radius R_S is much
greater than the wavelength lambda, we find that the cross section approaches
the classical result for a point particle. At intermediate couplings we find
oscillations around the classical limit whose precise form depends on the
particle mass. These oscillations give quantum violations of the equivalence
principle. At high energies the cross section converges on the geometric-optics
value of 27 \pi R_S^2/4, and at low energies we find agreement with an
approximation derived by Unruh. When the hole is much smaller than the particle
wavelength we confirm that the minimum possible cross section approaches \pi
R_S^2/2.Comment: 11 pages, 3 figure
Hawking radiation in an electro-magnetic wave-guide?
It is demonstrated that the propagation of electro-magnetic waves in an
appropriately designed wave-guide is (for large wave-lengths) analogous to that
within a curved space-time -- such as around a black hole. As electro-magnetic
radiation (e.g., micro-weaves) can be controlled, amplified, and detected (with
present-day technology) much easier than sound, for example, we propose a
set-up for the experimental verification of the Hawking effect. Apart from
experimentally testing this striking prediction, this would facilitate the
investigation of the trans-Planckian problem. PACS: 04.70.Dy, 04.80.-y,
42.50.-p, 84.40.Az.Comment: 4 pages RevTeX, 1 figur
Circular orbits and spin in black-hole initial data
The construction of initial data for black-hole binaries usually involves the
choice of free parameters that define the spins of the black holes and
essentially the eccentricity of the orbit. Such parameters must be chosen
carefully to yield initial data with the desired physical properties. In this
paper, we examine these choices in detail for the quasiequilibrium method
coupled to apparent-horizon/quasiequilibrium boundary conditions. First, we
compare two independent criteria for choosing the orbital frequency, the
"Komar-mass condition" and the "effective-potential method," and find excellent
agreement. Second, we implement quasi-local measures of the spin of the
individual holes, calibrate these with corotating binaries, and revisit the
construction of non-spinning black hole binaries. Higher-order effects, beyond
those considered in earlier work, turn out to be important. Without those,
supposedly non-spinning black holes have appreciable quasi-local spin;
furthermore, the Komar-mass condition and effective potential method agree only
when these higher-order effects are taken into account. We compute a new
sequence of quasi-circular orbits for non-spinning black-hole binaries, and
determine the innermost stable circular orbit of this sequence.Comment: 24 pages, 17 figures, accepted for publication in Physical Review D,
revtex4; Fixed error in computing proper separation and updated figures and
tables accordingly, added reference to Sec. IV.A, fixed minor error in Sec.
IV.B, added new data to Tables IV and V, fixed 1 reference, fixed error in
Eq. (A7b), included minor changes from PRD editin
Transgressing the horizons: Time operator in two-dimensional dilaton gravity
We present a Dirac quantization of generic single-horizon black holes in
two-dimensional dilaton gravity. The classical theory is first partially
reduced by a spatial gauge choice under which the spatial surfaces extend from
a black or white hole singularity to a spacelike infinity. The theory is then
quantized in a metric representation, solving the quantum Hamiltonian
constraint in terms of (generalized) eigenstates of the ADM mass operator and
specifying the physical inner product by self-adjointness of a time operator
that is affinely conjugate to the ADM mass. Regularity of the time operator
across the horizon requires the operator to contain a quantum correction that
distinguishes the future and past horizons and gives rise to a quantum
correction in the hole's surface gravity. We expect a similar quantum
correction to be present in systems whose dynamics admits black hole formation
by gravitational collapse.Comment: 32 pages, 1 eps figure. v2: references and comments adde
The Gravitational Hamiltonian in the Presence of Non-Orthogonal Boundaries
This paper generalizes earlier work on Hamiltonian boundary terms by omitting
the requirement that the spacelike hypersurfaces intersect the
timelike boundary orthogonally. The expressions for the action and
Hamiltonian are calculated and the required subtraction of a background
contribution is discussed. The new features of a Hamiltonian formulation with
non-orthogonal boundaries are then illustrated in two examples.Comment: 23 pages, 1 figure, LaTeX. The action is altered to include a corner
term which results in a different value for the non-orthogonal term. An
additional appendix with Euclidean results is included. To appear in Class.
Quant. Gra
The Jang equation, apparent horizons, and the Penrose inequality
The Jang equation in the spherically symmetric case reduces to a first order
equation. This permits an easy analysis of the role apparent horizons play in
the (non)existence of solutions. We demonstrate that the proposed derivation of
the Penrose inequality based on the Jang equation cannot work in the
spherically symmetric case. Thus it is fruitless to apply this method, as it
stands, to the general case. We show also that those analytic criteria for the
formation of horizons that are based on the use of the Jang equation are of
limited validity for the proof of the trapped surface conjecture.Comment: minor misprints correcte
Riemannian geometry of irrotational vortex acoustics
We consider acoustic propagation in an irrotational vortex, using the
technical machinery of differential geometry to investigate the ``acoustic
geometry'' that is probed by the sound waves. The acoustic space-time curvature
of a constant circulation hydrodynamical vortex leads to deflection of phonons
at appreciable distances from the vortex core. The scattering angle for phonon
rays is shown to be quadratic in the small quantity , where
is the vortex circulation, the speed of sound, and the impact
parameter.Comment: 4 pages, 2 figures, RevTex4. Discussion of focal length added; to
appear in Physical Review Letter
Counting a black hole in Lorentzian product triangulations
We take a step toward a nonperturbative gravitational path integral for
black-hole geometries by deriving an expression for the expansion rate of null
geodesic congruences in the approach of causal dynamical triangulations. We
propose to use the integrated expansion rate in building a quantum horizon
finder in the sum over spacetime geometries. It takes the form of a counting
formula for various types of discrete building blocks which differ in how they
focus and defocus light rays. In the course of the derivation, we introduce the
concept of a Lorentzian dynamical triangulation of product type, whose
applicability goes beyond that of describing black-hole configurations.Comment: 42 pages, 11 figure
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