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Towards a general description of the interior structure of rotating black holes
The purpose of this paper is to present a number of proposals about the
interior structure of a rotating black hole that is accreting slowly, but in an
arbitrary time- and space-dependent fashion. The proposals could potentially be
tested with numerical simulations. Outgoing and ingoing particles free-falling
in the parent Kerr geometry become highly focused along the principal outgoing
and ingoing null directions as they approach the inner horizon, triggering the
mass inflation instability. The original arguments of Barrabes, Israel &
Poisson (1990) regarding inflation in rotating black holes are reviewed, and
shown to be based on Raychauduri's equation applied along the outgoing and
ingoing null directions. It is argued that gravitational waves should behave in
the geometric optics limit, and consequently that the spacetime should be
almost shear-free. A full set of shear-free equations is derived. A specific
line-element is proposed, which is argued should provide a satisfactory
approximation during early inflation. Finally, it is argued that
super-Planckian collisions between outgoing and ingoing particles will lead to
entropy production, bringing inflation to an end, and precipitating collapse.Comment: 15 page
The interior structure of rotating black holes 3. Charged black holes
This paper extends to the case of charged rotating black holes the
conformally stationary, axisymmetric, conformally separable solutions presented
for uncharged rotating black holes in a companion paper. In the present paper,
the collisionless fluid accreted by the black hole may be charged. The charge
of the black hole is determined self-consistently by the charge accretion rate.
As in the uncharged case, hyper-relativistc counter-streaming between ingoing
and outgoing streams drives inflation at (just above) the inner horizon,
followed by collapse. If both ingoing and outgoing streams are charged, then
conformal separability holds during early inflation, but fails as inflation
develops. If conformal separability is imposed throughout inflation and
collapse, then only one of the ingoing and outgoing streams can be charged: the
other must be neutral. Conformal separability prescribes a hierarchy of
boundary conditions on the ingoing and outgoing streams incident on the inner
horizon. The dominant radial boundary conditions require that the incident
ingoing and outgoing number densities be uniform with latitude, but the charge
per particle must vary with latitude such that the incident charge densities
vary in proportion to the radial electric field. The sub-dominant angular
boundary conditions require specific forms of the incident number- and
charge-weighted angular motions. If the streams fall freely from outside the
horizon, then the prescribed angular conditions can be achieved by the charged
stream, but not by the neutral stream. Thus, as in the case of an uncharged
black hole, the neutral stream must be considered to be delivered ad hoc to
just above the inner horizon.Comment: Version 1: 12 pages, no figures. Version 2: Extensively revised,
logic tightened, derivation more elegant. 18 pages, no figures. Version 3:
Minor revisions to match published version. Mathematica notebook available at
http://jila.colorado.edu/~ajsh/rotatinginflationary/rotatinginflationary.n
The interior structure of rotating black holes 2. Uncharged black holes
(Abridged) A solution is obtained for the interior structure of an uncharged
rotating black hole that accretes a collisionless fluid. The solution is
conformally stationary, axisymmetric, and conformally separable, possessing a
conformal Killing tensor. Hyper-relativistic counter-streaming between
collisionless ingoing and outgoing streams drives inflation at (just above) the
inner horizon, followed by collapse. As ingoing and outgoing streams approach
the inner horizon, they focus into twin narrow beams directed along the ingoing
and outgoing principal null directions, regardless of the initial angular
motions of the streams. The radial energy-momentum of the counter-streaming
beams gravitationally accelerates the streams even faster along the principal
directions, leading to exponential growth in the streaming density and
pressure, and in the Weyl curvature and mass function. At exponentially large
density and curvature, inflation stalls, and the spacetime collapses. As the
spacetime collapses, the angular motions of the freely-falling streams grow.
When the angular motion has become comparable to the radial motion, which
happens when the conformal factor has shrunk to an exponentially tiny scale,
conformal separability breaks down, and the solution fails. The condition of
conformal separability prescribes the form of the ingoing and outgoing
accretion flows incident on the inner horizon. The dominant radial part of the
solution holds provided that the densities of ingoing and outgoing streams
incident on the inner horizon are uniform, independent of latitude; that is,
the accretion flow is "monopole." The sub-dominant angular part of the solution
requires a special non-radial pattern of angular motion of streams incident on
the inner horizon. The prescribed angular pattern cannot be achieved if the
collisionless streams fall freely from outside the horizon.Comment: Version 1: 30 pages, 1 figure. Version 2: Extensively revised, logic
tightened, derivation more elegant. 37 pages, 1 figure. Version 3: Minor
revisions to match published version. Mathematica notebook available at
http://jila.colorado.edu/~ajsh/rotatinginflationary/rotatinginflationary.n
The interior structure of rotating black holes 1. Concise derivation
This paper presents a concise derivation of a new set of solutions for the
interior structure of accreting, rotating black holes. The solutions are
conformally stationary, axisymmetric, and conformally separable.
Hyper-relativistic counter-streaming between freely-falling collisionless
ingoing and outgoing streams leads to mass inflation at the inner horizon,
followed by collapse. The solutions fail at an exponentially tiny radius, where
the rotational motion of the streams becomes comparable to their radial motion.
The papers provide a fully nonlinear, dynamical solution for the interior
structure of a rotating black hole from just above the inner horizon inward,
down to a tiny scale.Comment: Version 1: 8 pages, 3 figures. Version 2: Extensively revised to
emphasize the derivation of the solution rather than the solution itself. 11
pages, 4 figures. Version 3: Minor changes to match published version.
Mathematica notebook available at
http://jila.colorado.edu/~ajsh/rotatinginflationary/rotatinginflationary.n
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