264 research outputs found
Regular coordinate systems for Schwarzschild and other spherical spacetimes
The continuation of the Schwarzschild metric across the event horizon is
almost always (in textbooks) carried out using the Kruskal-Szekeres
coordinates, in terms of which the areal radius r is defined only implicitly.
We argue that from a pedagogical point of view, using these coordinates comes
with several drawbacks, and we advocate the use of simpler, but equally
effective, coordinate systems. One such system, introduced by Painleve and
Gullstrand in the 1920's, is especially simple and pedagogically powerful; it
is, however, still poorly known today. One of our purposes here is therefore to
popularize these coordinates. Our other purpose is to provide generalizations
to the Painleve-Gullstrand coordinates, first within the specific context of
Schwarzschild spacetime, and then in the context of more general spherical
spacetimes.Comment: 5 pages, 2 figures, ReVTeX; minor changes were made, new references
were include
Spherically Symmetric Black Hole Formation in Painlev\'e-Gullstrand Coordinates
We perform a numerical study of black hole formation from the spherically
symmetric collapse of a massless scalar field. The calculations are done in
Painlev\'e-Gullstrand (PG) coordinates that extend across apparent horizons and
allow the numerical evolution to proceed until the onset of singularity
formation. We generate spacetime maps of the collapse and illustrate the
evolution of apparent horizons and trapping surfaces for various initial data.
We also study the critical behaviour and find the expected Choptuik scaling
with universal values for the critical exponent and echoing period consistent
with the accepted values of and , respectively.
The subcritical curvature scaling exhibits the expected oscillatory behaviour
but the form of the periodic oscillations in the supercritical mass scaling
relation, while universal with respect to initial PG data, is non-standard: it
is non-sinusoidal with large amplitude cusps.Comment: 12 pages, 7 figure
Einstein's Real "Biggest Blunder"
Albert Einstein's real "biggest blunder" was not the 1917 introduction into
his gravitational field equations of a cosmological constant term \Lambda,
rather was his failure in 1916 to distinguish between the entirely different
concepts of active gravitational mass and passive gravitational mass. Had he
made the distinction, and followed David Hilbert's lead in deriving field
equations from a variational principle, he might have discovered a true (not a
cut and paste) Einstein-Rosen bridge and a cosmological model that would have
allowed him to predict, long before such phenomena were imagined by others,
inflation, a big bounce (not a big bang), an accelerating expansion of the
universe, dark matter, and the existence of cosmic voids, walls, filaments, and
nodes.Comment: 4 pages, LaTeX, 11 references, Honorable Mention in 2012 Gravity
Research Foundation Essay Award
A Secret Tunnel Through The Horizon
Hawking radiation is often intuitively visualized as particles that have
tunneled across the horizon. Yet, at first sight, it is not apparent where the
barrier is. Here I show that the barrier depends on the tunneling particle
itself. The key is to implement energy conservation, so that the black hole
contracts during the process of radiation. A direct consequence is that the
radiation spectrum cannot be strictly thermal. The correction to the thermal
spectrum is of precisely the form that one would expect from an underlying
unitary quantum theory. This may have profound implications for the black hole
information puzzle.Comment: First prize in the Gravity Research Foundation Essay Competition. 7
pages, LaTe
Acoustic analogues of black hole singularities
We search for acoustic analogues of a spherical symmetric black hole with a
pointlike source. We show that the gravitational system has a dynamical
counterpart in the constrained, steady motion of a fluid with a planar source.
The equations governing the dynamics of the gravitational system can be exactly
mapped in those governing the motion of the fluid. The different meaning that
singularities and sources have in fluid dynamics and in general relativity is
also discussed. Whereas in the latter a pointlike source is always associated
with a (curvature) singularity in the former the presence of sources does not
necessarily imply divergences of the fields.Comment: 9 pages, no figure
Embedding spherical spacelike slices in a Schwarzschild solution
Given a spherical spacelike three-geometry, there exists a very simple
algebraic condition which tells us whether, and in which, Schwarzschild
solution this geometry can be smoothly embedded. One can use this result to
show that any given Schwarzschild solution covers a significant subset of
spherical superspace and these subsets form a sequence of nested domains as the
Schwarzschild mass increases. This also demonstrates that spherical data offer
an immediate counter example to the thick sandwich `theorem'
Cauchy-perturbative matching revisited: tests in spherical symmetry
During the last few years progress has been made on several fronts making it
possible to revisit Cauchy-perturbative matching (CPM) in numerical relativity
in a more robust and accurate way. This paper is the first in a series where we
plan to analyze CPM in the light of these new results.
Here we start by testing high-order summation-by-parts operators, penalty
boundaries and contraint-preserving boundary conditions applied to CPM in a
setting that is simple enough to study all the ingredients in great detail:
Einstein's equations in spherical symmetry, describing a black hole coupled to
a massless scalar field. We show that with the techniques described above, the
errors introduced by Cauchy-perturbative matching are very small, and that very
long term and accurate CPM evolutions can be achieved. Our tests include the
accretion and ring-down phase of a Schwarzschild black hole with CPM, where we
find that the discrete evolution introduces, with a low spatial resolution of
\Delta r = M/10, an error of 0.3% after an evolution time of 1,000,000 M. For a
black hole of solar mass, this corresponds to approximately 5 s, and is
therefore at the lower end of timescales discussed e.g. in the collapsar model
of gamma-ray burst engines.
(abridged)Comment: 14 pages, 20 figure
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
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