291 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
Integrable dynamics of a discrete curve and the Ablowitz-Ladik hierarchy
We show that the following elementary geometric properties of the motion of a
discrete (i.e. piecewise linear) curve select the integrable dynamics of the
Ablowitz-Ladik hierarchy of evolution equations: i) the set of points
describing the discrete curve lie on the sphere S^3, ii) the distance between
any two subsequant points does not vary in time, iii) the dynamics does not
depend explicitly on the radius of the sphere. These results generalize to a
discrete context our previous work on continuous curves.Comment: LaTeX file, 14 pages + 4 figure
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
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
Sthenic incompatibilities in rigid bodies motion
When a rigid body slides with friction on a surface, hopping motion is observed: this is an everyday phenomenon. In rigid bodies mechanics, this phenomenon appears when it is no longer possible to compute the reaction contact forces. The difficulty is overcome by a motion theory involving velocity discontinuities. Velocity discontinuities may result either from an obstacle which makes impossible to compute the acceleration: this is a cinematic incompatibility or from the impossibility to compute the reaction forces: this is a sthenic incompatibility. We describe two examples: the Klein and Painlevé sthenic incompatibilities. Springer 2006
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
Movable algebraic singularities of second-order ordinary differential equations
Any nonlinear equation of the form y''=\sum_{n=0}^N a_n(z)y^n has a
(generally branched) solution with leading order behaviour proportional to
(z-z_0)^{-2/(N-1)} about a point z_0, where the coefficients a_n are analytic
at z_0 and a_N(z_0)\ne 0. We consider the subclass of equations for which each
possible leading order term of this form corresponds to a one-parameter family
of solutions represented near z_0 by a Laurent series in fractional powers of
z-z_0. For this class of equations we show that the only movable singularities
that can be reached by analytic continuation along finite-length curves are of
the algebraic type just described. This work generalizes previous results of S.
Shimomura. The only other possible kind of movable singularity that might occur
is an accumulation point of algebraic singularities that can be reached by
analytic continuation along infinitely long paths ending at a finite point in
the complex plane. This behaviour cannot occur for constant coefficient
equations in the class considered. However, an example of R. A. Smith shows
that such singularities do occur in solutions of a simple autonomous
second-order differential equation outside the class we consider here
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
Hawking radiation as tunneling from a Vaidya black hole in noncommutative gravity
In the context of a noncommutative model of coordinate coherent states, we
present a Schwarzschild-like metric for a Vaidya solution instead of the
standard Eddington-Finkelstein metric. This leads to the appearance of an exact
dependent case of the metric. We analyze the resulting metric in
three possible causal structures. In this setup, we find a zero remnant mass in
the long-time limit, i.e. an instable black hole remnant. We also study the
tunneling process across the quantum horizon of such a Vaidya black hole. The
tunneling probability including the time-dependent part is obtained by using
the tunneling method proposed by Parikh and Wilczek in terms of the
noncommutative parameter . After that, we calculate the entropy
associated to this noncommutative black hole solution. However the corrections
are fundamentally trifling; one could respect this as a consequence of quantum
inspection at the level of semiclassical quantum gravity.Comment: 19 pages, 5 figure
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