10,298 research outputs found
A proof of Price's law for the collapse of a self-gravitating scalar field
A well-known open problem in general relativity, dating back to 1972, has
been to prove Price's law for an appropriate model of gravitational collapse.
This law postulates inverse-power decay rates for the gravitational radiation
flux on the event horizon and null infinity with respect to appropriately
normalized advanced and retarded time coordinates. It is intimately related
both to astrophysical observations of black holes and to the fate of observers
who dare cross the event horizon. In this paper, we prove a well-defined (upper
bound) formulation of Price's law for the collapse of a self-gravitating scalar
field with spherically symmetric initial data. We also allow the presence of an
additional gravitationally coupled Maxwell field. Our results are obtained by a
new mathematical technique for understanding the long-time behavior of large
data solutions to the resulting coupled non-linear hyperbolic system of
p.d.e.'s in 2 independent variables. The technique is based on the interaction
of the conformal geometry, the celebrated red-shift effect, and local energy
conservation; we feel it may be relevant for the problem of non-linear
stability of the Kerr solution. When combined with previous work of the first
author (gr-qc/0307013) concerning the internal structure of charged black
holes, which assumed the validity of Price's law, our results can be applied to
the strong cosmic censorship conjecture for the Einstein-Maxwell-real scalar
field system with complete spacelike asymptotically flat spherically symmetric
initial data. Under Christodoulou's C^0 formulation, the conjecture is proven
to be false.Comment: 74 pages, 24 figures, v2: revised and expanded, v3: two misprints in
Theorem 1.2 correcte
Is there a connection between no-hair behavior and universality in gravitational collapse?
We apply linear perturbation theory to the study of the universality and
criticality first observed by Choptuik in gravitational collapse. Since these
are essentially nonlinear phenomena our attempt is only a rough approximation.
In spite of this, universal behavior of the final black hole mass is observed
with an exponent of 1/2, slightly higher than the observed value of 0.367. The
universal behavior is rooted in the universal form that in-falling
perturbations on black holes have at the horizon.Comment: RevTeX, 3 Pages, no figures, CGPG-94/9-
Smoothness for Simultaneous Composition of Mechanisms with Admission
We study social welfare of learning outcomes in mechanisms with admission. In
our repeated game there are bidders and mechanisms, and in each round
each mechanism is available for each bidder only with a certain probability.
Our scenario is an elementary case of simple mechanism design with incomplete
information, where availabilities are bidder types. It captures natural
applications in online markets with limited supply and can be used to model
access of unreliable channels in wireless networks.
If mechanisms satisfy a smoothness guarantee, existing results show that
learning outcomes recover a significant fraction of the optimal social welfare.
These approaches, however, have serious drawbacks in terms of plausibility and
computational complexity. Also, the guarantees apply only when availabilities
are stochastically independent among bidders.
In contrast, we propose an alternative approach where each bidder uses a
single no-regret learning algorithm and applies it in all rounds. This results
in what we call availability-oblivious coarse correlated equilibria. It
exponentially decreases the learning burden, simplifies implementation (e.g.,
as a method for channel access in wireless devices), and thereby addresses some
of the concerns about Bayes-Nash equilibria and learning outcomes in Bayesian
settings. Our main results are general composition theorems for smooth
mechanisms when valuation functions of bidders are lattice-submodular. They
rely on an interesting connection to the notion of correlation gap of
submodular functions over product lattices.Comment: Full version of WINE 2016 pape
Bogoliubov transformations for amplitudes in black-hole evaporation
The familiar approach to quantum radiation following collapse to a black hole
proceeds via Bogoliubov transformations, and yields probabilities for final
outcomes. In our (complex) approach, we find quantum amplitudes, not just
probabilities, by following Feynman's prescription. Initial and
final data for Einstein gravity and (say) a massless scalar field are specified
on a pair of asymptotically-flat space-like hypersurfaces and
; both are diffeomorphic to . Denote by the (real)
Lorentzian proper-time interval between the surfaces, as measured at spatial
infinity. Then rotate: .
The {\it classical} boundary-value problem is expected to be well-posed on a
region of topology , where is a closed interval. For a
locally-supersymmetric theory, the quantum amplitude should be dominated by the
semi-classical expression , where is the
classical action. One finds the Lorentzian quantum amplitude from the limit
. In the usual approach, the only possible such final surfaces
are in the strong-field region shortly before the curvature singularity. In our
approach one can put arbitrary smooth gravitational data on ,
provided that it has the correct mass -- the singularity is by-passed in
the analytic continuation. Here, we consider Bogoliubov transformations and
their possible relation to the probability distribution and density matrix in
the traditional approach. We find that our probability distribution for
configurations of the final scalar field cannot be expressed in terms of the
diagonal elements of some non-trivial density-matrix distribution
The instability of naked singularities in the gravitational collapse of a scalar field
One of the fundamental unanswered questions in the general theory of
relativity is whether ``naked'' singularities, that is singular events which
are visible from infinity, may form with positive probability in the process of
gravitational collapse. The conjecture that the answer to this question is in
the negative has been called ``cosmic censorship.'' The present paper, which is
a continuation previous work, addresses this question in the context of the
spherical gravitational collapse of a scalar field.Comment: 35 pages, published version, abstract added in migratio
Mutual coupling between circular apertures on an infinite conducting ground plane and radiating into a finite width slab
The problem of electromagnetic coupling between two horns is of interest for the Microwave Reflectometer Ionization Sensor (MRIS) that will be used in the Aeroassist Flight Experiment (AFE). Laboratory measurements of mutual coupling between conical horns (using a flat metallic reflector to simulate a critically dense plasma outside) have shown a strong dependence on the finite dimensions of the shuttle tile over the apertures. Since both, the dielectric tile and the plasma outside the tile reflect microwaves, a study should be done to isolate the two mechanisms so that the MRIS reentry flight data can be interpreted correctly. Once the coupling due to the tile itself is determined then the location of the critial electron number density layers can be determined. As a first attempt to tackle this problem the Geometrical Theory of Diffraction was used to modify the existing solution to mutual coupling between apertures with infinite dielectric sheets. By using the equivalent current method, aperture theory to determine the radiated fields inside the dielectric tiles, and ray tracing the contributions to mutual coupling were determined. Results from two cases with different tile thicknesses have indicated that the main contribution to mutual coupling is due to diffraction from the bottom and top (back and front) wedges
Choptuik scaling in null coordinates
A numerical simulation is performed of the gravitational collapse of a
spherically symmetric scalar field. The algorithm uses the null initial value
formulation of the Einstein-scalar equations, but does {\it not} use adaptive
mesh refinement. A study is made of the critical phenomena found by Choptuik in
this system. In particular it is verified that the critical solution exhibits
periodic self-similarity. This work thus provides a simple algorithm that gives
verification of the Choptuik results.Comment: latex (revtex), 6 figures included in the fil
A deterministic truthful PTAS for scheduling related machines
Scheduling on related machines () is one of the most important
problems in the field of Algorithmic Mechanism Design. Each machine is
controlled by a selfish agent and her valuation can be expressed via a single
parameter, her {\em speed}. In contrast to other similar problems, Archer and
Tardos \cite{AT01} showed that an algorithm that minimizes the makespan can be
truthfully implemented, although in exponential time. On the other hand, if we
leave out the game-theoretic issues, the complexity of the problem has been
completely settled -- the problem is strongly NP-hard, while there exists a
PTAS \cite{HS88,ES04}.
This problem is the most well studied in single-parameter algorithmic
mechanism design. It gives an excellent ground to explore the boundary between
truthfulness and efficient computation. Since the work of Archer and Tardos,
quite a lot of deterministic and randomized mechanisms have been suggested.
Recently, a breakthrough result \cite{DDDR08} showed that a randomized truthful
PTAS exists. On the other hand, for the deterministic case, the best known
approximation factor is 2.8 \cite{Kov05,Kov07}.
It has been a major open question whether there exists a deterministic
truthful PTAS, or whether truthfulness has an essential, negative impact on the
computational complexity of the problem. In this paper we give a definitive
answer to this important question by providing a truthful {\em deterministic}
PTAS
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