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 $n$ bidders and $m$ 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 $+i\epsilon$ 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 $\Sigma_I$ and $\Sigma_F$; both are diffeomorphic to ${\Bbb R}^3$. Denote by $T$ the (real) Lorentzian proper-time interval between the surfaces, as measured at spatial infinity. Then rotate: $T\to{\mid}T{\mid}\exp(-i\theta),0<\theta\leq \pi/2$. The {\it classical} boundary-value problem is expected to be well-posed on a region of topology $I\times{\Bbb R}^3$, where $I$ is a closed interval. For a locally-supersymmetric theory, the quantum amplitude should be dominated by the semi-classical expression $\exp(iS_{\rm class})$, where $S_{\rm class}$ is the classical action. One finds the Lorentzian quantum amplitude from the limit $\theta\to 0_+$. 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 $\Sigma_F$, provided that it has the correct mass $M$ -- 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

On the possibility of laboratory evidence for quantum superposition of geometries

We analyze the recent proposal of measuring a quantum gravity phenomenon in the lab by entangling two particles gravitationally. We give a generally covariant description of this phenomenon, where the relevant effect turns out to be a quantum superposition of proper times. We point out that measurement of this effect would count as evidence for quantum superposition of spacetime geometries. This interpretation addresses objections appeared in the literature. We observe that the effect sheds light on the Planck mass, and argue that it is very plausibly a real effect.Comment: 6 pages, 1 figur