Mathematics of random growing interfaces

We establish a thermodynamic limit and Gaussian fluctuations for the height and surface width of the random interface formed by the deposition of particles on surfaces. The results hold for the standard ballistic deposition model as well as the surface relaxation model in the off-lattice setting. The results are proved with the aid of general limit theorems for stabilizing functionals of marked Poisson point processes.Comment: 12 page

Strict inequalities of critical values in continuum percolation

We consider the supercritical finite-range random connection model where the points $x,y$ of a homogeneous planar Poisson process are connected with probability $f(|y-x|)$ for a given $f$. Performing percolation on the resulting graph, we show that the critical probabilities for site and bond percolation satisfy the strict inequality $p_c^{\rm site} > p_c^{\rm bond}$. We also show that reducing the connection function $f$ strictly increases the critical Poisson intensity. Finally, we deduce that performing a spreading transformation on $f$ (thereby allowing connections over greater distances but with lower probabilities, leaving average degrees unchanged) {\em strictly} reduces the critical Poisson intensity. This is of practical relevance, indicating that in many real networks it is in principle possible to exploit the presence of spread-out, long range connections, to achieve connectivity at a strictly lower density value.Comment: 38 pages, 8 figure

Thermal gravity, black holes and cosmological entropy

Taking seriously the interpretation of black hole entropy as the logarithm of the number of microstates, we argue that thermal gravitons may undergo a phase transition to a kind of black hole condensate. The phase transition proceeds via nucleation of black holes at a rate governed by a saddlepoint configuration whose free energy is of order the inverse temperature in Planck units. Whether the universe remains in a low entropy state as opposed to the high entropy black hole condensate depends sensitively on its thermal history. Our results may clarify an old observation of Penrose regarding the very low entropy state of the universe.Comment: 5 pages, 2 figures, RevTex. v4: to appear in Phys. Rev.

Linearized gravity and gauge conditions

In this paper we consider the field equations for linearized gravity and other integer spin fields on the Kerr spacetime, and more generally on spacetimes of Petrov type D. We give a derivation, using the GHP formalism, of decoupled field equations for the linearized Weyl scalars for all spin weights and identify the gauge source functions occuring in these. For the spin weight 0 Weyl scalar, imposing a generalized harmonic coordinate gauge yields a generalization of the Regge-Wheeler equation. Specializing to the Schwarzschild case, we derive the gauge invariant Regge-Wheeler and Zerilli equation directly from the equation for the spin 0 scalar.Comment: 24 pages, corresponds to published versio

The Symplectic Penrose Kite

The purpose of this article is to view the Penrose kite from the perspective of symplectic geometry.Comment: 24 pages, 7 figures, minor changes in last version, to appear in Comm. Math. Phys

A Causal Order for Spacetimes with C0C^0 Lorentzian Metrics: Proof of Compactness of the Space of Causal Curves

We recast the tools of ``global causal analysis'' in accord with an approach to the subject animated by two distinctive features: a thoroughgoing reliance on order-theoretic concepts, and a utilization of the Vietoris topology for the space of closed subsets of a compact set. We are led to work with a new causal relation which we call $K^+$, and in terms of it we formulate extended definitions of concepts like causal curve and global hyperbolicity. In particular we prove that, in a spacetime \M which is free of causal cycles, one may define a causal curve simply as a compact connected subset of \M which is linearly ordered by $K^+$. Our definitions all make sense for arbitrary $C^0$ metrics (and even for certain metrics which fail to be invertible in places). Using this feature, we prove for a general $C^0$ metric, the familiar theorem that the space of causal curves between any two compact subsets of a globally hyperbolic spacetime is compact. We feel that our approach, in addition to yielding a more general theorem, simplifies and clarifies the reasoning involved. Our results have application in a recent positive energy theorem, and may also prove useful in the study of topology change. We have tried to make our treatment self-contained by including proofs of all the facts we use which are not widely available in reference works on topology and differential geometry.Comment: Two small revisions to accomodate errors brought to our attention by R.S. Garcia. No change to chief results. 33 page

Self-Similar Collapse of Conformally Coupled Scalar Fields

A massless scalar field minimally coupled to the gravitational field in a simplified spherical symmetry is discussed. It is shown that, in this case, the solution found by Roberts, describing a scalar field collapse, is in fact the most general one. Taking that solution as departure point, a study of the gravitational collapse for the self-similar conformal case is presented.Comment: 9 pages, accepted for publication, Classical and Quantum Gravity. Available at http://dft.if.uerj.br/preprint/e-17.tex or at ftp://dft.if.uerj.br/preprint/e-17.tex . Figures can be obtained on request at [email protected]

Beyond the veil: Inner horizon instability and holography

We show that scalar perturbations of the eternal, rotating BTZ black hole should lead to an instability of the inner (Cauchy) horizon, preserving strong cosmic censorship. Because of backscattering from the geometry, plane wave modes have a divergent stress tensor at the event horizon, but suitable wavepackets avoid this difficulty, and are dominated at late times by quasinormal behavior. The wavepackets have cuts in the complexified coordinate plane that are controlled by requirements of continuity, single-valuedness and positive energy. Due to a focusing effect, regular wavepackets nevertheless have a divergent stress-energy at the inner horizon, signaling an instability. This instability, which is localized behind the event horizon, is detected holographically as a breakdown in the semiclassical computation of dual CFT expectation values in which the analytic behavior of wavepackets in the complexified coordinate plane plays an integral role. In the dual field theory, this is interpreted as an encoding of physics behind the horizon in the entanglement between otherwise independent CFTs.Comment: 40 pages, LaTeX, 3 eps figures, v2: references adde

Quasi-local energy-momentum and energy flux at null infinity

The null infinity limit of the gravitational energy-momentum and energy flux determined by the covariant Hamiltonian quasi-local expressions is evaluated using the NP spin coefficients. The reference contribution is considered by three different embedding approaches. All of them give the expected Bondi energy and energy flux.Comment: 14 pages, accepted by Phys.Rev.

Angular momentum of isolated systems

Penrose's twistorial approach to the definition of angular momentum at null infinity is developed so that angular momenta at different cuts can be meaningfully compared. This is done by showing that the twistor spaces associated with different cuts of scri can be identified as manifolds (but not as vector spaces). The result is a well-defined, Bondi-Metzner-Sachs-invariant notion of angular momentum in a radiating space-time; the difficulties and ambiguities previously encountered are attached to attempts to express this in special-relativistic terms, and in particular to attempts to identify a single Minkowski space of origins. Unlike the special-relativistic case, the angular momentum cannot be represented by a purely j=1 quantity M_{ab}, but has higher-j contributions as well. Applying standard kinematic prescriptions, these higher-j contributions are shown to correspond precisely to the shear. Thus it appears that shear and angular momentum should be regarded as different aspects of a single unified concept.Comment: 23 pages, to appear in GR