Energy extremality in the presence of a black hole

We derive the so-called first law of black hole mechanics for variations about stationary black hole solutions to the Einstein--Maxwell equations in the absence of sources. That is, we prove that $\delta M=\kappa\delta A+\omega\delta J+VdQ$ where the black hole parameters $M, \kappa, A, \omega, J, V$ and $Q$ denote mass, surface gravity, horizon area, angular velocity of the horizon, angular momentum, electric potential of the horizon and charge respectively. The unvaried fields are those of a stationary, charged, rotating black hole and the variation is to an arbitrary `nearby' black hole which is not necessarily stationary. Our approach is 4-dimensional in spirit and uses techniques involving Action variations and Noether operators. We show that the above formula holds on any asymptotically flat spatial 3-slice which extends from an arbitrary cross-section of the (future) horizon to spatial infinity.(Thus, the existence of a bifurcation surface is irrelevant to our demonstration. On the other hand, the derivation assumes without proof that the horizon possesses at least one of the following two (related)properties: ($i$) it cannot be destroyed by arbitrarily small perturbations of the metric and other fields which may be present, ($ii$) the expansion of the null geodesic generators of the perturbed horizon goes to zero in the distant future.)Comment: 30 pages, latex fil

Energy-momentum diffusion from spacetime discreteness

We study potentially observable consequences of spatiotemporal discreteness for the motion of massive and massless particles. First we describe some simple intrinsic models for the motion of a massive point particle in a fixed causal set background. At large scales, the microscopic swerves induced by the underlying atomicity manifest themselves as a Lorentz invariant diffusion in energy-momentum governed by a single phenomenological parameter, and we derive in full the corresponding diffusion equation. Inspired by the simplicity of the result, we then derive the most general Lorentz invariant diffusion equation for a massless particle, which turns out to contain two phenomenological parameters describing, respectively, diffusion and drift in the particle's energy. The particles do not leave the light cone however: their worldlines continue to be null geodesics. Finally, we deduce bounds on the drift and diffusion constants for photons from the blackbody nature of the spectrum of the cosmic microwave background radiation.Comment: 13 pages, 4 figures, corrected minor typos and updated to match published versio

The influence of boundaries on high pressure melting experiments

At low pressure, free surfaces play a crucial role in the melting transition. Under pressure, the surface of the sample is acted upon by some pressure transmitting medium. To examine the effect of this medium on melting, we performed Monte Carlo simulations of a system of argon atoms in the form of a slab with two boundaries. We examined two cases, one with a soft and the other with a rigid medium at the boundaries. We found that in the presence of a rigid medium, melting resembles the mechanical lattice instability found in a surface-free solid. With a soft medium at the boundary, melting begins at the surface and at a lower temperature. The relevance of these results to experiment is discussed.Comment: 4 pages, 5 figure

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

On critical collapse of gravitational waves

An axisymmetric collapse of non-rotating gravitational waves is numerically investigated in the subcritical regime where no black holes form but where curvature attains a maximum and decreases, following the dispersion of the initial wave packet. We find that near the threshold for black hole formation a curvature invariant with dimensions of length, scales as a power-law with the approximate exponent 0.38. In addition, the variation of the curvature in the critical limit is accompanied by increasing amount of echos, with nearly equal temporal and spatial periods. The scaling and the echoing patterns, and the corresponding constants are independent of the initial data and coordinate choices

Discreteness without symmetry breaking: a theorem

This paper concerns sprinklings into Minkowski space (Poisson processes). It proves that there exists no equivariant measurable map from sprinklings to spacetime directions (even locally). Therefore, if a discrete structure is associated to a sprinkling in an intrinsic manner, then the structure will not pick out a preferred frame, locally or globally. This implies that the discreteness of a sprinkled causal set will not give rise to ``Lorentz breaking'' effects like modified dispersion relations. Another consequence is that there is no way to associate a finite-valency graph to a sprinkling consistently with Lorentz invariance.Comment: 7 pages, laTe

Stable non-uniform black strings below the critical dimension

The higher-dimensional vacuum Einstein equation admits translationally non-uniform black string solutions. It has been argued that infinitesimally non-uniform black strings should be unstable in 13 or fewer dimensions and otherwise stable. We construct numerically non-uniform black string solutions in 11, 12, 13, 14 and 15 dimensions. Their stability is investigated using local Penrose inequalities. Weakly non-uniform solutions behave as expected. However, in 12 and 13 dimensions, strongly non-uniform solutions appear to be stable and can have greater horizon area than a uniform string of the same mass. In 14 and 15 dimensions all non-uniform black strings appear to be stable.Comment: 26 pages, 11 figures. V2: reference added, matches published versio