Extremal black holes, gravitational entropy and nonstationary metric fields

We show that extremal black holes have zero entropy by pointing out a simple fact: they are time-independent throughout the spacetime and correspond to a single classical microstate. We show that non-extremal black holes, including the Schwarzschild black hole, contain a region hidden behind the event horizon where all their Killing vectors are spacelike. This region is nonstationary and the time $t$ labels a continuous set of classical microstates, the phase space $[\,h_{ab}(t), P^{ab}(t)\,]$, where $h_{ab}$ is a three-metric induced on a spacelike hypersurface $\Sigma_t$ and $P^{ab}$ is its momentum conjugate. We determine explicitly the phase space in the interior region of the Schwarzschild black hole. We identify its entropy as a measure of an outside observer's ignorance of the classical microstates in the interior since the parameter $t$ which labels the states lies anywhere between 0 and 2M. We provide numerical evidence from recent simulations of gravitational collapse in isotropic coordinates that the entropy of the Schwarzschild black hole stems from the region inside and near the event horizon where the metric fields are nonstationary; the rest of the spacetime, which is static, makes no contribution. Extremal black holes have an event horizon but in contrast to non-extremal black holes, their extended spacetimes do not possess a bifurcate Killing horizon. This is consistent with the fact that extremal black holes are time-independent and therefore have no distinct time-reverse.Comment: 12 pages, 2 figures. To appear in Class. and Quant. Gravity. Based on an essay selected for honorable mention in the 2010 gravity research foundation essay competitio

Spontaneous breaking of conformal invariance in theories of conformally coupled matter and Weyl gravity

We study the theory of Weyl conformal gravity with matter degrees of freedom in a conformally invariant interaction. Specifically, we consider a triplet of scalar fields and SO(3) non-abelian gauge fields, i.e. the Georgi-Glashow model conformally coupled to Weyl gravity. We show that the equations of motion admit solutions spontaneously breaking the conformal symmetry and the gauge symmetry, providing a mechanism for supplying a scale in the theory. The vacuum solution corresponds to anti-de-Sitter space-time, while localized soliton solutions correspond to magnetic monopoles in asymptotically anti-de-Sitter space-time. The resulting effective action gives rise to Einstein gravity and the residual U(1) gauge theory. This mechanism strengthens the reasons for considering conformally invariant matter-gravity theory, which has shown promising indications concerning the problem of missing matter in galactic rotation curves.Comment: 20 pages, 1 figure, revised and added reference

Causal Structure of Vacuum Solutions to Conformal(Weyl) Gravity

Using Penrose diagrams the causal structure of the static spherically symmetric vacuum solution to conformal (Weyl) gravity is investigated. A striking aspect of the solution is an unexpected physical singularity at $r=0$ caused by a linear term in the metric. We explain how to calculate the deflection of light in coordinates where the metric is manifestly conformal to flat i.e. in coordinates where light moves in straight lines.Comment: 18 pages, 2 figures, title and abstract changed, contents essentially unaltered accepted for publication in General Relativity and Gravitatio

Casimir forces in Bose-Einstein condensates: finite size effects in three-dimensional rectangular cavities

The Casimir force due to {\it thermal} fluctuations (or pseudo-Casimir force) was previously calculated for the perfect Bose gas in the slab geometry for various boundary conditions. The Casimir pressure due to {\it quantum} fluctuations in a weakly-interacting dilute Bose-Einstein condensate (BEC) confined to a parallel plate geometry was recently calculated for Dirichlet boundary conditions. In this paper we calculate the Casimir energy and pressure due to quantum fluctuations in a zero-temperature homogeneous weakly-interacting dilute BEC confined to a parallel plate geometry with periodic boundary conditions and include higher-order corrections which we refer to as Bogoliubov corrections. The leading order term is identified as the Casimir energy of a massless scalar field moving with wave velocity equal to the speed of sound in the BEC. We then obtain the leading order Casimir pressure in a general three-dimensional rectangular cavity of arbitrary lengths and obtain the finite-size correction to the parallel plate scenario.Comment: 12 pages; no figures; v.2: version accepted for publication in JSTAT v.3: references adde

The Bright Side of Dark Matter

We show that it is not possible in the absence of dark matter to construct a four-dimensional metric that explains galactic observations. In particular, by working with an effective potential it is shown that a metric which is constructed to fit flat rotation curves in spiral galaxies leads to the wrong sign for the bending of light i.e. repulsion instead of attraction. Hence, without dark matter the motion of particles on galactic scales cannot be explained in terms of geodesic motion on a four- dimensional metric. This reveals a new bright side to dark matter: it is indispensable if we wish to retain the cherished equivalence principle.Comment: 7 pages, latex, no figures. Received an honorable mention in the 1999 Gravity research Foundation Essay Competition. Submitted to Phys. Rev. Let

Emergence of thin shell structure during collapse in isotropic coordinates

Numerical studies of gravitational collapse in isotropic coordinates have recently shown an interesting connection between the gravitational Lagrangian and black hole thermodynamics. A study of the actual spacetime was not the main focus of this work and in particular, the rich and interesting structure of the interior has not been investigated in much detail and remains largely unknown. We elucidate its features by performing a numerical study of the spacetime in isotropic coordinates during gravitational collapse of a massless scalar field. The most salient feature to emerge is the formation of a thin shell of matter just inside the apparent horizon. The energy density and Ricci scalar peak at the shell and there is a jump discontinuity in the extrinsic curvature across the apparent horizon, the hallmark that a thin shell is present in its vicinity. At late stages of the collapse, the spacetime consists of two vacuum regions separated by the thin shell. The interior is described by an interesting collapsing isotropic universe. It tends towards a vacuum (never reaches a perfect vacuum) and there is a slight inhomogeneity in the interior that plays a crucial role in the collapse process as the areal radius tends to zero. The spacetime evolves towards a curvature (physical) singularity in the interior, both a Weyl and Ricci singularity. In the exterior, our numerical results match closely the analytical form of the Schwarzschild metric in isotropic coordinates, providing a strong test of our numerical code.Comment: 24 pages, 10 figures. version to appear in Phys. Rev.

Casimir Energy of a BEC: From Moderate Interactions to the Ideal Gas

Considering the Casimir effect due to phononic excitations of a weakly interacting dilute {BEC}, we derive a re-normalized expression for the zero temperature Casimir energy $\mathcal{E}_c$ of a {BEC} confined to a parallel plate geometry with periodic boundary conditions. Our expression is formally equivalent to the free energy of a bosonic field at finite temperature, with a nontrivial density of modes that we compute analytically. As a function of the interaction strength, $\mathcal{E}_c$ smoothly describes the transition from the weakly interacting Bogoliubov regime to the non-interacting ideal {BEC}. For the weakly interacting case, $\mathcal{E}_c$ reduces to leading order to the Casimir energy due to zero-point fluctuations of massless phonon modes. In the limit of an ideal Bose gas, our result correctly describes the Casimir energy going to zero.Comment: 12 pages, 3 figures, accepted for publication in JPA. New version with corrected typos and an additional appendi

Casimir interaction: pistons and cavity

The energy of a perfectly conducting rectangular cavity is studied by making use of pistons' interactions. The exact solution for a 3D perfectly conducting piston with an arbitrary cross section is being discussed.Comment: 10 pages, 2 figures, latex2