Static solutions from the point of view of comparison geometry

We analyze (the harmonic map representation of) static solutions of the Einstein Equations in dimension three from the point of view of comparison geometry. We find simple monotonic quantities capturing sharply the influence of the Lapse function on the focussing of geodesics. This allows, in particular, a sharp estimation of the Laplacian of the distance function to a given (hyper)-surface. We apply the technique to asymptotically flat solutions with regular and connected horizons and, after a detailed analysis of the distance function to the horizon, we recover the Penrose inequality and the uniqueness of the Schwarzschild solution. The proof of this last result does not require proving conformal flatness at any intermediate step.Comment: 41 page

Holographic Bound From Second Law of Thermodynamics

A necessary condition for the validity of the holographic principle is the holographic bound: the entropy of a system is bounded from above by a quarter of the area of a circumscribing surface measured in Planck areas. This bound cannot be derived at present from consensus fundamental theory. We show with suitable {\it gedanken} experiments that the holographic bound follows from the generalized second law of thermodynamics for both generic weakly gravitating isolated systems and for isolated, quiescent and nonrotating strongly gravitating configurations well above Planck mass. These results justify Susskind's early claim that the holographic bound can be gotten from the second law.Comment: RevTeX, 8 pages, no figures, several typos correcte

On the Bartnik extension problem for the static vacuum Einstein equations

We develop a framework for understanding the existence of asymptotically flat solutions to the static vacuum Einstein equations with prescribed boundary data consisting of the induced metric and mean curvature on a 2-sphere. A partial existence result is obtained, giving a partial resolution of a conjecture of Bartnik on such static vacuum extensions. The existence and uniqueness of such extensions is closely related to Bartnik's definition of quasi-local mass.Comment: 33 pages, 1 figure. Minor revision of v2. Final version, to appear in Class. Quantum Gravit

Static Cosmological Solutions of the Einstein-Yang-Mills-Higgs Equations

Numerical evidence is presented for the existence of a new family of static, globally regular `cosmological' solutions of the spherically symmetric Einstein-Yang-Mills-Higgs equations. These solutions are characterized by two natural numbers ($m\geq 1$, $n\geq 0$), the number of nodes of the Yang-Mills and Higgs field respectively. The corresponding spacetimes are static with spatially compact sections with 3-sphere topology.Comment: 7 pages, 5 figures, LaTe

Gluing Initial Data Sets for General Relativity

We establish an optimal gluing construction for general relativistic initial data sets. The construction is optimal in two distinct ways. First, it applies to generic initial data sets and the required (generically satisfied) hypotheses are geometrically and physically natural. Secondly, the construction is completely local in the sense that the initial data is left unaltered on the complement of arbitrarily small neighborhoods of the points about which the gluing takes place. Using this construction we establish the existence of cosmological, maximal globally hyperbolic, vacuum space-times with no constant mean curvature spacelike Cauchy surfaces.Comment: Final published version - PRL, 4 page

Trapped Surfaces in Vacuum Spacetimes

An earlier construction by the authors of sequences of globally regular, asymptotically flat initial data for the Einstein vacuum equations containing trapped surfaces for large values of the parameter is extended, from the time symmetric case considered previously, to the case of maximal slices. The resulting theorem shows rigorously that there exists a large class of initial configurations for non-time symmetric pure gravitational waves satisfying the assumptions of the Penrose singularity theorem and so must have a singularity to the future.Comment: 14 page

Effects of distance dependence of exciton hopping on the Davydov soliton

The Davydov model of energy transfer in molecular chains is reconsidered assuming the distance dependence of the exciton hopping term. New equations of motion for phonons and excitons are derived within the coherent state approximation. Solving these nonlinear equations result in the existence of Davydov-like solitons. In the case of a dilatational soliton, the amplitude and width is decreased as a results of the mechanism introduced here and above a critical coupling strength our equations do not allow for localized solutions. For compressional solitons, stability is increased.Comment: RevTeX 13 pages, 3 Postscript figure

Multidimensional Gravity on the Principal Bundles

The multidimensional gravity on the total space of principal bundle is considered. In this theory the gauge fields arise as nondiagonal components of multidimensional metric. The spherically symmetric and cosmology solutions for gravity on SU(2) principal bundle are obtained. The static spherically symmetric solution is wormhole-like solution located between two null surfaces, in contrast to 4D Einstein-Yang-Mills theory where corresponding solution (black hole) located outside of event horizon. Cosmology solution (at least locally) has the bouncing off effect for spatial dimensions. In spirit of Einstein these solutions are vacuum solutions without matter.Comment: REVTEX, 13pages, 2 EPS figure

Late time behaviour of the maximal slicing of the Schwarzschild black hole

A time-symmetric Cauchy slice of the extended Schwarzschild spacetime can be evolved into a foliation of the $r>3m/2$-region of the spacetime by maximal surfaces with the requirement that time runs equally fast at both spatial ends of the manifold. This paper studies the behaviour of these slices in the limit as proper time-at-infinity becomes arbitrarily large and gives an analytic expression for the collapse of the lapse.Comment: 18 pages, Latex, no figure

A Gluing Construction Regarding Point Particles in General Relativity

We develop a gluing construction which adds scaled and truncated asymptotically Euclidean solutions of the Einstein constraint equations to compact solutions with potentially non-trivial cosmological constants. The result is a one-parameter family of initial data which has ordinary and scaled "point-particle" limits analogous to those of Gralla and Wald ("A rigorous derivation of gravitational self-force," Class. Quantum Grav. 2008). In particular, we produce examples of initial data which generalize Schwarzschild - de Sitter initial data and gluing theorems of IMP-type