Gauge Invariant Hamiltonian Formalism for Spherically Symmetric Gravitating Shells

The dynamics of a spherically symmetric thin shell with arbitrary rest mass and surface tension interacting with a central black hole is studied. A careful investigation of all classical solutions reveals that the value of the radius of the shell and of the radial velocity as an initial datum does not determine the motion of the shell; another configuration space must, therefore, be found. A different problem is that the shell Hamiltonians used in literature are complicated functions of momenta (non-local) and they are gauge dependent. To solve these problems, the existence is proved of a gauge invariant super-Hamiltonian that is quadratic in momenta and that generates the shell equations of motion. The true Hamiltonians are shown to follow from the super-Hamiltonian by a reduction procedure including a choice of gauge and solution of constraint; one important step in the proof is a lemma stating that the true Hamiltonians are uniquely determined (up to a canonical transformation) by the equations of motion of the shell, the value of the total energy of the system, and the choice of time coordinate along the shell. As an example, the Kraus-Wilczek Hamiltonian is rederived from the super-Hamiltonian. The super-Hamiltonian coincides with that of a fictitious particle moving in a fixed two-dimensional Kruskal spacetime under the influence of two effective potentials. The pair consisting of a point of this spacetime and a unit timelike vector at the point, considered as an initial datum, determines a unique motion of the shell.Comment: Some remarks on the singularity of the vector potantial are added and some minor corrections done. Definitive version accepted in Phys. Re

Geometry of the quantum universe

A universe much like the (Euclidean) de Sitter space-time appears as background geometry in the causal dynamical triangulation (CDT) regularization of quantum gravity. We study the geometry of such universes which appear in the path integral as a function of the bare coupling constants of the theory.Comment: 19 pages, 7 figures. Typos corrected. Conclusions unchange

Black hole formation from point-like particles in three-dimensional anti-de Sitter space

We study collisions of many point-like particles in three-dimensional anti-de Sitter space, generalizing the known result with two particles. We show how to construct exact solutions corresponding to the formation of either a black hole or a conical singularity from the collision of an arbitrary number of massless particles falling in radially from the boundary. We find that when going away from the case of equal energies and discrete rotational symmetry, this is not a trivial generalization of the two-particle case, but requires that the excised wedges corresponding to the particles must be chosen in a very precise way for a consistent solution. We also explicitly take the limit when the number of particles goes to infinity and obtain thin shell solutions that in general break rotational invariance, corresponding to an instantaneous and inhomogeneous perturbation at the boundary. We also compute the stress-energy tensor of the shell using the junction formalism for null shells and obtain agreement with the point particle picture.Comment: 42 pages, 9 figures; v2: fixed some typo

Archipelagian Cosmology: Dynamics and Observables in a Universe with Discretized Matter Content

We consider a model of the Universe in which the matter content is in the form of discrete islands, rather than a continuous fluid. In the appropriate limits the resulting large-scale dynamics approach those of a Friedmann-Robertson-Walker (FRW) universe. The optical properties of such a space-time, however, do not. This illustrates the fact that the optical and `average' dynamical properties of a relativistic universe are not equivalent, and do not specify each other uniquely. We find the angular diameter distance, luminosity distance and redshifts that would be measured by observers in these space-times, using both analytic approximations and numerical simulations. While different from their counterparts in FRW, the effects found do not look like promising candidates to explain the observations usually attributed to the existence of Dark Energy. This incongruity with standard FRW cosmology is not due to the existence of any unexpectedly large structures or voids in the Universe, but only to the fact that the matter content of the Universe is not a continuous fluid.Comment: 49 pages, 15 figures. Corrections made to description of lattice constructio

A covariant causal set approach to discrete quantum gravity

A covariant causal set (c-causet) is a causal set that is invariant under labeling. Such causets are well-behaved and have a rigid geometry that is determined by a sequence of positive integers called the shell sequence. We first consider the microscopic picture. In this picture, the vertices of a c-causet have integer labels that are unique up to a label isomorphism. This labeling enables us to define a natural metric $d(a,b)$ between time-like separated vertices $a$ and $b$. The time metric $d(a,b)$ results in a natural definition of a geodesic from $a$ to $b$. It turns out that there can be $n\ge 1$ such geodesics. Letting $a$ be the origin (the big bang), we define the curvature $K(b)$ of $b$ to be $n-1$. Assuming that particles tend to move along geodesics, $K(b)$ gives the tendency that vertex $b$ is occupied. In this way, the mass distribution is determined by the geometry of the c-causet. We next consider the macroscopic picture which describes the growth process of c-causets. We propose that this process is governed by a quantum dynamics given by complex amplitudes. At present, these amplitudes are unknown. But if they can be found, they will determine the (approximate) geometry of the c-causet describing our particular universe. As an illustration, we present a simple example of an amplitude process that may have physical relevance. We also give a discrete analogue of Einstein's field equations.Comment: 23 pages, 6 tables; new version corrects some typos in the proof of Theorem 6.

Quantum Singularities in Spacetimes with Spherical and Cylindrical Topological Defects

Exact solutions of Einstein equations with null Riemman-Christoffel curvature tensor everywhere, except on a hypersurface, are studied using quantum particles obeying the Klein-Gordon equation. We consider the particular cases when the curvature is represented by a Dirac delta function with support either on a sphere or on a cylinder (spherical and cylindrical shells). In particular, we analyze the necessity of extra boundary conditions on the shells.Comment: 7 page,1 fig., Revtex, J. Math. Phys, in pres

Black Hole Thermodynamics without a Black Hole?

In the present paper we consider, using our earlier results, the process of quantum gravitational collapse and argue that there exists the final quantum state when the collapse stops. This state, which can be called the ``no-memory state'', reminds the final ``no-hair state'' of the classical gravitational collapse. Translating the ``no-memory state'' into classical language we construct the classical analogue of quantum black hole and show that such a model has a topological temperature which equals exactly the Hawking's temperature. Assuming for the entropy the Bekenstein-Hawking value we develop the local thermodynamics for our model and show that the entropy is naturally quantized with the equidistant spectrum S + gamma_0*N. Our model allows, in principle, to calculate the value of gamma_0. In the simplest case, considered here, we obtain gamma_0 = ln(2).Comment: 20 pages, it will be submitted to Phys.Lett.

Time Discrete Geodesic Paths in the Space of Images

In this paper the space of images is considered as a Riemannian manifold using the metamorphosis approach, where the underlying Riemannian metric simultaneously measures the cost of image transport and intensity variation. A robust and effective variational time discretization of geodesics paths is proposed. This requires to minimize a discrete path energy consisting of a sum of consecutive image matching functionals over a set of image intensity maps and pairwise matching deformations. For square-integrable input images the existence of discrete, connecting geodesic paths defined as minimizers of this variational problem is shown. Furthermore, $\Gamma$-convergence of the underlying discrete path energy to the continuous path energy is proved. This includes a diffeomorphism property for the induced transport and the existence of a square-integrable weak material derivative in space and time. A spatial discretization via finite elements combined with an alternating descent scheme in the set of image intensity maps and the set of matching deformations is presented to approximate discrete geodesic paths numerically. Computational results underline the efficiency of the proposed approach and demonstrate important qualitative properties.Comment: 27 pages, 7 figure