Spatial Hypersurfaces Research Articles

Midi-superspace quantization of non-compact toroidally symmetric gravity

We consider the quantization of the midi-superspace associated with a class of spacetimes with toroidal isometries, but without the compact spatial hypersurfaces of the well-known Gowdy models. By a symmetry reduction, the phase space for the system at the classical level can be identified with that of a free massless scalar field on a fixed background spacetime, thereby providing a simple route to quantization. We are then able to study certain non-perturbative features of the quantum gravitational system. In particular, we examine the quantum geometry of the asymptotic regions of the spacetimes involved and find some surprisingly large dispersive effects of quantum gravity.

Lorentzian approach to black hole thermodynamics in the Hamiltonian formulation

In this work, we extend the analysis of Brown and York to find the quasilocal energy in a spherical box in the Schwarzschild spacetime. Quasilocal energy is the value of the Hamiltonian that generates unit magnitude proper-time translations on the box orthogonal to the spatial hypersurfaces foliating the Schwarzschild spacetime. We call this Hamiltonian the Brown-York Hamiltonian. We find different classes of foliations that correspond to time-evolution by the Brown-York Hamiltonian. We show that although the Brown-York expression for the quasilocal energy is correct, one needs to supplement their derivation with an extra set of boundary conditions on the interior end of the spatial hypersurfaces inside the hole in order to obtain it from an action principle. Replacing this set of boundary conditions with another set yields the Louko-Whiting Hamiltonian, which corresponds to time-evolution of spatial hypersurfaces in a different foliation of the Schwarzschild spacetime. We argue that in the thermodynamical picture, the Brown-York Hamiltonian corresponds to the internal energy whereas the Louko-Whiting Hamiltonian corresponds to the Helmholtz free energy of the system. Unlike what has been the usual route to black hole thermodynamics in the past, this observation immediately allows us to obtain the partition function of such a system without resorting to any kind of Euclideanization of either the Hamiltonian or the action. In the process, we obtain some interesting insights into the geometrical nature of black hole thermodynamics.

Semiclassical interpretation of the topological solutions for canonical quantum gravity.

Ashtekar's formulation for canonical quantum gravity is known to possess the topological solutions which have their supports only on the moduli space $\CN$ of flat $SL(2,C)$ connections. We show that each point on the moduli space $\CN$ corresponds to a geometric structure, or more precisely the Lorentz group part of a family of Lorentzian structures, on the flat (3+1)-dimensional spacetime. A detailed analysis is given in the case where the spacetime is homeomorphic to $R\times T^{3}$. Most of the points on the moduli space $\CN$ yield pathological spacetimes which suffers from singularities on each spatial hypersurface or which violates the strong causality condition. There is, however, a subspace of $\CN$ on which each point corresponds to a family of regular spacetimes.

Gravitational energy in spherical symmetry.

Various properties of the Misner-Sharp spherically symmetric gravitational energy E are established or reviewed. In the Newtonian limit of a perfect fluid, E yields the Newtonian mass to leading order and the Newtonian kinetic and potential energy to the next order. For test particles, the corresponding Hajicek energy is conserved and has the behaviour appropriate to energy in the Newtonian and special-relativistic limits. In the small-sphere limit, the leading term in E is the product of volume and the energy density of the matter. In vacuo, E reduces to the Schwarzschild energy. At null and spatial infinity, E reduces to the Bondi-Sachs and Arnowitt-Deser-Misner energies respectively. The conserved Kodama current has charge E. A sphere is trapped if E>r/2, marginal if E=r/2 and untrapped if E<r/2, where r is the areal radius. A central singularity is spatial and trapped if E>0, and temporal and untrapped if E<0. On an untrapped sphere, E is non-decreasing in any outgoing spatial or null direction, assuming the dominant energy condition. It follows that E>=0 on an untrapped spatial hypersurface with regular centre, and E>=r_0/2 on an untrapped spatial hypersurface bounded at the inward end by a marginal sphere of radius r_0. All these inequalities extend to the asymptotic energies, recovering the Bondi-Sachs energy loss and the positivity of the asymptotic energies, as well as proving the conjectured Penrose inequality for black or white holes. Implications for the cosmic censorship hypothesis and for general definitions of gravitational energy are discussed.

Averaging of a locally inhomogeneous realistic universe.

We present an averaging scheme in general relativity which allows us to study the effect of local inhomogeneity on the global behavior of the universe. The scheme uses 3+1 splitting of spacetime and introduces Isaacson averaging on the spatial hypersurface to get the averaged geometry. As a result of the averaging, the Friedmann-Robertson-Walker (FWR) geometry is derived in the first-order approximation for a wide class of inhomogeneous nonlinear matter distribution. The deviation from the FRW expansion is derived to the next order in terms of the anisotropic distribution of an effective stress-energy tensor. Using a simple model of inhomogeneity we show that the average effect of the inhomogeneity behaves like a negative spatial curvature term and thus has a tendency to extend the age of the universe. \textcopyright{} 1996 The American Physical Society.

Inhomogeneous cosmological models with homogeneous inner hypersurface geometry

Space-times which allow a slicing into homogeneous spatial hypersurfaces generalize the usual Bianchi models. One knows already that in these models the Bianchi type may change with time. Here we show which of the changes really appear. To this end we characterize the topological space whose points are the 3-dimensional oriented homogeneous Riemannian manifolds; locally isometric manifolds are considered as same.

Classical and quantum evolutions of the de Sitter and the anti-de Sitter universes in 2+1 dimensions.

Two canonical formulations of the Einstein gravity in 2+1 dimensions, namely, the ADM formalism and the Chern-Simons gravity, are investigated in the case of nonvanishing cosmological constant. General arguments for reducing phase spaces of the two formalisms are given when spatial hypersurface is compact. In particular when the space has the topology of a sphere $S^{2}$ or a torus $T^{2}$, the spacetimes constructed from these two formulations can be identified and the classical equivalence between the ADM and the CSG is shown. Moreover in the $g=1$ case the relations between their phase spaces, and therefore between their quantizations, are given in almost the same form as that in the case when the cosmological constant vanishes. There are, however, some modifications, the most remarkable one of which is that the phase space of the CSG is in 1 to 2 correspondence with the one of the ADM when the cosmological constant is negative.

Shear-free anisotropic cosmological models

A class of orthogonal, spatially homogeneous shear-free cosmological models with an imperfect liquid matter content is considered. It is shown that, for these models, the anisotropic stresses are related to the anisotropic curvature of the spatial hypersurfaces through the electric part of the Weyl tensor. This creates a counterbalance effect which has important consequences for the evolution of the cosmological models. Such features are discussed, and, as an application, some eternally anisotropic models with an imperfect fluid stress tensor which exhibit FRW-like behaviour are given.

Dirac equation in Robertson-Walker metric

We recast the Dirac relativistic equation within the theoretical framework of the Robertson-Walker metric, using spatial hypersurfaces that are essentially curved, and hence more general, as compared to the flat ones employed by Barut and Duru.

Joining conditions for cosmological perturbations at an equation-of-state transition.

Joining conditions for cosmological spatial irregularities at an equation-of-state transition spatial hypersurface are derived by requiring that the relativistic linear perturbation theory equations of motion remain singularity-free at the transition spatial hypersurface. The transition spatial hypersurface is taken to be a spatially homogeneous local energy-density spatial hypersurface, between two spacetime-dependent speed-of-sound fluid-dominated epochs of a spatially flat Friedmann-Lema\^{\i}tre-Robertson-Walker cosmological model. Since the spacetime-dependent speed-of-sound fluid model includes, as special cases, both the scalar field model and the ideal fluid model, these joining conditions may be used at the scalar-field-radiation and radiation-baryon transitions in simple inflation-modified hot big-bang models. These joining conditions differ from two earlier sets of joining conditions (which differ from each other).

Wilson loop observables in (2 + 1)-dimensional Chern-Simons supergravity

(2 + 1)-dimensional gravity and supergravity, without a cosmological constant, can be described as Chern-Simons gauge theories of the Poincaré and super-Poincaré groups respectively. The only gauge invariant observables are Wilson loops.In this paper, we consider N = 2 Poincaré supergravity on R × Σ, where R is “time” and the spatial hypersurface Σ is the plane with an arbitrary number, N, of isolated sources or punctures carrying super-Poincaré charges, and examine Wilson loops enclosing one or more of them. We describe how to compute the Wilson loop observables and give explicit expressions for the cases of one and two sources. We also discuss the Poisson brackets between two Wilson loops and derive a rule by which they might be obtained.

Extended inflation in closed cosmologies

We examine the recently proposed extended inflationary scenarios, which are based on Brans-Dicke rather than Einstein gravity. In particular we are interested in the initial conditions required for the universe to enter the inflationary epoque. We show that the evolution depends strongly on the geometry of the spatial hypersurfaces. For a closed universe we derive a new constraint, implying that the inflation scale coincides with the effective Planck scale. Thus there remains a fine tuning problem for such models.

Parametrization of the space of solutions of Einstein's equations.

An explicit parametrization is obtained of the set of all space-time solutions of Einstein's equations which are globally hyperbolic and contain a compact spatial hypersurface with constant mean curvature. This parametrization is based upon the conformal treatment of the initial-value problem for Einstein's equations, which is studied by the method of sub and super solutions for quasilinear elliptic partial differential equations. The Yamabe-Aubin-Trudinger-Schoen classification of conformal classes of Riemannian metrics plays a key role.

Dust universes with spatial spherical symmetry and euclidean initial hypersurfaces

The Tolman-Bondi equations of pressure-free spherically symmetric systems of particles in general relativity are applied to the special case of a euclidean initial hypersurface.

Positive mass theorems for black holes

We extend Witten's proof of the positive mass theorem at spacelike infinity to show that the mass is positive for initial data on an asymptotically flat spatial hypersurface Σ which is regular outside an apparent horizonH. In addition, we prove that if a black hole has electromagnetic charge, then the mass is greater than the modulus of the charge. These results are also valid for the Bondi mass at null infinity. Finally, in the case of the Einstein equation with a negative cosmological constant, we show that a suitably defined mass is positive for data on an asymptotically anti-de Sitter surface Σ which is regular outside an apparent horizon.

Novikov-type coordinates for HJ separable metrics

Synchronous frames are known to remove (at least partially) deficiencies inherent in Killing coordinates. Strangely enough, no constructive general method of finding these frames has been formulated, though it follows from the definition of synchronous coordinates; We propose here such a method and apply it to the Kerr metric to obtain a Novikov-type synchronous frame. The removal of nonphysical singularities is discussed in the context of 3-metric and 3-curvature on the spatial hypersurfaces in this frame.

Geodesic instability and internal time in relativistic cosmology

The concept of is applied to a cosmological model having spatial hypersurfaces of negative curvature. It is then possible to ascribe an irreversible evolution to the expanding universe without resorting to any coarse graining or loss of information. The key observation which enables this description to be used is that geodesic flow on a four-manifold can be reduced to geodesic flow on a three-manifold when the Robertson-Walker metric is used. If the three-surface is compactified in such a way as not to change the metric, and if it has negative curvature, the geodesic system is a Bernoulli flow a dynamical system which has the highest degree of instability. We draw various conclusions about mixing in the system pertinent to the microwave background, the observational consequences of negative curvature for objects moving with respect to the galaxies, and we show that the requirement of negative curvature always leads to a particle horizon, a conclusion which has some bearing on the physical spectrum of the internal time operator and on the possibility of removing the cosmological singularity to the infinite past. © 1982 The American Physical Society.

Some global properties of closed spatially homogeneous space-times

We consider some properties of the space-times which contain a spatially homogeneous domain of dependenceD(V), whereF is acompact achronal spatial hypersurface of homogeneity. For example, it is shown that ifV has a nonempty future Cauchy horizon then the timelike geodesies orthogonal toV are future incomplete and there is strong causality failure onH +(V). Also, conditions for the global hyperbolicity of such space-times are obtained.

Energy density and spatial curvature in general relativity

Positive energy density tends to limit the size of space. This effect is studied within several contexts. We obtain sufficient conditions (which involve the energy density in a crucial way) for the compactness of spatial hypersurfaces in space-time. We then obtain some results concerning static or, more generally, stationary space-times. The Schwarzchild solution puts an upper bound on the size of a static spherically symmetric fluid with density bounded from below. We derive a result of roughly the same nature which, however, requires no symmetry and allows for rotation. Also, we show that static or rotating universes with Λ = 0 require that the density along some spatial geodesic must fall off rapidly with distance from a point.

Formal quark binding and geometric strings

Geometric quantization is applied to a line bundle over spacetime with structure group GL(2,C)/SL(2,C) and curvature form determined by the Ricci tensor. The Ricci form is interpreted as the Hamiltonian form for a particle moving on a string associated with a harmonic oscillator spectrum. Strings are characterized as minimal surfaces, and quarks as the gradients of minimal immersions of surfaces in a three-dimensional spatial hypersurface, leading to a model of a baryon as the intersection of three such surfaces, and their associated quarks.