Compact Spatial Sections Research Articles

The Bianchi type Iminisuperspace model

The minisuperspace model of a Bianchi type I universe with compactspatial sections is investigated. The classical solutions are broughtinto a form where the difference between compact and infinite spatialsections is manifest. One of the features of the compact case is that ithas a non-trivial moduli space. The solution space of the compact Bianchitype I universe is 10-dimensional, whereas the Kasner solutions onlyhave a one-dimensional solution space. We also include the classicalsolutions with a dustand a cosmological constant. Solutions to the Wheeler-DeWitt equationare obtained in the light of the tunnelling boundary proposal by Vilenkin.Backreaction effects from a simple scalar field are also investigated.

Analytic approximation and an improved method for computing the stress-energy of quantized scalar fields in Robertson-Walker spacetimes

An improved method is given for the computation of the stress-energy tensor of a quantized scalar field using adiabatic regularization. The method works for fields with arbitrary mass and curvature coupling in Robertson-Walker spacetimes and is particularly useful for spacetimes with compact spatial sections. For massless fields it yields an analytic approximation for the stress-energy tensor that is similar in nature to those obtained previously for massless fields in static spacetimes.

Addendum to the paper “Closed Spaces in Cosmology”

This paper deals with two aspects of relativistic cosmologies with closed (compact and boundless) spatial sections. These spacetimes are based on the theory of General Relativity, and admit a foliation into space sections S(t), which are spacelike hypersurfaces satisfying the postulate of the closure of space: each S(t) is a 3-dimensional, closed Riemannian manifold. The discussed topics are: (1) A comparison, previously obtained, between Thurston's geometries and Bianchi-Kantowski-Sachs metrics for such 3-manifolds is here clarified and developed. (2) Some implications of global inhomogeneity for locally homogeneous 3-spaces of constant curvature are analyzed from an observational viewpoint.

Conditions for regular space-times with compact spatial sections

We apply the holonomy method to study the regularity of space-times whose initially open spatial sections are made compact. We restrict our attention to a set of different expressions of the anti-de Sitter space-time in (2+1)-dimensions. After we have computed the holonomy matrix to this set, we explicitly derive the regularity conditions for two elements of the set. The first expression, relevant to classical cosmology, will be regular if the periods of the compactified directions are 2π. The other expression, relevant to quantum cosmology, will be regular if the cosmological constant has a certain, well defined, discrete spectrum.

The large-distance limit of the gravitational effective action in hyperbolic backgrounds

The one-loop effective action for D-dimensional quantum gravity with negative cosmological constant is investigated in spacetimes with compact hyperbolic spatial section. The explicit expansion of the effective action is derived as a power series of the curvature on the hyperbolic background, making use of heat-kernel and zeta-regularization techniques. We discuss, at one-loop level, the Coleman--Weinberg-type suppression of the cosmological constant, proposed by Taylor and Veneziano.

The effective action in gauged supergravity on hyperbolic background and induced cosmological constant

The one-loop effective action for 4-dimensional gauged supergravity with negative cosmological constant, is investigated in space-times with compact hyperbolic spatial section. The explicit expansion of the effective action as a power series of the curvature on hyperbolic background is derived, making use of heat-kernel and zeta-regularization techniques. The induced cosmological and Newton constants are computed.

Reduced canonical quantization of the induced two-dimensional gravity.

The quantization of the induced two-dimensional gravity on a compact spatial section is carried out in three different ways. In the three approaches the supermomentum constraint is solved at the classical level but they differ in the way the Hamiltonian constraint is imposed. We compare these approaches establishing an isomorphism between the resulting Hilbert spaces.

Comments on closed Bianchi models

The authors show several kinematical properties that are intrinsic to the Bianchi models with compact spatial sections. Especially, with spacelike hypersurfaces being closed, (i) no anisotropic expansion is allowed for Bianchi type V and VII (A not=0) and (ii) type IV and VI (A not=0,1) do not exist. In order to show them, they put into geometric terms what is meant by spatial homogeneity and employ a mathematical result on 3-manifolds. They make clear the relation between the Bianchi-type symmetry of spacetime and spatial compactness, some parts of which seem to have gone unnoticed in the literature. Especially, it is shown under what conditions class-B Bianchi models do not possess compact spatial sections. Finally, they briefly describe how this study is useful in investigating global dynamics in (3+1)-dimensional gravity.

Biaxial Bianchi type IX quantum cosmology

We investigate the quantum cosmology of spatially homogeneous models with compact spatial sections admitting a u(2) isometry algebra. The metric ansatz in these models is that of Bianchi type IX with two scale factors set to be equal. We apply the Hartle-Hawking no-boundary path integral prescription and find the semi-classical contributions to the wave function. Exact formulae are obtainable for certain contributions and otherwise of large and small anisotropy (for the pure vacuum case) and large spatial volume or small anisotropy (for the case with a positive cosmological constant) are considered. For the pure vacuum case we find no rapidly oscillating semiclassical components in the wave function, and hence do recover lorentzian space-time as a prediction of the no-boundary proposal. For the case with a cosmological constant the wave function does contain rapidly oscillating components and thus predicts approximately lorentzian space-times; however, one of the parameters in these lorentzian solutions is formally constrained to be purely imaginary. The possible implications of this imaginary parameter for the breakdown of the lorentzian semiclassical approximation and for the predictions of the no-boundary proposal are discussed.

Acceleration radiation in a compact space

The authors study the response of a uniformly accelerated model particle detector in a spacetime with compact spatial sections. The basic thermal character of the response re-emerges, in spite of the fact that the spacetime does not possess event horizons. The model also permits a study of detector response to twisted field states.

Equivalence of massless boson and fermion theories in curved two-dimensional space-time: Sugawara stress tensor

It is shown that the well known equivalence of two-dimensional massless scalar and spinor quantum field theories extends to curved space-time. The vacuum Sugawara stress tensor for a spinor field is evaluated by a covariant point-separation procedure and found to be identical to the scalar vacuum stress, previously evaluated by Davies and Unruh (1977) to be equivalent to the conventional fermion stress tensor (in the absence of compact spatial sections).