Topology changes by quantum tunneling in four dimensional gravity

Nuclear Physics B, 1996

The point of this paper is see what light new results in hyperbolic geometry may throw on gravitational entropy and whether gravitational entropy is relevant for the quantum origin of the univeres. We introduce some new gravitational instantons which mediate the birth from nothing of closed universes containing wormholes and suggest that they may contribute to the density matrix of the universe. We also discuss the connection between their gravitational action and the topological and volumetric entropies introduced in hyperbolic geometry. These coincide for hyperbolic 4-manifolds, and increase with increasing topological complexity of the four manifold. We raise the questions of whether the action also increase with the topological complexity of the initial 3-geometry, measured either by its three volume or its Matveev complexity. We point out, in distinction to the non-supergravity case, that universes with domains of negative cosmological constant separated by supergravity domain walls cannot be born from nothing. Finally we point out that our wormholes provide examples of the type of Perpetual Motion machines envisaged by Frolov and Novikov.

Classical and Quantum Gravity, 1997

Quantum fields are investigated in the (2+1)-open-universes with non-trivial topologies by the method of images. The universes are locally de Sitter spacetime and anti-de Sitter spacetime. In the present article we study spacetimes whose spatial topologies are a torus with a cusp and a sphere with three cusps as a step toward the more general case. A quantum energy momentum tensor is obtained by the point stripping method. Though the cusps are no singularities, the latter cusps cause the divergence of the quantum field. This suggests that only the latter cusps are quantum mechanically unstable. Of course at the singularity of the background spacetime the quantum field diverges. Also the possibility of the divergence of topological effect by a negative spatial curvature is discussed. Since the volume of the negatively curved space is larger than that of the flat space, one see so many images of a single source by the non-trivial topology. It is confirmed that this divergence does not appear in our models of topologies. The results will be applicable to the case of three dimensional multi black hole [8].

2013

As shown in previous work, there is a well-defined nonperturbative gravitational path integral including an explicit sum over topologies in the setting of Causal Dynamical Triangulations in two dimensions. In this paper we derive a complete analytical solution of the quantum continuum dynamics of this model, obtained uniquely by means of a double-scaling limit. We show that the presence of infinitesimal wormholes leads to a decrease in the effective cosmological constant, reminiscent of the suppression mechanism considered by Coleman and others in the four-dimensional Euclidean path integral. Remarkably, in the continuum limit we obtain a finite spacetime density of microscopic wormholes without assuming fundamental discreteness. This shows that one can in principle make sense of a gravitational path integral which includes a sum over topologies, provided suitable causality restrictions are imposed on the path integral histories. 1 1

Phd Thesis 2008, 2008

In this thesis we analyze a very simple model of two dimensional quantum gravity based on causal dynamical triangulations (CDT). We present an exactly solvable model which indicates that it is possible to incorporate spatial topology changes in the nonperturbative path integral. It is shown that if the change in spatial topology is accompanied by a coupling constant it is possible to evaluate the path integral to all orders in the coupling and that the result can be viewed as a hybrid between causal and Euclidian dynamical triangulation. The second model we describe shows how a classical geometry with constant negative curvature emerges naturally from a path integral over noncompact manifolds. No initial singularity is present, hence the quantum geometry is naturally compatible with the Hartle Hawking boundary condition. Furthermore, we demonstrate that under certain conditions the quantum fluctuations are small! To conclude, we treat the problem of spacetime topology change. Although we are not able to completely solve the path integral over all manifolds with arbitrary topology, we do obtain results that indicate that such a path integral might be consistent, provided suitable causality restrictions are imposed.

Classical and Quantum Gravity, 2014

We consider the quantization of space-times which can possess different topologies within a symmetry reduced version of Wheeler-DeWitt theory. The quantum states are defined from a natural decomposition as an outer-product of a topological state, dictating the topology of the two-surfaces of the space-time, and a geometric state, which controls the geometry and is comprised of solutions to the Wheeler-DeWitt constraints. Within this symmetry reduced theory an eigenvalue equation is derived for the two-volume of spacetime, which for spherical topology is fixed to a value of 4π. However, for the other topologies it is found that the spectrum can be discrete and hence the universe, if in one of these other topological states, may only possess certain possible values for the two-volume, whereas classically all values are allowed. We analyze this result in the context of pure gravity (black holes).

Journal of High Energy Physics, Gravitation and Cosmology

For purposes of quantization, classical gravity is normally expressed by canonical variables, namely the metric g ab (x) and the momentum π cd (x). Canonical quantization requires a proper promotion of these classical variables to quantum operators, which, according to Dirac, the favored operators should be those arising from classical variables that formed Cartesian coordinates; sadly, in this case, that is not possible. However, an affine quantization features promoting the metric g ab (x) and the momentric π c d (x) [≡ π ce (x) g de (x)] to operators. Instead of these classical variables belonging to a constant zero curvature space (i.e., instead of a flat space), they belong to a space of constant negative curvatures. This feature may even have its appearance in black holes, which could strongly point toward an affine quantization approach to quantize gravity.

Physical Review D, 1991

We investigate the nucleation of universe in a (2+1)-dimensional gravity model with negative cosmological constant. There are a variety of universes born from nothing by quantum tunneling. Utilizing the powerful technique in hyperbolic geometry, we explicitly construct 3-manifolds which describe the nucleation of a higher genus universe. We calculate the wavefunction of the universe in the WKB approximation.

General Relativity and Gravitation - Proceedings of the 16th International Conference, 2002

Over the last three years, a number of fundamental physical issues were addressed in loop quantum gravity. These include: A statistical mechanical derivation of the horizon entropy, encompassing astrophysically interesting black holes as well as cosmological horizons; a natural resolution of the big-bang singularity; the development of spin-foam models which provide background independent path integral formulations of quantum gravity and 'finiteness proofs' of some of these models; and, the introduction of semi-classical techniques to make contact between the background independent, non-perturbative theory and the perturbative, low energy physics in Minkowski space. These developments spring from a detailed quantum theory of geometry that was systematically developed in the mid-nineties and have added a great deal of optimism and intellectual excitement to the field. The goal of this article is to communicate these advances in general physical terms, accessible to researchers in all areas of gravitational physics represented in this conference.

Nuclear Physics B, 1998

We study the fractal structure of space-time of two-dimensional quantum gravity coupled to c = ?2 conformal matter by means of computer simulations. We nd that the intrinsic Hausdor dimension d H = 3:58 0:04. This result supports the conjecture d H = ?2 1 = ?1 , where n is the gravitational dressing exponent of a spinless primary eld of conformal weight (n + 1; n + 1), and it disfavours the alternative prediction d H = 2=j j. On the other hand hl n i r 2n for n > 1 with good accuracy, i.e. the the boundary length l has an anomalous dimension relative to the area of the surface.

Physical Review D, 1990

If the topology and geometry of spacetime are quantum-mechanically variable, then the particular classical large-scale topology and geometry observed in our universe must be statistical predictions of its initial condition. This paper examines the predictions of the "no boundary" initial condition for the present large-scale topology and geometry. Finite-action real tunneling solutions of Einstein s equation are important for such predictions. These consist of compact Riemannian (Euclidean) geometries joined to a Lorentzian cosmological geometry across a spacelike surface of vanishing extrinsic curvature. The classification of such solutions is discussed and general constraints on their topology derived. For example, it is shown that, if the Euclidean Ricci tensor is positive, then a real tunneling solution can nucleate only a single connected Lorentzian spacetime (the unique conception theorem). Explicit examples of real tunneling solutions driven by a cosmological constant are exhibited and their implications for cosmic baldness described. It is argued that the most probable large-scale spacetime predicted by the real tunneling solutions of the "no-boundary" initial condition has the topology R X S' with the de Sitter metric.