It is well known that, in the first-order formalism, pure three-dimensional gravity is just the BF theory. Similarly, four-dimensional general relativity can be formulated as BF theory with an additional constraint term added to the Lagrangian. In this paper we show that the same is true also for higherdimensional Einstein gravity: in any dimension gravity can be described as a constrained BF theory. Moreover, in any dimension these constraints are quadratic in the B field. After describing in details the structure of these constraints, we scketch the "spin foam" quantization of these theories, which proves to be quite similar to the spin foam quantization of general relativity in three and four dimensions. In particular, in any dimension, we solve the quantum constraints and find the so-called simple representations and intertwiners. These exhibit a simple and beautiful structure that is common to all dimensions. * In three spacetime dimensions Einstein's general relativity becomes a beautiful and simple theory. There are no local degrees of freedom, and gravity is an example of topological field theory. Owing to this fact, a variety of techniques from TQFT can be used, and a great deal is known about quantization of the theory. More precisely, when written in the first order formalism, three-dimensional gravity is just the BF theory, whose action is given by:(1)Here M is the spacetime manifold, F is the curvature of the spin connection, and B is the frame field one-form. The trace is taken in the Lie algebra of the relevant gauge group, which in the case of 3D is given by SO(2, 1) for Lorentzian spacetimes and by SO (3) in the Euclidean case. The quantization of BF theory is well-understood, both canonically and by the path integral method, at least in the Euclidean case. This is one of the possible ways to construct quantum gravity in three spacetime dimensions: it exists as a topological field theory. 1 It is tempting to apply the beautiful quantization methods from TQFT to other, more complicated theories, including those with local degrees of freedom. An interesting proposal along these lines was made in a series of papers by Martellini and collaborators [2], who proposed to treat Yang-Mills theory as a certain deformation of the BF theory. This gives an interesting picture of the confining phase of Yang-Mills theory.Recently, a proposal was made suggesting a way to apply the ideas and methods from TQFT to four-dimensional gravity. The new approach to quantum gravity, for which the name "spin foam approach" was proposed in [3], lies on the intersection between TQFT and loop quantum gravity (see [4] for a recent review on "loop" gravity). As it was advocated by Rovelli and Reisenberger [5], the results of the "loop" approach suggest a possibility of constructing the partition function of 4D gravity as a "spin foam" model. The first "spin foam" model of 4D gravity was constructed by Reisenberger [6], and was intimately related to the self-dual canonical (loop) quantum gravity. Later, Barrett, Crane [7] and Ba...
The Einstein equations for a spacetime of the form can be reduced to a Hamiltonian system on the Teichmüller space. However, the resulting Hamiltonian is the square root of a quadratic form in the momenta. Trying to make sense of this as a quantum operator is problematic since the Hamiltonian operator would be non-polynomial and non-local. In the first half of this article, I will examine the eigenfunctions of this operator. These go under the name of Maass functions and have been studied extensively by number theorists. In the second half, I will show that the quantum evolution due to this Hamiltonian does not cause the spacetime to collapse in a finite lapse of mean curvature time. Because of the difficult number theory involved with the Maass functions, I will actually perform the calculation for a simplified problem---a model of Teichmüller space---and argue that the answer should not be sensitive to the simplifications made in my model.
In this paper we give a derivation for the allometric scaling relation between the metabolic rate and the mass of animals and plants. We show that the characteristic scaling exponent of 3/4 occurring in this relation is a result of the distribution of sources and sinks within the living organism. We further introduce a principle of least mass and discuss the kind of flows that arise from it.
This work consists of two distinct parts. In the first part we present a new method for solving the initial value problem of general relativity. Given any spatial metric with a surface orthogonal Killing field and two freely specified components of the extrinsic curvature we solve for extrinsic curvature's remaining components. For the second part, after noting that initial data for the Kerr spacetime can be derived within our formalism we construct data for axisymmetric configurations of spinning black holes. Though our method is limited to axisymmetry, it offers an advantage over the Bowen-York proceedure that our data approach those for Kerr holes in the limit of large separations and in the close limit.
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