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Register a new userOn the Action, Topology and Geometric Invariants in Quantum Gravity
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Authors
A. Coley
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The action in general relativity (GR), which is an integral over the manifold plus an integral over the boundary, is a global object and is only well defined when the topology is fixed. Therefore, to use the action in GR and in most approaches to quantum gravity (QG) based on a covariant Lorentzian action, there needs to exist a prefered (global) timelike vector, and hence a global topology $R times S^3$, for it to make sense. This is especially true in the Hamiltonian formulation of QG. Therefore, in order to do canonical quantization, we need to know the topology, appropriate boundary conditions and (in an open manifold) the conditions at infinity, which affects the fundamental geometrical scalar invariants of the spacetime (and especially those which may occur in the QG action).
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A short review of scalar curvature invariants in gravity theories is presented. We introduce how these invariants are constructed and discuss the minimal number of invariants required for a given spacetime. We then discuss applications of these invariants and focus on three topics that are of particular interest in modern gravity theories.
We study curvature invariants in a binary black hole merger. It has been conjectured that one could define a quasi-local and foliation independent black hole horizon by finding the level--$0$ set of a suitable curvature invariant of the Riemann tensor. The conjecture is the geometric horizon conjecture and the associated horizon is the geometric horizon. We study this conjecture by tracing the level--$0$ set of the complex scalar polynomial invariant, $mathcal{D}$, through a quasi-circular binary black hole merger. We approximate these level--$0$ sets of $mathcal{D}$ with level--$varepsilon$ sets of $|mathcal{D}|$ for small $varepsilon$. We locate the local minima of $|mathcal{D}|$ and find that the positions of these local minima correspond closely to the level--$varepsilon$ sets of $|mathcal{D}|$ and we also compare with the level--$0$ sets of $text{Re}(mathcal{D})$. The analysis provides evidence that the level--$varepsilon$ sets track a unique geometric horizon. By studying the behaviour of the zero sets of $text{Re}(mathcal{D})$ and $text{Im}(mathcal{D})$ and also by studying the MOTSs and apparent horizons of the initial black holes, we observe that the level--$varepsilon$ set that best approximates the geometric horizon is given by $varepsilon = 10^{-3}$.
We present a model of (modified) gravity on spacetimes with fractal structure based on packing of spheres, which are (Euclidean) variants of the Packed Swiss Cheese Cosmology models. As the action functional for gravity we consider the spectral action of noncommutative geometry, and we compute its expansion on a space obtained as an Apollonian packing of 3-dimensional spheres inside a 4-dimensional ball. Using information from the zeta function of the Dirac operator of the spectral triple, we compute the leading terms in the asymptotic expansion of the spectral action. They consist of a zeta regularization of a divergent sum which involves the leading terms of the spectral actions of the individual spheres in the packing. This accounts for the contribution of the points 1 and 3 in the dimension spectrum (as in the case of a 3-sphere). There is an additional term coming from the residue at the additional point in the real dimension spectrum that corresponds to the packing constant, as well as a series of fluctuations coming from log-periodic oscillations, created by the points of the dimension spectrum that are off the real line. These terms detect the fractality of the residue set of the sphere packing. We show that the presence of fractality influences the shape of the slow-roll potential for inflation, obtained from the spectral action. We also discuss the effect of truncating the fractal structure at a certain scale related to the energy scale in the spectral action.
In this paper, we analyse the well-posedness of the initial value formulation for particular kinds of geometric scalar-tensor theories of gravity, which are based on a Weyl integrable space-time. We will show that, within a frame-invariant interpretation for the theory, the Cauchy problem in vacuum is well-posed. We will analyse the global in space problem, and, furthermore, we will show that geometric uniqueness holds for the solutions. We make contact with Brans-Dicke theory, and by analysing the similarities with such models, we highlight how some of our results can be translated to this well-known context, where not all of these problems have been previously addressed.
EPR-type measurements on spatially separated entangled spin qubits allow one, in principle, to detect curvature. Also the entanglement of the vacuum state is affected by curvature. Here, we ask if the curvature of spacetime can be expressed entirely in terms of the spatial entanglement structure of the vacuum. This would open up the prospect that quantum gravity could be simulated on a quantum computer and that quantum information techniques could be fully employed in the study of quantum gravity.
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