A note on the Gannon–Lee theorem

We prove a Gannon–Lee theorem for non-globally hyperbolic Lorentzian metrics of regularity $$C^1$$ C 1 , the most general regularity class currently available in the context of the classical singularity theorems. Along the way, we also prove that any maximizing causal curve in a $$C^1$$ C 1 -spacetime is a geodesic and hence of $$C^2$$ C 2 -regularity.

Tunneling dynamics in cosmological bounce models

Quasiclassical methods are used to define dynamical tunneling times in models of quantum cosmological bounces. These methods provide relevant new information compared with the traditional treatment of quantum tunneling by means of tunneling probabilities. As shown here, the quantum dynamics in bounce models is not secure from reaching zero scale factor, re-opening the question of how the classical singularity may be avoided. Moreover, in the examples studied here, tunneling times remain small even for large barriers, highlighting the quantum instability of underlying bounce models.

Quantum phenomenological gravitational dynamics: a general view from thermodynamics of spacetime

In this work we derive general quantum phenomenological equations of gravitational dynamics and analyse its features. The derivation uses the formalism developed in thermodynamics of spacetime and introduces low energy quantum gravity modifications to it. Quantum gravity effects are considered via modification of Bekenstein entropy by an extra logarithmic term in the area. This modification is predicted by several approaches to quantum gravity, including loop quantum gravity, string theory, AdS/CFT correspondence and generalised uncertainty principle phenomenology, giving our result a general character. The derived equations generalise classical equations of motion of unimodular gravity, instead of the ones of general relativity, and they contain at most second derivatives of the metric. We provide two independent derivations of the equations based on thermodynamics of local causal diamonds. First one uses Jacobson's maximal vacuum entanglement hypothesis, the second one Clausius entropy flux. Furthermore, we consider questions of diffeomorphism and local Lorentz invariance of the resulting dynamics and discuss its application to a simple cosmological model, finding a resolution of the classical singularity.

“Time”-Covariant Schrödinger Equation and the Canonical Quantization of the Reissner–Nordström Black Hole

A “time”-covariant Schrödinger equation is defined for the minisuperspace model of the Reissner–Nordström (RN) black hole, as a “hybrid” between the “intrinsic time” Schrödinger and Wheeler–DeWitt (WDW) equations. To do so, a reduced, regular, and “time(r)”-dependent Hamiltonian density was constructed, without “breaking” the re-parametrization covariance r→f(r˜). As a result, the evolution of states with respect to the parameter r and the probabilistic interpretation of the resulting quantum description is possible, while quantum schemes for different gauge choices are equivalent by construction. The solutions are found for Dirac’s delta and Gaussian initial states. A geometrical interpretation of the wavefunctions is presented via Bohm analysis. Alongside this, a criterion is presented to adjudicate which, between two singular spacetimes, is “more” or “less” singular. Two ways to adjudicate the existence of singularities are compared (vanishing of the probability density at the classical singularity and semi-classical spacetime singularity). Finally, an equivalence of the reduced equations with those of a 3D electromagnetic pp-wave spacetime is revealed.

Classical and quantum equations of motion of a 4-dimensional Schwarzschild–AdS and Reissner–Nordström–AdS black hole

We investigate the gravitational collapse of both a massive (Schwarzschild–AdS) and a massive-charged (Reissner–Nordström–AdS) 4-dimensional domain wall in AdS space. Here, we consider both the classical and quantum collapse, in the absence of quasi-particle production and backreaction. For the massive case, we show that, as far as the asymptotic observer is concerned, the collapse takes an infinite amount of time to occur in both the classical and quantum cases. Hence, quantizing the domain wall does not lead to the formation of the black hole in a finite amount of time. For the infalling observer, we find that the domain wall collapses to both the event horizon and the classical singularity in a finite amount of proper time. In the region of the classical singularity, however, the wave function exhibits both nonlocal and nonsingular effects. For the massive-charged case, we show that, as far as the asymptotic observer is concerned, the details of the collapse depend on the amount of charge present; that is, the extremal, nonextremal and overcharged cases. In the overcharged case, the collapse never fully occurs since the solution is an oscillatory solution which prevents the formation of a naked singularity. For the extremal and nonextremal cases, it takes an infinite amount of time for the outer horizon to form. For the infalling observer in the nonextremal case, we find that the domain wall collapses to both the event horizon and the classical singularity in a finite amount of proper time. In the region of the classical singularity, the wave function also exhibits both nonlocal and nonsingular effects. Furthermore, in the large energy density limit, the wave function vanishes as the domain wall approaches classical singularity implying that the quantization does not rid the black hole of its singular nature.

Quantum no-singularity theorem from geometric flows

In this paper, we analyze the classical geometric flow as a dynamical system. We obtain an action for this system, such that its equation of motion is the Raychaudhuri equation. This action will be used to quantize this system. As the Raychaudhuri equation is the basis for deriving the singularity theorems, we will be able to understand the effects and such a quantization will have on the classical singularity theorems. Thus, quantizing the geometric flow, we can demonstrate that a quantum space–time is complete (nonsingular). This is because the existence of a conjugate point is a necessary condition for the occurrence of singularities, and we will be able to demonstrate that such conjugate points cannot occur due to such quantum effects.

Quantum collapse of a charged n-dimensional BTZ-like domain wall

We investigate both the classical and quantum gravitational collapse of a massive, charged, nonrotating [Formula: see text]-dimensional Bañados–Teitelboim–Zanelli (BTZ)-like domain wall in AdS space. In the classical picture, we show that, as far as the asymptotic observer is concerned, the details of the collapse depend on the amount of charge present in the domain wall; that is, if the domain wall is extremal, nonextremal or overcharged. In both the extremal and nonextremal cases, the collapse takes an infinite amount of observer time to complete. However, in the over-charged case, the collapse never actually occurs, instead one finds an oscillatory solution which prevents the formation of a naked singularity. As far as the infalling observer is concerned, in the nonextremal case, the collapse is completed within a finite amount of proper time. Thus, the gravitational collapse follows that of the typical formation of a black hole via gravitational collapse.Quantum mechanically, we take the absence of induced quasi-particle production and fluctuations of the metric geometry; that is, we ignore the effect of radiation and back-reaction. For the asymptotic observer, we find that, near the horizon, quantization of the domain wall does not allow the formation of the black hole in a finite amount of observer time. For the infalling observer, we are primarily interested in the quantum mechanical effect as the domain wall approaches the classical singularity. In this region, the main result is that the wave function exhibits nonlocal effects, demonstrated by the fact that the Hamiltonian depends on an infinite number of derivatives that cannot be truncated after a finite number of terms. Furthermore, in the large energy density limit, the wave function vanishes at the classical singularity implying that quantization does not rid the black hole of its singularity.

New singularities in unexpected places

Spacetime singularities have been discovered which are physically much weaker than those predicted by the classical singularity theorems. Geodesics evolve through them and they only display infinities in the derivatives of their curvature invariants. So far, these singularities have appeared to require rather exotic and unphysical matter for their occurrence. Here, we show that a large class of singularities of this form can be found in a simple Friedmann cosmology containing only a scalar-field with a power-law self-interaction potential. Their existence challenges several preconceived ideas about the nature of spacetime singularities and has an impact upon the end of inflation in the early universe.

A General Asymptotic Scheme for the Analysis of Partition Statistics

We consider statistical properties of random integer partitions. In order to compute means, variances and higher moments of various partition statistics, one often has to study generating functions of the form P(x)F(x), where P(x) is the generating function for the number of partitions. In this paper, we show how asymptotic expansions can be obtained in a quasi-automatic way from expansions of F(x) around x = 1, which parallels the classical singularity analysis of Flajolet and Odlyzko in many ways. Numerous examples from the literature, as well as some new statistics, are treated via this methodology. In addition, we show how to compute further terms in the asymptotic expansions of previously studied partition statistics.