On the volume inside old black holes

Black holes that have nearly evaporated are often thought of as small objects, due to their tiny exterior area. However, the horizon bounds large spacelike hypersurfaces. A compelling geometric perspective on the evolution of the interior geometry was recently shown to be provided by a generally covariant definition of the volume inside a black hole using maximal surfaces. In this article, we expand on previous results and show that finding the maximal surfaces in an arbitrary spherically symmetric spacetime is equivalent to a 1+1 geodesic problem. We then study the effect of Hawking radiation on the volume by computing the volume of maximal surfaces inside the apparent horizon of an evaporating black hole as a function of time at infinity: while the area is shrinking, the volume of these surfaces grows monotonically with advanced time, up to when the horizon has reached Planckian dimensions. The physical relevance of these results for the information paradox and the remnant scenarios are discussed.Comment: 9 pages, 5 figure

Light Cone Black Holes

When probed with conformally invariant matter fields, light cones in Minkowski spacetime satisfy thermodynamical relations which are the analog of those satisfied by stationary black holes coupled to standard matter fields. These properties stem from the fact that light cones are conformal Killing horizons stationary with respect to observers following the radial conformal Killing fields in flat spacetime. The four laws of light cone thermodynamics relate notions such as (conformal) temperature, (conformal) surface gravity, (conformal) energy and a conformally invariant notion related to area change. These quantities do not admit a direct physical interpretation in flat spacetime. However, they become the usual thermodynamical quantities when Minkowski is mapped, via a Weyl transformation, to a target spacetime where the conformal Killing field becomes a proper Killing field. In this paper we study the properties of such spacetimes. The simplest realisation turns out to be the Bertotti-Robinson solution, which is known to encode the near horizon geometry of near extremal and extremal charged black holes. The analogy between light cones in flat space and black hole horizons is therefore strengthened. The construction works in arbitrary dimensions; in two dimensions one recovers the Jackiv-Teitelboim black hole of dilaton gravity. Other interesting realisations are also presented.Comment: 23 pages, 7 figures; v2: typos corrected, matches published versio

Light Cone Thermodynamics

We show that null surfaces defined by the outgoing and infalling wave fronts emanating from and arriving at a sphere in Minkowski spacetime have thermodynamical properties that are in strict formal correspondence with those of black hole horizons in curved spacetimes. Such null surfaces, made of pieces of light cones, are bifurcate conformal Killing horizons for suitable conformally stationary observers. They can be extremal and non-extremal depending on the radius of the shining sphere. Such conformal Killing horizons have a constant light cone (conformal) temperature, given by the standard expression in terms of the generalisation of surface gravity for conformal Killing horizons. Exchanges of conformally invariant energy across the horizon are described by a first law where entropy changes are given by $1/(4\ell_p^2)$ of the changes of a geometric quantity with the meaning of horizon area in a suitable conformal frame. These conformal horizons satisfy the zeroth to the third laws of thermodynamics in an appropriate way. In the extremal case they become light cones associated with a single event; these have vanishing temperature as well as vanishing entropy.Comment: 30 pages, 5 pictures; V_2: a problem in the proof of the first law has been corrected. Results remain unchanged. Geometric interpretation and presentation improved; V_3: matches published versio

Self-assembly of short DNA duplexes: from a coarse-grained model to experiments through a theoretical link

Short blunt-ended DNA duplexes comprising 6 to 20 base pairs self-assemble into polydisperse semi-flexible chains due to hydrophobic stacking interactions between terminal base pairs. Above a critical concentration, which depends on temperature and duplex length, such chains order into liquid crystal phases. Here, we investigate the self-assembly of such double-helical duplexes with a combined numerical and theoretical approach. We simulate the bulk system employing the coarse-grained DNA model recently proposed by Ouldridge et al. [ J. Chem. Phys. 134, 08501 (2011) ]. Then we evaluate the input quantities for the theoretical framework directly from the DNA model. The resulting parameter-free theoretical predictions provide an accurate description of the simulation results in the isotropic phase. In addition, the theoretical isotropic-nematic phase boundaries are in line with experimental findings, providing a route to estimate the stacking free energy.Comment: 13 pages, 10 figure

Probing the Big Bang with quantum fields

By carrying out a systematic investigation of linear, test quantum fields $\hat{\phi}(x)$ in cosmological space-times, we show that $\hat{\phi}(x)$ remain well-defined across the big bang as operator valued distributions in a large class of Friedmann, Lema\^itre, Robertson, Walker space-times, including radiation and dust filled universes. In particular, the expectation values $\langle \hat{\phi}(x)\,\hat{\phi}(x')\rangle$ are well-defined bi-distributions in the extended space-time in spite of the big bang singularity. Interestingly, correlations between fields evaluated at spatially and temporally separated points exhibit an asymmetry that is reminiscent of the Belinskii, Khalatnikov, Lifshitz behavior. The renormalized products of fields $\langle \hat{\phi}^2(x)\rangle_{\rm ren}$ and $\langle \hat{T}_{ab}(x) \rangle_{\rm ren}$ also remain well-defined as distributions. Conformal coupling is not necessary for these considerations to hold. Thus, when probed with observables associated with quantum fields, the big bang (and the big crunch) singularities are quite harmless.Comment: 38 pages; Three small clarifications added and minor typos corrected. Version to appear in Advances in Mathematical and Theoretical Physic

Investigating Static and Dynamic Non-Singular Black Holes

Einstein’s General Relativity predicts the existence of black holes. In the deep interior of a black hole a singularity arises, that implies the breakdown of the theory itself. Such ill-defined regions are expected to be eliminate by quantum gravitational effects. While a complete and satisfactory theory of Quantum Gravity is not yet available, it is possible to start from a semi-classical approach. The basic idea is to mimic background-independent effects that allow the avoidance of the singularity, by building effective metrics solutions of the Einstein’s Equations such that the resulting SpaceTime is Schwarzschild-like in the outer region and, at the same time, singularity-free in the deep interior The main goal of this work is to analyze, thanks to classic and new tools, both static and dynamic properties of such solutions, in order to verify their physical plausibility. The analysis is divided in three Parts, each of which contains two Chapters. Part I: The First Part is devoted to a presentation of the principal features of classic black holes. Nevertheless, its aim is not to make a complete satisfactory review of the classical black hole physics, but to discuss those aspects that will be relevant in the following analysis. Therefore, the First Chapter starts with the presentation of the Schwarzschild’s solution to the Einstein’s Equations from where the study of black holes initially arose. The following more mathematical Section is dedicated to the proof of the Singularity Theorems that lay at the basis of the motivations of this thesis. Going beyond the purely classical properties, in Chapter 2 we introduce the machinery of Quantum Field Theory in Curved SpaceTime and Bogoliubov’s formalism for particle creation. The astonishing result of Hawking’s radiation naturally arises applying these formalisms to the Vaidya-Schwarzschild’s metric: black holes radiate energy away with a Planck spectrum. Their complete evaporation implies the so called information-loss paradox: the evolution of a pure quantum state propagating on Vaidya-Schwarzschild metric is not unitary. Moreover, a discussion on the importance of the Hawking’s result for the so called black hole thermodynamics introduces us to the concepts of black hole entropy and entanglement entropy. Since the latter plays as basis for the entire Chapter 6, we dedicate the last Section of this Part to its definition and basic features. The Second and the Third Part represent the core of the thesis. As said before, the goal is to analyze non-singular black holes metrics both in their static and dynamic behavior. Therefore, we dedicate the Second Part to the study of the properties of such objects settled by the gravitational collapse of a spherical body and remaining in their static configuration. In the Third Part Hawking’s radiation is turned on and the dynamic features are analyzed. Part II: The Second Part starts, in its first Chapter (Chapter 3), with the introduction of what is known about non-singular black holes. The first Section presents the proof of a theorem by Irina Dymnikova asserting that, if such non-singular black holes exist, they must have a rather universal causal structure. This structure is deeply studied in the second Section, focusing on the particular example proposed by Sean Hayward and recently reconsidered by many authors. Chapter 4, on the other hand, contains the first original results. We point out two physical requirements that are not satisfied by the current metrics, and show how to properly take them into account. Indeed, it seems physically unreasonable that a clock at the (regular) center of the star suffers no time delay with respect to a clock at infinity. Moreover, an effective metric that supposes to mimic quantum effects should capture the 1-loop quantum corrections to the Newton’s potential obtained by John Donoghue using effective field theory. In the last Section a relatively easy solution is proposed (Modified Hayward’s Metric), providing a more realistic description of a non-singular black hole. Part III: Static non-singular black holes form an event horizon. Therefore we expected Hawking’s radiation and consequent evaporation to take place and the after-formation system to become dynamic. In the introductory brief Chapter 5 we introduce some first insights in the problem considering the so called quasi-statical approximation to hold during the entire evaporation process. As in the original Hawking’s evaporation case, the dynamics will be simply encoded allowing the mass of the black hole to decrease in time. Different scenarios are shortly discussed. The main results, however, are presented in the last Chapter. Here the plausibility of evaporation processes is studied through the investigation of their entanglement entropy production, the so called Page’s curve. This analysis is made quantitative possible thanks to a new covariant definition of entanglement entropy developed by Eugenio Bianchi and Matteo Smerlak. From this definition follows the possibility to give a precise characterization of entanglement entropy production and to analytically compute the Page’s curve associated to any SpaceTime. In particular, applied to the Hayward’s metric, this analysis confirms the recover of unitarity, but at the same time shines a light on two non-easily solvable problems. Namely, (i) the total energy radiated by the hole turns out to be much bigger than the initial ADM mass, and (ii) the so called purification time does not satisfies a physical lower bound we can impose on it. These inconsistencies undermine the physical validity of the dynamic Hayward’s metric itself (and, because of the Dymnikova’s theorem, of almost all the metrics so far proposed) as a good semi-classical approach to the resolution of the singularity and of the information-loss paradox. Different ideas are needed. The new definition of entanglement entropy provides a powerful tool to analyze the physical plausibility of any semi-classical scenario of formation and consequent unitary evaporation that can be proposed, as for example the ‘black hole firework’ proposed by Hal Haggard and Carlo Rovelli studied in the last Section of this work. Up to now, however, no one of the proposal we encountered seems to satisfy all the requirements one can impose on it. The study of Hawking’s radiation and evaporating black holes is a very active and fascinating field of research to which this thesis can contribute with original ideas and results

On the Effective Metric of a Planck Star

Spacetime metrics describing `non-singular' black holes are commonly studied in the literature as effective modification to the Schwarzschild solution that mimic quantum gravity effects removing the central singularity. Here we point out that to be physically plausible, such metrics should also incorporate the 1-loop quantum corrections to the Newton potential and a non-trivial time delay between an observer at infinity and an observer in the regular center. We present a modification of the well-known Hayward metric that features these two properties. We discuss bounds on the maximal time delay imposed by conditions on the curvature, and the consequences for the weak energy condition, in general violated by the large transversal pressures introduced by the time delay.Comment: 10 pages, many figures; v2 added reference