Infinite Shannon entropy

Even if a probability distribution is properly normalizable, its associated Shannon (or von Neumann) entropy can easily be infinite. We carefully analyze conditions under which this phenomenon can occur. Roughly speaking, this happens when arbitrarily small amounts of probability are dispersed into an infinite number of states; we shall quantify this observation and make it precise. We develop several particularly simple, elementary, and useful bounds, and also provide some asymptotic estimates, leading to necessary and sufficient conditions for the occurrence of infinite Shannon entropy. We go to some effort to keep technical computations as simple and conceptually clear as possible. In particular, we shall see that large entropies cannot be localized in state space; large entropies can only be supported on an exponentially large number of states. We are for the time being interested in single-channel Shannon entropy in the information theoretic sense, not entropy in a stochastic field theory or QFT defined over some configuration space, on the grounds that this simple problem is a necessary precursor to understanding infinite entropy in a field theoretic context.Comment: 13 pages; V2: 4 references adde

Inertial frames without the relativity principle

Ever since the work of von Ignatowsky circa 1910 it has been known (if not always widely appreciated) that the relativity principle, combined with the basic and fundamental physical assumptions of locality, linearity, and isotropy, leads almost uniquely to either the Lorentz transformations of special relativity or to Galileo's transformations of classical Newtonian mechanics. Thus, if one wishes to entertain the possibility of Lorentz symmetry breaking within the context of the class of local physical theories, then it seems likely that one will have to abandon (or at the very least grossly modify) the relativity principle. Working within the framework of local physics, we reassess the notion of spacetime transformations between inertial frames in the absence of the relativity principle, arguing that significant and nontrivial physics can still be extracted as long as the transformations are at least linear. An interesting technical aspect of the analysis is that the transformations now form a groupoid/pseudo-group --- it is this technical point that permits one to evade the von Ignatowsky argument. Even in the absence of a relativity principle we can nevertheless deduce clear and compelling rules for the transformation of space and time, rules for the composition of 3-velocities, and rules for the transformation of energy and momentum. As part of the analysis we identify two particularly elegant and physically compelling models implementing "minimalist" violations of Lorentz invariance --- in the first of these minimalist models all Lorentz violations are confined to carefully delineated particle physics sub-sectors, while the second minimalist Lorentz-violating model depends on one free function of absolute velocity, but otherwise preserves as much as possible of standard Lorentz invariant physics.Comment: V1: 42 pages; V2: now 43 pages; added 8 references, added brief discussion of Carroll kinematics, added brief discussion of Robertson-Mansouri-Sexl framework, added various minor clarifications. V3: now 51 pages; added another 34 references; more discussion of DSR and relative locality; various clarifications and extensions; this version accepted for publication in JHE

Null Energy Condition violations in bimetric gravity

We consider the effective stress-energy tensors for the foreground and background sectors in ghost-free bimetric gravity. By considering the symmetries of the theory, we show that the foreground and background null energy conditions (NECs) are strongly anti-correlated. In particular, the NECs can only be simultaneously fulfilled when they saturate, corresponding to foreground and background cosmological constants. In all other situations, either the foreground or the background is subject to a NEC-violating contribution to the total stress-energy.Comment: v1: 16 pages; v2: 2 references adde

Gordon and Kerr-Schild ansatze in massive and bimetric gravity

We develop the "generalized Gordon ansatz" for the ghost-free versions of both massive and bimetric gravity, an ansatz which is general enough to include almost all spacetimes commonly considered to be physically interesting, and restricted enough to greatly simplify calculations. The ansatz allows explicit calculation of the matrix square root gamma = sqrt{g^{-1} f} appearing as a central feature of the ghost-free analysis. In particular, this ansatz automatically allows us to write the effective stress-energy tensor as that corresponding to a perfect fluid. A qualitatively similar "generalized Kerr-Schild ansatz" can also be easily considered, now leading to an effective stress-energy tensor that corresponds to a null fluid. Cosmological implications are considered, as are consequences for black hole physics. Finally we have a few words to say concerning the null energy condition in the framework provided by these ansatze.Comment: 22 page

The information recovery problem

The issue of unitary evolution during creation and evaporation of a black hole remains controversial. We~argue that some prominent cures are more troubling than the disease, demonstrate that their central element---forming of the event horizon before the evaporation begins---is not necessarily true, and describe a fully coupled matter-gravity system which is manifestly unitary.Comment: 7 pages +1 fig Published versio

Phenomena at the border between quantum physics and general relativity

In this thesis we shall present a collection of research results about phenomena that lie at the interface between quantum physics and general relativity. The motivation behind our research work is to find alternative ways to tackle the problem of a quantum theory of/for gravitation. In the general introduction, we shall briefly recall some of the characteristics of the well-established approaches to this problem that have been developed since the beginning of the middle of the last century. Afterward we shall illustrate why one would like to engage in alternative paths to better understand the problem of a quantum theory of/for gravitation, and the extent to which they will be able to shed some light into this problem. In the first part of the thesis, we shall focus on formulating physics without Lorentz invariance. In the introduction to this part we shall describe the motivations that are behind such a possible choice, such as the possibility that the physics at energies near Planck regime may violate Lorentz symmetry. In the following part we shall first consider a minimalist way of breaking Lorentz invariance by renouncing the relativity principle, that corresponds to the introduction of a preferred frame, the aether frame. In this case we shall look at the transformations between a generic inertial frame and the aether frame still requiring the transformations to be linear. The second step is to establish the transformations for the energy and momentum in order to define some dynamics and design possible experiments to test such assumptions. As an application we shall present two compelling models that minimally break Lorentz invariance, the first one only in the energy-momentum sector, the second one in the transformation between inertial frames. Following along the line of physics without Lorentz invariance, we shall next explore some threshold theorems in both scattering and decay processes by considering only the existence of some energy momentum relation E(p), without making any further assumption. We shall see that quite a lot can be said and that 3-momenta can behave in a complicated and counter-intuitive manner. In the second part of the thesis we shall address the thermodynamics of space-time and the important role played by entropy. In the introduction we shall outline the idea of induced gravity, which is the motivation behind this possible interpretation of general relativity as a mean field theory of some underlying microscopic degrees of freedom. In the next chapter we shall partially review Jacobson's thermodynamic derivation of the Einstein equations and generalise it to a generic birfucate null surface. The interesting result we shall see is that, given the construction of the thermodynamic system via some virtual constantly accelerating observers, we can assign a "virtual" definition of Clausius entropy to essentially arbitrary causal horizons. To conclude this part we shall present some of the mathematical properties of entropy. In particular we shall focus on the simpler case of single-channel Shannon entropy and study under which conditions it is infinite, even though the probability distribution is normalisable. In the last part, we shall describe a proposal for a space-base experiment to test the effects of acceleration and gravity of quantum physics. In principle, the results of such an experiment could shed some light on fundamental questions about the overlap of quantum theory and general relativity; at the same time, they may enable experimentalists interested to implement quantum communication into space based technology, to correct adverse gravitational effects. We conclude with a brief discussion of lessons learned from these different approaches