A key insight used in developing the theory of Causal Dynamical Triangulations (CDTs) is to use the causal (or light-cone) structure of Lorentzian manifolds to restrict the class of geometries appearing in the Quantum Gravity (QG) path integral. By exploiting this structure the models developed in CDTs differ from the analogous models developed in the Euclidean domain, models of (Euclidean) Dynamical Triangulations (DT), and the corresponding Lorentzian results are in many ways more "physical".In this paper we use this insight to formulate a Lorentzian signature model that is analogous to the Quantum Regge Calculus (QRC) approach to Euclidean Quantum Gravity. We exploit another crucial fact about the structure of Lorentzian manifolds, namely that certain simplices are not constrained by the triangle inequalities present in Euclidean signature. We show that this model is not related to QRC by a naive Wick rotation; this serves as another demonstration that the sum over Lorentzian geometries is not simply related to the sum over Euclidean geometries. By removing the triangle inequality constraints, there is more freedom to perform analytical calculations, and in addition numerical simulations are more computationally efficient.We first formulate the model in 1+1 dimensions, and derive scaling relations for the pure gravity path integral on the torus using two different measures. It appears relatively easy to generate "large" universes, both in spatial and temporal extent. In addition, loop-to-loop amplitudes are discussed, and a transfer matrix is derived. We then also discuss the model in higher dimensions.

Recent tentative experimental indications, and the subsequent theoretical speculations, regarding possible violations of Lorentz invariance have attracted a vast amount of attention. An important technical issue that considerably complicates detailed calculations in any such scenario, is that once one violates Lorentz invariance the analysis of thresholds in both scattering and decay processes becomes extremely subtle, with many new and naively unexpected effects. In the current article we develop several extremely general threshold theorems that depend only on the existence of some energy momentum relation E(p), eschewing even assumptions of isotropy or monotonicity. We shall argue that there are physically interesting situations where such a level of generality is called for, and that existing (partial) results in the literature make unnecessary technical assumptions. Even in this most general of settings, we show that at threshold all final state particles move with the same 3-velocity, while initial state particles must have 3-velocities parallel/anti-parallel to the final state particles. In contrast the various 3-momenta can behave in a complicated and counter-intuitive manner.

In a previous article [JHEP 1111[JHEP (2011 arXiv:1108.4965] we have developed a Lorentzian version of the Quantum Regge Calculus in which the significant differences between simplices in Lorentzian signature and Euclidean signature are crucial. In this article we extend a central result used in the previous article, regarding the realizability of Lorentzian triangles, to arbitrary dimension. This technical step will be crucial for developing the Lorentzian model in the case of most physical interest: 3 + 1 dimensions.We first state (and derive in an appendix) the realizability conditions on the edgelengths of a Lorentzian n-simplex in total dimension n = d + 1, where d is the number of space-like dimensions. We then show that in any dimension there is a certain type of simplex which has all of its time-like edge lengths completely unconstrained by any sort of triangle inequality. This result is the d + 1 dimensional analogue of the 1 + 1 dimensional case of the Lorentzian triangle.

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

We investigate inertial frames in the absence of Lorentz invariance, reconsidering the usual group structure implied by the relativity principle. We abandon the relativity principle, discarding the group structure for the transformations between inertial frames, while requiring these transformations to be at least linear (to preserve homogeneity). In theories with a preferred frame (aether), the set of transformations between inertial frames forms a groupoid/pseudogroup instead of a group, a characteristic essential to evading the von Ignatowsky theorems. In order to understand the dynamics, we also demonstrate that the transformation rules for energy and momentum are in general affine. We finally focus on one specific and compelling model implementing a minimalist violation of Lorentz invariance.Even though Lorentz symmetry seems to be very well established, the question of its possible breaking is still theoretically and experimentally engaging, and may have implications that span from quantum gravity to gravitational waves and cosmology. There exist several different approaches to this topic in the literature, highlighting different aspects of the problem.1-5 In this work, we focus on a modification of von Ignatowsky's 1910 argument that establishes a tight relation between the relativity principle and the group structure for the Lorentz transformations. 6In particular, assuming linearity for the set of spacetime transformations between inertial frames {M i,j }, isotropy of space time, and the relativity principle, the set {M i,j } will be either the Lorentz group for special relativity, or the Galilean group for classical Newtonian mechanics.Therefore, in order to explore the possibility of Lorentz symmetry breaking, we relax von Ignatowsky's argument by abandoning the relativity principle, while retaining the concept of inertial frames. This implies that the set {M i,j } is no longer a group and that we are de facto introducing a preferred frame, the aether. Since we want to preserve as much as possible of the definition of inertial frames, we want the transformations {M i,j } to map straight lines into straight lines, a condition that is satisfied if the transformations are linear. We therefore define the transformation from the aether frame F to any other inertial frameF as:Here M is in general a function of v but it can also depend on the orientation ofF with respect to F . From this, it can easily be deduced that the velocity v aether of the aether as seen from the inertial frameF , and the velocity v moving of the frameF as seen from the aether, are not equal-but-opposite and may not even be collinear.