Quantum Gravity Phenomenology Induced in the Propagation of UHECR, a Kinematical Solution in Finsler and Generalized Finsler Spacetime

It is well-known that the universe is opaque to the propagation of Ultra-High-Energy Cosmic Rays (UHECRs) since these particles dissipate energy during their propagation interacting with the background fields present in the universe, mainly with the Cosmic Microwave Background (CMB) in the so-called GZK cut-off phenomenon. Some experimental evidence seems to hint at the possibility of a dilation of the GZK predicted opacity sphere. It is well-known that kinematical perturbations caused by supposed quantum gravity (QG) effects can modify the foreseen GZK opacity horizon. The introduction of Lorentz Invariance Violation can indeed reduce, and in some cases making negligible, the CMB-UHECRs interaction probability. In this work, we explore the effects induced by modified kinematics in the UHECR lightest component phenomenology from the QG perspective. We explore the possibility of a geometrical description of the massive fermions interaction with the supposed quantum structure of spacetime in order to introduce a Lorentz covariance modification. The kinematics are amended, modifying the dispersion relations of free particles in the context of a covariance-preserving framework. This spacetime description requires a more general geometry than the usual Riemannian one, indicating, for instance, the Finsler construction and the related generalized Finsler spacetime as ideal candidates. Finally we investigate the correlation between the magnitude of Lorentz covariance modification and the attenuation length of the photopion production process related to the GZK cut-off, demonstrating that the predicted opacity horizon can be dilated even in the context of a theory that does not require any privileged reference frame.

Foundations of Finsler Spacetimes from the Observers’ Viewpoint

Physical foundations for relativistic spacetimes are revisited in order to check at what extent Finsler spacetimes lie in their framework. Arguments based on inertial observers (as in the foundations of special relativity and classical mechanics) are shown to correspond with a double linear approximation in the measurement of space and time. While general relativity appears by dropping the first linearization, Finsler spacetimes appear by dropping the second one. The classical Ehlers–Pirani–Schild approach is carefully discussed and shown to be compatible with the Lorentz–Finsler case. The precise mathematical definition of Finsler spacetime is discussed by using the space of observers. Special care is taken in some issues such as the fact that a Lorentz–Finsler metric would be physically measurable only on the causal directions for a cone structure, the implications for models of spacetimes of some apparently innocuous hypotheses on differentiability, or the possibilities of measurement of a varying speed of light.

Finsler spacetime geometry in physics

Finsler geometry naturally appears in the description of various physical systems. In this review, I divide the emergence of Finsler geometry in physics into three categories: dual description of dispersion relations, most general geometric clock and geometry being compatible with the relevant Ehlers–Pirani–Schild axioms. As Finsler geometry is a straightforward generalization of Riemannian geometry there are many attempts to use it as generalized geometry of spacetime in physics. However, this generalization is subtle due to the existence of non-trivial null directions. I review how a pseudo-Finsler spacetime geometry can be defined such that it provides a precise notion of causal curves, observers and their measurements as well as a gravitational field equation determining the Finslerian spacetime geometry dynamically. The construction of such Finsler spacetimes lays the foundation for comparing their predictions with observations, in astrophysics as well as in laboratory experiments.

Finsler spacetime in light of Segal’s principle

ISIM(2) symmetry group of Cohen and Glashow’s very special relativity is unstable with respect to small deformations of its underlying algebraic structure, and according to Segal’s principle cannot be a true symmetry of nature. However, like special relativity, which is a very good description of nature, thanks to the smallness of the cosmological constant, which characterizes the deformation of the Poincaré group, the very special relativity can also be a very good approximation, thanks to the smallness of the dimensionless parameter characterizing the deformation of ISIM(2).

SCHWARZSCHILD SOLUTION OF THE MODIFIED EINSTEIN FIELD EQUATIONS

A reformulation of the Schwarzschild solution of the linearized Einstein field equations in post-Riemannian Finsler spacetime is derived. The solution is constructed in three stages: the exterior solution, the event-horizon solution and the interior solution. It is shown that the exterior solution is asymptotically similar to Newtonian gravity at large distances implying that Newtonian gravity is a low energy approximation of the solution. Application of Eddington-Finklestein coordinates is shown to reproduce the results obtained from standard general relativity at the event horizon. Further application of Kruskal-Szekeres coordinates reveals that the interior solution contains maximally extensible geodesics.

On singular generalized Berwald spacetimes and the equivalence principle

The notion of singular generalized Finsler spacetime and singular generalized Berwald spacetime is introduced and their relevance for the description of classical gravity is discussed. A method to construct examples of such generalized Berwald spacetimes is sketched. The method is applied at two different levels of generality. First, a class of flat, singular generalized Berwald spacetimes is obtained. Then in an attempt of further generalization, a class of non-flat generalized Berwald spacetimes is presented and the associated Einstein field equations are discussed. In this context, an argument in favor of a small value of the cosmological constant is given. The physical significance of the models is briefly discussed in the last section.

Standard static Finsler spacetimes

We introduce the notion of a standard static Finsler spacetime [Formula: see text] where the base [Formula: see text] is a Finsler manifold. We prove some results which connect causality with the Finslerian geometry of the base extending analogous ones for static and stationary Lorentzian spacetimes.

The tangent bundle exponential map and locally autoparallel coordinates for general connections on the tangent bundle with application to Finsler geometry

We construct a tangent bundle exponential map and locally autoparallel coordinates for geometries based on a general connection on the tangent bundle of a manifold. As concrete application we use these new coordinates for Finslerian geometries and obtain Finslerian geodesic coordinates. They generalize normal coordinates known from metric geometry to Finsler geometric manifolds and it turns out that they are identical to the Douglas–Thomas normal coordinates introduced earlier. We expand the Finsler Lagrangian of a Finsler spacetime in these new coordinates and find that it is constant to quadratic order. The quadratic order term comes with the nonlinear curvature of the manifold. From physics these coordinates may be interpreted as the realisation of an Einstein elevator in Finslerian spacetime geometries.