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 {\em 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.
The radar experiment connects the geometry of spacetime with an observers measurement of spatial length. We investigate the radar experiment on Finsler spacetimes which leads to a general definition of radar orthogonality and radar length. The directions radar orthogonal to an observer form the spatial equal time surface an observer experiences and the radar length is the physical length the observer associates to spatial objects. We demonstrate these concepts on a forth order polynomial Finsler spacetime geometry which may emerge from area metric or pre-metric linear electrodynamics or in qua…
We briefly review two recently developed extensions of the Lorentzian geometry of spacetime and prove that they are in fact closely related. The first is the concept of observer space, which generalizes the space of Lorentzian observers, i.e., future unit timelike vectors, using Cartan geometry. The second is the concept of Finsler spacetimes, which generalizes the Lorentzian metric of general relativity to an observer-dependent Finsler metric. We show that every Finsler spacetime possesses a well-defined observer space that can naturally be equipped with a Cartan geometry. Conversely, we deri…
We consider a triality between the Zermelo navigation problem, the geodesic flow on a Finslerian geometry of Randers type, and spacetimes in one dimension higher admitting a timelike conformal Killing vector field. From the latter viewpoint, the data of the Zermelo problem are encoded in a (conformally) Painleve-Gullstrand form of the spacetime metric, whereas the data of the Randers problem are encoded in a stationary generalisation of the usual optical metric. We discuss how the spacetime viewpoint gives a simple and physical perspective on various issues, including how Finsler geometries wi…
We construct gravitational dynamics for Finsler spacetimes in terms of an action integral on the unit tangent bundle. These spacetimes are generalizations of Lorentzian metric manifolds which satisfy necessary causality properties. A coupling procedure for matter fields to Finsler gravity completes our new theory that consistently becomes equivalent to Einstein gravity in the limit of metric geometry. We provide a precise geometric definition of observers and their measurements, and show that the transformations by means of which different observers communicate form a groupoid that generalizes…
We present our Finsler spacetime formalism which extends the standard formulation of Finsler geometry to be applicable in physics. Finsler spacetimes are viable non-metric geometric backgrounds for physics; they guarantee well defined causality, the propagation of light on a non-trivial null structure, a clear notion of physical observers and the existence of physical field theories determining the geometry of space-time dynamically in terms of an extended gravitational field equation. Here we give the precise definition of Finsler spacetimes and the notion of well-defined observers, their mea…
The general notion of anisotropic connections $\nabla$ is revisited, including its precise relations with the standard setting of pseudo-Finsler metrics, i.e., the canonic nonlinear connection and the (linear) Finslerian connections. In particular, the vertically trivial Finsler connections are identified canonically with anisotropic connections. So, these connections provide a simple intrinsic interpretation of a part of any Finsler connection closer to the Koszul formulation in $M$. Moreover, a new covariant derivative and parallel transport along curves is introduced, taking first a self-pr…
We study the symmetries of the Lorentz violating Randers-Finsler spacetime. The privileged frame defined by the background vector diagonalises the deformed mass-shell and provides an anisotropic observer transformations. The particle Randers transformations are achieved through the Finslerian Killing equation which also shows the equivalence between observer and particle Randers transformation. For a constant background vector, the Randers transformations can be regarded as a deformed directional-dependent Poincaré transformation. The Randers algebra is a deformed Poincaré algebra whose struct…
We consider the geometry of spacetime based on a non-metric, Finslerian, length measure, which, in terms of physics, represents a generalized clock. Our defnition of Finsler spacetimes ensure a well defined notion of causality, a precise description of observers and a geometric background for field theories. Moreover we present our Finsler geometric extension of the Einstein equations, which determine the geometry of Finsler spacetimes dynamically.
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: as dual description of dispersion relations, as most general geometric clock and as geometry being compatible with the relevant Ehlers-Pirani-Schild axioms. As Finsler geometry is a straightforward generalisation of Riemannian geometry there are many attempts to use it as generalized geometry of spacetime in physics. However, this generalisation is subtle due to the existence of non-trivial null directions. I review how a…
We construct a cosmological toy model based on a Finslerian structure of space-time. In particular, we are interested in a specific Finslerian Lorentz violating theory based on a curved version of Cohen and Glashow's Very Special Relativity. The osculation of a Finslerian manifold to a Riemannian leads to the limit of Relativistic Cosmology, for a specified observer. A modified flat FRW cosmology is produced. The analogue of a zero energy particle unfolds some special properties of the dynamics. The kinematical equations of motion are affected by local anisotropies. Seeds of Lorentz Violations…
We introduce a definition of symmetry generating vector fields on manifolds which are equipped with a first-order reductive Cartan geometry. We apply this definition to a number of physically motivated examples and show that our newly introduced notion of symmetry agrees with the usual notions of symmetry of affine, Riemann-Cartan, Riemannian and Weizenböck geometries, which are conventionally used as spacetime models. Further, we discuss the case of Cartan geometries which can be used to model observer space instead of spacetime. We show which vector fields on an observer space can be interpr…
In the context of Lorentz-Finsler spacetime theories the relativity principle holds at a spacetime point if the indicatrix (observer space) is homogeneous. We point out that in four spacetime dimensions there are just three kinematical models which respect an exact form of the relativity principle and for which all observers agree on the spacetime volume. They have necessarily affine sphere indicatrices. For them every observer which looks at a flash of light emitted by a point would observe, respectively, an expanding (a) sphere, (b) tetrahedron, or (c) cone, with barycenter at the point. The…
We present a concise new definition of Finsler spacetimes that generalize Lorentzian metric manifolds and provide consistent backgrounds for physics. Extending standard mathematical constructions known from Finsler spaces we show that geometric objects like the Cartan non-linear connection and its curvature are well-defined almost everywhere on Finsler spacetimes, also on their null structure. This allows us to describe the complete causal structure in terms of timelike and null curves; these are essential to model physical observers and the propagation of light. We prove that the timelike dir…
As a prototypical massive field theory we study the scalar field on the recently introduced Finsler spacetimes. We show that particle excitations exist that propagate faster than the speed of light recognized as the boundary velocity of observers. This effect appears already in Finsler spacetime geometries with very small departures from Lorentzian metric geometry. It switches on for a sufficiently large ratio of the particle four-momentum and mass, and is the consequence of a modified version of the Coleman-Glashow velocity dispersion relation. The momentum dispersion relation on Finsler spac…
We revisit the physical arguments which lead to the definition of the stress-energy tensor $T$ in the Lorentz-Finsler setting $(M,L)$ starting at classical Relativity. Both the standard heuristic approach using fluids and the Lagrangian one are taken into account. In particular, we argue that the Finslerian breaking of Lorentz symmetry makes $T$ an anisotropic 2-tensor (i. e., a tensor for each $L$-timelike direction), in contrast with the energy-momentum vectors defined on $M$. Such a tensor is compared with different ones obtained by using a Lagrangian approach. The notion of divergence is r…
From general relativity we have learned the principles of general covariance and local Lorentz invariance, which follow from the fact that we consider observables as tensors on a spacetime manifold whose geometry is modeled by a Lorentzian metric. Approaches to quantum gravity, however, hint towards a breaking of these symmetries and the possible existence of more general, non-tensorial geometric structures. Possible implications of these approaches are non-tensorial transformation laws between different observers and an observer-dependent notion of geometry. In this work we review two differe…
We investigate spacetimes whose light cones could be anisotropic. We prove the equivalence of the structures: (a) Lorentz-Finsler manifold for which the mean Cartan torsion vanishes, (b) Lorentz-Finsler manifold for which the indicatrix (observer space) at each point is a convex hyperbolic affine sphere centered on the zero section, and (c) pair given by a spacetime volume and a sharp convex cone distribution. The equivalence suggests to describe {\em (affine sphere) spacetimes} with this structure, so that no algebraic-metrical concept enters the definition. As a result, this work shows how t…
By using a method improved with a generalized optical metric, the deflection of light for an observer and source at finite distance from a lens object in a stationary, axisymmetric and asymptotically flat spacetime has been recently discussed [Ono, Ishihara, Asada, Phys. Rev. D {\bf 96}, 104037 (2017)]. In this paper, we study a possible extension of this method to an asymptotically nonflat spacetime. We discuss a rotating global monopole. Our result of the deflection angle of light is compared with a recent work on the same spacetime but limited within the asymptotic source and observer [Jusu…
While the postulate of covariance of Maxwell's equations for all inertial observers led Einstein to special relativity, it was the further demand of general covariance -- form invariance under general coordinate transformations, including between accelerating frames -- that led to general relativity. Several lines of inquiry over the past two decades, notably the development of metamaterial-based transformation optics, has spurred a greater interest in the role of geometry and space-time covariance for electrodynamics in ponderable media. I develop a generally covariant, coordinate-free framew…
The gravitational deflection angle of light for an observer and source at finite distance from a lens object has been studied by Ishihara et al. [Phys. Rev. D, 94, 084015 (2016)], based on the Gauss-Bonnet theorem with using the optical metric. Their approach to finite-distance cases is limited within an asymptotically flat spacetime. By making several assumptions, we give an interpretation of their definition from the observer's viewpoint: The observer assumes the direction of a hypothetical light emission at the observer position and makes a comparison between the fiducial emission direction…
We investigate a foliation of Schwarzschild spacetime determined by observers freely falling in the radial direction. This is described using a generalisation of Gullstrand-Painlevé coordinates which allows for any possible radial velocity. This foliation provides a contrast with the usual static foliation implied by Schwarzschild coordinates. The $3$-dimensional spaces are distinct for the static and falling observers, so the embedding diagrams, spatial measurement, simultaneity, and time at infinity are also distinct, though the $4$-dimensional spacetime is unchanged. Our motivation is conce…
In this paper we discuss Internal Relativity, a recent program to address the problem of quantum gravity. In our approach we change the relationship between spacetime and matter. Currently we view matter as propagating on spacetime. Einstein's equations encode how spacetime curves due to the presence of matter and how spacetime, in turn, tells matter how to propagate. In internal realtivity matter and spacetime cease to exist as distinct entities, rather, they arise simultaneously from an underlying quantum system. It is through the emergent matter degrees of freedom that geometry is inferred.…
