Search

We obtain a new family of exact vacuum solutions to quadratic gravity that describe pp-waves with two-dimensional wave surfaces that can have any prescribed constant curvature. When the wave surfaces are flat we recover the Peres waves obtained by Madsen, a subset of which forms precisely the vacuum pp-waves of general relativity. If, on the other hand, the wave surfaces have non-vanishing constant curvature then all our solutions are non-Einstein (i.e. they do not solve Einstein's equations in vacuum, with or without cosmological constant) and we find that the curvature is linearly related to the value of the cosmological constant. We show that the vacuum field equations reduce to a simple linear biharmonic equation on the curved wave surfaces, and as consequence, the general solution can be written down. We also provide some simple explicit examples.

The (pseudo-)Riemann-metrizability and Ricci-flatness of Finsler spaces with $m$-Kropina metric $F = \alpha^{1+m}\beta^{-m}$ of Berwald type are investigated. We prove that the affine connection on $F$ can locally be understood as the Levi-Civita connection of some (pseudo-)Riemannian metric if and only if the Ricci tensor of the canonical affine connection is symmetric. We also obtain a third equivalent characterization in terms of the covariant derivative of the 1-form $\beta$. We use these results to classify all locally metrizable $m$-Kropina spaces whose 1-forms have a constant causal character. In the special case where the first de Rahm cohomology group of the underlying manifold is trivial (which is true of simply connected manifolds, for instance), we show that global metrizability is equivalent to local metrizability and hence, in that case, our necessary and sufficient conditions also characterize global metrizability. In addition, we further obtain explicitly all Ricci-flat, locally metrizable $m$-Kropina metrics in $(3+1)$D whose 1-forms have a constant causal character. In fact, the only possibilities are essentially the following two: either $\alpha$ is flat and $\beta$ is $\alpha$-parallel, or $\alpha$ is a pp-wave and $\beta$ is $\alpha$-parallel.

This PhD dissertation covers a range of topics in Finsler geometry and Finsler gravity, most notably: (i) the characterization of Berwald spaces, (ii) pseudo-Riemann (non-)metrizability of Berwald spaces, (iii) $(\alpha,\beta)$-metrics, (iv) exact solutions to Pfeifer and Wohlfarth's vacuum field equation in Finsler gravity, and (v) Finsler gravitational waves and their observational signature. An extended abstract can be found in the dissertation itself., Comment: PhD thesis. ISBN: 978-90-386-6026-4

We present a new family of exact vacuum solutions to Pfeifer and Wohlfarth's field equation in Finsler gravity, consisting of Finsler metrics that are Landsbergian but not Berwaldian, also known as unicorns due to their rarity. Interestingly we find that these solutions have a physically viable light cone structure, even though in some cases the signature is not Lorentzian but positive definite. We furthermore find a promising analogy between our solutions and classical FLRW cosmology. One of our solutions in particular has cosmological symmetry, i.e. it is spatially homogeneous and isotropic, and it is additionally conformally flat, with the conformal factor depending only on the timelike coordinate. We show that this conformal factor can be interpreted as the scale factor, we compute it as a function of cosmological time, and we show that it corresponds to a linearly expanding (or contracting) Finsler universe.

We introduce a new class of $(\alpha,\beta)$-type exact solutions in Finsler gravity closely related to the well-known pp-waves in general relativity. Our class contains most of the exact solutions currently known in the literature as special cases. The linearized versions of these solutions may be interpretted as Finslerian gravitational waves, and we investigate the physical effect of such waves. More precisely, we compute the Finslerian correction to the radar distance along an nterferometer arm at the moment a Finslerian gravitational wave passes a detector. We come to the remarkable conclusion that the effect of a Finslerian gravitational wave on an interferometer is indistinguishable from that of standard gravitational wave in general relativity. Along the way we also physically motivate a modification of the Randers metric and prove that it has some very interesting properties.

We investigate the local metrizability of Finsler spaces with $m$-Kropina metric $F = \alpha^{1+m}\beta^{-m}$, where $\beta$ is a closed null 1-form. We show that such a space is of Berwald type if and only if the (pseudo-)Riemannian metric $\alpha$ and 1-form $\beta$ have a very specific form in certain coordinates. In particular, when the signature of $\alpha$ is Lorentzian, $\alpha$ belongs to a certain subclass of the Kundt class and $\beta$ generates the corresponding null congruence, and this generalizes in a natural way to arbitrary signature. We use this result to prove that the affine connection on such an $m$-Kropina space is locally metrizable by a (pseudo-)Riemannian metric if and only if the Ricci tensor constructed form the affine connection is symmetric. In particular we construct all counterexamples of this type to Szabo's metrization theorem, which has only been proven for positive definite Finsler metrics that are regular on all of the slit tangent bundle.

In this work we study Randers spacetimes of Berwald type and analyze Pfeifer and Wohlfarth's vacuum field equation of Finsler gravity for this class. We show that in this case the field equation is equivalent to the vanishing of the Finsler Ricci tensor, analogously to Einstein gravity. This implies that the considered vacuum field equation and Rutz's equation coincide in this scenario. We also construct all exact solutions of Berwald-Randers type to vacuum Finsler gravity, which turn out to be composed of a CCNV (covariantly constant null vector) Lorentzian spacetime, commonly known as pp-wave, and a 1-form given by the pp-wave distinguished null vector. We therefore refer to the found solutions as Randers pp-waves., Comment: 11 pagers, updated to journal version

We investigate whether Szabo's metrizability theorem can be extended to Finsler spaces of indefinite signature. For smooth, positive definite Finsler metrics, this important theorem states that, if the metric is of Berwald type (i.e., its Chern-Rund connection defines an affine connection on the underlying manifold), then it is affinely equivalent to a Riemann space, meaning that its affine connection is the Levi-Civita connection of some Riemannian metric. We show for the first time that this result does not extend to Finsler spacetimes. More precisely, we find a large class of Berwald spacetimes for which the Ricci tensor of the affine connection is not symmetric. The fundamental difference from positive definite Finsler spaces that makes such an asymmetry possible, is the fact that generally, Finsler spacetimes satisfy certain smoothness properties only on a proper conic subset of the slit tangent bundle. Indeed, we prove that when the Finsler Lagrangian is smooth on the entire slit tangent bundle, the Ricci tensor must necessarily be symmetric. For large classes of Finsler spacetimes, however, the Berwald property does not imply that the affine structure is equivalent to the affine structure of a pseudo-Riemannian metric. Instead, the affine structure is that of metric-affine geometry with vanishing torsion., Comment: 12 pages, contribution to the Special Issue "Finsler Modification of Classical General Relativity" in the Journal Universe

Berwald geometries are Finsler geometries close to (pseudo)-Riemannian geometries. We establish a simple first order partial differential equation as necessary and sufficient condition, which a given Finsler Lagrangian has to satisfy to be of Berwald type. Applied to $(\alpha,\beta)$-Finsler spaces, respectively $(A,B)$-Finsler spacetimes, this reduces to a necessary and sufficient condition for the Levi-Civita covariant derivative of the defining $1$-form. We illustrate our results with novel examples of $(\alpha,\beta)$-Berwald geometries which represent Finslerian versions of Kundt (constant scalar invariant) spacetimes. The results generalize earlier findings by Tavakol and van den Bergh, as well as the Berwald conditions for Randers and m-Kropina resp. very special/general relativity geometries., Comment: 17 pages, results on $(\alpha,\beta)$-Finsler geometries extended, explicit examples added, updated to journal version

We investigate the relativistic properties under boost transformations of the $\kappa$-Poincar\'e model with multiple causally connected interactions, both at the level of its formulation in momentum space only and when it is endowed with a full phase space construction, provided by the relative locality framework. Previous studies focussing on the momentum space picture showed that in presence of just one interaction vertex the model is relativistic, provided that the boost parameter acting on each given particle receives a "backreaction" from the momenta of the other particles that participate in the interaction. Here we show that in presence of multiple causally-connected vertices the model is relativistic if the boost parameter acting on each given particle receives a backreaction from the total momentum of all the particles that are causally connected, even those that do not directly enter the vertex. The relative locality framework constructs spacetime by defining a set of dual coordinates to the momentum of each particle and interaction coordinates as Lagrange multipliers that enforce momentum conservation at interaction events. We show that the picture is Lorentz invariant if one uses an appropriate "total boost" to act on the particles' worldlines and on the interaction coordinates. The picture we develop also allows for a reinterpretation of the backreaction as the manifestation of the "total boost" action. Our findings provide the basis to consistently define distant relatively boosted observers in the relative locality framework., Comment: 23 pages, 3 figures