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The method of variational bootstrapping, based on canonical variational completion, allows one to construct a Lagrangian for a physical theory depending on two sets of field variables, starting from a guess of the field equations for only one such set. This setup is particularly appealing for the construction of modified theories of gravity since one can take lessons from GR for the "educated guess" of the metric field equations; the field equations for the other fields are then fixed by the obtained Lagrangian (up to terms completely independent from the metric tensor). In the present paper, we explore the applications of variational bootstrapping in the context of metric-affine theories of gravity. We find those metric-affine models which are, in a variational sense, closest to the $\Lambda$CDM model of cosmology. The method then allows to find "corrected" metric equations and to "bootstrap" the connection field equations. The Lagrangians obtained via this method, though imposing some restricting criteria, encompass a wide variety of metric-affine models. In particular, they allow for quadratic metric-affine theories only if these avoid non-linear terms in the curvature tensor., Comment: 15 pages + 1 page references + 3 pages appendix, 0 figure

For a torsion-free affine connection on a given manifold, which does not necessarily arise as the Levi-Civita connection of any pseudo-Riemannian metric, it is still possible that it corresponds in a canonical way to a Finsler structure; this property is known as Finsler (or Berwald-Finsler) metrizability. In the present paper, we clarify, for 4-dimensional SO(3)-invariant, Berwald-Finsler metrizable connections, the issue of the existence of an affinely equivalent pseudo-Riemannian structure. In particular, we find all classes of SO(3)-invariant connections which are not Levi-Civita connections for any pseudo-Riemannian metric - hence, are non-metric in a conventional way - but can still be metrized by SO(3)-invariant Finsler functions. The implications for physics, together with some examples are briefly discussed., Comment: 22 pages

The method of variational completion allows one to transform an (in principle, arbitrary) system of partial differential equations -- based on an intuitive ``educated guess'' -- into the Euler-Lagrange one attached to a Lagrangian, by adding a canonical correction term. Here, we extend this technique to theories that involve at least two sets of dynamical variables : we show that an educated guess of the field equations with respect to one of these sets of variables only is sufficient to variationally complete these equations and recover a Lagrangian for the full theory, up to terms that do not involve the respective variables. Applying this idea to natural metric-affine theories of gravity, we prove that, starting from an educated guess of the metric equations only, one can find the full metric equations, together with a generally covariant Lagrangian for the theory, up to metric-independent terms; the latter terms (which can only involve the distortion of the connection) are then completely classified over 4-dimensional spacetimes, by techniques pertaining to differential invariants., Comment: 16 pages + 1 page references & acknowledgments + 5 pages appendix, 0 figure

We prove that all Ricci-flat, spatially spherically symmetric Berwald spacetime structures are either pseudo-Riemannian (Lorentzian), or flat. This insight enables us to formulate a Jebsen-Birkhoff type theorem for Berwald spacetimes.

For the general class of pseudo-Finsler spaces with $(\alpha,\beta)$-metrics, we establish necessary and sufficient conditions such that these admit a Finsler spacetime structure. This means that the fundamental tensor has Lorentzian signature on a conic subbundle of the tangent bundle and thus the existence of a cone of future pointing timelike vectors is ensured. The identified $(\alpha,\beta)$-Finsler spacetimes are candidates for applications in gravitational physics. Moreover, we completely determine the relation between the isometries of an $(\alpha,\beta)$-metric and the isometries of the underlying pseudo-Riemannian metric $a$; in particular, we list all $(\alpha,\beta)$-metrics which admit isometries that are not isometries of $a$., Comment: 25 pages

We locally classify all SO(3)-invariant 4-dimensional pseudo-Finsler Berwald structures. These are Finslerian geometries which are closest to (spatially, or SO(3))-spherically symmetric pseudo-Riemannian ones - and serve as ansatz to find solutions of Finsler gravity equations which generalize the Einstein equations. We find that there exist six classes of non pseudo-Riemannian (i.e., non-quadratic in the velocities) SO(3)-spherically symmetric pseudo-Finsler Berwald functions, which have either: a power law, an exponential law, or a one- or two-variable dependence on the velocities., Comment: 24 pages

The paper introduces a general mathematical framework for action based field theories on Finsler spacetimes. As most often fields on Finsler spacetime (e.g., the Finsler fundamental function or the resulting metric tensor) have a homogeneous dependence on the tangent directions of spacetime, we construct the appropriate configuration bundles whose sections are such homogeneous fields; on these configuration bundles, the tools of coordinate free calculus of variations can be consistently applied to obtain field equations. Moreover, we prove that general covariance of natural Finsler field Lagrangians leads to an averaged energy-momentum conservation law which, in the particular case of Lorentzian spacetimes, is equivalent to the usual, pointwise energy-momentum covariant conservation law., Comment: 48 pages, 1 figure

General Relativity and the $\Lambda$CDM framework are currently the standard lore and constitute the concordance paradigm. Nevertheless, long-standing open theoretical issues, as well as possible new observational ones arising from the explosive development of cosmology the last two decades, offer the motivation and lead a large amount of research to be devoted in constructing various extensions and modifications. All extended theories and scenarios are first examined under the light of theoretical consistency, and then are applied to various geometrical backgrounds, such as the cosmological and the spherical symmetric ones. Their predictions at both the background and perturbation levels, and concerning cosmology at early, intermediate and late times, are then confronted with the huge amount of observational data that astrophysics and cosmology are able to offer recently. Theories, scenarios and models that successfully and efficiently pass the above steps are classified as viable and are candidates for the description of Nature. This work is a Review of the recent developments in the fields of gravity and cosmology, presenting the state of the art, high-lighting the open problems, and outlining the directions of future research. Its realization was performed in the framework of the COST European Action ``Cosmology and Astrophysics Network for Theoretical Advances and Training Actions''., Comment: 543 pages

Lepage equivalents of Lagrangians are a higher order, field-theoretical generalization of the notion of Poincare-Cartan form from mechanics and play a similar role: they give rise to a geometric formulation (and to a geometric understanding) of the calculus of variations. A long-standing open problem is the determination, for field-theoretical Lagrangians of order greater than one, of a Lepage equivalent with the so-called closure property: the Lepage equivalent is a closed differential form if and only if the Lagrangian has vanishing Euler-Lagrange expressions. The present paper proposes a solution to this problem, for general Lagrangians, of any order. The construction is a local one; yet, we show that in most of the cases of interest for physical applications, the obtained Lepage equivalent is actually globally defined. A variant of this construction, which is advantageous for reducible Lagrangians, is also introduced. In particular, for reducible Lagrangians of order two, the obtained Lepage equivalents are of order one., Comment: 15 pages

Recently, a proposal to obtain a finite contribution of second derivative order to the gravitational field equations in \(D = 4\) dimensions from a renormalized Gauss-Bonnet term in the action has received a wave of attention. It triggered a discussion whether the employed renormalization procedure yields a well-defined theory. One of the main criticisms is based on the fact that the resulting field equations cannot be obtained as the Euler-Lagrange equations from a diffeomorphism invariant action. In this work, we use techniques from the inverse calculus of variations to point out that the renormalized truncated Gauss-Bonnet equations cannot be obtained from any action at all (either diffeomorphism invariant or not), in any dimension. Then, we employ canonical variational completion, based on the notion of Vainberg-Tonti Lagrangian - which consists in adding a canonically defined correction term to a given system of equations, so as to make them derivable from an action. To apply this technique to the suggested $4$D renormalized Gauss-Bonnet equations, we extend the variational completion algorithm to some classes of PDE systems for which the usual integral providing the Vainberg-Tonti Lagrangian diverges. We discover that in $D>4$ the suggested field equations can be variationally completed, choosing either the metric or its inverse as field variables; both approaches yield consistently the same Lagrangian, whose variation leads to fourth order field equations. In $D=4$, the Lagrangian of the variationally completed theory diverges in both cases., Comment: 16 pages

We show that Lorentz-Finsler geometry offers a powerful tool in obtaining inequalities. With this aim, we first point out that a series of famous inequalities such as: the (weighted) arithmetic-geometric mean inequality, Acz\'el's, Popoviciu's and Bellman's inequalities, are all particular cases of a reverse Cauchy-Schwarz, respectively, of a reverse triangle inequality holding in Lorentz-Finsler geometry. Then, we use the same method to prove some completely new inequalities, including two refinements of Acz\'el's inequality., Comment: 24 pages

A description of many-particle systems, which is more fundamental than the fluid approach, is to consider them as a kinetic gas. In this approach the dynamical variable in which the properties of the system are encoded, is the distribution of the gas particles in position and velocity space, called 1-particle distribution function (1PDF). However, when the gravitational field of a kinetic gas is derived via the Einstein-Vlasov equations, the information about the velocity distribution of the gas particles is averaged out and therefore lost. We propose to derive the gravitational field of a kinetic gas directly from its 1PDF, taking the velocity distribution fully into account. We conjecture that this refined approach could possibly account for the observed dark energy phenomenology., Comment: 4 pages, Essay written for the Gravity Research Foundation competition 2020, Journal references updated

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

Applying the cosmological principle to Finsler spacetimes, we identify the Lie Algebra of symmetry generators of spatially homogeneous and isotropic Finsler geometries, thus generalising Friedmann-Lema\^{i}tre-Robertson-Walker geometry. In particular, we find the most general spatially homogeneous and isotropic Berwald spacetimes, which are Finsler spacetimes that can be regarded as closest to pseudo-Riemannian geometry. They are defined by a Finsler Lagrangian built from a zero-homogeneous function on the tangent bundle, which encodes the velocity dependence of the Finsler Lagrangian in a very specific way. The obtained cosmological Berwald geometries are candidates for the description of the geometry of the universe, when they are obtained as solutions from a Finsler gravity equation., Comment: 14 pages, contribution to the Special Issue "Finsler Modification of Classical General Relativity" in the Journal Universe

We propose a new model for the description of a gravitating multi particle system, viewed as a kinetic gas. The properties of the, colliding or non-colliding, particles are encoded into a so called one-particle distribution function, which is a density on the space of allowed particle positions and velocities, i.e. on the tangent bundle of the spacetime manifold. We argue that an appropriate theory of gravity, describing the gravitational field generated by a kinetic gas, must also be modeled on the tangent bundle. The most natural mathematical framework for this task is Finsler spacetime geometry. Following this line of argumentation, we construct a coupling between the kinetic gas and a recently proposed Finsler geometric extension of general relativity. Additionally, we explicitly show how the coordinate invariance of the action of the kinetic gas leads to a novel formulation of conservation of the energy-momentum distribution of the gas on the tangent bundle., Comment: 13 pages, updated to journal version

In the attempts to apply Finsler geometry to construct an extension of general relativity, the question about a suitable generalization of the Einstein equations is still under debate. Since Finsler geometry is based on a scalar function on the tangent bundle, the field equation which determines this function should also be a scalar equation. In the literature two such equations have been suggested: the one by Rutz and the one by one of the authors. Here we employ the method of canonical variational completion to show that Rutz equation can not be obtained from a variation of an action and that its variational completion yields the latter field equations. Moreover, to improve the mathematical rigor in the derivation of the Finsler gravity field equation, we formulate the Finsler gravity action on the positive projective tangent bundle. This has the advantage of allowing us to apply the classical variational principle, by choosing the domains of integration to be compact and independent of the dynamical variable. In particular in the pseudo-Riemannian case, the vacuum field equation becomes equivalent to the vanishing of the Ricci tensor., Comment: 22 pages + 11 pages auxiliary material, no figures, LaTeX; updated version accepted by Phys. Rev. D

The paper aims to initiate a systematic study of conformal mappings between Finsler spacetimes and, more generally, between pseudo-Finsler spaces. This is done by extending several results in pseudo-Riemannian geometry which are necessary for field-theoretical applications and by proposing a technique which reduces a series of problems involving pseudo-Finslerian conformal vector fields to their pseudo-Riemannian counterparts. Also, we point out, by constructing classes of examples, that conformal groups of flat (locally Minkowskian) pseudo-Finsler spaces can be much richer than both flat Finslerian and pseudo-Euclidean conformal groups.

The paper presents a general geometric approach to energy-momentum tensors in Lagrangian field theories, based on a Hilbert-type definition. The approach is consistent with the ones defining energy-momentum tensors in terms of hypermomentum maps given by the diffeomorphism invariance of the Lagrangian - and, in a sense, complementary to these, with the advantage of an increased simplicity of proofs and also, opening up new insights into the topic. A special attention is paid to the particular cases of metric and metric-affine theories.

The paper proposes extensions of the usual notions of Finslerian volume to time orientable Finsler spacetime manifolds. The basic idea is to replace, in the classical Busemann-Hausdorff and Holmes-Thompson definitions, integration on the indicatrices of the given metric (which are, in Lorentzian signature, non-compact, generally leading to infinite integrals) with integration on ellipsoids. Under certain conditions regarding the time orientations of the given spacetime, these ellipsoids can be attached to the given Lorentzian Finsler metric by means of a variational procedure. While the construction of the Holmes-Thompson volume form requires the determinant of the Finslerian metric tensor to be defined and smooth on the entire slit tangent bundle, the Busemann-Hausdorff-type volume form can be constructed even if the metric tensor is not defined or is degenerate along some directions - which is the case with the large majority of the known Lorentzian Finsler metrics. This feature makes it possible to build well-defined field-theoretical integrals having such metrics as a background., Comment: This is a corrected and completed version of the paper arXiv:1412.3838 [math.DG]

The paper proposes a notion of volume element for Finsler spaces with metrics of Lorentzian signature, equipped with a time orientation. This notion is based on a slight modification of the idea of Holmes-Thompson volume element working for positive definite Finsler metrics and can be used in field-theoretical applications on a Finslerian background. Though the approach assumes the existence of a time orientation, the volume element does not depend on its choice., Comment: This paper has been withdrawn by the author due to an error in the statement according to which the volume element does not depend on the choice of the time orientation

Given a non-variational system of differential equations, the simplest way of turning it into a variational one is by adding a correction term. In the paper, we propose a way of obtaining such a correction term, based on the so-called Vainberg-Tonti Lagrangian, and present several applications in general relativity and classical mechanics.

Biharmonic curves are a generalization of geodesics, with applications in elasticity theory and various branches of computer science. The paper proposes a first study of biharmonic curves in spaces with Finslerian geometry, covering the following topics: a deduction of their equations, existence of non-geodesic biharmonic curves for some classes of Finsler spaces and specific properties. We prove, for instance, that geodesic curvature is constant along Finslerian biharmonic curves. Integration of the biharmonic equation is presented for two concrete Finsler metrics.

The notions of bienergy of a smooth mapping and of biharmonic map between Riemannian manifolds are extended to the case when the domain is Finslerian. We determine the first and the second variation of the bienergy functional, the equations of Finsler-to-Riemann biharmonic maps and some specific examples. Two notable results in Riemannian geometry concerning the inexistence of nonharmonic biharmonic maps are extended without difficulty in our case.

In a previous paper, we have introduced a new unified description of the main equations of the gravitational and of the electromagnetic field, in terms of tidal tensors and connections on the tangent bundle TM of the space-time manifold. In the present work, we relate these equations to variational procedures on the tangent bundle. The Ricci scalar of the proposed connection is dynamically equivalent to the usual Einstein-Maxwell Lagrangian. Also, in order to be able to perform these variational procedures, we find an appropriate completion of the metric tensor (from the base manifold) up to a metric structure on TM.

In 2008-2009, F. Costa and C. Herdeiro proposed a new gravito-electromagnetic analogy, based on tidal tensors. We show that connections on the tangent bundle of the space-time manifold can help not only in finding a covnenient geometrization of their ideas, but also a common mathematical description of the main equations of gravity and electromagnetism., Comment: submitted to: Journal of Nonlinear Mathematical Physics

By using variational calculus and exterior derivative formalism, we proposed in two previous joint papers with S. Siparov a new geometric approach for electromagnetism in pseudo-Finsler spaces. In the present paper, we provide more details, especially regarding generalized currents, the domain of integration and gauge invariance. Also, for flat pseudo-Finsler spaces, we define a generalized energy-momentum tensor (consisting of two blocks), as the symmetrized Noether current corresponding to the invariance of the field Lagrangian with respect to spacetime translations. In curved spaces, one of the blocks of the generalized energy-momentum tensor is obtained by varying the field Lagrangian with respect to the metric tensor and the other one, by varying the same Lagrangian with respect to the nonlinear connection., Comment: 29 pages

We find the generalization of Einstein equations to Finsler spaces by variational means and, based on the invariance of the Finslerian Hilbert action to infinitesimal transformations, we find the analogue of the energy- momentum conservation law in these spaces., Comment: 12 pages