We develop the theory of weighted Ricci curvature in a weighted Lorentz-Finsler framework and extend the classical singularity theorems of general relativity. In order to reach this result, we generalize the Jacobi, Riccati and Raychaudhuri equations to weighted Finsler spacetimes and study their implications for the existence of conjugate points along causal geodesics. We also show a weighted Lorentz-Finsler version of the Bonnet-Myers theorem based on a generalized Bishop inequality.
We translate Penrose's singularity theorem to a Finsler spacetime. To that end, causal concepts in Lorentzian geometry are extended, including definitions and properties of focal points and trapped surfaces, with careful attention paid to the differences that arise in the Finslerian setting.
In a $(1+n)$-dimensional Lorentz--Finsler manifold with $N$-Bakry--Émery Ricci curvature bounded below for $N\in(n,\infty]$, using the Riccati equation techniques, we established the Bishop--Gromov volume comparison for the so-called standard sets for comparisons in Lorentzian volumes (SCLVs).We also established the Günther volume comparison for SCLVs when the flag curvature was bounded above.
In the current paper the Lagrangian of a classical, relativistic point particle is obtained whose conjugate momentum satisfies the dispersion relation of a quantum wave packet that is subject to Lorentz violation based on a particular coefficient of the nonminimal Standard-Model Extension (SME). The properties of this Lagrangian are analyzed and two corresponding Finsler structures are obtained. One structure describes a scaled Euclidean geometry, whereas the other is neither a Riemann nor a Randers or Kropina structure. The results of the article provide some initial understanding of classica…
The Hawking and Penrose singularity theorems for low regularity Finsler spacetimes
Dipartimento di Matematica, Università degli Studi di Pisa（比萨大学数学系）；Department of Mathematics, University of Toronto（多伦多大学数学系）；Department of Mathematics, University of Osaka（大阪大学数学系）；RIKEN Center for Advanced Intelligence Project (AIP)（理化学研究所先进智能研究中心）；Department of Mathematics, University of Vienna（维也纳大学数学系）
Bipartite Riemann-Finsler geometries with complementary Finsler structures are constructed. Calculable examples are presented based on a bilinear-form coefficient for explicit Lorentz violation.
Effective field theories with explicit Lorentz violation are intimately linked to Riemann-Finsler geometry. The quadratic single-fermion restriction of the Standard-Model Extension provides a rich source of pseudo-Riemann-Finsler spacetimes and Riemann-Finsler spaces. An example is presented that is constructed from a 1-form coefficient and has Finsler structure complementary to the Randers structure.
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, Aczél'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 Aczél's inequality.
The physics of classical particles in a Lorentz-breaking spacetime has numerous features resembling the properties of Finsler geometry. In particular, the Lagrange function plays a role similar to that of a Finsler structure function. A summary is presented of recent results, including new calculable Finsler structures based on Lagrange functions appearing in the Lorentz-violation framework known as the Standard-Model Extension.
A consistent theory of quantum gravity (QG) at Planck scale almost sure contains manifestations of Lorentz local symmetry violations (LV) which may be detected at observable scales. This can be effectively described and classified by models with nonlinear dispersions and related Finsler metrics and fundamental geometric objects (nonlinear and linear connections) depending on velocity/ momentum variables. We prove that the trapping brane mechanism provides an accurate description of gravitational and matter field phenomena with LV over a wide range of distance scales and recovering in a systema…
The optimal transport problem is studied in the context of Lorentz-Finsler geometry. For globally hyperbolic Lorentz-Finsler spacetimes the first Kantorovich problem and the Monge problem are solved. Further the intermediate regularity of the transport paths is studied. These results generalize parts of Bertrand & Puel and Brenier et al.
The correspondence between Riemann-Finsler geometries and effective field theories with spin-independent Lorentz violation is explored. We obtain the general quadratic action for effective scalar field theories in any spacetime dimension with Lorentz-violating operators of arbitrary mass dimension. Classical relativistic point-particle lagrangians are derived that reproduce the momentum-velocity and dispersion relations of quantum wave packets. The correspondence to Finsler structures is established, and some properties of the resulting Riemann-Finsler spaces are investigated. The results prov…
Using a navigation process with the datum $(F,V)$, in which $F$ is a Finsler metric and the smooth tangent vector field $V$ satisfies $F(-V(x))>1$ everywhere, a Lorentz Finsler metric $\tilde{F}$ can be induced. Isoparametric functions and isoparametric hypersurfaces with or without involving a smooth measure can be defined for $\tilde{F}$. When the vector field $V$ in the navigation datum is homothetic, we prove the local correspondences between isoparametric functions and isoparametric hypersurfaces before and after this navigation process. Using these correspondences, we provide some exampl…
We show an analogue of the Lorentzian splitting theorem for weighted Lorentz-Finsler manifolds: If a weighted Berwald spacetime of nonnegative weighted Ricci curvature satisfies certain completeness and metrizability conditions and includes a timelike straight line, then it necessarily admits a one-dimensional family of isometric translations generated by the gradient vector field of a Busemann function. Moreover, our formulation in terms of the $ε$-range introduced in our previous work enables us to unify the previously known splitting theorems for weighted Lorentzian manifolds by Case and Wo…
We obtain some results in both Lorentz and Finsler geometries, by using a correspondence between the conformal structure (Causality) of standard stationary spacetimes on $M=\R\times S$ and Randers metrics on $S$. In particular, for stationary spacetimes, we give a simple characterization of when they are causally continuous or globally hyperbolic (including in the latter case, when $S$ is a Cauchy hypersurface), in terms of an associated Randers metric. Consequences for the computability of Cauchy developments are also derived. Causality suggests that the role of completeness in many results o…
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…
We develop area and volume comparison theorems for the evolution of spacelike, acausal, causally complete hypersurfaces in Lorentzian manifolds, where one has a lower bound on the Ricci tensor along timelike curves, and an upper bound on the mean curvature of the hypersurface. Using these results, we give a new proof of Hawking's singularity theorem.
We consider the Hawking-Penrose singularity theorems and the Lorentzian splitting theorem under the weaker curvature condition of nonnegative Bakry-Emery-Ricci curvature $Ric_f^m$ in timelike directions. We prove that they still hold when $m$ is finite, and when $m$ is infinite, they hold under the additional assumption that $f$ is bounded from above.
Finsler space is differentiable manifold for which Minkowski space is the fiber of the tangent bundle. To understand structure of the reference frame in Finsler space, we need to understand the structure of orthonormal basis in Minkowski space. In this paper, I considered the definition of orthonormal basis in Minkowski space, the structure of metric tensor relative to orthonormal basis, procedure of orthogonalization. Linear transformation of Minkowski space which preserves the scalar product is called a motion. Linear transformation which maps the orthonormal basis into an infinitely close o…
In some recent papers, the relations existing between the metric properties of Randers spaces and the conformal geometry of stationary Lorentzian manifolds were discovered and investigated. In this note, we focus on the equality between the index of a geodesic in a Randers space and that of its lightlike lift in the associated conformal stationary spacetime. Moreover we make some remarks about regularity of the energy functional of a Finsler metric on the infinite dimensional manifold of $H^1$ curves between two points, in connection with infinite dimensional techniques in Morse Theory.
In this paper we study the flag curvature of a particular class of Finsler metrics called general $(α,β)$-metrics, which are defined by a Riemannian metric $α$ and a $1$-form $β$. The classification of such metrics with constant flag curvature are completely determined under some suitable conditions, which make them be locally projectively flat. As a result, we construct some new projectively flat Finsler metrics with flag curvature $1$, $0$ and $-1$, all of which are of singularity at some directions.
We show that the existence of noncontractible periodic orbits for compactly supported time-dependent Hamiltonian on the disk cotangent bundle of a Finsler manifold provided that the Hamiltonian is sufficiently large over the zero section. We generalize the BPS capacities and earlier constructions of Weber (2006 Duke Math. J. 133, 527-568) and other authors Biran et al (2003 Duke Math. J. 119, 65-118) to the Finsler setting. We then obtain a number of applications including: (1) generalizing the main theorem of Xue (2017 J. Symplectic Geom. 15, 905-936) to the Lie group setting, (2) preservatio…
Continuing recent efforts in extending the classical singularity theorems of General Relativity to low regularity metrics, we give a complete proof of both the Hawking and the Penrose singularity theorem for $C^1$-Lorentzian metrics - a regularity where one still has existence but not uniqueness for solutions of the geodesic equation. The proofs make use of careful estimates of the curvature of approximating smooth metrics and certain stability properties of long existence times for causal geodesics. On the way we also prove that for globally hyperbolic spacetimes with a $C^1$-metric causal ge…
Starting from the Hawking--Page solutions, we consider the corresponding Lorentzian cone metrics. These represent cone interior scale-invariant vacuum solutions, defined in the chronological past of the scaling origin. We extend the Lorentzian Hawking--Page solutions to the cone exterior region in the class of $(4+1)$-dimensional scale-invariant vacuum solutions with an $SO(3)\times U(1)$ isometry, using the Kaluza--Klein reduction and the methods of Christodoulou. We prove that each Lorentzian Hawking--Page solution has extensions with a null curvature singularity, extensions with a spacelike…
We analyze Lorentzian spacetimes subject to curvature-dimension bounds using the Bakry-Émery-Ricci tensor. We extend the Hawking-Penrose type singularity theorem and the Lorentzian timelike splitting theorem to synthetic dimensions $N\le 1$, including all negative synthetic dimensions. The rigidity of the timelike splitting reduces to a warped product splitting when $N=1$. We also extend the null splitting theorem of Lorentzian geometry, showing that it holds under a null curvature-dimension bound on the Bakry-Émery-Ricci tensor for all $N\in (-\infty, 2]\cup (n,\infty)$ and for the $N=\infty$…
In the early 80's S.-T. Yau posed the problem of establishing the rigidity of the Hawking-Penrose singularity theorems. Approaches to this problem have involved the introduction of Lorentzian Busemann functions and the study of the geometry of their level sets - the horospheres. The regularity theory in the Lorentzian case is considerably more complicated and less complete than in the Riemannian case. In this paper we introduce a broad generalization of the notion of horosphere in Lorentzian geometry and take a completely different (and highly geometric) approach to regularity. These generaliz…
The maximal analytic Schwarzschild spacetime is manifestly inextendible as a Lorentzian manifold with a twice continuously differentiable metric. In this paper, we prove the stronger statement that it is even inextendible as a Lorentzian manifold with a continuous metric. To capture the obstruction to continuous extensions through the curvature singularity, we introduce the notion of the spacelike diameter of a globally hyperbolic region of a Lorentzian manifold with a merely continuous metric and give a sufficient condition for the spacelike diameter to be finite. The investigation of low-reg…
