Critical remarks on Finsler modifications of gravity and cosmology by Zhe Chang and Xin Li

I do not agree with the authors of papers arXiv:0806.2184 and arXiv:0901.1023v1 (published in [Zhe Chang, Xin Li, Phys. Lett. B 668 (2008) 453] and [Zhe Chang, Xin Li, Phys. Lett. B 676 (2009) 173], respectively). They consider that “In Finsler manifold, there exists a unique linear connection – the Chern connection … It is torsion freeness and metric compatibility …”. There are well-known results (for example, presented in monographs by H. Rund and R. Miron and M. Anastasiei) that in Finsler geometry there exist an infinite number of linear connections defined by the same metric structure and that the Chern and Berwald connections are not metric compatible. For instance, the Chern's one (being with zero torsion and “weak” compatibility on the base manifold of tangent bundle) is not generally compatible with the metric structure on total space. This results in a number of additional difficulties and sophistication in definition of Finsler spinors and Dirac operators and in additional problems with further generalizations for quantum gravity and noncommutative/string/brane/gauge theories. I conclude that standard physics theories can be generalized naturally by gravitational and matter field equations for the Cartan and/or any other Finsler metric compatible connections. This allows us to construct more realistic models of Finsler spacetimes, anisotropic field interactions and cosmology.     

Exact Scalar Field Inflationary Solution in Rainbow Universe

Using the Friedmann equation in rainbow Universe, we obtain an exact scalar field Inflationary Solution, which is a modification of the exact scalar field with negative potential −V ₀+m ² φ ²/2. Because the rainbow metric is Finsler metric, the result in this paper implies that the research of Finsler geometry in Cosmology should lead to several new physics theories.     

Black Holes, Ellipsoids, and Nonlinear Waves in Pseudo-Finsler Spaces and Einstein Gravity

We model pseudo-Finsler geometries, with pseudo-Euclidean signatures of metrics, for two classes of four dimensional nonholonomic manifolds: (a) tangent bundles with two dimensional base manifolds and (b) pseudo-Riemannian/Einstein spaces. Such spacetimes are enabled with nonholonomic distributions and theirs metrics are solutions of the field equations in general relativity and/or generalizations. We rewrite the Schwarzschild metric in Finsler variables and use it for generating new classes of black hole objects with stationary deformations to ellipsoidal configurations. The conditions are analyzed when such metrics describe embedding of black hole solutions into nontrivial solitonic backgrounds.     

Beyond the speed of light on Finsler spacetimes

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 spacetimes is shown to be the same as on metric spacetimes, which differs from many quantum gravity models. If similar relations resulted for fermions on Finsler spacetimes, these generalized geometries could explain the potential observation of superluminal neutrinos claimed by the Opera Collaboration.     

Einstein gravity in almost Kähler and Lagrange–Finsler variables and deformation quantization

A geometric procedure is elaborated for transforming (pseudo) Riemannian metrics and connections into canonical geometric objects (metric and nonlinear and linear connections) for effective Lagrange, or Finsler, geometries which, in turn, can be equivalently represented as almost Kähler spaces. This allows us to formulate an approach to quantum gravity following standard methods of deformation quantization. Such constructions are performed not on tangent bundles, as in usual Finsler geometry, but on spacetimes enabled with nonholonomic distributions defining 2+2$2+2$ splitting with associate nonlinear connection structure. We also show how the Einstein equations can be written in terms of Lagrange–Finsler variables and corresponding almost symplectic structures and encoded into the zero-degree cohomology coefficient for a quantum model of Einstein manifolds.     

Riemann-Finsler geometry and Lorentz-violating kinematics

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.     

Fermat principle in Finsler spacetimes

It is shown that, on a manifold with a Finsler metric of Lorentzian signature, the lightlike geodesics satisfy the following variational principle. Among all lightlike curves from a point q (emission event) to a timelike curve γ (worldline of receiver), the lightlike geodesics make the arrival time stationary. Here “arrival time” refers to a parametrization of the timelike curve γ. This variational principle can be applied (i) to the vacuum light rays in an alternative spacetime theory, based on Finsler geometry, and (ii) to light rays in an anisotropic non-dispersive medium with a general-relativistic spacetime as background.     

Injectivity Radius of Lorentzian Manifolds

Motivated by the application to general relativity we study the geometry and regularity of Lorentzian manifolds under natural curvature and volume bounds, and we establish several injectivity radius estimates at a point or on the past null cone of a point. Our estimates are entirely local and geometric, and are formulated via a reference Riemannian metric that we canonically associate with a given observer (p, T) –where p is a point of the manifold and T is a future-oriented time-like unit vector prescribed at p only. The proofs are based on a generalization of arguments from Riemannian geometry. We first establish estimates on the reference Riemannian metric, and then express them in terms of the Lorentzian metric. In the context of general relativity, our estimate on the injectivity radius of an observer should be useful to investigate the regularity of spacetimes satisfying Einstein field equations.     

En route to Background Independence: Broken split-symmetry,and how to restore it with bi-metric average actions

The most momentous requirement a quantum theory of gravity must satisfy is Background Independence, necessitating in particular an ab initio derivation of the arena all non-gravitational physics takes place in, namely spacetime. Using the background field technique, this requirement translates into the condition of an unbroken split-symmetry connecting the (quantized) metric fluctuations to the (classical) background metric. If the regularization scheme used violates split-symmetry during the quantization process it is mandatory to restore it in the end at the level of observable physics. In this paper we present a detailed investigation of split-symmetry breaking and restoration within the Effective Average Action (EAA) approach to Quantum Einstein Gravity (QEG) with a special emphasis on the Asymptotic Safety conjecture. In particular we demonstrate for the first time in a non-trivial setting that the two key requirements of Background Independence and Asymptotic Safety can be satisfied simultaneously. Carefully disentangling fluctuation and background fields, we employ a ‘bi-metric’ ansatz for the EAA and project the flow generated by its functional renormalization group equation on a truncated theory space spanned by two separate Einstein–Hilbert actions for the dynamical and the background metric, respectively. A new powerful method is used to derive the corresponding renormalization group (RG) equations for the Newton- and cosmological constant, both in the dynamical and the background sector. We classify and analyze their solutions in detail, determine their fixed point structure, and identify an attractor mechanism which turns out instrumental in the split-symmetry restoration. We show that there exists a subset of RG trajectories which are both asymptotically safe and split-symmetry restoring: In the ultraviolet they emanate from a non-Gaussian fixed point, and in the infrared they loose all symmetry violating contributions inflicted on them by the non-invariant functional RG equation. As an application, we compute the scale dependent spectral dimension which governs the fractal properties of the effective QEG spacetimes at the bi-metric level. Earlier tests of the Asymptotic Safety conjecture almost exclusively employed ‘single-metric truncations’ which are blind towards the difference between quantum and background fields. We explore in detail under which conditions they can be reliable, and we discuss how the single-metric based picture of Asymptotic Safety needs to be revised in the light of the new results. We shall conclude that the next generation of truncations for quantitatively precise predictions (of critical exponents, for instance) is bound to be of the bi-metric type.     

Post-Newtonian limit of Finsler space theories of gravity and solar system tests

Finsler geometry is considered as a wider framework for analysing solar system tests of theories of gravity than is afforded by Riemannian geometry. The post-Newtonian limit for the spherically symmetric one-body problem is examined by expanding the Finsler metric about the Minkowski space of Special Relativity for those Finsler spaces whose null surface is Riemannian. In such a framework there are five PPN parameters instead of the three in Riemannian geometry. The classical solar system tests can readily be satisfied leaving two arbitrary parameters. These parameters could be determined from measurements of the second order gravitational red-shift and periodic perturbations in particle orbits, thus providing a consistency check on the Riemannian metric hypothesis of General Relativity. Such an experiment is possible on a satellite on an orbit with perihelion of a few solar radii.     

The generalized Taub-NUT congruence in Minkowski spaces

With any shear-free congruence of null geodesics in a Lorentzian geometry there is associated a Cauchy-Riemann three-space; and in certain spacetimes including the Ricci-flat spacetimes with expanding null shear-free (n.s.f.) congruences the deviation form of the congruence picks out an integrable distribution of complex two-spaces in the CR geometry. Conversely, given a CR geometry with an integrable distribution of two-spaces one can construct an associated family of spacetimes with a null, shear-free congruence. The interesting problem is the restrictionR _ab =0. We consider the case of n.s.f. congruences in Minkowski spacetime constructed from CR geometries of maximal symmetry. The special two-spaces are here taken to be those associated with either the Taub-NUT geometry or, as a limiting case, those associated with the Hauser twisting typeN solution. We obtain the most general solution for these cases.     

A Spherical Symmetrical Spacetime Solution in Finsler Gravity

Several physical principles of Finsler gravity are proposed in this paper, and I apply the principles to construct a Finsler gravity action, which satisfy the condition that the action can be reduced to the General Relativity action once the metric is independent from the tangent vector. I also get a spacetime solution in Finsler spacetime with the tangent vector y _φ , moreover the solution indicates that the metric relies on the property of test particle in Finsler spacetime.     

Existence and stability of circular orbits in general static and spherically symmetric spacetimes

The existence and stability of circular orbits (CO) in static and spherically symmetric (SSS) spacetime are important because of their practical and potential usefulness. In this paper, using the fixed point method, we first prove a necessary and sufficient condition on the metric function for the existence of timelike COs in SSS spacetimes. After analyzing the asymptotic behavior of the metric, we then show that asymptotic flat SSS spacetime that corresponds to a negative Newtonian potential at large r will always allow the existence of CO. The stability of the CO in a general SSS spacetime is then studied using the Lyapunov exponent method. Two sufficient conditions on the (in)stability of the COs are obtained. For null geodesics, a sufficient condition on the metric function for the (in)stability of null CO is also obtained. We then illustrate one powerful application of these results by showing that three SSS spacetimes whose metric function is not completely known will allow the existence of timelike and/or null COs. We also used our results to assert the existence and (in)stabilities of a number of known SSS metrics.     

On 1-form finsler spaces

By definition, the metric of N-dimensional 1-form Finsler space is constructed proceeding from, in general arbitrary, Minkowskian metric by means of an N-dimensional reference frame, i.e., in other words, by means of N independent linear 1-forms. In the present paper, the attention is drawn to the fact that a preferred osculating Riemannian space with simple but interesting properties exists for a 1-form Finsler space. Then, some significant simplifications are emphasized which are imposed by the conditions C_i=0 in a 1-form Finsler space. The T- condition and S3-likeness give rise to additional simplifications. Since the 1-form Berwald- Moór's metric exhibits all three properties mentioned above, a fairly simple expression for the curvature scalar H = H^ij_ij of this metric may be calculated. In conclusion, the attention is drawn to the fact that the assumption of a 1-form structure of the space-time is certainly expedient in obtaining a deeper insight into the Finslerian relativity.     

Confronting Finsler space–time with experiment

Within all approaches to quantum gravity small violations of the Einstein Equivalence Principle are expected. This includes violations of Lorentz invariance. While usually violations of Lorentz invariance are introduced through the coupling to additional tensor fields, here a Finslerian approach is employed where violations of Lorentz invariance are incorporated as an integral part of the space–time metrics. Within such a Finslerian framework a modified dispersion relation is derived which is confronted with current high precision experiments. As a result, Finsler type deviations from the Minkowskian metric are excluded with an accuracy of 10⁻¹⁶.     

Congruences of Fluids in a Finslerian Anisotropic Space-Time

We derive the generalized Raychauduri equation concepts of expansion, shear and vorticity. We give the Ricci tensor of a constant-curvature Randers–Finsler space metric whose first term is the Robertson–Walker metric.Dedicated to the memory of Professor Nikolaos Danikas.     

Nonlinear Connections and Nearly Autoparallel Maps in General Relativity

We apply the method of moving anholonomic frames with associated nonlinear connections to the (pseudo) Riemannian space geometry and examine the conditions when locally anisotropic structures (Finsler like and more general ones) could be modeled in the general relativity theory and/or Einstein–Cartan–Weyl extensions [1]. New classes of solutions of the Einstein equations with generic local anisotropy are constructed. We formulate the theory of nearly autoparallel (na) maps generalizing the conformal transforms and formulate the Einstein gravity theory on na–backgrounds provided with a set of na–map invariant conditions and local conservation laws. There are illustrated some examples when vacuum Einstein fields are generated by Finsler like metrics and chains of na–maps.     

Super-Luminal Effects for Finsler Branes as a Way to Preserve the Paradigm of Relativity Theories

Using Finsler brane solutions [see details and methods in: S. Vacaru, Class. Quant. Grav. 28:215001, 2011], we show that neutrinos may surpass the speed of light in vacuum which can be explained by trapping effects from gravity theories on eight dimensional (co) tangent bundles on Lorentzian manifolds to spacetimes in general and special relativity. In nonholonomic variables, the bulk gravity is described by Finsler modifications depending on velocity/momentum coordinates. Possible super-luminal phenomena are determined by the width of locally anisotropic brane (spacetime) and induced by generating functions and integration functions and constants in coefficients of metrics and nonlinear connections. We conclude that Finsler brane gravity trapping mechanism may explain neutrino super-luminal effects and almost preserve the paradigm of Einstein relativity as the standard one for particle physics and gravity.     

Symmetries of cosmological Cauchy horizons

We consider analytic vacuum and electrovacuum spacetimes which contain a compact null hypersurface ruled byclosed null generators. We prove that each such spacetime has a non-trivial Killing symmetry. We distinguish two classes of null surfaces, degenerate and non-degenerate ones, characterized by the zero or non-zero value of a constant analogous to the “surface gravity” of stationary black holes. We show that the non-degenerate null surfaces are always Cauchy horizons across which the Killing fields change from spacelike (in the globally hyperbolic regions) to timelike (in the acausal, analytic extensions). For the special case of a null surface diffeomorphic toT ³ we characterize the degenerate vacuum solutions completely. These consist of an infinite dimensional family of “plane wave” spacetimes which are entirely foliated by compact null surfaces. Previous work by one of us has shown that, when one dimensional Killing symmetries are allowed, then infinite dimensional families of non-degenerate, vacuum solutions exist. We recall these results for the case of Cauchy horizons diffeomorphic toT ³ and prove the generality of the previously constructed non-degenerate solutions. We briefly discuss the possibility of removing the assumptions of closed generators and analyticity and proving an appropriate generalization of our main results. Such a generalization would provide strong support for the cosmic censorship conjecture by showing that causality violating, cosmological solutions of Einstein's equations are essentially an artefact of symmetry.     

Spacetime symplectic extension

It is conjectured that in the origin of spacetime there lies a symplectic rather than metric structure. The complex symplectic symmetry Sp(2l, C), l≥1 instead of the pseudoorthogonal one SO(1, d?1), d≥4 is proposed as the spacetime local structure group. A discrete sequence of the metric spacetimes of the fixed dimensionalities d=(2l)² and signatures, with l(2l?1) timelike and l(2l+1) spacelike directions, defined over the set of Hermitian second-rank spin tensors, is considered as an alternative to the pseudo-Euclidean extra dimensional spacetimes. The basic concepts of the symplectic framework are developed in general, and the ordinary and next-to-ordinary spacetime cases with l=1, 2, respectively, are elaborated in more detail. In particular, the scheme provides the rationale for the four-dimensionality and 1+3 signature of the ordinary spacetime.