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.     

The local geometry of compact homogeneous Lorentz spaces

In 1995, S. Adams and G. Stuck as well as A. Zeghib independently provided a classification of non-compact Lie groups which can act isometrically and locally effectively on compact Lorentzian manifolds. In the case that the corresponding Lie algebra contains a direct summand isomorphic to the two-dimensional special linear algebra or to a twisted Heisenberg algebra, Zeghib also described the geometric structure of the manifolds. Using these results, we investigate the local geometry of compact homogeneous Lorentz spaces whose isometry groups have non-compact connected components. It turns out that they all are reductive. We investigate the isotropy representation and curvatures. In particular, we obtain that any Ricci-flat compact homogeneous Lorentz space is flat or has compact isometry group.     

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.     

Generalized contact geometry and T-duality

We study generalized almost contact structures on odd-dimensional manifolds. We introduce a notion of integrability and show that the class of these structures is closed under symmetries of the Courant–Dorfman bracket, including T-duality. We define a notion of geometric type for generalized almost contact structures, and study its behavior under T-duality.     

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.     

Traffic noise and the hyperbolic plane

We consider the problem of sound propagation in a wind. We note that the rays, as in the absence of a wind, are given by Fermat’s principle and show how to map them to the trajectories of a charged particle moving in a magnetic field on a curved space. For the specific case of sound propagating in a stratified atmosphere with a small wind speed, we show that the corresponding particle moves in a constant magnetic field on the hyperbolic plane. In this way, we give a simple ‘straightedge and compass’ method to estimate the intensity of sound upwind and downwind. We construct Mach envelopes for moving sources. Finally, we relate the problem to that of finding null geodesics in a squashed anti-de Sitter spacetime and discuss the SO(3,1)×R symmetry of the problem from this point of view.     

Comments on the unification of electromagnetism and gravitation through “Generalized Einstein manifolds”

Akbar-Zadeh [J. Geom. Phys. 17 (1995) 342] has recently proposed a new geometric formulation of Einstein-Maxwell system with source in terms of what are called “Generalized Einstein manifolds”. We show that, contrary to the claim, Maxwell equations have not been derived in this formulation, and that the assumed equations can be identified only as source-free Maxwell equations in the proposed geometric setup. A genuine derivation of source-free Maxwell equations is presented within the same framework. We draw a conclusion that the proposed unification scheme can pertain only to source-free situations.     

On the integration of Poisson homogeneous spaces

We study a reduction procedure for describing the symplectic groupoid of a Poisson homogeneous space obtained by quotient of a coisotropic subgroup. We perform it as a reduction of the Lu–Weinstein symplectic groupoid integrating Poisson Lie groups, that is suitable even for the non-complete case.     

A possible scenario of the Pioneer anomaly in the framework of Finsler geometry

The weak field approximation of geodesics in Randers–Finsler space is investigated. We show that a Finsler structure of Randers space corresponds to the constant and sunward anomalous acceleration demonstrated by the Pioneer 10 and 11 data. The additional term in the geodesic equation acts as “electric force”, which provides the anomalous acceleration.     

Connection Considerations of Gravitational Field in Finsler Spaces

Some alternative connection structures of the Finslerian gravitational field are considered by modifying the independent variables (x,y) (x: point and y: vector) in various ways. For example, (x ^k ,y ⁱ ) (k,i = 1,2,3,4) are changed to (x ^k ,y ⁰) (y ⁰: scalar) or (x ⁰,y ⁱ ) (x ⁰: time axis); (x ^k ,y ⁱ ) are generalized to (x ^k ,y ⁱ ,p _i ) (p _i : covector dual to y ⁱ ) or (x ^k ,y ⁱ ,q _a ) (q _a : covector different from p _i ); (x ^k ,y ⁱ ) are further generalized to (x ^k ,y ^(a)i ) (a = 1,2,…,m), (y ^(a): (a)th vector), etc.