Covariant kinematics and gravitational bounce in Finsler space-times

Abstract

The similarity between Finsler and Riemann geometry is an intriguing starting point to extend general relativity. The lack of quadratic restriction over the line element (color) naturally generalize the Riemannian case and breaks the local symmetries of general relativity. In addition, the Finsler manifold is enriched with new geometric entities and all the classical identities are suitably extended. We investigate the covariant kinematics of a medium formed by a time-like congruence. After a brief view in the general case we impose particular geometric restrictions to get some analytic insight. Central role to our analysis plays the Lie derivative where even in case of irrotational Killing vectors the bundle still deforms. We demonstrate an example of an isotropic and exponentially expanding cross-section that finally deflates or forms a caustic. Furthermore, using the 1+3 covariant formalism we investigate the expansion dynamics of the congruence. For certain geometric restrictions we retrieve the Raychaudhuri equation where a color-curvature coupling is revealed. The condition to prevent the focusing of neighboring particles is given and is more likely to fulfilled in highly curved regions. Then, we introduce the Levi-Civita connection for the osculating Riemannian metric and develop a (spatially) isotropic and homogeneous dust-like model with a non-singular bounce.

Cosmic magnetization in curved and Lorentz violating space–times

“…of relation (23), since Cartan's tensor (2) is involved. On the other hand, using conservation arguments similar to GR we can relate the curvature tensor R abcd with the energy-momentum tensor (see for instance [104,105,110]) by the following formula:…”

“…For our case, where the supporting direction is parallel transported and the electromagnetic field is treated as a first-order quantity, the Raychaudhuri equation [110] for shear and vorticity free geodesics can be writteṅ…”

“…The dependence on the fiber coordinates y a directly reflects the Lorentz violating structure of the Finslerian space-time. They may be physically interpreted as an arbitrary direction at each tangent space induced by the breaking of Lorentz invariance (see for example [110,112]). In fact, Finsler geometry is encountered in Lorentz violating branches of quantum gravity [86][87][88][89][90][91][92][93][94][95][96][97][98][99][100][101] and also effectively describes motion in anisotropic media [113,114].…”

“…where X = X a u a is the time-like part and X a is the projection on the instantaneous rest space (for further details see [107,110]). Using the time derivative (4) and the spatial derivative (5) together with the Finslerian curvature theory we can investigate the evolution of physical fields in a full covariant way.…”

“…Central role to our analysis plays the 1+3 covariant formalism [107,108] that we can extend in Finsler geometry [110]. Firstly, we develop the 4-vector potential representation of electromagnetism in curved space-times and we discuss the FRW inflationary scenario.…”