Weak gravitational field in Finsler–Randers space and Raychaudhuri equation

Abstract

The linearized form of the metric of a Finsler -Randers space is studied in relation to the equations of motion, the deviation of geodesics and the generalized Raychaudhuri equation are given for a weak gravitational field. This equation is also derived in the framework of a tangent bundle.By using Cartan or Berwald-like connections we get some types "gravito -electromagnetic" curvature. In addition we investigate the conditions under which a definite Lagrangian in a Randers space leads to Einstein field equations under the presence of electromagnetic field.Finally, some applications of the weak field in a generalized Finsler spacetime for gravitational waves are given. * pstavrin@math.uoa.gr

Relativistic Finsler geometry

“…†† ††For a discussion of the deformable kinematics of the tangent bundle using the horizontal and vertical split of , see . Also, see for a 1 + 3 covariant treatment of Finsler flows in an arbitrary direction with respect to the supporting element.…”

“…where in the Riemannian limit, R abcd depends solely on the position coordinates. From a relativistic standpoint, relation (22) describes the relative acceleration between neighboring observers. The acceleration is in the foreground manifold where the gravitational force may not be the only tidal effect [1].…”

“…and along the supporting direction is trivial to prove that B ab D r b l a . Then, by taking the time derivative of (30) and substituting in (22), we recover the evolution of the tensorial distortion…”

Weak field equations and generalized FRW cosmology on the tangent Lorentz bundle

“…In the framework of Finslerian extensions, Raychaudhuri equations and energy conditions have been studied in [13,[52][53][54]. Raychaudhuri equation was developed on the tangent bundle of a spacetime and has been derived for a timelike congruence in [13]. Adapting a tangent field Y on a timelike geodesic congruence, the h-space Raychaudhuri equation gives:…”

Classical-physics applications for Finsler b space

“…In particular, 0 ≤ θ(x) ≤ 90 • and dμ(x) lies in the plane spanned by B(x) and the tangent dx to the curve x(λ). 3 In this stable configuration, we therefore have cos θ(x) = sin φ(x) = + 1 − cos 2 φ(x) , (19) where φ(x) is the angle between B(x) and dx. Since the beads on our magnetic chain are identical, we can take the linear magneticmoment density ζ = dμ/ds to be constant and write dμ(x) = ζ ds for the magnitude dμ of the magnetic moment.…”