Finslerian Fields in the Spacetime Tangent Bundle

The Extended Relativity Theory in Born-Clifford Phase Spaces with a Lower and Upper Length Scales and Clifford Group Geometric Unification

“…In this process the speed of light remains the maximum attainable speed (it takes an infinite energyto reach it) and in our scheme the Planck scale is never surpassed. The effects of a minimal length can be clearly seen in Finsler geometries [12] having both a maximum four acceleration c 2 /Λ (maximum tidal forces) and a maximum speed . The Riemannian limit is reached when the maximum four acceleration goes to infinity; i.e.…”

“…The deep connection between the physics of maximal acceleration and Finsler geometry has been analyzed by [12]. The action is real-valued if, and only if, m 2 g 2 < m 2 P a 2 in the same fashion that the action in Minkowski spacetime is real-valued if, and only if, v 2 < c 2 .…”

“…This is because the rotor R includes the extra generator θ t12 γ t ∧γ 1 ∧γ 2 which will bring extra terms into the transformations ; i.e. it will rotate the E [12] bivector-basis , that couples to the holographic coordinates x 12 , into the E t direction which is being contracted with the E t element in the definition of L t M . There are extra terms in the C-space boosts because the poly-particle dynamics is taking place in C-space and all coordinates X M which contain the t, x 1 , x 12 directions will contribute to the C-space boosts in D = 3, since one is projecting down the dynamics from C-space onto the (t, x 1 ) plane when one studies the motion of the tachyon in M 4 .…”

“…The dynamical consequences of the minimal-length in Newtonian dynamics have been recently reviewed by [38]. The idea of minimal length ( the Planck scale L P ) can be incorporated within the context of the maximal acceleration Relativity principle [20] a max = c 2 /L P in Finsler Geometries [12]. A different approach than the one based on Finsler Geometries is the pseudo-complex Lorentz group description by Schuller [13] related to the effects of maximal acceleration in Born-Infeld models that also maintains Lorentz invariance, in contrast to the approaches of Double Special Relativity (DSR) [22] where the Lorentz symmetry is deformed .…”

“…In 6 we analyze different geometric actions, like the Clifford-space extensions of Maxwell's EM; Brandt's action [12] related the 8D spacetime tangent-bundle involving coordinates and velocities ( Finsler geometries ) and, finally, we discuss in further detail why Einstein's gravity in m + n dimensions is equivalent to an m-dim Yang-Mills-like theory of diffemorphisms of an internal n-dim space which admits a holographic reduction to lower dimensions in the case of AdS m × S n and dS m × H n backgrounds. Finally in 7 we outline the reasons why a Clifford-Space Geometric Unification of all forces is a reasonable avenue to take and propose an Einstein-Hilbert type action in the Clifford-Phase space (associated with the 8D Phase space), as the Unified Field theory action that should reduce to the Standard Model plus Gravity in the low energy limit.…”