Kähler spacetime tangent bundle

Requirements are delineated for the spacetime tangent bundle to be Kählerian. In particlar, an almost complex structure is constructed in the case of a Finsler spacetime, and its covariant derivative in terms of the bundle connection is shown to be vanishing, provided the gauge curvature field is vanishing. The Levi-Civita connection coefficients and the Riemann curvature scalar are also specified for the Kähler spacetime tangent bundle.     

Riemann curvature scalar of spacetime tangent bundle

The maximum possible proper acceleration relative to the vacuum determines much of the differential geometric structure of the space-time tangent bundle. By working in an anholonomic basis adapted to the spacetime affine connection, one derives a useful expression for the Riemann curvature scalar of the bundle manifold. The explicit documentation of the proof is important because of the central role of the curvature scalar in the formulation of an action with resulting field equations and associated solutions to physical problems.     

Finsler-spacetime tangent bundle

The Levi-Civita connection coefficients of the spacetime tangent bundle, for the case of a Finsler spacetime, are reduced to the form given by Yano and Davies for a generic tangent bundle of a Finsler manifold. A useful expression is also obtained for the Riemann curvature scalar of a Finsler-spacetime tangent bundle.     

Structure of spacetime tangent bundle

The universal upper limit on attainable proper acceleration relative to the vacuum imposes restrictions on possible structures in the spacetime tangent bundle. Various features of the differential geometry of the spacetime tangent bundle are presented here. Also, a modified Schwarzschild solution is obtained, and the associated gravitational red shift is calculated.     

Spacetime tangent bundle with torsion

It is demonstrated explicitly that the bundle connection of the Finslerspacetime tangent bundle can be made compatible with Cartan's theory of Finsler space by the inclusion of bundle torsion, and without the restriction that the gauge curvature field be vanishing. A component of the contorsion is made to cancel the contribution of the gauge curvature field to the relevant component of the bundle connection. Also, it is shown that the bundle manifold remains almost complex, and that the almost complex structure can be made to have a vanishing covariant derivative if additional conditions on the torsion are satisfied. However, the Finsler-spacetime tangent bundle remains complex only if the gauge curvature field vanishes.     

Solution to torsion relations in finsler-spacetime tangent bundle

In the Finsler-spacetime tangent bundle, a simple solution is determined to the torsion relations that were obtained previously to maintain (1) compatibility with Cartan's theory of Finsler space, (2) the almost complex structure, and (3) the vanishing of the covariant derivative of the almost complex structure.     

Quantum Fields in the Spacetime Tangent Bundle

Maximal-acceleration invariant quantum fields are formulated in terms of the differential geometric structure of the spacetime tangent bundle. The simple special case is considered of a flat Minkowski space-time for which the bundle is also flat. The field is shown to have a physically based Planck-scale effective regularization and a spectral cutoff at the Planck mass.     

CAUSAL DOMAIN OF MINKOWSKI-SPACETIME TANGENT BUNDLE

The geometry of the causal domain of the spacetime tangent bundle is examined for Finslerian quantum fields in Minkowski spacetime. The Planck-scale structure of the boundary of the causal domain is elaborated. The geometry indicates that at the Planck scale, causal connectivity of Finslerian quantum fields may occur between spacelike separated points, and also at larger scales for extremely large relative four-velocities.     

FINSLERIAN QUANTUM FIELDS AND MICROCAUSALITY

Microcausality is addressed for a class of Finslerian quantum fields in Minkowski spacetime by the calculation of the appropriate field commutators. It is demonstrated that, provided the adjoint field is consistently generalized, the necessary commutators are vanishing, and the field is microcausal. There are, however, Planck-scale modifications of the causal domain, but they only become significant for extremely large relative four-velocities at the separated spacetime points. For vanishing relative four-velocities, the causal domain is canonical.