On Born’s Deformed Reciprocal Complex Gravitational Theory and Noncommutative Gravity

Born’s reciprocal relativity in flat spacetimes is based on the principle of a maximal speed limit (speed of light) and a maximal proper force (which is also compatible with a maximal and minimal length duality) and where coordinates and momenta are unified on a single footing. We extend Born’s theory to the case of curved spacetimes and construct a deformed Born reciprocal general relativity theory in curved spacetimes (without the need to introduce star products) as a local gauge theory of the deformed Quaplectic group that is given by the semi-direct product of U(1, 3) with the deformed (noncommutative) Weyl-Heisenberg group corresponding to noncommutative generators [Za, Zb] 6= 0. The Hermitian metric is complex-valued with symmetric and nonsymmetric components and there are two different complex-valued Hermitian Ricci tensors Rμν ,Sμν . The deformed Born’s reciprocal gravitational action linear in the Ricci scalars R,S with Torsion-squared terms and BF terms is presented. The plausible interpretation of Zμ = E a μ Za as noncommuting p-brane background complex spacetime coordinates is discussed in the conclusion, where E μ is the complex vielbein associated with the Hermitian metric Gμν = g(μν) + ig[μν] = E a μ Ē b ν ηab. This could be one of the underlying reasons why string-theory involves gravity. Born’s reciprocal (”dual”) relativity [1] was proposed long ago based on the idea that coordinates and momenta should be unified on the same footing, and consequently, if there is a limiting speed (temporal derivative of the position coordinates) in Nature there should be a maximal force as well, since force is the temporal derivative of the momentum. A curved phase space case scenario has been analyzed by Brandt [2] within the context of the Finsler geometry of the 8D tangent bundle of spacetime where there is a limiting value to the proper acceleration and such that generalized 8D gravitational equations reduce