On Born's deformed reciprocal complex gravitational theory and noncommutative gravity

Abstract

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 [ Z a , Z b ] ≠ 0 . The Hermitian metric is complex-valued with symmetric and nonsymmetric components and there are two different complex-valued Hermitian Ricci tensors R μ ν , S μ ν . 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 Z μ = E μ a Z a as noncommuting p-brane background complex spacetime coordinates is discussed in the conclusion, where E μ a is the complex vielbein associated with the Hermitian metric G μ ν = g ( μ ν ) + i g [ μ ν ] = E μ a E ¯ ν b η a b . This could be one of the underlying reasons why string-theory involves gravity.