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)U(1,3) with the deformed (noncommutative) Weyl–Heisenberg group corresponding to noncommutative   generators [Za,Zb]≠0[Za,Zb]≠0. The Hermitian metric is complex-valued with symmetric and nonsymmetric components and there are two   different complex-valued Hermitian Ricci tensors Rμν,SμνRμν,Sμν. The deformed Born's reciprocal gravitational action linear in the Ricci scalars R,SR,S with Torsion-squared terms and BF   terms is presented. The plausible interpretation of Zμ=EμaZa as noncommuting p  -brane background complex spacetime coordinates is discussed in the conclusion, where Eμa is the complex vielbein associated with the Hermitian metric Gμν=g(μν)+ig[μν]=EμaE¯νbηab. This could be one of the underlying reasons why string-theory involves gravity.