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 Quaplecric group that is given by the semi-direct product of U(1, 3) with the deformed (noncommutative) Weyl-Heisenberg group corresponding to noncommutative generators vertical bar Z(a), Z(b)vertical bar not equal 0. The Hermitian metric is complex-valued with symmetric and nonsymmetric components and there are two different complex-valued Hermitian Ricci tensors R-mu nu, S-mu nu.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(mu) = E(mu)(a)Z(a) as noncommuting p-brane background complex spacetime coordinates is discussed in the conclusion, where E-mu(a), is the complex vielbein associated with the Hermitian metric G(mu nu) = g(mu nu) + ig([mu nu]) = E-mu(a)(E) over bar (b)(t),eta(ab). This could be one of the underlying reasons why string-theory involves gravity. (C) 2008 Elsevier B.V. All rights reserved.