**Authors:** Carlos Castro

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_{μν}, S_{μν}. 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_{μν} = g_{(μν)} + ig_{[μν]} = E_{μ}^{a} Ē_{ν}^{b}
This could be one of the underlying reasons why string-theory involves gravity.

**Comments:** 11 pages, This article appeared in Phys Letts B 675, (2009) 226-230

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[v1] 25 Aug 2009

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