Quantum Gravity and String Theory

   

Complex Gravitational Theory and Noncommutative Gravity

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 [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, 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 = Eμa Za 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] = Eμa Ēνb 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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Submission history

[v1] 25 Aug 2009

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