Quantum Gravity and String Theory


Gravity in Curved Phase-Spaces and Two-Times Physics

Authors: Carlos Castro

The generalized (vacuum) field equations corresponding to gravity on curved $2d$-dimensional tangent bundle/phase spaces and associated to the geometry of the (co)tangent bundle $ TM_{ d -1, 1} (T^*M_{d-1,1}) $ of a $d$-dim spacetime $ M_{d - 1, 1}$ are investigated following the strict formalism of Lagrange-Finsler and Hamilton-Cartan geometry. It is found that there is $no$ mathematical equivalence with Einstein's vacuum field equations in spacetimes of $2d$-dimensions, with $two$ times, after a $ d + d $ Kaluza-Klein-like decomposition of the $ 2d$-dim scalar curvature $ {\bf R } $ is performed and involving the introduction of a nonlinear connection $ A_\mu^a ( x^\mu, y^b ) $. The physical applications of the $ 4$-dim phase space metric solutions found in this work, corresponding to the cotangent space of a $ 2$-dim spacetime, deserve further investigation. The physics of $two$ times may be relevant in the solution to the problem of time in Quantum Gravity and in the explanation of Dark Matter. Finding nontrivial solutions of the generalized gravitational field equations corresponding to the $ 8$-dim cotangent bundle (phase space) of the $4$-dim spacetime remains a challenging task.

Comments: 30 Pages. Submitted the IJMPA.

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Submission history

[v1] 26 Aug 2011
[v2] 2012-03-19 04:07:21

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