International Journal of Modern Physics A Vol. 27, No. 12 (2012) 1250069 (19 pages) c  World Scientific Publishing Company DOI: 10.1142/S0217751X12500698 GRAVITY IN CURVED PHASE-SPACES, FINSLER GEOMETRY AND TWO-TIMES PHYSICS CARLOS CASTRO Center for Theoretical Studies of Physical Systems, Clark Atlanta University, Atlanta, Georgia, 30314, USA perelmanc@hotmail.com Received 26 August 2011 Revised 25 March 2012 Accepted 25 March 2012 Published 5 May 2012 The generalized (vacuum) field equations corresponding to gravity on curved 2d- dimensional (dim) tangent bundle/phase spaces and associated with the geometry of the (co)tangent bundle TM d-1,1 (T * M d-1,1 ) of a d-dim space–time M d-1,1 are investi- gated following the strict distinguished d-connection formalism of Lagrange–Finsler and Hamilton–Cartan geometry. It is found that there is no mathematical equivalence with Einstein’s vacuum field equations in space–times of 2d dimensions, with two times, after a d + d Kaluza–Klein-like decomposition of the 2d-dim scalar curvature R is performed and involving the introduction of a nonlinear connection A a µ (x µ ,y b ). The physical appli- cations of the 4-dim phase space metric solutions found in this work, corresponding to the cotangent space of a 2-dim space–time, 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 space–time remains a challenging task. Keywords : Gravity; phase space; Finsler geometry; Hamilton–Carten spaces. 1. Introduction The geometry of phase space (in contrast to space–time) might be very relevant to solve the problem of quantum gravity. Born’s reciprocal (“dual”) relativity 1,2 was proposed long ago based on the idea that coordinates and momenta should be unified on the same footing and consequently, if there is a limiting speed (temporal derivative of the position coordinates) in Nature there should be a maximal force as well, since force is the temporal derivative of the momentum. A maximal speed limit (speed of light) must be accompanied with a maximal proper force (which is also compatible with a maximal and minimal length duality). The generalized velocity and acceleration boosts (rotations) transformations of the 8-dim phase space, where 1250069-1