Gravity in two-time physics

Abstract

The field theoretic action for gravitational interactions in $d+2$ dimensions is constructed in the formalism of two-time (2T) physics. General relativity in $d$ dimensions emerges as a shadow of this theory with one less time and one less space dimensions. The gravitational constant turns out to be a shadow of a dilaton field in $d+2$ dimensions that appears as a constant to observers stuck in $d$ dimensions. If elementary scalar fields play a role in the fundamental theory (such as Higgs fields in the standard model coupled to gravity), then their shadows in $d$ dimensions must necessarily be conformal scalars. This has the physical consequence that the gravitational constant changes at each phase transition (inflation, grand unification, electroweak, etc.), implying interesting new scenarios in cosmological applications. The fundamental action for pure gravity, which includes the spacetime metric ${G}_{MN}(X)$, the dilaton $\ensuremath{\Omega}(X)$, and an additional auxiliary scalar field $W(X)$, all in $d+2$ dimensions with two times, has a mix of gauge symmetries to produce appropriate constraints that remove all ghosts or redundant degrees of freedom. The action produces on-shell classical field equations of motion in $d+2$ dimensions, with enough constraints for the theory to be in agreement with classical general relativity in $d$ dimensions. Therefore this action describes the correct classical gravitational physics directly in $d+2$ dimensions. Taken together with previous similar work on the standard model of particles and forces, the present paper shows that 2T physics is a general consistent framework for a physical theory. Furthermore, the 2T-physics approach reveals more physical information for observers stuck in the shadow in $d$ dimensions in the form of hidden symmetries and dualities, that are largely concealed in the usual one-time formulation of physics.