From Relation to Time. Constraint-Based Geometry of Bilocal Relational Dynamics.

The aim of this work is to show that the notion of time can be derived from theconstraint geometry of a bilocal phase space. The starting point is neither timenor a Hamiltonian, but rather the relation between two configurations, its phasespace, the symplectic form, and the constraint. We show locally that the transition to relational variables leads to the decomposition \[Δ^a=t n^a,^an_a=1,\] and in phase space to the dual decomposition \[Π^2=p_t^2+L^2/t^2.\] The bilocal constraint then takes the form \[C=Qₑₓₜ(X,P)-p_t^2-L^2/t^2=0.\] After restricting the symplectic form to the constraint surface, we obtain apresymplectic form whose kernel is one-dimensional. Consequently, the constraintsurface is foliated by one-dimensional orbits, and the variable \(t\) can beinterpreted as a parameter of these orbits, that is, as relational time.