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 \ᵃ=t nᵃ, ᵃnₐ=1, \ and in phase space to the dual decomposition \²=pₜ²+L²t². \ The bilocal constraint then takes the form =Q₄ₗₓ (X, P) -pₜ²-L²t²=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.