A Conformal Co-Symplectic Structure on the Space of Pseudo-Riemannian Geodesics

Abstract

The classical construction of the symplectic structure on the space of geodesic trajectories via Hamiltonian reduction fails in the pseudo-Riemannian setting due to a dimensional mismatch created by the null geodesics. This paper proposes a new, unified approach. We first construct the space of all geodesic trajectories $\mathcal{G}_\text{traj}$ directly as the quotient of the space of geodesics curves $\mathcal{G}_\text{curv}$ by the affine reparametrization group. The analysis of the orbits of this group action reveals a key geometric distribution. To describe this distribution globally, we introduce a canonical object, the "conformal co-symplectic structure" $\sigma$, defined by pushing forward the conformal class of the inverse $\omega^{-1}$ of the original symplectic form $\omega$. We prove that the image of this structure coincides with the geometric distribution identified previously. On the subspace of time-like and space-like geodesics, this structure is non-degenerate and defines a conformal class of symplectic forms. On the null subspace, its image is a codimension-$1$ distribution that we prove is the canonical contact structure on the space of light rays.