Researchers introduce a new geometric framework that rewrites the motion of relativistic particles so it can include dissipative effects such as particle decay. Instead of the usual phase space of positions and momenta, they work on a nine-dimensional “extended phase space” made of four position coordinates, four momentum coordinates, and one extra variable ϕ. That extra variable plays the role of a geometric version of the particle’s proper time — the time measured by a clock moving with the particle — and it lets the authors write evolution in a fully geometric way.

The main idea is to use contact geometry, a close cousin of the symplectic geometry normally used for conservative (non-dissipative) systems. Contact geometry naturally handles irreversible or energy-dissipating processes. The authors pick a contact Hamiltonian that encodes the usual relativistic mass–shell condition, but with the mass allowed to depend on ϕ. The resulting evolution is generated by an “evolution contact vector field.” This field preserves the Hamiltonian on its trajectories, so if a motion starts on the mass shell it stays there, but it generally does not preserve the usual volume form — a mathematical sign of dissipation.

A technical problem in relativistic mechanics is that the particle’s proper time τ is the only intrinsic time, while Hamiltonian flows are often written with an arbitrary parameter. The new framework removes that arbitrariness by promoting ϕ to an independent variable and using it to parametrize motion. Projecting the contact flow onto the usual position–momentum space and reparametrizing with ϕ gives a reduced set of evolution equations. On the mass shell those equations relate changes in ϕ to the particle’s proper time, so ϕ acts as a geometrically defined clock. A useful consequence is that the evolution of massless particles (like photons), which are hard to parametrize by proper time, becomes well defined without extra reparametrization tricks.