Fermat's Principle in General Relativity via Herglotz Variational Formalism
Joanna Piwnik, Joanna Gonera, Piotr Kosiński
TL;DR
This work generalizes Fermat's principle to arbitrary gravitational fields using the Herglotz variational formalism, where an autonomous, velocity-homogeneous Lagrangian leads to a reduced, action-dependent dynamics via an energy constraint. By eliminating one generalized velocity, the authors show that the remaining coordinates obey a Herglotz principle with an effective Lagrangian $\mathcal{L}(oldsymbol{q},oldsymbol{\dot{q}},S;E)$, enabling a coordinate-free formulation that applies to both light (null geodesics) and massive particles. The paper derives a general Fermat-like equation for gravity, discusses gauge issues for the massless case, and demonstrates the approach with the Schwarzschild–de Sitter metric, where Noether-type integrals yield a local constant of motion and enable quadrature-based integration. Overall, the framework provides a flexible, non-stationary generalization of Fermat's principle applicable to diverse spacetimes and practical for gravitational lensing/time-delay analyses.
Abstract
New form of Fermat's principle for light propagation in arbitrary (i.e. in general neither static nor stationary) gravitational field is proposed. It is based on Herglotz extension of canonical formalism and simple relation between the dynamics described by the Lagrangians homogeneous in velocities and the reduced dynamics on lower-dimensional configuration manifold. This approach is more flexible as it allows to extend immediately the Fermat principle to the case of massive particles and to eliminate any space-time coordinate, not only $x^0$.