Generalized Fermat's principle and Snell's law for cone structures and applications

TL;DR

The work unifies Fermat's principle and Snell's law in a spacetime setting by introducing two interacting cone structures (Lorentz-Finsler lightcones) separated by a smooth interface. A lightlike trajectory is Fermat-critical exactly when it is orthogonal to the source, follows cone geodesics on each side, and satisfies a generalized Snell's law at the interface, with reflection emerging as a special case; the authors also classify existence, uniqueness, and minimization properties, including total reflection and double refraction. The framework supports time-dependent, anisotropic, and discontinuous media and offers applications to Zermelo navigation and discretized spacetimes, providing tools for wave propagation analysis and numerical relativity. The results connect local refractive behavior to global extremality through Snell cone geodesics and supply practical guidance for computing lightlike geodesics in discretized spacetimes, with implications for causality, horismos, and and geodesic structure in Finsler-type spacetimes.

Abstract

Fermat's principle is fully generalized to the case where a smooth interface separates two cone structures -- Lorentz-Finsler lightcones -- representing wave propagation in a potentially inhomogeneous, anisotropic, time-dependent and discontinuous medium. The interface, wave source and receiver are assumed to be a hypersurface, a submanifold and a curve in the spacetime, respectively, of any causal character. For a trajectory to fullfil Fermat's principle -- i.e., to be a critical point of the arrival time functional -- its direction must change at the interface, obeying a precise condition that generalizes Snell's law of refraction when the wave crosses the interface, or the law of reflection when it remains in a single medium. Both laws are analyzed in detail to establish the conditions ensuring the existence and uniqueness of refracted and reflected trajectories, and to determine whether they actually minimize the arrival time. Applications to Zermelo's navigation problem and the determination of geodesics in discretized spacetimes are also emphasized.

Figures (9)

• Figure 1: A simple representation of the setting we are working on. We highlight that $P$, $\eta$ and $\alpha$ can have, in principle, any causal character (in the figure, $P$ is $\mathcal{C}^1$-spacelike, $\eta$ is timelike for both cone structures and $\alpha$ is $\mathcal{C}^2$-timelike).
• Figure 2: Given $r \in \mathcal{C}_p^{\mu}$ such that $r^{\perp_{\mathcal{C}^{\mu}}} = T_r\mathcal{C}_p^{\mu}$ is transverse to $T_p\eta$, $\Pi^{\mu}_r \coloneqq r^{\perp_{\mathcal{C}^{\mu}}} \cap T_p\eta$ is a codimension 1 subspace of $T_p\eta$ ($\mu = 1$ when $r$ is an incident or reflected direction, $\mu = 2$ when it is a refracted one). We highlight that $r$ might be tangent to $\eta$.
• Figure 3: Case (A) of Theorem \ref{['thm:existence_refraction']}, when $T_p\eta$ is $\mathcal{C}^2$-timelike (and also $\mathcal{C}^1$-timelike in this figure). On the left, $\Pi_u^1$ is $\mathcal{C}^2$-spacelike and $v \in Q^2_p$ is the unique refracted direction satisfying Snell's law $\Pi_u^1 = \Pi_v^2$. On the right, $\Pi_u^1$ is $\mathcal{C}^2$-lightlike and $v \in T_p\eta$ is the unique refracted direction satisfying Snell's law.
• Figure 4: Case (B) of Theorem \ref{['thm:existence_refraction']}, when $T_p\eta$ is $\mathcal{C}^2$-lightlike and tangent to $\mathcal{C}^2_p$ along the direction $\hat{v}$. On the left, $\Pi_u^1$ is $\mathcal{C}^2$-spacelike and $v \in Q^2_p$ is the unique refracted direction satisfying Snell's law $\Pi_u^1 = \Pi_v^2$. On the right, $\Pi_u^1$ is $\mathcal{C}^2$-lightlike and $v = \hat{v} \in T_p\eta$ is not a refracted direction because $v^{\perp_{\mathcal{C}^2}} = T_p\eta$ (although it can be regarded as an exceptional refraction; see Remark \ref{['rem:exceptional']}).
• Figure 5: Case (C) of Theorem \ref{['thm:existence_refraction']}, when $T_p\eta$ is $\mathcal{C}^2$-spacelike (and also $\mathcal{C}^1$-spacelike in this figure). There are two refracted directions $v, \hat{v} \in Q^2_p$ satisfying Snell's law $\Pi_u^1 = \Pi_v^2 = \Pi_{\hat{v}}^2$.

Theorems & Definitions (71)

• Definition 2.1
• Definition 2.2
• Definition 2.3
• Definition 2.4
• Definition 2.5
• Proposition 2.6
• Remark 2.7
• Definition 2.8
• Definition 2.9
• Remark 2.10