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Principle of Reversibility

The assumption that light rays are directionless and can be traced in either the forward or backward direction without altering their path.
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The statement of the theorem

Let the medium be defined by a static refractive index n(x):R3→R+n(\mathbf{x}): \mathbb{R}^3 \to \mathbb{R}^+. The path r(s)\mathbf{r}(s) of a light ray, parameterized by arc length ss, is a geodesic curve in the Riemannian manifold (R3,g)(\mathbb{R}^3, g), where the metric tensor gg is given by gij=n2(x)δijg_{ij} = n^2(\mathbf{x}) \delta_{ij}. The path r(s)\mathbf{r}(s) must satisfy the geodesic equation: \begin{equation*} \frac{d^2 x^k}{d s^2} + \Gamma^k_{ij} \frac{d x^i}{d s} \frac{d x^j}{d s} = 0 \end{equation*} where Γijk\Gamma^k_{ij} are the Christoffel symbols associated with gijg_{ij}. The Principle of Reversibility asserts that if r(s)\mathbf{r}(s) is a solution to this equation for s∈[0,L]s \in [0, L], then the curve r′(s)=r(L−s)\mathbf{r}'(s) = \mathbf{r}(L-s) is also a solution, provided the metric gijg_{ij} is independent of time tt. Specifically, the tangent vector k(s)=drds\mathbf{k}(s) = \frac{d\mathbf{r}}{ds} satisfies k(L−s)=−k(s)\mathbf{k}(L-s) = -\mathbf{k}(s) and the path remains a geodesic.