Hyperbolic metamaterials have dielectric tensors with opposite signs along different axes — permittivity is positive in one direction and negative in another. Light propagation in these materials follows hyperbolic dispersion curves rather than elliptical ones, enabling sub-wavelength imaging and exotic refraction. But designing lenses from hyperbolic materials is intractable with standard wave optics because the anisotropy couples all spatial directions.
The solution: embed the problem in Minkowski spacetime. The hyperbolic dispersion relation has the same mathematical structure as the Lorentzian metric of special relativity — one timelike and two spacelike directions. Light rays in the hyperbolic material follow geodesics in the Lorentzian geometry, and lens design reduces to computing geodesics in a curved Minkowski space.
The reformulation makes the problem analytically solvable. Geodesic equations in Lorentzian geometry have well-developed solution methods — the entire apparatus of general relativity becomes available for optics design. Flat hyperbolic lenses, Luneburg-type concentrators, and Maxwell fisheye designs all emerge as solutions to specific spacetime metric prescriptions.
The key insight is that material anisotropy IS geometric curvature in the Lorentzian embedding. Changing the material's permittivity tensor along the lens is equivalent to curving the Minkowski space. The lens design problem becomes: what spacetime curvature produces the desired ray trajectories? This is the inverse problem of general relativity, applied to optics.
A photonics problem, solved by pretending light lives in spacetime. The mathematics of curved spacetime, repurposed for flat metamaterial optics.