Fully metallic geodesic lenses as analog electromagnetic models of static and spherically symmetric gravitational fields

We demonstrate that a fully metallic and air-filled geodesic waveguide can be employed as an analog electromagnetic model of a static and spherically symmetric gravitational field. By following the Plebanski formalism, a space-time metric of the aforementioned type is firstly encoded into the electromagnetic properties of a flat space-time region in the form of an isotropic and radially varying refractive index distribution. Then, a three-dimensional, air-filled, and axially symmetric waveguide, composed of two equally spaced and curved metallic surfaces, is employed. Its shape is tailored such that the effective paths, followed by transverse electromagnetic beams of microwave radiation within this waveguide, result equivalent to null-geodesics taking place in the aforementioned refractive medium. This strategy avoids the need for a refractive medium, although it only allows to reproduce the space-time metric on the invariant plane. Two analog electromagnetic models of gravity, using the proposed approach, are designed to reproduce the metric of both a Schwarzschild black hole and a Morris-Thorne wormhole. The results from full-wave simulations demonstrate that a one-dimensional Gaussian beam faithfully follows a path completely equivalent to general relativistic null geodesics with a mean relative error within 4%.