Transformation cosmology

Recent observation of black hole and gravitational wave has stirred up great interest in Einstein's general relativity. In an optical system, the "optical black hole" has also been a key topic in mimicking black holes. Another good way to study or mimic general relativity effects is based on transformation optics. In this paper, we propose a way by utilizing transformation optics theory to directly obtain the equivalent isotropic refractive index profiles which are the analogies of some static spaces of general relativity, such as de Sitter space, anti-de Sitter space, and Schwarzschild black hole. We find that the analog of de Sitter space is the Poincare disk, while anti-de Sitter space is equivalent to Maxwell's fish-eye lens. In particular, we prove that the optical black hole actually has infinite number of photon spheres, while our black hole only has a single one, which is closer to the real black hole. We study the effect from both geometric optics and wave optics. It can also be generalized to mimic any kind of metrics. Furthermore, with the isotropic refractivity index profile, we visualize the gravitational lensing effect of black hole from our software Dr TIM. The image not only recovers the donutlike halo of black hole, but also shows other phenomena.