Fermat’s principle and the variational analysis of an optical model for light propagation exhibiting a critical radius

Abstract

Fermat’s principle is used to analyze the trajectories of light propagating in a radially inhomogeneous medium with a singularity in the center. It is found that the light trajectories are similar to those around a black hole, in the sense that there exists a critical radius within which the light cannot escape, but spirals into the singularity.

Stability of Circular Orbits in General Relativity: a Phase Space Analysis

“…This result is quite consistent with the nature of various trajectories analyzed in Ref. [16]. However, a pathological solution ofẏ = 0 may be imagined by taking ε = 0 ⇒ r 2 0 = r 2 i .…”

“…From the original set of (15), (16), we see that they lead to the same Hamiltonian set of (51), (52) with the difference that α is now to be replaced by b. The equilibrium points then are P 2 : (0, 0) and Q 2 : (− b c , 0).…”

“…Stability of circular orbits can be easily studied directly once a form for n(r) is given. A plausible form for n(r) simulating a static dielectric medium of optical black holes has been studied by Marklund, Anderson, Cattani, Lisak and Lundgren [16]. It is given by…”

“…To examine stability, we first note that the autonomous system is of the same type as in (15), (16), only the coefficients are different. The next step is to follow the same procedure as in Sect.…”

“…Eliminating the parameter a, the autonomous system (15), (16) can be reduced to the following set of equationṡ…”