A refractive index of a kink in curved space

Abstract

The refractive index and curvature relation is formulated using the Riemann-Christoffel curvature tensor. As a consequence of the fourth rank tensor of the Riemann-Christoffel curvature tensor, we found that the refractive index should be a second rank tensor. The second rank tensor of the refractive index describes a linear optics. It implies naturally that the Riemann-Christoffel curvature tensor is related to the linear optics. In case of a non-linear optics, the refractive index is a sixth rank tensor, if susceptibility is a fourth rank tensor. The Riemann-Christoffel curvature tensor can be formulated in the non-linear optics but with a reduction term. The relation between the (linear and non-linear) refractive index and a (linear and non-linear) mass in curved space are formulated. Related to the Riemann-Christoffel curvature tensor, we formulate "the (linear and non-linear) generalized Einstein field equations". Sine-Gordon model in curved space is shown, where the Lagrangian is the total energy. This total energy is the mass of a kink (anti-kink) associated with a topological charge (a winding number). We formulate the relation between the (linear and non-linear) refractive index of the kink (anti-kink) and the topological charge-the winding number. Deflection of light is discussed in brief where the (linear and non-linear) angle of light deflection are formulated in relation with the mass (the topological charge, the winding number) of the kink (anti-kink).