Finite element analysis of propagation modes in a waveguide:
Effect of gravitational field

Arti Vaish^* and Harish Parthasarathy

The law of Electromagnetic wave propagation inside a waveguide is governed by the Helmholtz equation. This equation is derived from the standard wave equation by assuming sinusoidal time dependence. Such an equation is valid if the space-time manifold is flat, i.e., Minkowskian. In the presence of a gravitational field, according to Einstein's general theory of relativity, the space-time manifold becomes curved and the geometry of such a curved manifold is described by a Riemannian metric. Consequently, the wave equation in such a curved space-time needs to be modified to account for the curvature. In addition, the assumption of sinusoidal time dependence gives a modified Helmholtz equation. Since the metric in a curved space-time manifold can be expressed in terms of the gravitational potential, the gravitational potential will enter into the curved space-time wave equation. Such an equation can be derived from a variational principle. In this paper, this variational problem has been solved using a finite element method on rectangular waveguide. The shifts in the modes of propagation induced by the gravitational potential are obtained and compared by considering a numerical example.

Keywords: General theory of relativity, electromagnetism, metric tensor, generalized Laplace operator, test functions, finite element method.