IJPAM: Volume 76, No. 2 (2012)

MAXWELL'S EQUATIONS IN A CURVED SPACE TIME

K. Ghosh
Department of Physics
St. Xavier's College
30 Mother Teresa Sarani, Kolkata, 700016, INDIA


Abstract. In this article we will discuss a few mathematical aspects of classical electrodynamics with particular reference to the Robertson-Walker-Friedmann cosmological model with closed spatial geometry. We will first give an alternate derivation of the Gauss' law for the electric field of a point charge in the flat space-time not using the Gauss' divergence theorem. We will thereafter derive an exact expression for the electrostatic energy. The expression for the electrostatic energy obtained in this article agrees with the standard expression when the sources are not point particles. However we will show that the electrostatic self-energy of a point charge may be zero. We will also consider the consistency of the Maxwell's equations for the electromagnetic potentials in the Robertson-Walker-Friedmann universe with closed spatial geometry. We find that the contravariant electromagnetic fields are invariant under the gauge transformations and so also are the inhomogeneous Maxwell's equations when they are expressed in terms of the contravariant electromagnetic fields. However the inhomogeneous Maxwell's equations are not invariant under the gauge transformations when they are expressed in terms of the electromagnetic potentials. This may also remain valid in an arbitrary curved space-time. We will give a naive interpretation of this aspect related with the formulation of the differential geometry of the curved spaces.

Received: November 3, 2011

AMS Subject Classification: 00A79, 78MXX, 53ZXX, 53BXX

Key Words and Phrases: Poisson's equation, electrostatic self-energy, gauge invariance, covariant derivatives, curved spaces

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Source: International Journal of Pure and Applied Mathematics
ISSN printed version: 1311-8080
ISSN on-line version: 1314-3395
Year: 2012
Volume: 76
Issue: 2