# IJPAM: Volume 76, No. 2 (2012)

**MAXWELL'S EQUATIONS IN A CURVED SPACE TIME**

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