Electrodynamics in curved spacetime: 3 + 1 formulation

Abstract

This paper develops the mathematical foundations for a companion paper on Black-Hole Electrodynamics'. More specifically, it re-expresses the equations of curved-spacetime electrodynamics in terms of a 3 + 1 (space + time) split, in which the key quantities are three-dimensional vectors (electric field E, magnetic field B etc.) that lie in hypersurfaces of constant time t. Three-dimensional vector analysis is used to express Maxwell's equations, the Gauss, Faraday and Ampere laws, the Lorentz force law, and the laws of energy and momentum conservation in forms closely resembling their flat-spacetime counterparts. After developing the 3 + 1 formalism for general spacetimes, this paper specializes to the spacetime outside a stationary but rotating black hole. The Znajek-Damour boundary conditions at the hole's horizon are re-expressed in 3 + 1 language. Because the black hole's hypersurfaces of constant time all have identical three-dimensional geometries, one can abandon entirely Einstein's view of spacetime and return to Galileo's: The electric and magnetic fields E and B can be regarded as living in an absolute (but curved) three-dimensional space, and as evolving in this space with the passage of universal time t. This viewpoint and associated mathematics are the foundation for the companion paper.