What is the Poynting "vector" mathematically?

[greypilgrim]

Hi.

The Poynting vector is a 3-tuple of real or complex numbers (depending on the respective formulation of electrodynamics) times a unit. It may be pictured as an arrow with some length and direction in IR^3 or IC^3. But is it a "vector" in the strict mathematical sense, i.e. an element of a vector space? If yes, what exactly is that vector space? As far as I know, Poynting vectors of two sources can only be added if the sources are completely decoherent, i.e. if there's no (time-averaged?) interference.

 

[vanhees71]

The Poynting "vector" is the momentum or energy-flow density of the electromagnetic field. It's a vector in the sense of Euclidean rotations in a fixed inertial reference frame. Relativitistically it's the space-time component of the symmetric (Belinfante) energy-momentum tensor, which are defined by the Noether currents of space-time translation invariance, which is a subgroup of the Poincare group, i.e., the homogeneity of Minkowski space. As the name says, the energy-momentum tensor defines energy and momentum, which are conserved for any closed system.