Poynting vector, super-Poynting vector, and principal observers in electromagnetism and general relativity

Abstract

In electromagnetism, the concept of Poynting vector as measured by an observer is well known. A mathematical analogue in general relativity is the super-Poynting vector of the Weyl tensor. Observers for which the (super-)Poynting vector vanishes are called principal. When, at a given point, the electromagnetic field is non-null, or the gravitational field is of Weyl-Petrov type I or D, principal observers instantaneously passing through that point always exist. We survey characterizations of such observers and study their relation to arbitrary observers. In the non-null electromagnetic case it is known that, given any observer, there is a principal observer which moves relative to the first in the direction of his Poynting vector. Replacing Poynting by super-Poynting yields a possible gravitational analogue; we show that this analogy indeed holds for any observer when the Petrov type is D, but only for a one-dimensional variety of observers when the Petrov type is I. We provide algorithms to obtain the principal observers directly from the electric and magnetic fields (in the electromagnetic case) or electric and magnetic parts of the Weyl tensor (in the gravitational case) relative to an arbitrary observer. It is found that in Petrov type D doubly aligned non-null Einstein-Maxwell fields (which include all classical charged black hole solutions) the Poynting and super-Poynting vectors are aligned, at each point and for each observer, and the principal observers coincide. Our results are illustrated in simple examples.

Vanishing Poynting Observers and Electromagnetic Field Classification in Kerr and Kerr-Newman Spacetimes

“…Using the locally nonrotating Wylleman observers, we have found that the electromagnetic Poynting vector and the gravitational super-Poynting vector vanish for these observers in the region between horizons. These observers also measure parallel electric and magnetic vectors in the nonpure regions, confirming results in [52]. Also, in this example, we found how the rotating Carter observers and locally rotating observers become comoving ones in the region between the horizons, due to the effect of frame dragging.…”

“…As the last example shows, observers measuring a null Poynting vector are not unique. In a recent work by Wylleman et al [52], it is shown how to find all these observers. Now, using the presented formalism and without considering any concrete space-time, we can generalize our previous results to show that these observers can always be introduced for any nonnull electromagnetic field.…”

“…Red lines are for ρ = 0:0008, blue for ρ = −0:0008, magenta for ρ = 0:0001, and green for ρ = − 0:0001; the black curve represents the black hole's horizon. 9 Advances in High Energy Physics Wylleman et al [52]. The latter group found that Carter observers, i.e., those observers which measure a vanishing gravitational super-Poynting vector, also measure a vanishing electromagnetic Poynting vector.…”

“…As a final remark, it has to be noticed that Carter observers are not the only ones which measure vanishing Poynting and super-Poynting vectors. In fact, as Wylleman et al have shown [52], any radially moving observer with respect to the Carter observers also measures the vanishing of both Poynting vectors (these authors use the term Weyl observers instead of the Carter observer used here).…”

Observer-based invariants for cosmological models

“…The Petrov types can now be characterized as follows: [7,20] • Petrov type with I 3 − 6J 2 = 0: I (algebraically general type);…”