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 Einstein's general relativity theory 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 EM 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 (partially new) characterizations of such observers and study their relation to arbitrary observers. In the nonnull electromagnetic case it is known that, given any observer, there is a principal observer which travels instantaneously 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. As an application, we find that in Petrov type D doubly aligned non-null Einstein-Maxwell fields (which include the classical charged black hole solutions) the Poynting and super-Poynting vectors are aligned, at each point and for each observer. We also 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. Our results are illustrated in simple examples.

Observer-based invariants for cosmological models

“…Next, the Weyl tensor is put into a canonical form in accord with the Petrov type; see table 4.2 of [7]. In terms of ON frames (e a 0 , e a i ) realizing this form and for 'diagonal' types I and D, this implies that e a 0 is a (timelike) Weyl principal vector u a C ; such a vector is characterized by the fact that the relative electric and magnetic Weyl operators E a b and H a b on u ⊥ C commute, and then (e a i ) is a eigentriad for both, which thus diagonalizes Q ab ; u a C is unique up to reflection in the type I case and is any unit timelike vector lying in the plane Σ C spanned by the Weyl principal null directions in the type D case [7,20]. For Petrov type I Q a b has distinct eigenvalues, such that the resulting ON frame (u a C , E a i ) is essentially unique (i.e., determined up to reflections) and called the Weyl principal frame [7].…”

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

Obtaining the multiple Debever null directions

“…N can also be found in [6], and the main expressions that we use for type D have also been recently obtained in [7].…”

“…The interest of the Bel-Robinson tensor in analyzing radiative gravitational states has been widely reported (see [6,7,8,9,10] and references therein). Here we use it as a mathematical tool to obtain the fundamental direction of the Bel radiative fields.…”