  We consider electromagnetic fields having an angular momentum density in a
locally non-rotating reference frame in Schwarzschild, Kerr, and Kerr-Newman
spacetimes. The nature of such fields is assessed with two families of
observers, the locally non-rotating ones and those of vanishing Poynting flux.
The velocity fields of the vanishing-Poynting observers in the locally
non-rotating reference frames are determined using the 3+1 decomposition
formalism. From a methodological point of view, and considering a
classification of the electromagnetic field based on its invariants, it is
convenient to separate the consideration of the vanishing-Poynting observers
into two cases corresponding to the pure and non-pure fields, additionally if
there are regions where the field rotates with the speed of light (light
surfaces) it becomes necessary to split these observers into two subfamilies.
We present several examples of relevance in astrophysics and general
relativity, such as pure rotating dipolar-like magnetic fields and the
electromagnetic field of the Kerr-Newman solution. For the latter example, we
see that vanishing-Poynting observers also measure a vanishing super-Poynting
vector, confirming recent results in the literature. Finally, for all non-null
electromagnetic fields, we present the 4-velocity fields of vanishing Poynting
observers in an arbitrary spacetime.
