Motivation: All the information on distant celestial objects comes to us by way of electromagnetic radiation. Therefore, studying the influence of the gravitational field on electromagnetic fields is of fundamental importance for our understanding of astrophysical objects such as, e.g., Black Holes (BH) and in particular neutron stars. For some applications, also the back-reaction of the electromagnetic field on the spacetime geometry is of relevance.

The interplay of gravity and electromagnetism can be treated at various levels of approximation:

(a) Ray optics on a background spacetime: The propagation of electromagnetic radiation can be modelled in terms of rays if the frequency is sufficiently high and if the back-reaction of the electromagnetic field can be neglected. For the standard Maxwell theory in vacuo, the rays are the lightlike geodesics of the underlying space-time metric. This is the formalism in which most work on, e.g., gravitational lensing is done. If the influence of matter, e.g. a plasma, is taken into account, the equation for the rays is modifyed. The same is true if non-standard vacuum theories of electrodynamics are considered, e.g. the non-linear Plebanski theories which include the Born-Infeld theory and the Heisenberg-Euler theory as special cases. The basic equations for the rays in such theories are well known, but many possible applications to astrophysics are still to be worked out. Also, the transport law for the polarisation plane along a ray has not been worked out for the Plebanski theories so far.

(b) Wave optics on a background spacetime: If the high-frequency approximation is not valid and the back-reaction can still be ignored, one has to deal with electromagnetic fields on a background space-time. One may consider the standard vacuum Maxwell theory or a modification thereof, taking matter into account or assuming a modified vacuum theory, e.g. of the Plebanski class. For BHs or other compact objects, this allows the treatment of scattering of electromagnetic waves or quasi normal modes. Details have been worked out only for the standard Maxwell theory on simple spacetimes such as Schwarzschild.

(c) Electrodynamics fully coupled to gravity: In (a) and (b) the coupling is one-way in the sense that the influence of the gravitational field on the electromagnetic field is taken into account but not the other way round. For very strong electromagnetic fields this approximation may not be valid and one has to deal with the electrodynamical equations coupled to Einstein's equations. The electrodynamical equations can be the standard vacuum Maxwell equations or some modifications. In the case of non-linear electrodynamics, the most intriguing known solutions to the coupled system are regular BHs.

In this Research Topic we plan:

(i) Calculation of the shadows of BHs: It is one of our aims to extend our previous work by taking the influence of a medium (like plasma) into account and possibly by considering perturbed BH space-times. A major goal is to work analytically, as far as possible, and to investigate if the parameters of the BH can be extracted from the boundary curve of the shadow. Here it is essential to consider rotating or deformed (“bumpy") BHs. For the spherically symmetric case, the influence on the shadow of a non-magnetised cold plasma was already be shown by us.

(ii) Tests of non-linear electrodynamics in astrophysics: Several tests of nonlinear electrodynamical theories in the laboratory have been suggested; e.g., a test of birefringence in vacuo, as predicted by the Heisenberg-Euler theory. Here it is our goal to complement these efforts with theoretical investigations on possible astrophysical tests of non-linear electrodynamics. To that end we have to study light propagation in a region of space-time with a strong electromagnetic field, e.g. the neighbourhood of a magnetar. The goal is to work out the effects on the path of light rays, on the polarisation plane or on other observable features as predicted by non-linear electrodynamics (Born-Infeld, Heisenberg-Euler, ... ) and to determine if they are big enough for providing a viable test. Up to now, only some rather rough estimates for the effects on the path exist.

(iii) Calculation of electromagnetic waves in regular BH spacetimes: We want to study electromagnetic waves in solutions to non-linear electrodynamics coupled to gravity, in particular regular BH solutions, and to discuss scattering, quasi-normal modes and related effects. For consistency, one has to assume that the electromagnetic waves do not satisfy the standard vacuum Maxwell equations on a curved space-time but rather a wave equation that results from linearising the underlying non-linear electrodynamical theory around the background electrodynamical field. Here a major goal will be to specify observable features that could be used to identify such regular BHs, if they exist in nature.