Image generation in Kerr geometry. I. Analytical investigations on the stationary emitter-observer problem

The stationary emitter-observer problem in Kerr geometry requires the systematic study of all lightlike geodesics connecting a given emitter with a given observer when emission is independent of coordinate time. To this aim, the separated first integrals of Carter governing geodesic motion of lightlike particles in Kerr geometry are removed from the uncertainties of sign. This enables a thorough discussion of the image generation of arbitrary axisymmetric, emitting objects at arbitrary observer locations, yielding a classification of all possible image orders by two parameters with only discrete values: one of them counts transits of the light path through the equatorial plane of the rotating hole, the second distinguishes two possible directions of the initial latitudinal emission. The properties of each image type are examined in detail, and it is shown that there are two principally distinct types of image generation, depending on whether or not the extended emitting object has surface elements with larger declination relative to the equatorial plane than the observer. It is demonstrated that the assumption of infinitesimal thinness of the emitting object as used for the general relativistic flux transformation until now, is a questionable approximation. Indeed, the distinction between the side turned towards and the side turned away from the observer of the emitting object is not made in this approximation; yet depending on the zenith angle the observer may see both sides: If emission takes place at larger declinations than observation, the side turned away from the observer is mapped in the direct image. Otherwise, this side is seen as part of the first indirect image. The first indirect image may continuously join end to end to the direct one; the innermost emission regime may be mapped twice in the observer's image, each order with considerable apparent angular size and not hidden from view. The contribution of at least the first indirect image to the total flux should therefore not be neglected. These investigations have resulted in a fast, flexible, and high precision transformation code between synthetic spectra on arbitrary spatially extended emitting surfaces, and fluxes received by arbitrary observers. As results of this code, local images of vertically extended objects like thick accretion discs as seen by different observers are presented. They are composed of the direct and first indirect image order. The code that calculates these images is based on the quasi-analytic solution of the geodesic problem by elliptic integrals.