Optics in the Schwarzschild space - time

The first light detected from a relativistic region about a black hole was discovered by the ASCA satellite by Tanaka et al. (1995); Yaqoob et al. (1998). The now accepted theoretical model describing the broad X-ray emission lines is that of an accretion disk around either a Kerr or a Schwarzscild black hole (Laor 1991; Reynolds et al. 1995; Turner, George, Nandra, & Mushotzky 1997; Nandra et al. 1997; Fanton et al. 1997; Bromley et al. 1997; Dabrowski & Lasenby 2001; Čadež et al. 1998; Martocchia et al. 2000). At the beginning rather crude methods were used in calculating theoretical line profiles expected from accretion disks. Ray tracing was done by direct numerical integration of geodesic equations, although analytic solutions of geodesic equations were given in a rather complicated form involving elliptic integrals by Chandrasekhar (1992), Rauch & Blandford (1994). Čadež, Fanton, & Calvani (1998) inverted the expressions given by Chandrasekhar, Rauch & Blandford and expressed the radial coordinate r as a function of θ using elliptic functions, which gives an orbit equation solution quite similar to that of the Kepler problem. This form of the solution allows a direct analytic solution of the basic ray tracing problem: connecting two points with a light-like geodesic.