Figure 16

Figure 16: Radial angular diameter distance D+(Q), tangential angular diameter distance D -(Q) and travel time T (Q) in the Schwarschild spacetime. The data are the same as in Figure 15. For the definition of D+ and D - see Figure 7. D ±(Q) can be calculated from P(Q) with the help of Equation (94View Equation) and Equation (95View Equation). For the Schwarzschild case, the resulting formulas are due to [84] (cf. [85, 119]). Zeros of D - indicate Einstein rings. If D + and D - have different signs, the observer sees a side-inverted image. The travel time T(Q) (= Fermat arclength) can be calculated from Equation (85View Equation). One sees that, over the logarithmic Q-axis used here, the graph of T approaches a straight line. This illustrates that T (Q) diverges logarithmically if Q approaches its limiting value d. This can be verified analytically and is characteristic of all cases where light rays approach an unstable light sphere [40].