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