List of Footnotes

1 As stated above. Only very recently it was shown that Einstein had derived these equations as early as 1912, but did not bother to publish them [153].
2 A detailed description of optics in curved spacetimes and a derivation of the lens equation from Einstein’s field equations can be found in Chapter 3 and 4 of [167Jump To The Next Citation Point].
3 In cosmology, the various methods to define distance diverge (see, e.g. Chapters 14.4 of [202Jump To The Next Citation Point] or 3.5 of [167Jump To The Next Citation Point]). The relevant distances for gravitational lensing are the angular diameter distances, see [126Jump To The Next Citation Point].
4 Due to the fact that physical objects have a finite size, and also because at some limit wave optics has to be applied, in reality the magnification stays finite.
5 This does not violate energy conservation, since this is the magnification relative to an “empty” universe and not relative to a “smoothed out” universe. This issue is treated in detail in, e.g., [163] or in Chapter 4.5 of [167Jump To The Next Citation Point].
6 There exists a theorem that gravitational lenses should produce an odd number of images (e.g., [120]).
7 This can be seen very simply: Imagine a lens situation like the one displayed in Figure 2View Image. If now all length scales are reduced by a factor of two and at the same time all masses are reduced by a factor of two, then for an observer the angular configuration in the sky would appear exactly identical. But the total length of the light path is reduced by a factor of two. Now, since the time delay between the two paths is the same fraction of the total lengths in either scenario, a measurement of this fractional length allows to determine the total length, and hence the Hubble constant, the constant of proportionality between distance and redshift.
8 Similarly, one cannot determine the temperature of a black body by measuring the energy of a single photon emitted by the black body, but one needs to measure a large number of them and compare with some underlying theory.
9 A well-known exception is the light deflection at the solar limb, where the difference between the lensed and the unlensed positions of stars was used to confirm General Relativity, see Chapter 2.