Gravitational Lensing 71- 98 ′′± 0. 12 ′′at Sobral and 1. 61 ′′± 0. 30 ′′at Principe. This confirmed the predicted
value of 1. 750 ′′with reasonable confidence and excluded the Newtonian value of - 875 ′′. The measurements have been repeated many times since then during later
solar eclipses, with superior results confirming the general relativistic prediction.
The case of starlight passing near the Sun is a special case of a general lensing
event, shown in Figure 4.1. For the Sun the distance from lens to observer is small so
that the angular size distances are퐷LS≈퐷S, which implies that the actual deflection
equals the observed deflection and훼=휃I−휃S. In the general case, simple geometry
gives the relation between the deflection and the observed displacement as
훼=퐷S
퐷LS
(휃I−휃S), (4.5)
For a lens composed of an ensemble of point masses, the deflection angle is, in this
approximation, the vectorial sum of the deflections of the individual point lenses.
When the light bending can be taken to be occurring instantaneously (over a short
distance relative to퐷LSand퐷L), we have a geometrically thin lens, as assumed in
Figure 4.1. Thick lenses are considerably more complicated to analyze.
Strong Lensing. The termsweak lensingandstrong lensingare not defined very
precisely. In weak lensing the deflection angles are small and it is relatively easy
to determine the true positions of the lensed objects in the source plane from their
displaced positions in the observer plane. Strong lensing implies deflection through
larger angles by stronger potentials. The images in the observer plane can then become
quite complicated because there may be more than one null geodesic connecting
source and observer, so that it is not even always possible to find a unique mapping
onto the source plane. Strong lensing is a tool for testing the distribution of mass in
the lens rather than purely a tool for testing general relativity.
If a strongly lensing object can be treated as a point mass and is positioned exactly
on the straight line joining the observer and a spherical or pointlike lensed object, the
lens focuses perfectly and the lensed image is a ring seen around the lens, called an
Einstein ring. The angular size can be calculated by setting the two expressions for훼,
Equations (4.4) and (4.5), equal, noting that휃S=0andsolvingfor휃I:
휃I=
√
4 GM퐷LS
푐^2 퐷L퐷S
. (4.6)
For small푀the image is just pointlike. In general, the lenses are galaxy clusters or
(more rarely) single galaxies that are not spherical and the geometry is not simple, so
that the Einstein ring breaks up into an odd number of sections of arc. Each arc is a
complete but distorted picture of the lensed object.
In general, the solution of the lensing equation and the formation of multiple
images can be found by jointly solving Equations (4.4) and (4.5). Equation (4.4) gives
the bend angle훼g(푀b)as a function of the gravitational potential for a (symmetric)
mass푀bwithin a sphere of radius푏, or the mass seen in projection within a cir-
cle of radius푏.FromFigure4.1wecanseethat푏=휃I×퐷LS, so inserting this into