ABSTRACT
The right-hand side of (10.8) is a relativistic correction to the corresponding
Newtonian equation.
Assuming mu 1, we seek approximate solutions of (10.7) of the form
u = u0 + u1, (10.9)
where is of order mu. Substituting (10.9) into (10.8) and collecting terms
of the same order yields
d2(mu0)
dφ2 +mu0 = 0, (10.10)
d2(mu1)
dφ2 +mu1 = 3(mu0)
2. (10.11)
The solution of the first equation is the straight line
u = 1
R sinφ, (10.12)
where R denotes the distance of closest approach to the origin, so that we
could have taken
= m
R . (10.13)
A particular solution of the second equation is then
u1 = m
R2 (1 + cos2 φ) (10.14)
so that our approximate solution for u takes the form
u = 1
R sinφ+
m
R2 (1 + cos2 φ). (10.15)
So our null geodesic is almost straight (the first term), and should in any
case be nearly straight for r large, that is for u small. Noting that (10.15)
implies that sinφ < 0, the (small, positive) asymptotic values φ1 = 2π− φ (on the right) and φ2 = φ − π (on the left) as r approaches ∞ give the (asymptotic) orientation of the line, as shown in Figure 10.1. Since u
approaches zero as r approaches ∞, we must have
−φi R
+ 2 m
R2 = 0 (10.16)
and the deflection angle is just the sum
δ = φ1 + φ2 = 4 m
= 4
GM
2 . (10.17)
Since the mass of the sun is
m = 1.9891× 1033 g (10.18)
and its radius is
R = 6.955× 105 km, (10.19)
a light ray that just grazes the surface of the sun would be deflected by 1.75
seconds of arc, as shown schematically in Figure 10.2. Since this deflection
is difficult to measure when the sun is shining, the best time to test this
prediction of general relativity is during a total eclipse of the sun. The
first expedition to attempt this was led by Sir Arthur Eddington in 1919;
his observations confirmed Einstein’s prediction and helped make Einstein
famous.