9.3 Deflection of Light
135
L = g μν
dx
μ
ds
dx
ν
ds
=
ds
2
ds 2 = 1, particles.
(9.44)
If instead d p = ds/α, is used the Lagrangian has the value
L = g μν
dx
μ
d p
dx
ν
d p
=
ds
2
d p 2 = α
2
, particles or light.
(9.45)
It is thus clear how we may analyze null geodesics: we take the limit α
2
→ 0 in
the above so as to force the line element to zero, the function L to zero, and use a
parameter dp proportional to ds in the geodesic equation.
Let us do this explicitly for the Schwarzschild metric. Most of the analysis of
the preceding section goes through unchanged for the null geodesic, except that the
curve parameter is dp instead of ds, and the left side of (9.22c) is 0 and not 1. We
then repeat the previous planetary orbit analysis and obtain an equation like (9.27)
except that the constant term on the right side is absent, so that
u
+ u = 3mu
2
.
(9.46)
We now solve this as in the planetary problem. Let the distance of closest approach
of the photon to the body be r c = 1/u c , which we take to be much greater than m, and
define as in the planetary problem a small dimensionless parameter ε = 3m/r c =
3mu c . Then the orbit equation reads
u
+ u = ε
u
2
u c
.
(9.47)
As before we solve this by perturbation theory, and set
u = u 0 + εu 1 ,
(9.48)
so that we have from (9.47) a zeroth order and a first order equation
u
0 + u 0 = 0, u
1 + u 1 =
u
2
0
u c
.
(9.49)
The zeroth order equation is trivial, and the desired solution with arbitrary constant
C is
u 0 = C sin ϕ, or r 0 sin ϕ =
1
C
= r c .
(9.50)
This describes an undeflected straight line path as shown in Fig. 9.2, just as we should
expect.
135
L = g μν
dx
μ
ds
dx
ν
ds
=
ds
2
ds 2 = 1, particles.
(9.44)
If instead d p = ds/α, is used the Lagrangian has the value
L = g μν
dx
μ
d p
dx
ν
d p
=
ds
2
d p 2 = α
2
, particles or light.
(9.45)
It is thus clear how we may analyze null geodesics: we take the limit α
2
→ 0 in
the above so as to force the line element to zero, the function L to zero, and use a
parameter dp proportional to ds in the geodesic equation.
Let us do this explicitly for the Schwarzschild metric. Most of the analysis of
the preceding section goes through unchanged for the null geodesic, except that the
curve parameter is dp instead of ds, and the left side of (9.22c) is 0 and not 1. We
then repeat the previous planetary orbit analysis and obtain an equation like (9.27)
except that the constant term on the right side is absent, so that
u
+ u = 3mu
2
.
(9.46)
We now solve this as in the planetary problem. Let the distance of closest approach
of the photon to the body be r c = 1/u c , which we take to be much greater than m, and
define as in the planetary problem a small dimensionless parameter ε = 3m/r c =
3mu c . Then the orbit equation reads
u
+ u = ε
u
2
u c
.
(9.47)
As before we solve this by perturbation theory, and set
u = u 0 + εu 1 ,
(9.48)
so that we have from (9.47) a zeroth order and a first order equation
u
0 + u 0 = 0, u
1 + u 1 =
u
2
0
u c
.
(9.49)
The zeroth order equation is trivial, and the desired solution with arbitrary constant
C is
u 0 = C sin ϕ, or r 0 sin ϕ =
1
C
= r c .
(9.50)
This describes an undeflected straight line path as shown in Fig. 9.2, just as we should
expect.