134
9 Spherically Symmetric Gravitational Fields
δϕ ∼ = 6π
m
r c
.
(9.41b)
To evaluate the precession more precisely for an elliptic orbit we need to evaluate
more accurately the constant ˜
A in (9.41a). Astronomers routinely measure a planet’s
eccentricity e and its semimajor axis, which is defined according to Fig. 9.1 as
a =
1
2
1
A(1 + e)
+
1
A(1 − e)
=
1
A
1 − e 2
, so A =
1
a
1 − e 2
.
(9.42)
Thus we can express the precession in terms of the parameters a and e, which can be
found in astronomy textbooks, as
δϕ =
6π
1 − e 2
m
a
.
(9.43)
This can now be conveniently compared with the observations of planets.
The orbit precession is most easily measured for the planet Mercury since the
semimajor axis is least for Mercury, and also since the perihelion position is accurately measurable for Mercury’s fairly eccentric orbit. Indeed it was well-known
before the development of general relativity that Mercury’s orbit precesses by about
43
per century more than predicted by classical theory (Le Verrier 1859; Newcomb
1895). Equation (9.43) gives about 43
per century for Mercury, which provided
a historically important verification of general relativity theory in its earliest days.
Indeed, Einstein himself calculated the perihelion shift and was aware of this agreement (Einstein 1923). We will return to the question of the observational verification
of general relativity in more detail in Sect. 9.4.
9.3 Deflection of Light
We have already noted how the equivalence principle predicts that light will fall in a
gravitational field. In this section we will explicitly calculate the orbit of a light ray
as it passes by a star such as the sun. The calculation is much like that for the orbit
of a planet, so we can rely heavily on the analysis of the previous section.
We first consider the nature of the orbit of a light ray or photon. In special relativity
the path of light is characterized by a null line element or ds
2
= 0. We naturally
carry this over to general relativity as a fundamental assumption. We also carry over
the geodesic motion of a particle and assume that light also follows a geodesic. Thus
we make the well-justified assumption that light follows a null geodesic. Recall that
the geodesic equation may use as an invariant curve parameter the line element ds
or a parameter proportional to it, d p = ds/α, where α is a constant. Recall also that
the function L that plays the role of a Lagrangian for the motion of bodies has the
value 1 if ds is used as a curve parameter,
9 Spherically Symmetric Gravitational Fields
δϕ ∼ = 6π
m
r c
.
(9.41b)
To evaluate the precession more precisely for an elliptic orbit we need to evaluate
more accurately the constant ˜
A in (9.41a). Astronomers routinely measure a planet’s
eccentricity e and its semimajor axis, which is defined according to Fig. 9.1 as
a =
1
2
1
A(1 + e)
+
1
A(1 − e)
=
1
A
1 − e 2
, so A =
1
a
1 − e 2
.
(9.42)
Thus we can express the precession in terms of the parameters a and e, which can be
found in astronomy textbooks, as
δϕ =
6π
1 − e 2
m
a
.
(9.43)
This can now be conveniently compared with the observations of planets.
The orbit precession is most easily measured for the planet Mercury since the
semimajor axis is least for Mercury, and also since the perihelion position is accurately measurable for Mercury’s fairly eccentric orbit. Indeed it was well-known
before the development of general relativity that Mercury’s orbit precesses by about
43
per century more than predicted by classical theory (Le Verrier 1859; Newcomb
1895). Equation (9.43) gives about 43
per century for Mercury, which provided
a historically important verification of general relativity theory in its earliest days.
Indeed, Einstein himself calculated the perihelion shift and was aware of this agreement (Einstein 1923). We will return to the question of the observational verification
of general relativity in more detail in Sect. 9.4.
9.3 Deflection of Light
We have already noted how the equivalence principle predicts that light will fall in a
gravitational field. In this section we will explicitly calculate the orbit of a light ray
as it passes by a star such as the sun. The calculation is much like that for the orbit
of a planet, so we can rely heavily on the analysis of the previous section.
We first consider the nature of the orbit of a light ray or photon. In special relativity
the path of light is characterized by a null line element or ds
2
= 0. We naturally
carry this over to general relativity as a fundamental assumption. We also carry over
the geodesic motion of a particle and assume that light also follows a geodesic. Thus
we make the well-justified assumption that light follows a null geodesic. Recall that
the geodesic equation may use as an invariant curve parameter the line element ds
or a parameter proportional to it, d p = ds/α, where α is a constant. Recall also that
the function L that plays the role of a Lagrangian for the motion of bodies has the
value 1 if ds is used as a curve parameter,