106
7 Classical Gravity and Geometry
It is worth pondering for a moment the conceptual view of gravity that is provided
by the above results. For a weak gravitational field bodies follow geodesics in the
spacetime of special relativity with a small correction: the metric is modified so that
their internal clocks tick at a slightly different rate depending on their proximity to
matter, with time intervals dτ = ds/c determined from (7.23) as
dτ =
1 +
φ
c 2
dt.
(7.28)
The potential φ is taken to be zero far from all sources of gravity.
Exercises
7.1 Using Poisson’s equation of classical gravitational theory (7.6) calculate the
potential φ and the field
g for a space filled with constant density matter. Assume
that the field is spherically symmetric about some arbitrary origin. Notice that the
uniform distribution of the matter appears to have a greater degree of symmetry
than the gravitational field. Does this bother you?
7.2 A common theme in science fiction is negative matter which falls upwards in a
gravitational field. How much general relativity do you need to know in order
to be very dubious of such a notion?
7.3 What is the gravitational redshift between a point on the earth’s surface and a
point on the sun’s surface? What is it between two points separated by a vertical
100 m on the surface of the earth? What is it between the earth’s surface and a
point at 10,000 km altitude? See Vessot (1980).
7.4 Does an experimental test of the redshift really test general relativity theory?
What if the measurement is extremely accurate? What would happen if g 00 were
zero at the point of emission? We will discuss just this situation in Chap. 10.
7.5 In our low velocity and weak field discussion the combination of velocity
squared and field strength that appears in (7.16) is h 00 − β
2 . Show that h 00
and β
2 are related and comparable for planets in circular orbit around the sun.
7.6 When we studied the Newtonian limit of (7.15) we only considered the space
parts, with μ = i. Show that the time equation, μ = 0, is consistent but does
not give us any interesting new information.
7.7 We obtained the gravitational redshift formula using two different methods.
Add a third by considering a photon moving upward in the field of the earth and
losing energy as it rises. You can do this heuristically by assigning the Planck
energy E = hv to the photon, with a corresponding effective mass m eff = E/c
2 .
7.8 In obtaining the equation of motion (7.22) we assumed that the metric was
diagonal. Repeat the derivation without this assumption; specifically, allow the
h 0 j to be nonzero so that the second equation in (7.21) no longer holds and a
velocity dependent force is added to (7.22).
7.9 Continue studying the velocity dependent force of Exercise 7.8. In classical
electromagnetism the Lorentz force on a particle moving at
v in a magnetic
field
B is proportional to
v ×
B. The magnetic field is related to a vector
potential
A by
B = ∇ ×
B, so the force is proportional to
v ×
∇ ×
A
. Show
7 Classical Gravity and Geometry
It is worth pondering for a moment the conceptual view of gravity that is provided
by the above results. For a weak gravitational field bodies follow geodesics in the
spacetime of special relativity with a small correction: the metric is modified so that
their internal clocks tick at a slightly different rate depending on their proximity to
matter, with time intervals dτ = ds/c determined from (7.23) as
dτ =
1 +
φ
c 2
dt.
(7.28)
The potential φ is taken to be zero far from all sources of gravity.
Exercises
7.1 Using Poisson’s equation of classical gravitational theory (7.6) calculate the
potential φ and the field
g for a space filled with constant density matter. Assume
that the field is spherically symmetric about some arbitrary origin. Notice that the
uniform distribution of the matter appears to have a greater degree of symmetry
than the gravitational field. Does this bother you?
7.2 A common theme in science fiction is negative matter which falls upwards in a
gravitational field. How much general relativity do you need to know in order
to be very dubious of such a notion?
7.3 What is the gravitational redshift between a point on the earth’s surface and a
point on the sun’s surface? What is it between two points separated by a vertical
100 m on the surface of the earth? What is it between the earth’s surface and a
point at 10,000 km altitude? See Vessot (1980).
7.4 Does an experimental test of the redshift really test general relativity theory?
What if the measurement is extremely accurate? What would happen if g 00 were
zero at the point of emission? We will discuss just this situation in Chap. 10.
7.5 In our low velocity and weak field discussion the combination of velocity
squared and field strength that appears in (7.16) is h 00 − β
2 . Show that h 00
and β
2 are related and comparable for planets in circular orbit around the sun.
7.6 When we studied the Newtonian limit of (7.15) we only considered the space
parts, with μ = i. Show that the time equation, μ = 0, is consistent but does
not give us any interesting new information.
7.7 We obtained the gravitational redshift formula using two different methods.
Add a third by considering a photon moving upward in the field of the earth and
losing energy as it rises. You can do this heuristically by assigning the Planck
energy E = hv to the photon, with a corresponding effective mass m eff = E/c
2 .
7.8 In obtaining the equation of motion (7.22) we assumed that the metric was
diagonal. Repeat the derivation without this assumption; specifically, allow the
h 0 j to be nonzero so that the second equation in (7.21) no longer holds and a
velocity dependent force is added to (7.22).
7.9 Continue studying the velocity dependent force of Exercise 7.8. In classical
electromagnetism the Lorentz force on a particle moving at
v in a magnetic
field
B is proportional to
v ×
B. The magnetic field is related to a vector
potential
A by
B = ∇ ×
B, so the force is proportional to
v ×
∇ ×
A
. Show