7.2 The Equivalence Principle
101
Fig. 7.5 The Doppler shift or redshift experimental layout in the accelerated lab
Example 7.1 We analyze the gravitational redshift using the equivalence
principle.
Consider on the floor of an accelerated lab a light source which emits a
pulse of light o of wavelength λ, as shown in Fig. 7.5. This moves upward a
distance z to be by a detector in about time = z/c. During this time the
lab has accelerated and the detector is moving upward slightly faster than the
source was moving when the light was emitted, with = g Thus there
will be a Doppler shift in the light, given by (1.27), of approximately
λ obs − λ = =
c
λ,
λ
=
c
=
g
c
=
gz
c 2 .
(7.11)
The equivalence principle tells us that we see the same effect in the earth
lab, the light redshifted to longer wavelength. Moreover in the earth-based lab
= gz is the difference between the gravitational potential at the source and
the receiver, so we may write
λ
=
c 2 .
(7.12)
This prediction has been tested experimentally. A very accurate test used a
microwave system in a high-altitude rocket sent to about 10
4 km, which gave
a result in agreement with the above prediction to better than about a part in
10
4 (Vessot 1980).
It is worth noting that any redshift experiment only tests the equivalence
principle and the general ideas of general relativity, but since we have not yet
presented the field equations it is clearly not a test of the field equations and
the full relativity theory.
101
Fig. 7.5 The Doppler shift or redshift experimental layout in the accelerated lab
Example 7.1 We analyze the gravitational redshift using the equivalence
principle.
Consider on the floor of an accelerated lab a light source which emits a
pulse of light o of wavelength λ, as shown in Fig. 7.5. This moves upward a
distance z to be by a detector in about time = z/c. During this time the
lab has accelerated and the detector is moving upward slightly faster than the
source was moving when the light was emitted, with = g Thus there
will be a Doppler shift in the light, given by (1.27), of approximately
λ obs − λ = =
c
λ,
λ
=
c
=
g
c
=
gz
c 2 .
(7.11)
The equivalence principle tells us that we see the same effect in the earth
lab, the light redshifted to longer wavelength. Moreover in the earth-based lab
= gz is the difference between the gravitational potential at the source and
the receiver, so we may write
λ
=
c 2 .
(7.12)
This prediction has been tested experimentally. A very accurate test used a
microwave system in a high-altitude rocket sent to about 10
4 km, which gave
a result in agreement with the above prediction to better than about a part in
10
4 (Vessot 1980).
It is worth noting that any redshift experiment only tests the equivalence
principle and the general ideas of general relativity, but since we have not yet
presented the field equations it is clearly not a test of the field equations and
the full relativity theory.