7.3 Gravity as a Geometric Phenomenon
105
Example 7.2 Let us return to the gravitational redshift. We have already estimated the redshift using the equivalence principle, but with the above result
relating the metric to the gravitational potential we may derive it in a more
precise and general geometric way. Consider a stationary emitter of radiation,
such as an atom, at position e and a detector at d, as shown in Fig. 7.6.
Suppose the beginning of a cycle number 1 leaves the emitter at coordinate
time x
0 and the beginning of another cycle number 2 leaves a very short coordinate time x
0 later. The paths of these travel through 3-space as a function
of time; whatever determines the path of number 1 it is clear that number 2 will
encounter very nearly the same conditions since it left a very short time later,
so 1 and 2 will follow the same path but with number 2 displaced uniformly
upwards by x
0 , as shown in the figure. Thus the coordinate time period of the
radiation will be the same at e and d. But the coordinates are merely markers
or labels for points in spacetime and have no direct physical meaning. As in
special relativity the proper time τ = s/c is what has physical meaning.
The relations between proper and coordinate time at the stationary emitter and
detector are
cτ e =
g 00 (e)x
0
, cτ d =
g 00 (d)x
0
.
(7.24)
Thus we obtain a relation between the period of the radiation at the emitter and
at the detector,
τ d
τ e
=
√
g 00 (d)
√
g 00 (e)
.
(7.25)
This is quite general and holds for widely separated emitter and detector, unlike
the equivalence principle derivation. You should think about the implication
of (7.25) when g 00 at the emitter or detector is very small or zero.
To show that this is consistent with the equivalence principle result (7.12)
we use the relation between g 00 and the gravitational potential in (7.23) and
expand, assuming a weak field, to get
τ d =
√
g 00 (d)
√
g 00 (e)
τ e =
1 + 2φ(d)/c 2
1 + 2φ(e)/c 2
τ e =
1 +
φ
c 2
τ e , (7.26)
or in terms of the wavelength
τ d − τ e
τ e
=
λ
λ
=
φ
c 2 .
(7.27)
Thus the results in (7.12) and (7.25) are consistent.
105
Example 7.2 Let us return to the gravitational redshift. We have already estimated the redshift using the equivalence principle, but with the above result
relating the metric to the gravitational potential we may derive it in a more
precise and general geometric way. Consider a stationary emitter of radiation,
such as an atom, at position e and a detector at d, as shown in Fig. 7.6.
Suppose the beginning of a cycle number 1 leaves the emitter at coordinate
time x
0 and the beginning of another cycle number 2 leaves a very short coordinate time x
0 later. The paths of these travel through 3-space as a function
of time; whatever determines the path of number 1 it is clear that number 2 will
encounter very nearly the same conditions since it left a very short time later,
so 1 and 2 will follow the same path but with number 2 displaced uniformly
upwards by x
0 , as shown in the figure. Thus the coordinate time period of the
radiation will be the same at e and d. But the coordinates are merely markers
or labels for points in spacetime and have no direct physical meaning. As in
special relativity the proper time τ = s/c is what has physical meaning.
The relations between proper and coordinate time at the stationary emitter and
detector are
cτ e =
g 00 (e)x
0
, cτ d =
g 00 (d)x
0
.
(7.24)
Thus we obtain a relation between the period of the radiation at the emitter and
at the detector,
τ d
τ e
=
√
g 00 (d)
√
g 00 (e)
.
(7.25)
This is quite general and holds for widely separated emitter and detector, unlike
the equivalence principle derivation. You should think about the implication
of (7.25) when g 00 at the emitter or detector is very small or zero.
To show that this is consistent with the equivalence principle result (7.12)
we use the relation between g 00 and the gravitational potential in (7.23) and
expand, assuming a weak field, to get
τ d =
√
g 00 (d)
√
g 00 (e)
τ e =
1 + 2φ(d)/c 2
1 + 2φ(e)/c 2
τ e =
1 +
φ
c 2
τ e , (7.26)
or in terms of the wavelength
τ d − τ e
τ e
=
λ
λ
=
φ
c 2 .
(7.27)
Thus the results in (7.12) and (7.25) are consistent.