158 Chapter 4
The Slowing of Time in General Relativity
Let’s use some equations to look at the “slowing of time” in general relativity based on the
explanation in the manga.
As in the manga, we assume that Ms. A is at the top of a tall tower, Ms. B is at the bot-
tom of the tower, and Mr. C is inside the elevator next to the tower, as shown in Figure 4-1.
We also assume that each of the three people has the same clock. However, since
space-time is warped by gravity, we don’t know whether the time of each and the rate at
which each one passes (tempo) are the same.
Therefore, we will check the rate at which time passes due to gravity under the follow-
ing three conditions:
1. Inside the free-falling elevator, there is a state of weightlessness.
2. Since special relativity holds there, the clock inside the elevator advances with constant
time intervals.
3. Ms. As clock at the top of the tower and Ms. B’s clock at the bottom of the tower each
advance with different constant time intervals.
In addition, we will use the following procedure to check the rate at which time passes
due to gravity.
1. Align the rates at which time passes for Mr. C’s clock inside the elevator and Ms. A’s clock
at the beginning of the descent.
2. Compare the rates at which time passes for Mr. C’s clock inside the elevator and Ms. B’s
clock at the end of the descent.
At first, since Ms. A and Mr. C are at the same height, they are affected by the same
gravity.
Let z denote the height direction at that location, and let f
1
denote the gravitational
potential. The gravitational potential is the quotient of the potential energy divided by the
mass of an object. For example, the potential energy of gravity near the surface of the Earth
is mgh, and the gravitational potential is gh.
Therefore, we will align Ms. As and Mr. C’s times and the rates at which time passes.
Let Dt
1
denote the time that passes at Ms. As location, and let Dt
2
denote the time
that passes at Ms. B’s location.
Now let’s assume that the cable that is holding the elevator is cut and the elevator starts
to free-fall. Since the falling velocity immediately after the cable is cut (the velocity at which
Ms. A is flying upward when viewed from Mr. C’s perspective) is v = 0, the tempos of Ms. As
and Mr. C’s clocks are the same.
u
The elevator is pulled by gravity, and its velocity gradually increases. The elevator
passes alongside Ms. B at a certain velocity (v).
If Ms. B were viewed at that time by Mr. C inside the elevator, he would observe her to
be moving upward toward himself, which is the reverse of his own motion (falling from the
top of the tower toward the bottom) viewed from his surroundings (see Figure 4-2).
τ τ
1 3
=
RELATIVITY_03.indb 158 3/15/2011 3:28:12 PM
What Is General Relativity? 159
Figure 4-1: Aligning the rates at which time passes for Mr. C’s clock inside the elevator and Ms. A’s clock at the
beginning of the descent
Let φ
1
denote Ms. As
gravitation potential.
∆τ
1
of Ms. As clock
∆τ
3
of Mr. C’s clock
inside the elevator
Elevator
Ms. As and Mr. C’s time intervals (tempos)
are the same.
Let φ
2
denote Ms. B’s
gravitation potential.
∆τ
2
of Ms. B’s clock
Figure 4-2: Comparing the rates at which time passes for the clocks of Mr. C inside the elevator and Ms. B at the
end of the descent
∆τ
2
of Ms. B’s clock
When observed by Mr. C,
Ms. A, Ms. B, and the
tower appear to be
moving upward.
∆τ
1
of Ms. As clock
Ms. As and Mr. C’s time intervals (tempos) differ.
Let φ
2
denote Ms. B’s
gravitation potential.
∆τ
3
of Mr. C’s clock
inside the free-falling
elevator
RELATIVITY_03.indb 159 3/15/2011 3:28:13 PM

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