7 0 D O C . 1 3 D I A L O G U E A B O U T R E L A T I V I T Y T H E O R Y
One must carefully keep in focus that both columns, left and right, describe
exactly the same process, but the description on the left refers to the coordinate
system , the one on the right to the coordinate system . According to both
descriptions, it is the clock which lags by a certain amount behind clock at
the end of the process considered. With respect to the coordinate system K' the
phenomenon is explained in the following manner: During procedural steps 2 and
4, clock moving at velocity v, has indeed a slower rate than clock which is
at rest. But the time lag gets overcompensated by the faster rate of during pro-
cedural step 3. Because, according to the general theory of relativity, a clock has a
more accelerated rate the higher the gravitational potential is at the clock’s loca-
tion; and during procedural step 3, is indeed at a location of higher gravita-
tional potential than Calculation shows that this running-ahead amounts to
precisely twice as much as the lag-behind during the procedural steps 2 and 4.
This analysis clarifies completely the paradox you referred to.
Krit.: I see, indeed, you extricated yourself very skillfully, but I would be
lying if I declared myself completely satisfied. The cause of contention is not
removed, only pushed to another place. Because your analysis of the difficulty
shows only its connection with another one, which also has been raised repeat-
edly. You solved the paradox by taking into account the influence on clocks of the
gravitational field, which is relative to But isn’t this gravitational field only
fictitious? Its existence is, I should say, only simulated by the choice of coordi-
nates. After all, real gravitational fields are always generated by masses and can-
not be made to vanish by a suitable choice of coordinates. How should one believe
that a merely fictitious field could influence the rate of clocks?
Rel.: First I have to point out that the distinction of real versus non-real is not
very productive. With respect to the gravitational field “exists” just as any
other physical object that can only be defined in relation to a system of coordi-
nates, even though it may not be there relative to the system . There is nothing
particularly strange in this, as can easily be seen from the following example from
classical mechanics. Nobody doubts the “reality” of kinetic energy because, other-
wise, one would have to deny the reality of energy per se. But it is clear that the
kinetic energy of a body depends upon the state of motion of the coordinate sys-
tem; with a suitable choice of the latter, one obviously can cause the kinetic
energy of a body in translatory movement to take on, at a certain moment, any
given positive value or, say, the value zero. In the special case where all masses
have velocities equal in size and direction, a suitable choice of the coordinate sys-
tem can bring the entire kinetic energy to zero. I think the analogy is a complete
one.
K K′
U2 U1
U1, U2
U1
U2
[8]
U1.
[9]
K′.
[10]
K′
K
[11]