In a 2006 article in this journal,1 Drake discusses the effect of the Earth's gravitational field on the dilation of time as measured by clocks located on the surface of the Earth. In particular, he compares the time as measured by a clock at the equator to the time measured by an identical clock at the pole. This was a comparison already made by Einstein in his1905 relativity paper where he wrote, “Thence we conclude that a balance clock at the equator must go more slowly, by a very small amount, than a precisely similar clock situated at one of the poles under otherwise identical conditions.”2 It is interesting that Einstein specified the type of clock. If by a balance clock he meant a pendulum then his statement is correct, though not for the reasons he thought (he did not yet know about the effect of gravity on time). On the other hand, using an atomic clock and including the effect of gravity the statement is false. In his 2006 article, Drake does not specify the clocks to which his arguments apply, and that is the purpose of this comment.

At any point on the Earth's surface, there is both a gravitational and a centrifugal force, the latter being zero at the poles and maximal at the equator. Hence, there is an effective potential that takes into account both forces, though it is quite complex because the Earth is not a sphere. In the paper cited this potential is given by3

$Φeff=−GMer−J2GMea2(1−3 cos2θ)2r3−12ω2r2 sin2θ,$
(1)

where $GMe=3.986 004 42×1014 m3/s2$ is the product of the gravitational constant and Earth's mass, $J2=1.082 636×10−3$ is a measure of Earth's equatorial bulge, $a=6 378 137 m$ is Earth's equatorial radius, and $ω=7.292 116×10−5 rad/s$ is Earth's rotation rate. As complicated as this potential may look, it has the same value at all points on the Earth's surface as the reader can verify by studying the value at the pole and the equator where the polar radius is $6 356 760 m$ with a small error. This is as it must be because otherwise Earth's crust would move.

The time dilation generated by the potential, if it is weak, is given by4

$dtdτ=1−Φeffc2,$
(2)

with τ being the proper time. This is the formula that applies to atomic clocks where the time is defined in terms of the frequency of an atomic transition; it tells us that the time measured this way would be the same at the pole and the equator. But what about the pendulum clock?

The key point about the pendulum is that when it is in free fall it no longer oscillates—there is no force of gravity. Thus, the period of the pendulum must vary inversely as the gravitational acceleration $g$ to some power. But this power must be 1/2 because $g$ is the only parameter in the problem that has a time dimension ($g$ has dimensions $m/s2$). If the length of the pendulum is $L$ then the period must be proportional to $L/g$. A standard calculation shows that the constant of proportionality is $2π$, so the period $T$ is given approximately by

$T=2πLg .$
(3)

We have argued that the gravitational potential is constant on Earth's surface, which means the total derivative with respect to $r$ must vanish, but not the partial derivative. Indeed, the acceleration in the radial direction is given by

$g=∂∂rΦeff .$
(4)

Taking the derivative of Eq. (1), we are led to

$g=GMer2+3J2GMea2(1−3 cos2θ)2r4−ω2r sin2θ ,$
(5)

and plugging in the numbers, we find that

$gpole=9.896 551 34 m/s$
(6)

while

$geq=9.839 650 901 m/s.$
(7)

So Einstein was right after all, if by a balance clock he meant a pendulum clock. A pendulum balance clock does go slower at the equator than the pole!

The author is grateful to Jim Peebles, Wolfgang Rindler, Luis Alvarez-Gaume, and an anonymous referee for very helpful comments.

1.
S. M.
Drake
, “
The equivalence principle as a stepping stone from special to general relativity: A Socratic dialog
,”
Am. J. Phys
,
74
,
22
25
(
2006
).
2.
A.
Einstein
,
The Principle of Relativity with notes by A. Sommerfeld
(
Dover Publications
,
New York
,
1952
), pp.
49
50
. The German term he uses isunruhuhr, which is better translated as “spring wound clock.” I have chosen the pendulum because of the effects of gravity. The spring wound clock would read the same at the pole as the equator. As the rotational speed of the Earth at the equator is only about 478 m/s, I have neglected the effects of special relativity in this note, which are of order 10−12 and thus much smaller than the general relativistic effects I am considering.
3.

Implicit in this discussion is the underlying metric $−c2 dτ2=−(1+2Φc2)c2 dt2+(1−2Φc2)dr2+r2 dθ2+r2 sin2θ dϕ2$. In a private communication, Luis Alvarez-Gaume has remarked that this metric is not technically correct because the Earth is rotating so that $θ$ is a function of time. Thus, this diagonal form of the metric is only an approximation (although in this case a very good one).

4.

See appendix of Ref. 1.