Re: Gravitational redshift and clocks

[John Mallinckrodt <ajm@CSUPOMONA.EDU>]

> I understand that the frequency of light (or any electromagnetic
> wave) decreases when it "climbs" higher from the ground. This means
> that period of the wave motion increases. Is there a way to "see"
> that this (decrease in f, increase in T) implies that time passes
> faster at higher altitude? For some reason I can't see it :-). Could
> you come up with an explanation which would be suitable at the high
> school level?

Assume for a moment that both clocks run at the same rate and, to be
needlessly but probably helpfully concrete, suppose one "wave" leaves
the ground every second.

These waves travel upward and eventually pass the higher altitude
observer.  If they pass that observer at a rate less than one per
second, then the number of waves in the intervening space would have
to be monotonically increasing with time.  (The argument ruling out
that possibility is not completely devoid of subtlety and does
involve a couple of usually tacit assumptions about the static nature
of space, but is "left as an exercise" and can probably be left to
take up only in the event that a student asks.)  The argument is the
same but more easily justified in the case that the waves pass the
higher observer at a rate of more than one per second.

Thus, if the clocks run at the same rate (i.e., if time passes at the
same rate in both locations), then the observed frequencies will be
the same.  The fact that the observed frequencies are not the same
then demonstrates that the clocks can't run at the same rate.

Having established an operational procedure for determining the
relative frequency, it should now be pretty easy to see that the way
to get a lower frequency at the higher altitude is to have that
observer's clock run faster so that fewer crests pass in any given
"amount of time."

--
John Mallinckrodt        mailto:ajm@csupomona.edu
Cal Poly Pomona          http://www.csupomona.edu/~ajm