Re: Gravitational redshift and clocks

[Stephen Speicher <sjs@COMPBIO.CALTECH.EDU>]

On Mon, 1 Sep 2003, Savinainen Antti wrote:

> General Relativity predicts that when a light ray leaves the
> ground and rises higher its frequency gets smaller (i.e., light
> is redshifted).

No. General relativity does not predict a gravitational effect on
the frequency itself, i.e., the frequency does not itself "get[s]
smaller." In general relativity the frequency is not intrinsic to
the object -- it does not itself get smaller or get larger -- but
rather it is a _relationship_ between observer and object.  The
observed redshift is a consequence of the nonlocal path through
the curvature of the spacetime manifold, not due to any change in
the local frequency itself.

> The Round & Rebka (1960) experiment confirmed the prediction.
      ^^^^^
      Pound

> On the other hand clocks run more slowly in the precence of
> gravity.

Again, clocks themselves do not "run more slowly in the
precence[sic] of gravity." Consider the tower used in the
referenced Pound and Rebka experiment. Synchronize two clocks and
place one clock at the lower section of the tower, and one clock
at the top. Compare the tick rate of the lower clock with a
collocated standard clock and determine that their tick rates are
the same. Collocate the same standard clock to the top of the
tower and determine that the tick rates of the upper clock and
the standard clock are the same.  Neither of the clocks "run more
slowly."

However, via exchanging signals either of the lower or upper
clocks will measure a difference in tick rate between it and the
other. But this measured difference is not a consequence of an
intrinsic change in tick rate of either of the clocks -- any
local measurements reveal no change in tick rate -- but rather it
is a global effect due to the curvature of the spacetime
manifold.

> This is related to the gravitational redshift since atomic
> oscillations which emit the radiation can be viewed as accurate
> clocks.

This confuses the issue: the transition energies _are_ intrinsic
to the atom, but the frequency of light is not an intrinsic
property, but rather is dependent on a relationship between the
source and the observer.

> Here comes my question: 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 :-).

The reason you cannot "see it" is because, as outlined above, it
is not true in general relativity that "time passes faster." That
notion is an old-fashioned, loosely worded interpretation which
has been largely discarded in modern formulations precisely
because it is a poor inference which tends to obfuscate, rather
than clarify.

> Could you come up with an explanation which would be suitable
> at the high school level?

The only explanation general relativity provides for any time
differences, is the differing proper time integrated along the
worldlines of the lower and upper portions at the tower. This
difference is not due to any change in the tick rate, but rather
is due to geometry -- the curvature of the spacetime manifold in
the vicinity of the tower. On a high school level one needs to
first introduce the proper conceptual foundation -- proper time,
worldline, curvature of spacetime, etc -- before making an
attempt at an integrated explanation.  Two very good books for
this conceptula foundation, on a high school level, are Taylor
and Wheeler, "Spacetime Physics," _W.H. Freeman and Co_, 1992,
and Robert Geroch, "General Relativity From A to B," _The
University of Chicago Press_, 1978.

--
Stephen
speicher@caltech.edu

Ignorance is just a placeholder for knowledge.

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