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Re: Gravitational redshift and clocks



On 09/01/2003 09:02 AM, 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). The Round & Rebka (1960) experiment
> confirmed the prediction.

Right.

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

It's not 100% clear what that means. If I assume
*) presence of gravity --> local gravitational field
(as opposed to gravitational potential)
*) run more slowly --> more slowly than an instantaneously
comoving colocated clock

... then I don't think it's true.

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

True as stated. It's related. But of course there is more
to the story.

> 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.

OK.

> 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 :-).

Again, the notion of "faster" is ambiguous. Faster
than what?

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

Let's start by reviewing what we know about curved space.
In particular, consider a town such as Tucson where they
have tried to lay out the streets on a grid, north/south and
east/west, with continuous major streets 1 mile apart. Alas
that's not possible; the problem is overconstrained. It
would work in a flat space, but it does not work on the
curved surface of the real world. It would work in a
sufficiently small town, but not in a large one. In order
to keep the major north/south streets 1 mile apart, they
have to introduce discontinuities. In Tucson, far from
downtown, the offsets are quite perceptible.

__|____|_____|_____|__
| | | |
| | | | Mercator
__|____|_____|_____|__ projection
|Down| | |
|town| | |
__|____|____|____|__
| | | |


Now there is nothing special about the terrain on the east
side of town. The whole area can be modelled as a spherical
surface for present purposes. All north/south lines on a sphere
are equivalent. You could lay out a non-distorted grid over a
small area on the east side just as easily as you could lay
one out downtown. The only problem comes when you try to
splice them together. A north-pointing vector in one place
is not parallel to a north-pointing vector in another place,
but you can't say that one place is "right" or the other
place "wrong".

===========

Now, rather than answering directly in terms of
gravitational fields, I'm going to answer in terms of
accelerations. We are taking the equivalence principle for
granted.

This converts the gravitational redshift problem into a
slightly messy version of the travelling twins problem.
For an analysis, including a useful diagram, see
http://www.av8n.com/physics/twins.htm

Pedagogical remark: I wouldn't start with gravitational
redshifts. For most high-school students, I would
try to morph such questions into a discussion of the
travelling twins. After (!) they thoroughly understand
the twins, you can revisit the redshift problem.

One of the twins (Joe) stays home, while the other twin
(Moe) goes on a trip, out and back. The special thing is
that Moe must turn around, and the turnaround must involve
an acceleration. This (red)shifts Moe's opinion of what
time it is back home chez Joe.

In the special case of a sudden impulsive acceleration, you
can analyze the problem using three reference frames:
-- Joe (frame j; time tj)
-- Moe outbound (frame o; time to)
-- Moe inbound (frame i; time ti)

During the entire trip, even during (!) the turnaround, there
is no reason why Moe's clock should run fast or slow relative
to an instantaneously comoving *colocated* freely-falling
clock. Indeed if necessary, we can use the latter to *define*
"time" in accelerated frames.

Consider some particular event (X), for instance the
event that Joe considers the midpoint of his world-line,
halfway between Moe's departure and return. In different
frames, this event will have different time-coordinates.
That is, there will be a discrepancy between to(X) and
ti(X).

It is crucial to see that the magnitude of the discrepancy
depends not merely on the velocity difference between frame i
and frame o, but also on the fact that the turnaround occurs
at a location far away from X. There is a breakdown of
simultaneity at a distance (but simultaneity is perfectly
OK locally).

So we see that the gravitational time discrepancy depends
not simply on the local gravitational field, but rather on
the difference in gravitational _potential_ between the two
clocks. Potential depends on the magnitude of the acceleration
*and* on the distance.

Imagine a uniform gravitational field over an extended
region. The earth's field in a medium-large laboratory will
do nicely. A clock on a high shelf will get out of phase
with a clock on a low shelf.

However, creatures on each shelf will consider their life to
be perfectly normal. Suppose you have some mayflies that
live exactly one day. On the low shelf, they live one day
as measured by the clock on the low shelf. On the high
shelf, they live one day as measured by the clock on the
high shelf. The two clocks disagree, but you can't say that
one shelf is "right" and the other one is "wrong". If
creatures on one shelf try to schedule their affairs
according to a clock on the other shelf, they will have
big problems (unless they understand relativity). By
analogy, in Tucson the intrinsic geometry of the east side
of town is just like the geometry of downtown. Offsets
arise only if you try to patch the two regions together.

People talk about curved space. But it would be better to
talk about curved spacetime. Gravitation creates nontrivial
curvature in the *timelike* direction. If you try to patch
together timekeeping systems over an extended region,
something is going to get broken. Roughly, you can visualize
the cuts in an orange-peel map. More quantitatively, you can
build a tabletop model of geodesics in curved spactime:
http://www.av8n.com/physics/geodesics.htm