This is a curious problem. It looks basic enough to have been considered before,
but I find nothing related anywhere.

Background: The gravitational Doppler shift formula is straightforward when the
energy change is proportionately small. Tracking energy E, we take
E_received = E0 (1 - g dot h/c^2).
Following the equivalence principle, we expect this to apply in general to
relative gravity fields, whether "real" or caused by relative motion, or even
the *combination of both*.

However, if you directly work out the change in relative energy for a photon
traversing a frame moving at constant velocity in a g-field (ie, elevator), the
result is not as given by the Eq. applied to local values of g and h for the
elevator K'. This problem would not appear in Newtonian physics. (Let us know if
you've heard of anything like this problem, especially with references.) I think
this effect is testable with current equipment. Critique is fine, but it won't
help unless you show your work. I think sci.astro is appropriate given the GR
crowd, and the problem could have cosmological consequences.

The point can be summarized as: First, assume you are in a "rest frame" K which
follows the normal rules given above, and that once light is emitted it follows
consistent rules. Then, consider a moving frame and track the paths and energy
changes of photons emitted from one place to another, combining the effects of
both gravity and local relative velocity. Photons emitted from the trailing end
P1 of the elevator travel farther along the "rest" frame of the gravity field to
catch up, and thus suffer greater gravitational Doppler shift in transit than
would normally apply with that elevator height. Photons emitted from the leading
end P2 of the elevator travel less rest-frame distance than if the elevator were
at rest, and show less GD shift. Since the velocities of the ends are locally
the same, the velocity part of Doppler shift (which acts at the moments of
emission and reception) cancels out, leaving the travel-based discrepancy.
(Putting the math simply: the emission and reception Doppler formulae give
inverse values, leaving the proportional change from gravitational effect to be
solely that determined by the distance the photon travels in K.) Consider also
reflecting a photon from the other end of the elevator: the photon is received
at a different potential (in K) than when emitted, but combined velocity effects
all cancel out. Therefore, it must show a net energy change, which wouldn't
happen in "normal" gravity fields. It perplexes me too, but this is the result
of directly working things out.

I'll work it out in more detail, but using the simple case of light going
parallel to g. If we combine Lorentz contraction (which does locally apply in a
g-field) and "catch-up" calculations, we get the following for the values of h
as actually moved in K, in terms of proper elevator height L0 and its velocity
v, which is locally consistent:

h = gamma*L0 (1 + v/c), with v signed negative when sent from a leading end and
L0 signed negative when light moves down. Things may get more complicated when v
approaches c, but at low v it is clear that the top of the elevator will move
very nearly this extra margin before receiving a photon, etc. Hence, when we
plug this formula into the GD shift we get

E_received = E0 [1 - gamma*g L (1 + v/c)/c^2]


Since the g' felt in K' is multiplied by gamma (check with transformations), the
actual discrepancy versus relative g' is
E_received = E0 [1 - g' L (1 + v/c)/c^2]. This seems like it would violate
energy conservation, since the photon's energy change does not correspond to the
work doing moving the mass-energy in the local g', but remember that when we
move the elevator, the impulse from photon emission and reception must be
accounted for.

This problem raises questions about the equivalence principle also, since we'd
expect things to work out normally for an "elevator" attached to an accelerating
reference frame - after all, the elevator is just "accelerating" at some rate,
albeit refined by hyperbolic motion, and relative signals should follow the
usual (?) rule. OTOH, such an elevator moves through regions of increasing or
decreasing proper acceleration, per Born motion. But the really big problem is
this: if we let a small box free fall within the elevator, the Doppler shifts
from one end to the other won't cancel out, since they are asymmetrical. (That
is, the increments of velocity from falling will not cancel out the asymmetrical
net shifts within the elevator.) Another problem I've thought of about the EP:
if light is sent obliquely from one part of a system in hyperbolic motion
towards a higher region, it should take very long to arrive, and be subject to
great motional Doppler shift - more than the amount appropriate to the
equivalent potential change.

"Neil" wrote in message
...
This is a curious problem. It looks basic enough to have been considered
before,
but I find nothing related anywhere.

Background: The gravitational Doppler shift formula is straightforward when
the
energy change is proportionately small. Tracking energy E, we take
E_received = E0 (1 - g dot h/c^2).
Following the equivalence principle, we expect this to apply in general to
relative gravity fields, whether "real" or caused by relative motion, or even
the *combination of both*.
snip
Just so there isn't any confusion about proper versus measured length, here's
the corrected portion referencing proper elevator height L0:

E_received = E0 [1 - gamma*g L0 (1 + v/c)/c^2]


Since the g' felt in K' is multiplied by gamma (check with transformations), the
actual discrepancy versus relative g' is
E_received = E0 [1 - g' L0 (1 + v/c)/c^2]. This seems like it would violate
energy conservation, since the photon's energy change does not correspond to the
work doing moving the mass-energy in the local g', but remember that when we
move the elevator, the impulse from photon emission and reception must be
accounted for.

Neil wrote:
This is a curious problem. It looks basic enough to have been considered before,
but I find nothing related anywhere.

Background: The gravitational Doppler shift formula is straightforward when the
energy change is proportionately small. Tracking energy E, we take
E_received = E0 (1 - g dot h/c^2).
Following the equivalence principle, we expect this to apply in general to
relative gravity fields, whether "real" or caused by relative motion, or even
the *combination of both*.
snip

I've been waiting since this morning for one of the regulars to answer
this, it has made me very curious. I think you should consider the
problem in a homogeneous gravitational field; the strong equivalence
principle requires uniform acceleration/fields if I'm not mistaken.

If the field is uniform, should the "rest frame" photon shift less than
the "inertial frame" photon, is that your question? The observer in the
rest frame will see a bigger energy change in the photon in the inertial
frame thanks to the longer path length through the gravitational field.
Isn't this expected? If different observers measure different
momenta, this result shouldn't be a surprise.

Nevertheless, I hope someone who knows more than me can answer this, so
I may sleep soundly.

Andrew
--
http://nuclear.gla.ac.uk/~andrew
Remove wizard to send email.

Andrew wrote in message ...
Neil wrote:
This is a curious problem. It looks basic enough to have been considered before,
but I find nothing related anywhere.

Background: The gravitational Doppler shift formula is straightforward when the
energy change is proportionately small. Tracking energy E, we take
E_received = E0 (1 - g dot h/c^2).
Following the equivalence principle, we expect this to apply in general to
relative gravity fields, whether "real" or caused by relative motion, or even
the *combination of both*.
snip

I've been waiting since this morning for one of the regulars to answer
this, it has made me very curious. I think you should consider the
problem in a homogeneous gravitational field; the strong equivalence
principle requires uniform acceleration/fields if I'm not mistaken.

If the field is uniform, should the "rest frame" photon shift less than
the "inertial frame" photon, is that your question? The observer in the
rest frame will see a bigger energy change in the photon in the inertial
frame thanks to the longer path length through the gravitational field.
Isn't this expected? If different observers measure different
momenta, this result shouldn't be a surprise.

Nevertheless, I hope someone who knows more than me can answer this, so
I may sleep soundly.

Andrew

Wow - that final sentiment shows that you really care about physics.
I'd like to tell you that you can, but there is an ironic twist -
solving this particular problem might make another one worse. That
other problem is a big deal, actually, and outlined at the end. First,
some background clarifications. We often hear of "uniform
gravitational fields" in discussions of GR and the Equivalence
Principle. That is a convenient approximation, however: the relative
gravity field around an observer in constant proper (co-moving
standards) acceleration is not really uniform. This follows from Born
"rigid" motion, such that g = -c^2/X, where X is a co-moving Rindler
coordinate of effective proper distance for this observer. (ie, the
accelerating metric standard must Lorentz contract ever more as it
gets faster, and some parts more than others, weird as that sounds)
Hence, gravity is more intense if one moves farther down in the
acceleration field. (I accept all this - it works out nicely with
relative Doppler shift, time dilation, etc.)

My problem does not really depend on the subtle way in which gravity
varies from point to point. In either a Rindler field or a
hypothetical true uniform field, my paradox depends on the fact that
an emitted photon has to travel further to catch up with the leading
end of the elevator, less to reach the trailing end (uniform velocity,
not to be confused with Einstein's free-falling or accelerating
elevator example.) Given my initial assumption (both ends of the
elevator always travel at the same *locally-defined* velocity v), the
problem is real. However, I now think that assumption is wrong. To
maintain it's "rigidity" in a gravity field, I think that a moving
extended body maintains the same "universal" velocity relative to
*synchronized clocks*, rather than in terms of local standards. IOW,
both ends of the body move at say 0.6c as measured by such clocks. If
the lower end moves locally at 0.6c, then the higher end is going only
0.6(1 - gy/c^2) by ordinary local time because the special
synchronized clocks run slower (to keep up with lower, red-shifted
ones) by local standards. [I use "y" now for height change.] Without
going into the math, this compensates for the different "absolute"
distances traveled by photons moving across the elevator, at least at
modest speeds (I didn't generalize beyond v c.) This paradox is
then solved.

However, there is another problem, which ironically I thought the
first effect could resolve were it real! This is not the same as the
old question about whether an accelerating charge radiates, when the
acceleration is inertial and requires external force. The other
problem is this: Let a charged body Q (macroscopic, to avoid particle
physics problems) undergo harmonic oscillation with a diameter-tunnel
through the earth. Certainly, this motion must cause radiation,
however weak, even if distorted by the earth's gravity (or maybe not
really; given the spherical symmetry it might just be red-shifted a
bit.) Radiation carries energy, of course. The trouble is, Q is
floating in a free-fall inertial space environment. There is a
symmetrical tidal field around it, but not the sort of directional
correction (?) that could provide the "radiative reaction" drag force
that it needs, to oppose it's motion and require work or deceleration
in proportion to radiated energy. (Away from gravitational fields,
accelerating charges have locally non-inertial environments, such that
field lines can be piled denser away from acceleration, etc. This is
equivalent to the trailing end moving into the field of the leading
end before the field catches up to source motion, etc., as discussed
by Feynman et al.) It should oscillate unimpeded, providing radiation
energy without work input. Hence, a paradox, since we can't find a
natural drag on an emitter of energy. I am posting this same problem
to sci.physics.relativity to see what I get.

The tie-in is: if there was a non-isotropy of the gravity in frames
moving in gravity fields, it might allow for asymmetric electrical
fields between the leading and trailing ends of the moving charge,
which is what we would need to solve the free-falling charge problem.
However, that paradox now looks solved, leaving the other one to
perplex!

Neil Bates

"Neil" wrote in message ...
This is a curious problem. It looks basic enough to have been considered before,
but I find nothing related anywhere.

Background: The gravitational Doppler shift formula is straightforward when the
energy change is proportionately small. Tracking energy E, we take
E_received = E0 (1 - g dot h/c^2).
Following the equivalence principle, we expect this to apply in general to
relative gravity fields, whether "real" or caused by relative motion, or even
the *combination of both*.

However, if you directly work out the change in relative energy for a photon
traversing a frame moving at constant velocity in a g-field (ie, elevator), the
result is not as given by the Eq. applied to local values of g and h for the
elevator K'. This problem would not appear in Newtonian physics. (Let us know if
you've heard of anything like this problem, especially with references.) I think
this effect is testable with current equipment. Critique is fine, but it won't
help unless you show your work. I think sci.astro is appropriate given the GR
crowd, and the problem could have cosmological consequences.

The point can be summarized as: First, assume you are in a "rest frame" K which
follows the normal rules given above, and that once light is emitted it follows
consistent rules. Then, consider a moving frame and track the paths and energy
changes of photons emitted from one place to another, combining the effects of
both gravity and local relative velocity. Photons emitted from the trailing end
P1 of the elevator travel farther along the "rest" frame of the gravity field to
catch up, and thus suffer greater gravitational Doppler shift in transit than
would normally apply with that elevator height. Photons emitted from the leading
end P2 of the elevator travel less rest-frame distance than if the elevator were
at rest, and show less GD shift. Since the velocities of the ends are locally
the same, the velocity part of Doppler shift (which acts at the moments of
emission and reception) cancels out, leaving the travel-based discrepancy.
(Putting the math simply: the emission and reception Doppler formulae give
inverse values, leaving the proportional change from gravitational effect to be
solely that determined by the distance the photon travels in K.) Consider also
reflecting a photon from the other end of the elevator: the photon is received
at a different potential (in K) than when emitted, but combined velocity effects
all cancel out. Therefore, it must show a net energy change, which wouldn't
happen in "normal" gravity fields. It perplexes me too, but this is the result
of directly working things out.

I'll work it out in more detail, but using the simple case of light going
parallel to g. If we combine Lorentz contraction (which does locally apply in a
g-field) and "catch-up" calculations, we get the following for the values of h
as actually moved in K, in terms of proper elevator height L0 and its velocity
v, which is locally consistent:

h = gamma*L0 (1 + v/c), with v signed negative when sent from a leading end and
L0 signed negative when light moves down. Things may get more complicated when v
approaches c, but at low v it is clear that the top of the elevator will move
very nearly this extra margin before receiving a photon, etc. Hence, when we
plug this formula into the GD shift we get

E_received = E0 [1 - gamma*g L (1 + v/c)/c^2]


Since the g' felt in K' is multiplied by gamma (check with transformations), the
actual discrepancy versus relative g' is
E_received = E0 [1 - g' L (1 + v/c)/c^2]. This seems like it would violate
energy conservation, since the photon's energy change does not correspond to the
work doing moving the mass-energy in the local g', but remember that when we
move the elevator, the impulse from photon emission and reception must be
accounted for.

This problem raises questions about the equivalence principle also, since we'd
expect things to work out normally for an "elevator" attached to an accelerating
reference frame - after all, the elevator is just "accelerating" at some rate,
albeit refined by hyperbolic motion, and relative signals should follow the
usual (?) rule. OTOH, such an elevator moves through regions of increasing or
decreasing proper acceleration, per Born motion. But the really big problem is
this: if we let a small box free fall within the elevator, the Doppler shifts
from one end to the other won't cancel out, since they are asymmetrical. (That
is, the increments of velocity from falling will not cancel out the asymmetrical
net shifts within the elevator.) Another problem I've thought of about the EP:
if light is sent obliquely from one part of a system in hyperbolic motion
towards a higher region, it should take very long to arrive, and be subject to
great motional Doppler shift - more than the amount appropriate to the
equivalent potential change.

In any situation where velocity (direction) is ARBITRARILY given +/-,
you may come across this contradiction, as velocity, force etc are
ALWAYS +. They may be 'less than', but not reliant on direction for
sign, and NOT 0. The Lorentz Transforms, SRelativity therefore
BS......

Jim G