Length contraction of falling things

Q-reeus

Maybe not as precisely presented as some demand, but taken in context the intention of jartsa's points above are more than reasonable, even logically necessary. The first is simply saying conservation of energy requires 'trapped light' to act gravitationally just as for rest matter. Cyclic process of hoisting+free-fall should be a zero-sum game. And that requires no locally observed frequency change when lifting the box (=perfectly reflective cavity resonator). Which is also the second point. One might argue hoisting process could somehow convert between the trapped EM field energy and overall box PE, but how?

Say for a cubical cavity resonator, resolve cavity field into three mutually orthogonal sets of counterpropagating travelling waves. Two sets propagating transverse to radial direction, one along that axis. Sets need not be equal amplitude. It should be obvious the two transverse sets have no appreciable coupling to the only conceivable factor that hauling-up process could bring to bear as locally measured - a changing gravity/tidal gravity. In the radial set case, it is equally obvious whatever is conceivably gained/lost by a travelling wave component in one direction is reversed during the other half trip. Perhaps for an exceedingly rapid 'snatch' process a tiny disequilibrium would occur, but up to critics to prove that matters here imo. And incidentally this kind of thing can be extended to say a box full of hot gas or whatever - there is no reason to suppose any such form of trapped energy acts differently to a dead lump of rest mass. Assuming varying gravity/tidal gravity is all there is to consider. And yet......

There is a possible caveat, stemming from my own earlier considerations in a thread dealing with angular momentum of a flywheel lowered into a potential well. We had assumed the flywheel's own angular momentum was invariant wrt potential, a reasonable assumption in keeping with conservation of angular momentum. Yet from a Machian pov that seems suspect. Time dilation demands that the coordinate determined spin-rate declines with lowering potential, and this in turn naively implies an increased coordinate flywheel mass to compensate. Problem is that would conflict with the two assumptions that rest mass/energy declines with lower potential and that EP holds true (inertial mass = active gravitational mass = passive gravitational mass). A redefinition of angular momentum in gravity well is thus necessary to at all reconcile these factors.
There is however an additional possibility - that angular momentum becomes progressively redistributed between flywheel proper and the rest of the gravitating mass system. Which is in keeping with the Machian position that inertia is a mutual effect between all mass/energy. Thus build a shell of mass around a flywheel and it's coordinate spin-rate declines, but one must expect an action of flywheel's motion back on the shell. And that seems at least roughly in line with a strictly GR 'frame dragging'. One might then conclude a varying gravitational and/or gravitational tidal force is not the only possible influence on 'light in a box'.

So yes I'm having it a bit both ways here!

PeterDonis

It depends on how you define "energy" and how you define "change". Different people have different definitions they like, and they often will say (as the Okun paper that harrylin mentioned does--btw, harrylin, do you have a link to the paper itself?) that other definitions besides theirs are "misleadinng" or some such. The key thing IMHO is to remember that in GR, "energy" is not a fundamental concept; it's a way of interpreting the physics, but you don't need it to actually calculate what happens. You can calculate everything using covariant geometric objects like vectors and tensors, which have unambiguous definitions and behave in well-defined, unambiguous ways.

For example, take the photon free-falling radially towards a gravitating body. It has a 4-momentum vector [itex]p^{a}[/itex] which is parallel transported along its worldline. Since parallel transport is one common way in GR of defining what it means for a geometric object to "not change" along a curve, this implies that the photon's 4-momentum does not change. And since the norm of the 4-momentum can be defined as the "energy at infinity" of the photon, some people interpret this as saying that the energy of the photon does not change.

However, the energy that the photon will be *observed* to have depends on the 4-velocity of the observer, via the equation:

[tex]E_{obs} = g_{ab} p^{a} u^{b}[/tex]

Even if we interpret what I said above as [itex]p^{a}[/itex] not changing as the photon falls, [itex]g_{ab} u^{b}[/itex] *will* change, in general, for different observers. In particular, it will change for static observers at different altitudes, who will therefore observe the photon to have different energies, and therefore different frequencies, i.e., gravitational redshift/blueshift. Some people interpret this as saying that the energy of the photon *does* change.

Which interpretation is "right"? Does it matter? It depends on what you are using the interpretation for. Both interpretations agree on the actual physical observables, so as far as physics is concerned, they're equivalent. One may lead to more understanding and less confusion for some people, in some situations. But IMHO that's not a question of physics.

vin300

Isn't the norm of photon 4-momentum zero?

PeterDonis

Oops, yes, you're right, I was confusing the photon case with the timelike particle case. I should have just said that some people interpret the photon 4-momentum being parallel transported as its energy not changing.

harrylin

I gave a link to the paper itself
Perhaps you meant a link to a freely downloadable (=pirated) version, but no I don't have that. On the other hand, a personal message can have good results (hint).
This issue may indeed be a matter of language. If you change your reference system from for example the ECI frame to the rest frame of a rocket to the moon, does that also change the distance Earth-moon? I will say no, your change of perspective cannot affect a distance. But some people might say yes to that question.
Only if it's just a matter of formulation, and not if it's a case of truly disagreeing interpretation (it may be the one for some, and the other for some others!). Gravitational time dilation is observable independent of redshift; see posts #26 and #32.

PeterDonis

I think in large part it often is. For example:

That's because the two of you are using the word "distance Earth-Moon" to refer to two different invariants; they are the invariant lengths of two different spacelike curves between the Earth's worldline and the Moon's worldline. Changing your perspective doesn't change either invariant, but it may change which one you think should be called "the distance Earth-Moon". But that's a question of terminology, not physics. When I said everyone agrees on all actual observables, I meant everyone agrees on the values of all invariants. They may disagree on what to call them.

Austin0

No,,,,my interpretation is the same as yours.You are confusing the quote of DaleSpams I inserted with my thoughts. The bolded above is his interpretation along with his assessment of my logic in reaching my conclusion.

Austin0

The spin rate declines according to which observer? The local or the observer at the initial elevation?

Could you elaborate on the assumption that "rest mass/energy declines with lower potential " ?

harrylin

It's good to see that you agree with my example. What is still not clear to me, if some people here actually have been misled by such language or not. You seem to think that this was clear for everyone - but I doubt that. Let's hope that it's at least now clear for everyone.

Perhaps it's better to give a crystal clear example:

Put an atomic clock in a high tower and synchronise¹ it with the ECI frame, as is commonly done. Tune a crystal oscillator to exactly 1.000 GHz with that atomic clock as reference, and use it for a radio emitter. There is a similarly tuned atomic clock and a radio receiver with frequency analyser on the ground. Send during exactly 24h a 1.000 GHz radio signal to the ground, 500 m below. According to GR the received signal as read on the frequency analyser will be exactly 1.000 GHz and the emission time exactly 24 h (in an ideal² situation). That should be clear and obvious to everyone.

Harald

ADDENDUM, notes:
1. tune it to run in sync, as is commonly done for time keeping
2. also overlooking that the ECI frame isn't perfectly inertial

Naty1

PeterDonis:
Peter: It was this issue that caused me to in my earlier post try to get the original poster back to basics first. He did not seem interested in that approach. I am still unsure what he wanted to accomplish by 'putting photons in a box'. Maybe he thought they would stay the same color forever if left in place??? who knows.

In any case, I had been thinking that slowly raising photons from varying gravitational potential depths would result in a different characteristic of light at the surface versus them following null geodesics as when freely emitted; that some changes in characteristics would result. Can you comment about what you think happens?

harrylin

Apparently you assume that no light ray reflects off the bottom or the top during motion. That's not likely IMHO. And thus its effect should be analysed. Is then work done or not? And equal but opposite?
I would say trapped radiation energy - not field energy!
As discussed in parallel, analyses with "local" measurements easily complicates analysis. It can be useful to develop a theory but when you already have the theory, needless frame jumping is often counter productive.
I find Machian POV's themselves suspect.
And I think that we have largely covered the topic of this thread.

DaleSpam

That is incorrect, and this contradicts both the Pound Rebka experiment and also current experiments done with atomic clocks at different heights.

DaleSpam

I cannot tell from this comment nor from your follow-up comments whether or not you have any disagreement or any confusion from my quote.

jartsa

So here's what happens in the box that contains light and is being lifted:


Let us examine light in a mirror lined box, which is being lifted at a constant coordinate speed. The coordinate speeds of the ceiling mirror and the floor mirror are the same. The coordinate speeds of the light near the ceiling and the light near the floor are different.

Now we must study Doppler shift. When an EM-wave pulse and a mirror collide, the mirror receives wave crests at average frequency:
the number of crests in the wave / time that the collision lasts

Now, in the Doppler shift effect there is no change of number of wave crests in a wave pulse, so the change of the duration of the collision between the wave and the mirror is the thing that determines the Doppler shift.

An observer who is staying at constant altitude, and observing our box being lifted, says that near the ceiling the light has a larger coordinate speed, and therefore the duration of the collision between the ceiling mirror and the light is determined by the motion of the light in a greater degree as compared to the collision between the light and the floor mirror, so near the floor mirror the duration of the light-mirror collision is determined by the coordinate speed of the mirror in a greater degree, as compared to the situation near the ceiling.

So this observer at constant altitude says that light blue shifts, when being reflected from the floor mirror, more than it red shifts when reflected from the ceiling mirror, and therefore the observer observes a frequency increase of the light.

DaleSpam

Can you quantify that?

PeterDonis

As DaleSpam's comment on this post shows, it still isn't.

First of all, when you say "synchronise it with the ECI frame", I assume you mean not just adopting the standard of simultaneity of the ECI frame, but also its clock rate, correct? In other words, you are artificially setting the frequency of *both* oscillators, at different heights, to different clock rates than they would normally run at, similar to what is done with the clocks on board the GPS satellites. Only if you do this does your prediction of unchanged "frequency" with height make sense.

But of course this "frequency" that you are saying does not change with height is really a "coordinate frequency", not a "proper frequency"; it's not the "natural" frequency that at least one of the oscillators (and probably both) would measure if you hadn't artificially set them to the ECI clock rate. In other words, you are artifically tuning each oscillator so that the "frequency" that it measures is *not* the "natural" frequency that would be measured by an observer with the same 4-velocity. The "effective" 4-velocity of the oscillator for purposes of measuring frequency, what you would contract with the 4-momentum of the photon to compute its observed frequency, is some *other* 4-vector, a 4-vector that is associated with "coordinate time" in the ECI frame.

All of which illustrates that it's tough to be clear, even when you are really trying to be.

PeterDonis

In a simple model where the light inside the box travels freely except when it hits a wall of the box, the periods of free travel can still be modeled as they would be if the box weren't there. But the collisions with the wall of the box change things.

The simplest assumption is that the walls of the box are perfectly reflecting mirrors; this means that each collision of a photon with a box wall can be modeled as perfectly elastic, with the photon's momentum perpendicular to the box wall reversing direction (and a corresponding change in the momentum of the box). In another thread I used this type of model to describe how one could extract work from a box filled with photons by slowly lowering it instead of letting it freely fall, and how extracting the work would lower the "photon temperature" inside the box, whereas a freely falling box of photons would have the same "photon temperature" inside as it fell (relative to the box, in both cases). But in either case, the "average" motion of the photons inside the box is no longer null, but timelike--it's just the motion of the center of mass of the box+photons.

One could try more complicated models, but I'm not sure what the point would be. The key point is already clear from the above simple model: the behavior of light inside a box, when the interaction with the box is included, does change.