High School Does E=mc^2 apply to gravitational potential energy?

[Dale]

The energy accounting for photon out of the box vs. photon in the box should be:

Photon out of the box: photon energy Ep plus box energy Eb.

I agree it should be that way, but the whole point was to treat the box and the light separately. If you do it that way then you have to consider that the box is different in those cases

In any case, there are now at least three possible explanations, so it is not clear that there actually is any problem. And the conservation of energy makes it clear that there is not a problem.

 

[vanhees71]

I don't think this is correct. The energy accounting for photon out of the box vs. photon in the box should be:

Photon out of the box: photon energy $E_\text{p}$ plus box energy $E_\text{b}$.

Photon in the box: photon energy $E_\text{p}$ plus box energy $E_\text{b}$ plus pressure exerted by photon on box wall $p$ minus tension in box wall $p$. As long as we treat the box with the photon inside as static, the pressure/tension terms cancel and the total energy is unchanged.

In other words, putting the photon into the box, or taking it out of the box, can be done, in the idealized case, without any work having to be done either by the environment on the photon/box, or by the photon/box on the environment, as long as we consider everything to happen at constant altitude.

It depends a bit on, what you mean by "photon in a box". Take a box with ideally reflecting walls. Then you have the usual cavity modes. A "photon in the box" means that the em. field state is the corresponding one-photon Fock state, i.e., one of the eigenmodes is occupied by one photon with energy $\hbar \omega$ (as measured in the rest frame of the box). The total mass of the box then is $m_{1\gamma}=m_0+\hbar \omega/c^2>E_{0\gamma}/c^2=m_{\text{empty box}}$.

 

[PeterDonis]

The total mass of the box then is $m_{1\gamma}=m_0+\hbar \omega/c^2>E_{0\gamma}/c^2=m_{\text{empty box}}$.

Yes, and that is equivalent to saying that no work needed to be done in either direction to put the photon in the box, because the mass of the box is just the sum of the empty box mass plus the mass-equivalent of the photon's energy. If work needed to be done in either direction to put the photon in the box, the mass of the box with the photon in it would be different from what you give, by the amount of work done.

 

[vanhees71]

Of course you need the energy $\hbar \omega$ to excite the corresponding mode.

 

[jartsa]

So you guys have solved my perceived problem in three different ways. But I did not mean to say that there is a problem. I meant to say: "does this solve your problem @snoopies622".

Snoopies622 thought that a falling object loses potential energy and gains kinetic energy, which would keep total energy of a falling object constant, which would be a problem, because the energy of a falling object may be turned into light, which redshifts when it climbs.

A simplified version of Snoopie622's problem: Falling light blueshifts (gains energy), falling rock has constant energy.

How do we correctly solve that problem?

 

[Dale]

Falling light blueshifts (gains energy), falling rock has constant energy.

How do we correctly solve that problem?

The problem statement is false. The falling rock gains energy just as the light does. (The PE does not belong to the rock)

 

[Ibix]

How do we correctly solve that problem?

The PE does not belong to the rock

...as discussed in posts #3, #5, and #11 at least.

 

[pervect]

It's really tough to discuss energy in GR, which is a rather advanced topic, with someone who does not have the proper background.

I recommend https://math.ucr.edu/home/baez/physics/Relativity/GR/energy_gr.html "Is Energy conserved in General relativity" as an I-level introduction.

Note that what they say in the FAQ is not "how is energy conserved" but "IS energy conserved". As the FAQ says quite correctly but at a very general level " In special cases, yes. In general, it depends on what you mean by "energy", and what you mean by "conserved". "

So, asking about "potential energy" in GR a bit hasty, as it's already assumed to much - it's assuming that such a thing always exists.

Note that I've given only the very first few sentences of the above FAQ. I encourage people to read it in its entirety if hey are able to. I am hopeful that these few sentences will encourage people to track down and read the FAQ, but I'm a bit skeptical that that will hapen. I should probably try to be more postive, but - meh,

At a more advanced level than the above FAQ, one might start looking at the various differeing notions of energy that GR has. At the most basic level, pointing out that there are several different possibilities is probably all that can be expected to communicate, though I'll drop a few names - there is the basic stress-energy tensor, which is only local, and then some global notions in the approriate special case domains such as there the ADM, Bondi, and Komar treatments.

At an A level, I rather like Wald's treatment, which goes into a few of the above in more detail. Many older texts (such as MTW) only give the pseudotensor treatment.

 

[Dale]

So, asking about "potential energy" in GR a bit hasty, as it's already assumed to much - it's assuming that such a thing always exists.

That is a good point to keep in mind. The context of this thread is dropping an object from a tower, which usually implies either a uniform gravitational field or a Schwarzschild gravitational field.

Both of those do have a gravitational potential, so this doesn’t change the above discussion. But it does mean that the concepts do not generalize to other spacetimes. In particular, they do not generalize to FLRW spacetime in cosmology nor to pp-wave spacetimes.

 

[snoopies622]

I meant to say: "does this solve your problem @snoopies622".

Oh yes I'm quite satisfied, thank you.

(For some reason I stopped receiving email notifications about this thread days ago.)

 

[H_A_Landman]

... an idealized experiment in which an object is dropped from a tower, then turned into a photon and sent back up to its original height.

Yeah that's Einstein's 1911 derivation of gravitational time dilation and gravitational redshift. (He first derived GTD a completely different way, from SR and EP, in 1907.)

Let's ignore the kinetic energy, and consider just raising or lowering the mass slowly, while doing work or extracting work respectively. Then it's clear (in all reference frames) that the mass at the top of the tower has more energy and that work can be extracted from its descent.

If I'm at the bottom of the tower, and you're at the top at height $z$, we evaluate the energies differently. To me, the mass at the bottom has energy $E(0) = mc^2$ and quantum phase frequency $\nu(0) = E(0)/h$, and the mass at the top has energy $E(z) = mc^2 + mgz$ and quantum phase frequency $\nu(z) = E(z)/h$. But to you at the top, the mass at the top only has energy $E(0) = mc^2$ and quantum phase frequency $\nu(0) = E(0)/h$, which is only possible if your clock is running faster than mine by a factor of $$T_d = \frac{\nu(z)}{\nu(0)} = \frac{E(z)}{E(0)} = \frac{mc^2 + mgz}{mc^2} = 1 + \frac{gz}{c^2}$$ which is just the linear "weak-field" approximation to gravitational time dilation. So GTD can be viewed as a purely quantum effect, except that the value of $h$ cancels out so that the classical ($h \rightarrow 0$) limit is exactly the same as the "quantum" result.

(I expect that some people will not like the above approach. Tough. It is not only forced by QM, but has also been experimentally confirmed by the Colella-Overhauser-Werner neutron interferometry experiment (1975) and others.)

Similarly, from this viewpoint there is no actual change in frequency from gravitational redshift. It is merely that an emitter deeper in a gravity well is running slower and a receiver higher up is running faster.

At any rate, this shows that it is not entirely unreasonable to ascribe a slightly increased mass to an object when it is higher up in a gravitational potential. But you have to be careful how you use that increased mass, or you can get incorrect answers.

 

[PeterDonis]

A simplified version of Snoopie622's problem: Falling light blueshifts (gains energy), falling rock has constant energy.

How do we correctly solve that problem?

By not incorrectly stating the facts. There is no consistent interpretation of the term "energy" which makes your statement true.

If we interpret "energy" as "energy at infinity", then a falling rock has constant energy, and so does falling light; energy at infinity is a constant of any geodesic (free-falling) motion.

If we interpret "energy" as "energy measured by static observers at different altitudes", then falling light gains energy, but so does a falling rock.

I'm quite satisfied, thank you.

With what? What do you think is the resolution of the perceived problem you stated in the OP?

If it's what I said above, good.

If not, then you shouldn't be satisfied yet.

 

[PeterDonis]

GTD can be viewed as a purely quantum effect,

No, it can be viewed as a purely classical effect. There is no need to bring QM into it at all.

(I expect that some people will not like the above approach. Tough. It is not only forced by QM, but has also been experimentally confirmed by the Colella-Overhauser-Werner neutron interferometry experiment (1975) and others.)

That experiment does not say what you claim. It says that the gravitational potential can be included as a potential in the Schrodinger equation just like any other potential (e.g., the Coulomb potential). That is not the same as what you claim.

Please do not hijack someone else's thread with questionable claims.

 

[snoopies622]

What do you think is the resolution of the perceived problem you stated in the OP?

Well, in the OP I wasn't sure where potential energy manifested itself in the hypothetical situation. (Was it in the falling object or not?) Answer: It's not in the object, but "in the field", hard to locate exactly. The mousetrap in entry #13 is a good example: the mass of the entire mousetrap increases when it is set, but one cannot measure the mass increase in this or that individual part of the mousetrap.

 

[Nugatory]

Then it's clear (in all reference frames) that the mass at the top of the tower has more energy and that work can be extracted from its descent.

That is not at all clear. It is clear that energy had to be added to the system consisting of the mass and the Earth to lift the mass... but why does that mean that the mass “has more energy”? Why must it be attached to the mass? Even in classical physics that claim is dubious, although the problem is more apparent if we consider situations in which the masses of two gravitationally attracting bodies are the same.

 

[PeterDonis]

in the OP I wasn't sure where potential energy manifested itself in the hypothetical situation

If you want to view things in terms of "potential energy", yes, that can't really be localized. But if you view things in terms of energy at infinity--which is basically the sum of kinetic energy relative to static observers and potential energy--then you can view that as a property of the object, and a constant of the object's motion as long as that motion is geodesic (free-fall).

the mass of the entire mousetrap increases when it is set, but one cannot measure the mass increase in this or that individual part of the mousetrap.

Actually, you can, because in the mousetrap, the stored energy will be in something tangible, like a spring. The reason gravitational potential energy can't be localized is that you can't point to anything tangible that it is "stored" in. That makes it different from the mousetrap example. (Note that the "potential energy" stored in the mousetrap when it is set is not gravitational.)

 

[pervect]

Well, in the OP I wasn't sure where potential energy manifested itself in the hypothetical situation. (Was it in the falling object or not?) Answer: It's not in the object, but "in the field", hard to locate exactly. The mousetrap in entry #13 is a good example: the mass of the entire mousetrap increases when it is set, but one cannot measure the mass increase in this or that individual part of the mousetrap.

Well, in those cases in which there is a conserved energy, one may have a desire for the energy to "be somewhere", but it turns out there isn't any consistent observer-independent way of saying exactly where the energy is in General Relativity. This is in contrast to, say , electromagnetism, where one can talk about the energy density being proportional to E^2 + B^2, with the suitable conversion factors if one isn't using geometric units. GR does not have any such formula for energy density in any soft of field. The Lagrangian density can be defined - which is interesting, though I'm not sure if the OP will find it relevant or is even familiar with Lagrangian mechanics.

But there are reasonably common cases where there isn't a conserved energy in GR, at least not any known conserved energy, such as, say, an expanding FRW universe. Such as, for instance, the one we live in.

So one is led to wonder, if something (energy in GR) doesn't always exist, and cannot be localized in those cases where we think it should exist, does it really exist?

However, there are special cases where energy is conserved globally in General relativity, some of which seem to be specifically of interest. Locally (as opposed to globally), there is not only no problem, as energy conservation in some sense is built into the theory - the differential form of energy of local conservation is built into Einstein's field equations. This was mentioned in the FAQ I linked to earlier.

There are some good insights to be had looking at the history of the problem though. Noether's theorem is especially relevant and interesting. Some very famous mathematicians in the early days of GR - Hilbert, and Klein (and possibly a third person, I forget) had some similar concerns about energy in General Relativity, as we have been discussing. Hilbert wound up consulting with Emily Noether about the issue, as some work she was doing seemed relevant to the problem. There's some interesting, though not relevant, history there, with the role of women in science. The result of this is known as Noether's theorem in physics. Mathematicans know her for lot of her other theorems in abstract algebra, so they are a bit surrised to see "Noether's theorem" be singular, rather than plural.

Noether's theorem relates energy conservation to time translation symmetries. And there are difference in how this works when one has a finite symmetry group, and an inifinite symmetry group. GR is the later case.

Anyway, I would say that the problem turns out to be quite deep and interesting, and if one is fully satisfied with the answer, it's likely that one doesn't really appreciate the full problem. There's really quite a lot to consider. Reading what's available on the issue and not making up one's mind too soon is a good first step. However, it may take a fair bit of background to really get into the depths :(.

 

[Dale]

But there are reasonably common cases where there isn't a conserved energy in GR, at least not any known conserved energy, such as, say, an expanding FRW universe.

I think historically that this issue was the motivation for Noether’s theorem

 

[pervect]

I think historically that this issue was the motivation for Noether’s theorem

I agree.

 

[snoopies622]

Thanks pervect, deep subject!

 

[vanhees71]

I think historically that this issue was the motivation for Noether’s theorem

Yes, Noether analyzed the topic thoroughly in carefully analyzing both what we nowadays call global symmetries and local (gauge) symmetries. The latter are the relevant case for the issue with energy in GR, and that makes it so complicated. The particular problem of GR which makes it more complicated than in SR is that the global symmetry, responsible for "time-translation invariance", only leads to pseudotensors for the corresponding Noether charge-current density, not tensors in the sense of the generally covariant formalism.

 

[H_A_Landman]

No, it can be viewed as a purely classical effect. There is no need to bring QM into it at all.

I said "can", not "must be". What you're saying does not contradict anything I said.

That experiment does not say what you claim. It says that the gravitational potential can be included as a potential in the Schrodinger equation just like any other potential (e.g., the Coulomb potential). That is not the same as what you claim.

Please do not hijack someone else's thread with questionable claims.

I am claiming that the COW experiment confirms that a neutron higher in a gravitational potential has a higher frequency of phase oscillation, exactly as predicted by QM. Are you really calling that questionable?

 

[PeterDonis]

I said "can", not "must be".

You're quibbling over words. Gravitational time dilation is not a "purely quantum" effect. Unless you have a theory of quantum gravity lurking somewhere that the rest of us don't know about.

I am claiming that the COW experiment confirms that a neutron higher in a gravitational potential has a higher frequency of phase oscillation

No, that's not what you claimed. You claimed that this result of the COW experiment somehow proves that gravitational time dilation is a "purely quantum" effect. Which is a different claim than the (true) claim that the COW experiment shows that gravitational potential is included in the Schrodinger equation just like any other classical potential.

You also claimed:

there is no actual change in frequency from gravitational redshift

Which is correct: energy at infinity is a constant of geodesic motion. And therefore, if you want to apply $E = \hbar \omega$ for a quantum object, so is "frequency at infinity". Which means, as you also correctly said, that the apparent difference in frequency of quantum objects at different heights is due to the difference in the observer's clocks at those heights, not to any difference in the quantum object being observed. But all these things are not the same as "gravitational time dilation is a purely quantum effect".

 

[vanhees71]

The experiment with neutrons in the gravitational field of the Earth (over a "reflecting ground") has been done with high precision. The result for the measured energy levels is precisely as expected from the corresponding standard problem treated in the QM 1 lecture:

https://www.nature.com/articles/415297a (I can't find a legal freely available link for this)

Here's another paper on the same measurement, available also from arXiv:

https://doi.org/10.1103/PhysRevD.67.102002
https://arxiv.org/abs/hep-ph/0306198

 

[PeterDonis]

The result for the measured energy levels

Note that these energy levels are energy levels of bound states in a "potential well" created with a mirror. So measurements of these energy levels are not measurements of "gravitational time dilation" for neutrons. They are measurements of whether the gravitational potential works like any other potential in the Schrodinger equation to determine bound state energy levels.

 

[H_A_Landman]

My main point in answer to the original question was merely "it is not entirely unreasonable to ascribe a slightly increased mass to an object when it is higher up in a gravitational potential" than you are, with the caveat that you need to be careful how you use that mass. The negative of that would be "it is entirely unreasonable", even with caveats, which I don't think is a defensible position. I never claimed (and do not think) that it is the best way to formulate the problem, but even though it's clunky, you can make it work.

Gravitational time dilation is not a "purely quantum" effect.

My claim was "GTD can be viewed as a purely quantum effect", so if you're disputing that then your claim must be "GTD can not be viewed as a purely quantum effect". But it's just high school algebra to see that the frequency shift predicted by QM is exactly the same (to first order) as the frequency shift predicted by GTD (in the low-speed weak-field "Newtonian" limit). If you assume that those are independent and unrelated effects, then you need to apply both, which gives the wrong answer by a factor of 2. But if you see them as different ways of describing the same effect, then you can either use the time-dependence of the Schrödinger equation, or you can use the GTD formula; they each give the same (correct) answer. If you do the former, it looks like a purely quantum effect (which oddly does not depend on the magnitude of h); if you do the latter, it looks like a purely classical effect. Your claim is that only the latter is valid, which leaves you responsible to explain why the former gets the right answer even though you claim it's completely wrong.

Unless you have a theory of quantum gravity lurking somewhere that the rest of us don't know about.

That's not required. All this stuff pops up in published semi-classical unified theories going back 40+ years. They're widely ignored, but as far as I can tell their fundamentals are correct. I don't know whether they would be easier or harder to quantize than GR; no one has tried.

the COW experiment shows that gravitational potential is included in the Schrodinger equation just like any other classical potential.

I completely agree, but treating all classical potentials equally inescapably implies that there is a time dilation associated with every classical potential (not just gravity), which is not something that most people accept. Mainstream physics is self-contradictory on this point, so it can't possibly be 100% right. Experiments to test for the predicted EM time dilation were first proposed in 1979 but have never been performed; I'm trying to get one run at PSI in late 2021 (if COVID permits and I don't get laughed out of the review process). So maybe we'll know for sure in a year or two.

If you want to continue this, we should take it offline. We've already "hijacked" too much and are wandering farther off-topic.

 

[Dale]

My claim was "GTD can be viewed as a purely quantum effect", so if you're disputing that then your claim must be "GTD can not be viewed as a purely quantum effect". But it's just high school algebra to see that the frequency shift predicted by QM is exactly the same (to first order) as the frequency shift predicted by GTD

This doesn’t work. Without a quantum theory of gravity it doesn’t make sense to claim that gravitational time dilation can be derived purely quantum mechanically.

We don’t even know if a quantum theory of gravity will obey the equivalence principle. I suspect that the high school algebra derivation you allude to would require that, although without an actual reference it is impossible to know. IMO, bringing the equivalence principle without a quantum theory of gravity that obeys the equivalence principle already makes it semi-classical, not purely quantum.

Your claim is that only the latter is valid, which leaves you responsible to explain why the former gets the right answer even though you claim it's completely wrong

It is up to you to support your claim, not up to others to refute it.

Have you a professional scientific reference that makes the claim “GTD can be viewed as a purely quantum effect”?

 

[PeterDonis]

treating all classical potentials equally inescapably implies that there is a time dilation associated with every classical potential

It implies no such thing. The Schrodinger equation says nothing whatever about time dilation. It's a limited, non-relativistic model.

All this stuff pops up in published semi-classical unified theories going back 40+ years.

Please give references.

Mainstream physics is self-contradictory on this point

Nonsense. Please stop posting misinformation.

Experiments to test for the predicted EM time dilation were first proposed in 1979

Please give a reference.

I'm trying to get one run at PSI in late 2021 (if COVID permits and I don't get laughed out of the review process). So maybe we'll know for sure in a year or two.

If you get a paper published giving the results, then we can discuss it here.

If you want to continue this, we should take it offline. We've already "hijacked" too much

You mean you have hijacked too much. I have no desire to continue this subthread because I didn't start it and have no interest in it. The only reason I have responded to your posts is to correct misstatements (and now to ask for references to back up unsupported statements). So if you stop posting in this subthread, I will too.

 

[H_A_Landman]

Fine. If you want the full machinery of this class of theories, then in my opinion the best references are:

• Apsel, D., “Gravitational, electromagnetic, and nuclear theory.”, Int. J. of Theoretical Physics 17 643-649 (1978)
• Apsel, D.,“Gravitation and electromagnetism.”, General Relativity and Gravitation 10 297-306 (1979)
• Apsel, D., “Time dilations in bound muon decay.”, General Relativity and Gravitation 13 605-607 (1981)
• Rodrigues Jr., W.A., “The Standard of Length in the Theory of Relativity and Ehrenfest Paradox.”, Il Nuovo Cimento 74 B 199-211 (1983).
• Ryff, L.C.B., “The Lifetime of an Elementary Particle in a Field.”, General Relativity and Gravitation 17 515-519 (1985).
• Beil, R.G., “Electrodynamics from a Metric.”, Int. J. of Theoretical Physics 26 189-197 (1987).

Apsel (1979) contains the first experiment proposal.

https://www.researchgate.net/publication/333878154_The_magnitude_of_electromagnetic_time_dilation just strips that stuff down to a very elementary level, and may be of less interest. But, it does include other references and some historical analysis. Feel free to rip it to shreds, but not here. You have my contact info.

 

[PeterDonis]

in my opinion the best references are

Thank you for the references. That will close out this subthread.