Energy Conservation Paradox: Is It True or Not?

[Dadface]

Thank you DaleSpam. As I said in post eleven I'm primarily interested in energy exchanges involving particle interactions and I just chose the electron as an example of a suitable particle to consider. I am interested in things such as particle collisions and potential/kinetic energy change events.

If it's true that energy is not conserved in some situations then I want to know what experimental evidence there is to prove that, but my searches have found nothing as of yet.

Particle experiments involving energy exchanges here involve several measurements, particle mass being just one of those measurements. I just wonder if we can carry out similar experiments with the same sort of precision at places where the geometry is not Miskownian.

That doesn't necessarily mean that we must go to such places. Perhaps we could gather some evidence here as did Eddington with his particular investigation.

 

[wabbit]

Maybe another perspective : my understanding is that
- GR is the general theory of spacetime.
- SR is the special case of GR when there's no gravity.
- Energy is conserved in the absence of gravity, but not always conserved in the presence of gravity.
Someone shoot this down if I'm talking rubbish : )

 

[wabbit]

In connection with what has been described before I would like to ask another question which may be so naive or perhaps even meaningless that I'm reluctant to send it. But here goes anyway:

If I'm now sitting in a place where the geometry of spacetime can be described as being approximately Minkowskian, at what places can I go to (perhaps as thought experiments) where the geometry of spacetime is different?
Thank you.

Anywhere close to a massive object.

Edit : please remember that spacetime is not flat on earth. It is so only to the extent that gravity is negligible for what you're studying.

 

[PeterDonis]

Particle experiments involving energy exchanges here involve several measurements particle mass being just one of those measurements. I just wonder if we can carry out similar experiments with the same sort of precision at places where the geometry is not Miskownian.

These experiments take place within a single local inertial frame (they happen so fast, and are confined to a small enough region of space, that the effects of spacetime curvature are negligible). They would give the same results in any local inertial frame anywhere in spacetime, even if spacetime as a whole is curved. For example, if your particle accelerator were freely falling through the horizon of a black hole, or were located in a distant galaxy that was receding from ours with a very large redshift, you would still get the same results for particle collision experiments.

The failure of energy conservation in a non-static spacetime is a global phenomenon, not a local one. It's not that particle collisions or other phenomena fail to conserve energy or momentum locally; it's that there is no well-defined "total energy" for spacetime as a whole. For example, there is no well-defined "total energy" for the universe as a whole. So there's no way to even assess the question of whether energy is conserved for the universe as a whole, since "energy" isn't well-defined to begin with for the universe as a whole.

If it's true that energy is not conserved in some situations then I want to know what experimental evidence there is to prove that

There isn't any evidence of the kind you seem to be looking for, because, as above, failure of energy conservation is a global phenomenon, not a local one. You will never find a local experiment, such as something happening in a particle accelerator or a lab, that fails to conserve energy. The only evidence you will be able to find is global evidence: for example, the evidence that our universe is not static, but expanding, and global effects arising from that expansion.

One such effect is the temperature of the CMBR: the universe is filled with photons at a current temperature of 2.7 degrees Kelvin, but when these photons were created, from the combination of electrons and ions into neutral atoms when the universe was about 300,000 years old, their temperature was a few thousand degrees Kelvin (since that's the temperature at which electrons and ions combine to form atoms as a plasma is cooling). The number of photons in the CMBR has not changed, so the only way their temperature can change is if they have lost energy as the universe expands (since "temperature" is just the energy per photon). This loss of energy is a manifestation of energy globally not being conserved in a non-static spacetime.

 

[Evan Maxwell]

There was an answer I saw on quora recently that delved into this: https://www.quora.com/In-what-respect-does-general-relativity-leave-open-the-possibility-that-energy-is-not-conserved-across-the-entire-universe

I am not sure if maybe it will shed some light on this but it does discuss the concept of the 4 component tensor in terms of momentum and energy and highlights why energy itself is not always conserved in GR.

 

[Dale]

. As I said in post eleven I'm primarily interested in energy exchanges involving particle interactions

So, to get energy non conservation in a particle exchange you would need a spacetime which is non static and so strongly curved that the time and distance scale of particle interaction would not be considered local. I cannot think of anything that would do that other than a pair of small black holes orbiting each other very closely.

We have no experimental evidence anywhere close to this regime.

 

[wabbit]

You make me wonder, what examples outside of cosmology do we have of non conservation ? A single BH isn't enough due to asymptotic flatness. Is it the case that say in the fusion of two stellar mass black holes energy is not conserved, or ill defined ? I vaguely remember that the latter might be the case due to the emission of gravitational waves but I'm unsure...

 

[PeterDonis]

Is it the case that say in the fusion of two stellar mass black holes energy is not conserved, or ill defined ?

Not really. You can still model a black hole merger as an asymptotically flat spacetime; it just won't be stationary. But asymptotic flatness is sufficient to define the ADM energy and the Bondi energy, which have an obvious physical interpretation; see below.

I vaguely remember that the latter might be the case due to the emission of gravitational waves but I'm unsure...

No, GW emission doesn't present a problem in this case, because the waves carry energy, and this energy exactly balances the mass loss in the merger process (i.e., the mass of the two original holes minus the mass of the final hole). The difference between the ADM energy and the Bondi energy for the spacetime is equal to the energy carried away by GWs, and those two energies are, respectively, the energy of the original system (the two holes) and the energy of the final system after all radiation has escaped to infinity (the final hole).

 

[PAllen]

You make me wonder, what examples outside of cosmology do we have of non conservation ? A single BH isn't enough due to asymptotic flatness. Is it the case that say in the fusion of two stellar mass black holes energy is not conserved, or ill defined ? I vaguely remember that the latter might be the case due to the emission of gravitational waves but I'm unsure...

Cosmology is the main example where total energy is generally considered undefined, thus not conserved [edit: to try to balace redshift of CMB, one would want to introduce gravitational PE in some form; it is the lack of any acceptable way of doing this that leads to the problem of definition. Plus theoretical arguments that such a definition should not be expected for a cosmologicall solution]. Even for inspiralling BHs, it is accurate to any plausible precision to consider them as if embedded in an asymptotically flat spacetime (because we are observing them over a time and distance scale for which expansion is not relevant for the GW observation [assuming we can do this]; and for the bound BH's themselves, expansion is irrelevant because it is a bound system]. Even at scales of galaxies including dynamics of the central BH, over millions of years, one would expect energy conservation to any achievable precision - assuming one could measure GW as well as EM radiation.

 

[wabbit]

Thanks. This should be FAQ, When explaining energy (non-)conservation in GR, Starting with "FAPP Yes except in some cosmological situations" would put the issue in perspective (not ditching in any way the usenet faq mentioned early in the thread, which is definitely a great resource)

 

[wabbit]

No, GW emission doesn't present a problem in this case, because the waves carry energy, and this energy exactly balances the mass loss in the merger process (i.e., the mass of the two original holes minus the mass of the final hole).

Thanks for clearing that up.

 

[PAllen]

Thanks. This should be FAQ, When explaining energy (non-)conservation in GR, Starting with "FAPP Yes except in some cosmological situations" would put the issue in perspective (not ditching in any way the usenet faq mentioned early in the thread, which is definitely a great resource)

It's not quite so simple, because the following questions have rather different answers:

1) How well is energy conserved in our universe? (essentially exactly up to cosmological distance and time scales)

2) How much of a principle is conserved total energy in the theoretical framework of GR? (very weak, because it can only be defined for very special spacetimes*)

*In generally accepted ways. There are a few theorists who devise general constructions (e.g. Phillip Gibbs). Others have characterized such constructions as a fancy way of demonstrating that a tensor constructed to vanish does so invariantly. Another approach besides the standard Asymptotically flat approaches is pseudo-tensors, e.g. Nakanishi, referenced earlier in this thread by Pervect, still require an asymptotically Lorentz transform to exist to make invariant statements; this is nearly the same as asymptotic flatness.

 

[Dale]

Even for inspiralling BHs, it is accurate to any plausible precision to consider them as if embedded in an asymptotically flat spacetime

Oops, yes, you are right. My comments were not accurate above. Not even a pair of orbiting black holes does it.

 

[wabbit]

It's not quite so simple, because the following questions have rather different answers:

1) How well is energy conserved in our universe? (essentially exactly up to cosmological distance and time scales)

2) How much of a principle is conserved total energy in the theoretical framework of GR? (very weak, because it can only be defined for very special spacetimes*)

Agreed. I was unclear, what I meant to say is, introductory discussions of energy conservation in GR that I've seen tend to focus on (2), and mentionning (1) also would be helpful - and your post I was replying to gives a clear and concise way of doing that.

 

[bhobba]

Hi Guys

I have been reading this thread and I think some are missing the point.

The modern definition of energy is by Noether's theorem. The conserved charge from time translational invariance is the definition of energy. So by definition its conserved. The issue with GR is time translational invariance breaks down hence the definition of energy breaks down. If energy is conserved or not in GR depends on your definition because the usual definition doesn't apply. This was all sorted out by Noether ages ago.

There are a number of reputable sites on the internet that explain this eg:
http://math.ucr.edu/home/baez/physics/Relativity/GR/energy_gr.html
'In general — it depends on what you mean by "energy", and what you mean by "conserved"'

Because its definition dependant checking it experimentally will depend entirely on your definition.

Thanks
Bill

 

[wabbit]

If energy is conserved or not in GR depends on your definition because the usual definition doesn't apply

Very true. I didn't see responses arguing against that here, and the site you link to was mentionnned in post 5 - not saying that a reminder is a bad idea, I would in fact suggest to OP to read it again.

At the same time, when "Energy" enters as a key term is the fundamental equation of the theory, saying "but of course Energy is not well defined in GR" can use some elaboration :)

 

[bhobba]

Very true. I didn't see responses arguing against that here,

Nothing said in that regard has been wrong.

I just felt this was a key point that wasn't mentioned.

Thanks
Bill

 

[Dadface]

Hi Guys

I have been reading this thread and I think some are missing the point.

The modern definition of energy is by Noether's theorem. The conserved charge from time translational invariance is the definition of energy. So by definition its conserved. The issue with GR is time translational invariance breaks down hence the definition of energy breaks down. If energy is conserved or not in GR depends on your definition because the usual definition doesn't apply. This was all sorted out by Noether ages ago.

There are a number of reputable sites on the internet that explain this eg:
http://math.ucr.edu/home/baez/physics/Relativity/GR/energy_gr.html
'In general — it depends on what you mean by "energy", and what you mean by "conserved"'

Because its definition dependant checking it experimentally will depend entirely on your definition.

Thanks
Bill

Hello bhobba,
when I'm finding the time I'm looking again at many of the other responses in this thread and reading through other sources. There's a lot to take in so I am a bit slow with it all.
My prime interest here is how conservation laws apply to interactions involving charged particles. As an example when an electron approaches a positively charged macroscopic object there is a conversion between PE and KE and when everything is measured and taken into account it seems that energy is conserved. That's the sort of energy I'm interested in, it's basic high school stuff and defined in terms of work done. But I will try to find out if other definitions are more appropriate.
One thing that seems common to all the sources that I have looked at so far is that there seems to be no references to particle interactions. I knew that GR was to do with gravity but I assumed it could encompass other areas of physics as well. I will get back to it. Thank you very much

 

[PAllen]

Hi Guys

I have been reading this thread and I think some are missing the point.

The modern definition of energy is by Noether's theorem. The conserved charge from time translational invariance is the definition of energy. So by definition its conserved. The issue with GR is time translational invariance breaks down hence the definition of energy breaks down. If energy is conserved or not in GR depends on your definition because the usual definition doesn't apply. This was all sorted out by Noether ages ago.

There are a number of reputable sites on the internet that explain this eg:
http://math.ucr.edu/home/baez/physics/Relativity/GR/energy_gr.html
'In general — it depends on what you mean by "energy", and what you mean by "conserved"'

Because its definition dependant checking it experimentally will depend entirely on your definition.

Thanks
Bill

Except it is not so simple at all what Noether's theorem says for GR. It requires an action and global time coordinate (equiv. global foliation into spatial hypersurfaces connected by a timelike congruence). Note that this suggests it SHOULD apply to FLRW spacetimes, just as it DOES apply to and justify well known results for asymptotically flat spacetimes. I see MANY misapplicatons of Noether to GR, where the a dynamic spacetime is considered automatically to violate Noether. As I understand it, that is wrong, and what Noether requires is that there be a global time space foliation such that you can ask about time symmetry of a Lagrangian physical law. To me, this suggests the possibility of conservation of energy to an extent greater than has been currently accepted by consensus.

 

[PeterDonis]

It requires an action and global time coordinate (equiv. global foliation into spatial hypersurfaces connected by a timelike congruence).

More precisely, the formalism in which the theorem is formulated requires an action and a global foliation. But the conditions for the theorem itself to be true are more stringent than that; see below.

this suggests it SHOULD apply to FLRW spacetimes

No, it doesn't, because the theorem itself requires a timelike Killing vector field, and FLRW spacetimes don't have one. In terms of your description of the formalism, quoted above, for the theorem to be true, the metric of the spatial hypersurfaces in the foliation would have to be independent of the global time coordinate. In FLRW spacetimes, it isn't, because the scale factor is a function of the time coordinate.

 

[Dale]

My prime interest here is how conservation laws apply to interactions involving charged particles.

No need to worry about GR then. You can assume flat spacetime and all of the standard conservation laws.

 

[Dale]

No, it doesn't, because the theorem itself requires a timelike Killing vector field

This is what led to my mistaken comments about pairs of orbiting black holes. A binary black-hole spacetime doesn't have a timelike Killing vector field, therefore a straight application of Noether's theorem says no conserved energy. But obviously there are definitions of energy in GR that are not limited by that and can be applied to asymptotically flat spacetimes, which I forgot.

 

[wabbit]

On a side note, http://www.physics.ucla.edu/~cwp/articles/noether.asg/noether.html that Noether's Theorems were originally developped for the purpose of resolving the issue of energy conservation in GR!

 

[bhobba]

My prime interest here is how conservation laws apply to interactions involving charged particles. As an example when an electron approaches a positively charged macroscopic object there is a conversion between PE and KE and when everything is measured and taken into account it seems that energy is conserved.

One would require some VERY strong gravitational fields for it to not be an extremely good approximation to an inertial frame. At that strength the electron would couple to the gravitational field and the whole situation would be far from simple to analyse.

Thanks
Bill

 

[PAllen]

More precisely, the formalism in which the theorem is formulated requires an action and a global foliation. But the conditions for the theorem itself to be true are more stringent than that; see below.
No, it doesn't, because the theorem itself requires a timelike Killing vector field, and FLRW spacetimes don't have one. In terms of your description of the formalism, quoted above, for the theorem to be true, the metric of the spatial hypersurfaces in the foliation would have to be independent of the global time coordinate. In FLRW spacetimes, it isn't, because the scale factor is a function of the time coordinate.

Yes, you are right about the simple form of the theorem that applies, e.g. to the Poincare group. I was mis-remembering the distinctions bertween the simple and the more general versions of the theorem. The more general form can be used to theoretically justify pseudo-tensor conserved energy formulations, as explained here:

http://www.physics.ucla.edu/~cwp/articles/noether.asg/noether.html

 

[macrobbair]

one simply defines a new type of energy to make it conserved...
unless one is in gr where one can change frame,
then one needs more information...

 

[phinds]

one simply defines a new type of energy ...

Uh ... "defines a new type of energy" ?