Energy Conservation Paradox: Is It True or Not?

[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" ?