Conservation of Energy Down The Drain?

[dm4b]

Okay, my title may have been a bit overdramatic ... but my question is still about something a bit troubling I read in Sean Carroll's GR text.

Basically, he claims that the total energy in an expanding Universe is not typically conserved. See pages 137-138 and 344.

This is because expansion means the metric is changing in time, and therefore there is no isometry in this direction. This, in turn, means there are no time-like Killing vectors. Killing vectors generate the symmetries in a curved spacetime. No time-like Killing vectors means no symmetry in time and, as we all remember, conservation of energy is related to symmetry in time.

He goes on elsewhere, IIRC, to claim that energy is conserved locally. It's just that globally the total energy is no longer conserved (in an expanding curved spacetime)

So, is this viewpoint generally regarded as true?

If so, what are the ramifications?

 

[JesseM]

See also here for some more details:

Is Energy Conserved in General Relativity? (from the Usenet Physics FAQ)

 

[bcrowell]

FAQ: How does conservation of energy work in general relativity, and how does this apply to cosmology? What is the total mass-energy of the universe?

Conservation of energy doesn't apply to cosmology. General relativity doesn't have a conserved scalar mass-energy that can be defined in all spacetimes.[MTW] There is no standard way to define the total energy of the universe (regardless of whether the universe is spatially finite or infinite). There is not even any standard way to define the total mass-energy of the *observable* universe. There is no standard way to say whether or not mass-energy is conserved during cosmological expansion.

Note the repeated use of the word "standard" above. To amplify further on this point, there is a variety of possible ways to define mass-energy in general relativity. Some of these (Komar mass, ADM mass [Wald, p. 293], Bondi mass [Wald, p. 291]) are valid tensors, while others are things known as "pseudo-tensors" [Berman 1981]. Pseudo-tensors have various undesirable properties, such as coordinate-dependence.[Weiss] The tensorial definitions only apply to spacetimes that have certain special properties, such as asymptotic flatness or stationarity, and cosmological spacetimes don't have those properties. For certain pseudo-tensor definitions of mass-energy, the total energy of a closed universe can be calculated, and is zero.[Berman 2009] This does not mean that "the" energy of the universe is zero, especially since our universe is not closed.

One can also estimate certain quantities such as the sum of the rest masses of all the hydrogen atoms in the observable universe, which is something like 10^54 kg. Such an estimate is not the same thing as the total mass-energy of the observable universe (which can't even be defined). It is not the mass-energy measured by any observer in any particular state of motion, and it is not conserved.

MTW: Misner, Thorne, and Wheeler, Gravitation, 1973. See p. 457.

Berman 1981: M. Berman, unpublished M.Sc. thesis, 1981.

Berman 2009: M. Berman, Int J Theor Phys, http://www.springerlink.com/content/357757q4g88144p0/

Weiss and Baez, "Is Energy Conserved in General Relativity?," http://math.ucr.edu/home/baez/physics/Relativity/GR/energy_gr.html

Wald, General Relativity, 1984

 

[pervect]

Okay, my title may have been a bit overdramatic ... but my question is still about something a bit troubling I read in Sean Carroll's GR text.

Basically, he claims that the total energy in an expanding Universe is not typically conserved. See pages 137-138 and 344.

This is because expansion means the metric is changing in time, and therefore there is no isometry in this direction. This, in turn, means there are no time-like Killing vectors. Killing vectors generate the symmetries in a curved spacetime. No time-like Killing vectors means no symmetry in time and, as we all remember, conservation of energy is related to symmetry in time.

He goes on elsewhere, IIRC, to claim that energy is conserved locally. It's just that globally the total energy is no longer conserved (in an expanding curved spacetime)

So, is this viewpoint generally regarded as true?

If so, what are the ramifications?

It's disturbing, but true - energy isn't conserved in GR, at least not the same manner as in other theories. If you read up on some of the history, the problem has been around for some time. Hilbert asked Emily Noether to investigate the apparent ill-behavior of energy in General Relativity, and Emily Noether came up with what physicists call Noether's theorem as a result (the result you mentioned about conserved energies being associated with time-translation symmetries). See

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

Though the general theory of relativity was completed in 1915, there remained unresolved problems. In particular, the principle of local energy conservation was a vexing issue. In the general theory, energy is not conserved locally as it is in classical field theories - Newtonian gravity, electromagnetism, hydrodynamics, etc.. Energy conservation in the general theory has been perplexing many people for decades. In the early days, Hilbert wrote about this problem as 'the failure of the energy theorem '. In a correspondence with Klein [3], he asserted that this 'failure' is a characteristic feature of the general theory, and that instead of 'proper energy theorems' one had 'improper energy theorems' in such a theory. This conjecture was clarified, quantified and proved correct by Emmy Noether.

While Energy isn't conserved in GR in the same manner as it is in other theories, there are some useful results related to energy conservation that do apply to GR. One result is that the divergence of the stress-energy tensor is always zero. This is a form of energy conservation, though it's not strong enough to come up with a number that represents "the energy of the universe" for instance.

If an exact time-translation symmetry exists, it yields a conserved Komar mass via Noether's theorem. It's possible, however, to have some useful notions of conserved energy even in the lack of such an exact symmetry. One may still have an asymptotic time-translation symmetry which is sufficient to give a conserved Bondi or ADM energy - notions which are very useful, though they require asymptotic flatness to be applied, something that our universe seems to lack.

 

[dm4b]

Thanks for the info guys. I'll have to read through all the links provided.

one additional question related to this:

If the vacuum energy throughout space is constant and space is expanding, wouldn't that indicate the total energy of the Universe is increasing (with time)?

 

[bcrowell]

If the vacuum energy throughout space is constant and space is expanding, wouldn't that indicate the total energy of the Universe is increasing (with time)?

No, because the total energy of the universe isn't well defined. See #3.

 

[dm4b]

I'm not sure the simple answer is a plain "no", is it?

I understand energy is not defined globally per my OP and the lack of time-like killing vectors.

But, it can be defined locally, at least approximately.

If you consider my question again locally, and then extrapolate to cosmological scales, it seems like an energy increase is implied, at the very least, doesn't it?

Besides, if energy truly isn't conserved on a global scale, it seems to me, it could increase, or decrease, or do whatever it wants, as far as we know and can predict. Unless, of course, the lack of it being defined is an artifact of GR, and/or a limitation of the theory. This is why I originally asked what are the ramifications of it not being conserved under GR?

 

[bcrowell]

If you consider my question again locally, and then extrapolate to cosmological scales, it seems like an energy increase is implied, at the very least, doesn't it?

Once you extrapolate to cosmological scales, it isn't local anymore.

Besides, if energy truly isn't conserved on a global scale, it seems to me, it could increase, or decrease, or do whatever it wants, as far as we know and can predict.

It doesn't make sense to talk about it increasing or decreasing. It just isn't well defined.

Unless, of course, the lack of it being defined is an artifact of GR, and/or a limitation of the theory.

It certainly can be described as a limitation of the theory.

-Ben

 

[dm4b]

It doesn't make sense to talk about it increasing or decreasing. It just isn't well defined.

Agreed, that's sort of my point in these two sentences you made. How can you state anything definitive about something that is undefined?

If we are unable to define something I don't think we can rightly make any absolute statements about that something.

It certainly can be described as a limitation of the theory.
-Ben

I guess this relates to my 2nd point. If local considerations, which seem to be valid, imply "something" when extrapolated, and our theory cannot explain this in any way, maybe you're bumping up against a limitation of the theory.

And, if you are, we cannot completely rule out was is being implied by the extrapolation, if it indeed falls outside the domain of validity of the theory.

It seems sort of hard to escape the conclusion that energy would increase globally when you're adding up all the local increases. Or, at the very least, just claiming it isn't defined seems like an answer that leaves something desired.

So:

(1) Is energy truly not defined on a global scale?

(2) or, are we unable to define it, given our current understanding?

 

[bcrowell]

It seems sort of hard to escape the conclusion that energy would increase globally when you're adding up all the local increases.

This type of addition isn't defined. Energy is the timelike component of a four-vector, and in GR addition of vectors isn't defined when the vectors are at different locations. To add them, you have to parallel-transport them into the same location, and parallel transport is path-dependent.

 

[dm4b]

Hmmm, energy consideration using 4-Vectors is more like SR, and it is not the total energy we are talking about. Under GR, energy is usually considered in a more sophisticated way within the energy-momentum tensor, which is divergence free and can be interpreted as a "local" energy conservation. Also, as part of a stationary spacetime, the Komar Integral can be interpreted as the total energy. But, with an expanding space, total energy is undefined under GR. All this is besides the point I was trying to make. So, leaving all that aside ...

I repeat:

(1) Is total energy truly not defined on a global scale?

(2) or, are we unable to define it, given our current understanding?

If you answer #2, the statement in your last post doesn't mean much for the question at hand. It could be trying to address a problem in the language of a theory that may be ill equipped for the task at hand. That's the main gist of the Point I wanted to make for #2.

If you answer #1, well, then all sorts of fun, interesting questions can get raised, which would be a great topic in its own right. (although, one likely to get banned on this forum)

Anyhow, this is why the main point of my OP was ... "What are the ramifications of total energy NOT being conserved under GR?"

 

[bcrowell]

Hmmm, energy consideration using 4-Vectors is more like SR, and it is not the total energy we are talking about. Under GR, energy is usually considered in a more sophisticated way within the energy-momentum tensor, which is divergence free and can be interpreted as a "local" energy conservation.

You're right about local conservation but wrong about it not being related to the SR 4-vector. When you integrate the energy-momentum tensor to get the flux, the flux is the net transport of the 4-vector.

(1) Is total energy truly not defined on a global scale?

(2) or, are we unable to define it, given our current understanding?

I think at this point you can make your own interpretation.

 

[dm4b]

You're right about local conservation but wrong about it not being related to the SR 4-vector. When you integrate the energy-momentum tensor to get the flux, the flux is the net transport of the 4-vector.

I said the SR 4-Vector component is not related to the total energy of the Universe that we were talking about, which you also stated when you claimed the total energy is undefined.

I think at this point you can make your own interpretation.

Don't know why folks have such a hard time saying this, but I think a better, more honest, answer might be to just say we don't know ... yet.

I personally suspect total energy will be definable at some point, and if it isn't conserved, we'll be able to say why. Perhaps, quantum gravity will help out here, whenever it gets figured out.

 

[JesseM]

I repeat:

(1) Is total energy truly not defined on a global scale?

(2) or, are we unable to define it, given our current understanding?

Are you just asking about what's true according to the theory of GR, or what might be true in some future ultimate theory? If you're asking about GR I get the impression the answer is 1 (i.e. there's no hope that in the future someone will come up with a coordinate-independent definition of global energy that works in GR), but of course there's no way of knowing for sure what might be true in a future theory.

 

[pervect]

Agreed, that's sort of my point in these two sentences you made. How can you state anything definitive about something that is undefined?

You can't. At first, I thought that's what you were doing - but on closer inspection, you seemed to be offering a definition of a number, which I understood to be as follows:

Divide space-time up into a bunch of small pieces. In each small piece:

Take the energy density rho in some local frame
multiply it by the volume element in said local frame

Sum them all together.

One of the problems with this idea is that it isn't in general independent of how one cuts up spacetime into pieces. But, we can take a specific case, a static geometry, where there is an "obvious" way to cut up space-time into pieces - and see that said number we compute by this method STILL fails to give us the total energy. This gives us some insight into why the number one computes in this manner isn't energy.

What you'd compute by this technique in a static space-time is what MTW calls the "energy before assembly". If you have this text, see chapter 23 on "spherical stars".

The actual energy, which MTW also computes, is lower than this number - we can say "actual energy" because the geometry is static so we have a defined energy to compare your number to. MTW calls the difference "gravitational binding energy", and demonstrates how the difference approaches the expected Newtonian value in the Newtonian limit

So, since the actual energy is lower than your number, showing that your number increases doesn't demonstrate anything, since we have reason be believe by example that the actual energy is lower than your number because of the "gravitational binding energy" in a simple situation where we can define both your number and energy.

(1) Is energy truly not defined on a global scale?

(2) or, are we unable to define it, given our current understanding?

The safest remark is probably 2). However, Noether's theorem, which was created to address this issue, suggests that if we did have a way of defining energy on a global scale, we'd have as a result some "preferred" frame. (At least, that's my understanding). We'd have a preferred four-parameter subgroup of space-time.

Preferred frame theories are possible, but are out of vogue - and are not GR. For instance, Self Creation Cosmology would be an example of a non-GR theory that would have a conserved energy, getting around Noether's theorem by having a scalar field with an associated Jordan Frame, a bit like Branse-Dicke theory.

Unfortunately, both Branse-Dicke theories and Self-Creation Cosmology (as originally proposed) appear to be inconsistent with experiment - the later predicted different results for Gravity probe B.

So, unless my understanding is wrong, GR won't have a conserved energy because it doesn't have a preferred frame. Examples of theories that have a conserved energy also have preferred frames.

 

[bcrowell]

The safest remark is probably 2). However, Noether's theorem, which was created to address this issue, suggests that if we did have a way of defining energy on a global scale, we'd have as a result some "preferred" frame. (At least, that's my understanding). We'd have a preferred four-parameter subgroup of space-time.

Preferred frame theories are possible, but are out of vogue - and are not GR. For instance, Self Creation Cosmology would be an example of a non-GR theory that would have a conserved energy, getting around Noether's theorem by having a scalar field with an associated Jordan Frame, a bit like Branse-Dicke theory.

Unfortunately, both Branse-Dicke theories and Self-Creation Cosmology (as originally proposed) appear to be inconsistent with experiment - the later predicted different results for Gravity probe B.

So, unless my understanding is wrong, GR won't have a conserved energy because it doesn't have a preferred frame. Examples of theories that have a conserved energy also have preferred frames.

I'm not sure that the discussion of Brans-Dicke gravity is quite right here. BD gravity doesn't have a preferred frame. I'm pretty sure it can't have a conserved energy scalar that applies in all spacetimes, because GR is a special case of BD, and GR doesn't have such a thing. Re consistency with experiment, since GR is a special case of BD, experiments can never disprove BD entirely without disproving GR -- but it's true that the $\omega$ parameter is constrained by solar-system measurements (the Cassini probe) to be so high that it makes BD lose its interest in most people's eyes. But maybe you were really just referring to BD gravity in order to explain that SCC is related to it...?

I read the WP article on SCC. I don't see anything in it about global conservation of energy, only local conservation. Of course, reading a WP article doesn't make me an expert, but are you sure that it's the case that there's a conserved global energy in all spacetimes? For example, SCC is supposed to be equivalent to GR in the case of vacuum solutions, but GR doesn't have a conserved global energy that applies to all vacuum solutions.

 

[dm4b]

Are you just asking about what's true according to the theory of GR, or what might be true in some future ultimate theory?

What's true, in general.

Take for example singularities. GR, more or less, predicts them, but then offers no explanation for them. This is mainly because we're now outside the scope of the theory.

Well, GR also "predicts" that the total energy of the Universe is undefined. Is it really? Or, are we outside the domain of the theory again?

Is it just a mathematical artifact? Many seem to think that spacetime is NOT really curved, and is just a convenient analogy to describe a mathematical model that happens to work, as far as giving predictions. Well, could an undefined total energy be an unfortunate consequence of using a model that doesn't truly physically represent reality? Or, could it just be the theory is incomplete and quantum gravity will fill in some holes? Or, has energy been conclusively proven to be undefined in GR? I think the answer to the last question is a "no". But, the jury seems to be out on the other two.

I personally have a hard time imagining that total energy is truly undefined. Everything in the Universe is essentially energy. We are not special in our little corner of the Universe. If we would argue that energy is defined locally and energy "conservation" is valid locally, well, then, so would everybody else in their little corner of the Universe, no matter where they are. How then can it be undefined globally in an absolute sense?

Anyhow, don't take my last paragraph as a scientific theory and analyze it. It's just meant as a simple analogy. ;-)

 

[dm4b]

You can't. At first, I thought that's what you were doing - but on closer inspection, you seemed to be offering a definition of a number, which I understood to be as follows:

Divide space-time up into a bunch of small pieces. In each small piece:

Take the energy density rho in some local frame
multiply it by the volume element in said local frame

Sum them all together ...

That was actually just meant as a pictorial analogy (and a bad one!) to represent where I was coming from, similar to the one in my last post, in an attempt to talk about something that we don't really have the math to handle, at least as far as I can tell.

sorry for the confusion

The safest remark is probably 2). However, Noether's theorem, which was created to address this issue, suggests that if we did have a way of defining energy on a global scale, we'd have as a result some "preferred" frame. (At least, that's my understanding). We'd have a preferred four-parameter subgroup of space-time.

Preferred frame theories are possible, but are out of vogue - and are not GR. For instance, Self Creation Cosmology would be an example of a non-GR theory that would have a conserved energy, getting around Noether's theorem by having a scalar field with an associated Jordan Frame, a bit like Branse-Dicke theory.

Unfortunately, both Branse-Dicke theories and Self-Creation Cosmology (as originally proposed) appear to be inconsistent with experiment - the later predicted different results for Gravity probe B.

So, unless my understanding is wrong, GR won't have a conserved energy because it doesn't have a preferred frame. Examples of theories that have a conserved energy also have preferred frames.

Thanks, this getting along the lines of what I was looking for.

Yes, preferred frames definitely don't seem like the right way to go.

Could it be possible to not necessarily use a perferred frame, but rather use an entirely coordinate-independent way to define total energy, as JesseM also stated above? I guess we don't know the answer to that though. Have there been any attempts to do so, though?

Coordinate systems and reference frames are man-made mental constructs. Perhaps they are getting in the way here?