Energy Conservation Paradox: Is It True or Not?

[Dale]

It works very well here but I want to know does it work equally well in other geometries.

Asked and answered.

Does that mean that laws and equations involving positions, for example Coulomb's law, work only in this approximately flat geometry.

Yes

 

[pervect]

The following quote from MTW's "Gravitation" $20.4 "Why the energy of the gravitational field cannot be localized" may be of some help.

Omitted is an introduction in which the autors introduce a 'straw-man' approach to gravitational energy using pseudotensors that the authors proceed to shoot down.

To ask for the amount of electromagnetic energy and momentum in an element of 3-volume makes sense. First, there is one and only one formula for this quantity. Second, and more important, this energy-momentum in principle "has weight." It curves space. It serves as a source term on the righthand side of Einstein's field equations. It produces a relative geodesic deviation of two nearby world lines that pass through the region of space in question. It is observable. Not one of these properties does "local gravitational energy-momentum" possess. There is no unique formula for it, but a multitude of quite distinct formulas. The two cited are only two among an infinity. Moreover, "local gravitational energy-momentum" has no weight. It does not curve space. It does not serve as a source term on the righthand side of Einstein's field equations. It does not produce any relative geodesic deviation of two nearby world lines that pass through the region of space in question. It is not observable.

Anybody who looks for a magic formula for "local gravitational energy-momentum" is looking for the right answer to the wrong question. Unhappily, enormous time and effort were devoted in the past to trying to "answer this question" before investigators realized the futility of the enterprise. Toward the end, above all mathematical arguments, one came to appreciate the quiet but rock-like strength of Einstein's equivalence principle. One can always find in any given locality a frame of reference in which all local "gravitational fields" (all ChristofTel symbols disappear. No Christoffel symbols means no "gravitational field" and no local gravitational field means no "local gravitational energy-momentum."

Let me paraphrase the argument. If you consider some 1m^3 volume on the surface of the Earth, if you use usually coordinates of static objects, there is a gravitational field in the form of the Christoffel symbols present. I will try to explain the techincal language by saying the Christoffel symbols represent, among other things, the weight you read on a scale, or the weight you feel on your rear when you sit in a chair. In short, what most people think of as "gravity", the same notion that Newton had, of gravity as a force.

However, we can equally imagine a free-falling observer. This observer won't feel any "gravitational field" - they will feel second-order tidal forces, but, being in free fall, they won't feel any weight pulling them down in their chair.

In Newton's theory, the force of gravity is an actual force, and you can look at what work this force does. In GR, gravity is curved space-time, and you can always find an observer, moving along a geodesic, who doesn't experience any force. One of the important features of the theory is its observer independence, there point is there isn't any formulation of the "gravitational field" in the sense of forces you feel on your backside that is observer independent, so trying to leverage off the Newtonian ideas doesn't get anywhere. The basic issue is that GR is observer independent, while the concept of weight (technically, Christoffel symbols) is not.

Back to MTW:

Nobody can deny or wants to deny that gravitational forces make a contribution to the mass-energy of a gravitationally interacting system. The mass-energy of the Earth-moon system is less than the mass-energy that the system would have if the two objects were at infinite separation. The mass-energy of a neutron star is less than the mass-energy of the same number of baryons at infinite separation. Surrounding a region of empty space where there is a concentration of gravitational waves, there is a net attraction, betokening a positive net mass-energy in that region of space (see Chapter 35). At issue is not the existence of gravitational energy, but the localizability of gravitational energy. It is not localizable. The equivalence principle forbids.

This is the part that says even though we can't come up with an observer independent notion of "the gravitational field", much less any way to come up with "how much energy the gravitational field has", we can't ignore the whole idea of "gravitational energy" and get an overall conserved quantity.

What we are left with is that there are ways to get a conserved quantity, but they numbers we get are not observer independent when we consider some specific locatoin - we can't assign the gravitational energy (that we need to include to have a conserved quantity) any specific location, the best we can do is come up with an overall number. This is the case, at least, without specifying some particular "preferred class" of observers. We need to include it to have the books balance, but the detailed assignment of energy we get when we do this is different for different observers.

There are some interesting ideas which I would loosely describe as specifying a preferred class of observers, the so-called "De-donder gauge". See for instance http://ptp.oxfordjournals.org/content/75/6/1351.full.pdf. I'm not sure how popular this idea is, I suspect not very because it seems to be limited to advanced papers rather than something you read in your average GR textbook. If you read a textbook on GR, you'll probably see something about ADM, Bondi, and Komar masses, but not little or nothing on the DeDonder gauge. But on the plus side, my understanding is that you get a notion of energy that's defined without the special requiremcan coents (of asymptotic flatness or stationary space times. Furthermore, when you do have these special requirements met, you get comparable numbers to the ADM, Bondi, and Komar formulae. I'm afraid I'm not quite sure if the "comparable numbers" are exactly the same. There are a few details of the comparsion process that need to be specified, at a minimum one would need to compensate for the fact that the Bondi approach doesn't include the energy stored in gravitational waves.

 

[bhobba]

Hi Guys

Nice thread.

One thing I want to mention is the modern conception of energy is based on Noethers theorem and the symmetries of the system which is usually associated with the symmetries of an assumed inertial frame it resides in. Since GR is based on space-time curvature Noether's theorem lacks applicability and you can't define energy using it so of course conservation is problematical:
http://motls.blogspot.com.au/2010/08/why-and-how-energy-is-not-conserved-in.html
'The main lesson here is that general relativity is not a theory that requires physical objects or fields to propagate in a pre-existing translationally invariant spacetime. That's why the corresponding energy conservation law justified by Noether's argument either fails, or becomes approximate, or becomes vacuous, or survives exclusively in spacetimes that preserve their "special relativistic" structure at infinity. At any rate, the status of energy conservation changes when you switch from special relativity to general relativity.'

Thanks
Bill

 

[bhobba]

My understanding at present is that if a system of interacting particles is analysed using classical physics or special relativity energy is conserved, but if that same system is analysed using general relativity energy is not conserved. So is it conserved or not?

See my post above.

Its due to the modern definition of energy which is based on symmetries and Noether's beautiful theorem. Those symmetries are lacking in GR.

In fact it was while investigating this very issue ie local energy conservation in GR, she discovered her very important theorem:
http://www.physics.ucla.edu/~cwp/articles/noether.asg/noether.html
'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 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. In the note to Klein he reports that had requested that Emmy Noether help clarify the matter.'

Thanks
Bill

 

[Dadface]

This calculation does make an assumption about the type of spacetime the experiment is carried out in. It assumes that spacetime is flat, at least to a good enough approximation in the region of spacetime in which the experiment took place. (At least, it does if you are doing the calculation the way I strongly suspect you are, since it's the way that is taught in electrodynamics class.) You just didn't realize that you were making that assumption. But if you show me the mathematical formula you use to calculate the increase of energy, I will show you where it makes that assumption, even if you didn't realize it.

(Hint: you said you "manually increase the separation of the plates". "Separation" means "distance between", and "distance" involves the geometry of spacetime. The formula I strongly suspect you would use to make the calculation implicitly assumes that spacetime, at least the region of it covered by the capacitor, is flat. It's easy to miss this because flat spacetime means that a lot of coefficients that would be present in a general curved spacetime are equal to 1, so they drop out of the formulas and it can seem like there's no spacetime at all in there, when actually there is.)

Yes and thank you .Things are beginning to clarify now. I think I know where you're going with the capacitor calculation.

 

[Dadface]

It is a reasonable question and you have received the correct answer and didn't like it.

I understand that you don't know this stuff and may not have the time to read up on the background material, but then don't try to argue with the people who have. Also, you cannot expect to ask a question about a theory and then exclude answers because they use that theory. The problem isn't your question, it is your refusal to listen to the answers.

All of that is very unfair because it's untrue.

I like every answer in this thread which has been relevant. Each one has contributed to the level of understanding I have now. Even the answers that I have yet to fully understand have taught me things such as the necessity to get familiar with the terminology used.
I have asked additional questions for example to get greater clarification but I have not argued. I have not excluded a single answer and I have not refused to "listen" to the answers. In fact I have read through this thread several times and researched on the net trying to get a better grasp of some of the answers given.

 

[Dadface]

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.

 

[jimgraber]

Hi Dadface,
As I hinted before, your best choice is to go back to the beginning of the universe, if it is not flat.
The next best choice is to go really close to a black hole, especially a small one.

Jim Graber

 

[jimgraber]

A second thought:
There are really (at least) two issues here;
First, where is spacetime non trivially curved?
Second, where is energy not conserved, or at least hard to define?
These issues are related in a complicated way, but they are not the same.
Best again,
Jim Graber

 

[jimgraber]

A third thought:
If you go really close to a small black hole, you can also worry about loss of information, Hawking radiation and firewalls.
Still more fun, for those who care.

 

[Dale]

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?

This depends on the sensitivity of your experiment. "Local" means "a small enough region of spacetime that tidal effects can be neglected". The more sensitive your measurement, the smaller the allowed region.

To worry about energy non conservation you need a spacetime that is not static with large tidal forces. For example around a binary pair of black holes or neutron stars.

 

[Dadface]

Hi Dadface,
As I hinted before, your best choice is to go back to the beginning of the universe, if it is not flat.
The next best choice is to go really close to a black hole, especially a small one.

Jim Graber

A second thought:
There are really (at least) two issues here;
First, where is spacetime non trivially curved?
Second, where is energy not conserved, or at least hard to define?
These issues are related in a complicated way, but they are not the same.
Best again,
Jim Graber

A third thought:
If you go really close to a small black hole, you can also worry about loss of information, Hawking radiation and firewalls.
Still more fun, for those who care.

That's really helpful jimgraber so thank you very much. You have mentioned places whose structures differ enormously from the sructure of this place. As a result of thinking about your examples I'm assuming that it is the structure of the place that determines the geometry of the spacetime in the vincinity of that place. Is my assumption correct? If so I should have realized it earlier.

 

[PAllen]

It might also be helpful to say things like "ADM energy and Bondi energy are defined and conserved in asymptotically flat space-times, while Komar energy is defined and conserved in static space-times.
...

You allude to this a little later, but I thought it worth pointing out that Bondi energy (when defined) is not meant to be a conserved quantity. Instead, it specifically does not include radiation reaching asymptotic infinity. It thus decreases slowly for most non-stationary systems. Meanwhile, ADM energy (when defined) is conserved, and the delta between ADM and Bondi energy is taken to be radiation escaping to asymptotic infinity.

 

[Dadface]

This depends on the sensitivity of your experiment. "Local" means "a small enough region of spacetime that tidal effects can be neglected". The more sensitive your measurement, the smaller the allowed region.

To worry about energy non conservation you need a spacetime that is not static with large tidal forces. For example around a binary pair of black holes or neutron stars.

Thank you DaleSpam. jimgraber mentioned places which can be visited by the means of a thought experiment. I wonder if you or anyone else can tell me if there are places where you can actually go to even if just in principle. As an example suppose we wanted to measure the mass of the electron. At present this is measured to ten decimal places.
m = 9.10938291(40) The uncertainty is highlighted by the use of brackets

 

[Dale]

I wonder if you or anyone else can tell me if there are places where you can actually go to even if just in principle. As an example suppose we wanted to measure the mass of the electron. At present this is measured to ten decimal places.
m = 9.10938291(40)

I am confused now. I thought that you wanted to know about the conservation of energy. What is the mass of the electron thing about?

 

[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.