Is energy conserved in cosmology according to the laws of thermodynamics?

[TrickyDicky]

I've just read the FAQ about this and IMO it is "not even wrong" to say that energy conservation doesn't apply to Cosmology. The fact is energy conservation is clearly stated in the EFE by the vanishing divergence of the stress-energy tensor and since GR is the theory we currently use in Cosmology, cerrtainly energy is conserved in cosmology.
There seems to be some silly confusion in that FAQ answer, the fact that currently the global energy can't be defined doesn't imply "energy is not conserved in cosmology", just like we don't say that since there is no "standard" way to define the total entropy of the universe it follows that the 2nd law of thermodynamics does't apply in cosmology.
It can be said that since we can't get out of the system called "universe", there is no easy way to define the universe energy unless we introduce a time-symmetry (Noether theorem). But that is a feature of the way we define energy. It doesn't have anything to do with GR.

 

[WannabeNewton]

\bigtriangledown _{\nu }T^{\mu \nu } = 0 is only local though so how could it justify global energy conservation under the friedmann metric which doesn't have \frac{\partial }{\partial t} as a killing field?

 

[pervect]

Where are the relativity FAQ's located? I thought the should be at https://www.physicsforums.com/forumdisplay.php?f=210 , but I don't see the FAQ in question (on energy conservation in GR)
there.

I'll defer detailed comments until I re-read the FAQ. I'm pretty sure I've read it before,and I doubt there is anything that a minor tweak wouldn't fix, but I'd like to re-read it again more carefully.

 

[WannabeNewton]

Where are the relativity FAQ's located? I thought the should be at https://www.physicsforums.com/forumdisplay.php?f=210 , but I don't see the FAQ in question (on energy conservation in GR)
there.

I'll defer detailed comments until I re-read the FAQ. I'm pretty sure I've read it before,and I doubt there is anything that a minor tweak wouldn't fix, but I'd like to re-read it again more carefully.

https://www.physicsforums.com/showthread.php?t=506985

 

[TrickyDicky]

\bigtriangledown _{\nu }T^{\mu \nu } = 0 is only local though so how could it justify global energy conservation under the friedmann metric which doesn't have \frac{\partial }{\partial t} as a killing field?

That \bigtriangledown _{\nu }T^{\mu \nu } = 0 is only local is debatable, chiefly when locally means as big a system as one can define, surely it assures local conservation, and by being in covariant form it opens the way to global conservation by introduction of symmetries.
I'm not trying to justify anything, my point was that energy conservation applies to cosmology if we strictly use the GR equations without further assumptions.

 

[TrickyDicky]

Maybe it would be more correct if it specified that the lack of energy conservation is a feature of our current cosmological model rather than something intrinsic to the general relativity equations.

 

[pervect]

Let's compare it to the sci.physics.faq- which addresses a similar question.

In special cases, yes. In general — it depends on what you mean by "energy", and what you mean by "conserved".

In flat spacetime (the backdrop for special relativity) you can phrase energy conservation in two ways: as a differential equation, or as an equation involving integrals (gory details below). The two formulations are mathematically equivalent. But when you try to generalize this to curved spacetimes (the arena for general relativity) this equivalence breaks down. The differential form extends with nary a hiccup; not so the integral form.

I've always thought "It depends on what you mean by energy and what you mean by conserved" was a bit weasel-worded. The bit about the differential and integral formulations is very helpful to the advanced reader, but it's not so helpful to the less advanced reader.

Note that the sci.phsics FAQ is concerned with whether or not energy is conserved in general realtivity, while the FAQ in question is in the cosmology section and applies to cosmology.

While we might want to mention somewhere in the FAQ that we have definitions of energy that work in certain situations, it's accurate and IMO helpful to state right-up-front that none of these situations appear to apply to our universe - by which I mean the universe as a whole, not some infinitesimally small piece of it.

 

[WannabeNewton]

Maybe it would be more correct if it specified that the lack of energy conservation is a feature of our current cosmological model rather than something intrinsic to the general relativity equations.

Right ok, I think I see what you're getting at. So you're not disputing anything, your're just saying this isn't something that should be uniquely attributed to GR?

 

[TrickyDicky]

I really thing it would pay off to clarify that we know from 1917 thanks to Noether theorem that energy conservation is just the result of time symmetry, and therefore the only reason we state energy is not conserved in the universe as a whole is the introduction of time asymmetry by the FRW metric, not something in the EFE or due to the way we define the total energy of the universe.

 

[TrickyDicky]

Right ok, I think I see what you're getting at. So you're not disputing anything, your're just saying this isn't something that should be uniquely attributed to GR?

Yes, and also that the phrase "energy conservation doesn't apply to cosmology" is misleading and should be rephrased.

 

[Q-reeus]

Given the claim coe holds 'locally', are we talking about first or higher order curvature terms re 'global' departure? Is the lack of a gravitational contribution to the stress-energy tensor seen as the problem here? Shouldn't there be some relatively simple thought experiment capable of showing any violation if it exists? One would expect specific scenarios have been worked through over the last 95+ years, to leave no doubt! :zzz:

 

[Dale]

Shouldn't there be some relatively simple thought experiment capable of showing any violation if it exists? One would expect specific scenarios have been worked through over the last 95+ years, to leave no doubt! :zzz:

The CMBR is a good example that leaves no doubt. The CMBR photons that we receive now have lost a lot of energy from when they were last scattered.

 

[Q-reeus]

The CMBR is a good example that leaves no doubt. The CMBR photons that we receive now have lost a lot of energy from when they were last scattered.

No doubting the fact of enormous CMBR redshift and thus reduction in net radiant energy, but is that a sufficient proof? Why could that not be a case of energy transfer - from CMBR to expansion rate for instance (ie matter gets an added 'kick')? As many are aware the prevailing view amongst cosmologists is that the total energy of the universe (inclusive of radiation, matter, and 'dark energy') is an invariant zero; from beginning to end. As I recall it this assumption strictly relies on an overall zero 4-curvature (flat spacetime), which studies have apparently shown to either be true or a good approximation.
If on the other hand the claim of a global failure of coe is valid, this cannot magically only appear on the scale of the visible universe. One should be able to model some perfectly finite 'average' volume containing the requisite uniform distribution of radiation, matter, and maybe DE, and show that it's time evolution leads to an excess or deficit of total energy. Certainly not my province, but would be extremely surprised such has not been done. Anyone aware of such studies, and how that then tallies or not with the zero energy universe concept?

 

[Dale]

As many are aware the prevailing view amongst cosmologists is that the total energy of the universe (inclusive of radiation, matter, and 'dark energy') is an invariant zero; from beginning to end.

Do you have any reference for this? I was not aware that the FLRW spacetime admitted a globally conserved energy.

 

[TrickyDicky]

As many are aware the prevailing view amongst cosmologists is that the total energy of the universe (inclusive of radiation, matter, and 'dark energy') is an invariant zero; from beginning to end. As I recall it this assumption strictly relies on an overall zero 4-curvature (flat spacetime), which studies have apparently shown to either be true or a good approximation.

Hi, Q-reeus.
AFAIK, that is not the prevailing view among cosmologists, might you be referring to zero spatial 3-curvature? Zero 4-curvature usually refers to the SR model.

 

[Q-reeus]

Do you have any reference for this? I was not aware that the FLRW spacetime admitted a globally conserved energy.

Sur - here's one I had saved: http://arxiv.org/abs/astro-ph/0212574 I tend to look at the conclusions part.

 

[TrickyDicky]

Sur - here's one I had saved: http://arxiv.org/abs/astro-ph/0212574 I tend to look at the conclusions part.

Ok, I had never heard of this, I'll take a look at it. However I still wouldn't say this is the prevailing view.

 

[Q-reeus]

Hi, Q-reeus.
AFAIK, that is not the prevailing view among cosmologists, might you be referring to zero spatial 3-curvature? Zero 4-curvature usually refers to the SR model.

TrickyDicky - I may be taking a bit of liberty claiming it is for sure the majority view, but the flat spacetime bit is correct as the article linked in #16 explains. Lawrence Krauss makes a big thing of zero universe in an entertaining (~ 1 hr long) Youtube presentation:
http://www.freelecturevideos.com/ph...lawrence-krauss-aai-2009-video_243dcb1d9.html Here's another link re flat spacetime and zeu http://arxiv.org/abs/gr-qc/0605063

 

[TrickyDicky]

TrickyDicky - I may be taking a bit of liberty claiming it is for sure the majority view, but the flat spacetime bit is correct as the article linked in #16 explains. Lawrence Krauss makes a big thing of zero universe in an entertaining (~ 1 hr long) Youtube presentation:
http://www.freelecturevideos.com/ph...lawrence-krauss-aai-2009-video_243dcb1d9.html Here's another link re flat spacetime and zeu http://arxiv.org/abs/gr-qc/0605063

I just read the paper and find it very interesting. And it further supports my criticism of the energy conservation FAQ, I really think that as it stands it is not only misleading but wrong and should be modified.

 

[Dale]

Sur - here's one I had saved: http://arxiv.org/abs/astro-ph/0212574 I tend to look at the conclusions part.

Interesting. I need to read this in more detail.

 

[Q-reeus]

I just read the paper and find it very interesting. And it further supports my criticism of the energy conservation FAQ, I really think that as it stands it is not only misleading but wrong and should be modified.

In the article of that last link I gave, Berman in part 3 makes the point that working in the wrong coordinate system can introduce fictitious forces then wrongly interpreted as gravitational, and that this throws out the calculations. I'll leave it for you experts to debate!

 

[TrickyDicky]

Certainly it makes sense if you look at it like this: at every infinitesimal point (aka "locally") in the universe the energy is conserved (according to both SR and GR), and at the same time at every infinitesimal point the total energy is zero. Integrating for the total of points in the manifold volume it leads to consider that in the universe the energy is conserved and it totals zero energy.
This only seems to need for the Equivalence principle, SR, and the Bianchi identities to hold true in order to be a viable conclusion.

 

[Q-reeus]

...at every infinitesimal point (aka "locally") in the universe the energy is conserved (according to both SR and GR), and at the same time at every infinitesimal point the total energy is zero...

I take it you will agree this bit can only correspond to a perfectly homogenized universe. Obviously in our real 'lumpy' universe local energy content will have big ups and downs, but averaged out on the universal scale, corresponds to the homogenized 'point' result.

 

[TrickyDicky]

I take it you will agree this bit can only correspond to a perfectly homogenized universe. Obviously in our real 'lumpy' universe local energy content will have big ups and downs, but averaged out on the universal scale, corresponds to the homogenized 'point' result.

Those big ups and downs always involve finite dimensions.
A different story is if you want to include singularity points, but since singularities (i.e. BB at t=0 or BH singularities) by definition are out of the realm of known physics and I don't want to speculate I left them out.

 

[Q-reeus]

Those big ups and downs always involve finite dimensions.
A different story is if you want to include singularity points, but since singularities (i.e. BB at t=0 or BH singularities) by definition are out of the realm of known physics and I don't want to speculate I left them out.

Ahh, my bad - was interpreting that previously quoted bit to mean zero energy density rather than just energy (ie you were just saying infinitesimal volume -> zero net energy in the limit as v -> 0). OK so you were actually making a connection argument there, not that energy is everywhere zero just because it is overall. Should have known better.

 

[TrickyDicky]

Ahh, my bad - was interpreting that previously quoted bit to mean zero energy density rather than just energy (ie you were just saying infinitesimal volume -> zero net energy in the limit as v -> 0). OK so you were actually making a connection argument there, not that energy is everywhere zero just because it is overall. Should have known better.

No sweat. I posted my connection argument so it can be torn apart anyway

 

[pervect]

Lets say you have a perfect fluid, and you try to define the energy of the fluid as the integral of T_00 in the rest frame of the fluid.

In curved space-time, it's my understanding that you can have \nabla \dot T_{ij} = 0[/itex], and have the above energy vary with time. For instance, you could have X number of watts per m^3. If you take the limit to zero volume, the "energy" creation rate is zero. But for any finite volume, the energy creation rate is finite.

Example: in a FRW space-time, there's a natural background frame for the "cosmological fluid" as defined above. But you'll find that the integral of T_00 in this frame is NOT constant with time, even though the FRW solution obeys Einstein's eqauations. If I get more time, I'll post the relevant section from MTW about this.

 

[PAllen]

The following reviews the 'mainstream' range of opinion on bridging the gap from local, differential statements (where strong conclusions are possible) to statements at infinity in asymptotically flat spacetime (strong statements possible). The focus is on definition and properties of so called quasi-local energy, momentum, and angular momentum. Not much is said about properties at infinity for non- asymptotically flat spacetimes.

http://relativity.livingreviews.org/Articles/lrr-2009-4/

 

[Dale]

This is an interesting take.

The authors are specifically considering universes where the energy becomes infinitely dilute, i.e. approaching Minkowski metric for sufficiently large time. I wonder if this is, in a sense, an asymptotically flat spacetime where energy is indeed conserved. I never though of going asymptotically in the time dimension, but it makes a certain amount of intuitive sense to me.

Of course, my intuition has been wrong in GR many times

 

[Q-reeus]

Wow - very first sentence in the Introduction to http://relativity.livingreviews.org/Articles/lrr-2009-4/ (link given by PAllen in #28) is a bit of a bombshell: "Over the last 30 years, one of the greatest achievements in classical general relativity has certainly been the proof of the positivity of the total gravitational energy, both at spatial and null infinity."
Right, well if so then that throws into limbo the apparently secure findings I linked to in #16 & #18 re invariant zero energy universe. Sure as hell there's no way to add up a net positive gravitational energy and a necessarily positive matter+radiation+DE and get zero! So one school of thought is dead wrong here, and I'm not game to put money on either. What that review article makes clear is that after nearly 100 years of intensive effort, important things are still not clear. I depart this thread!

 

[bcrowell]

I really thing it would pay off to clarify that we know from 1917 thanks to Noether theorem that energy conservation is just the result of time symmetry, and therefore the only reason we state energy is not conserved in the universe as a whole is the introduction of time asymmetry by the FRW metric, not something in the EFE or due to the way we define the total energy of the universe.

Noether's theorem is actually a bunch of different theorems, since it's been generalized in various ways. All the versions are specific mathematical results that make specific assumptions and derive specific conclusions from them. None of them leads to the result you claim.

Certainly it makes sense if you look at it like this: at every infinitesimal point (aka "locally") in the universe the energy is conserved (according to both SR and GR), and at the same time at every infinitesimal point the total energy is zero. Integrating for the total of points in the manifold volume it leads to consider that in the universe the energy is conserved and it totals zero energy.

This is incorrect, because the integration step doesn't work the way you believe it does. The basic problem is that energy-momentum is a vector, and when you want to add up vector quantities that occur at different points in spacetime, you get an ambiguity due to the path-dependence of parallel transport.

 

[Q-reeus]

Back again. Asked earlier for any specific model of a finite time evolving gravitating system that can be shown to violate conservation of energy. let's generalize to include momentum/angular momentum. This should be unambiguously determinable by referencing to coordinate measure - ie assume a flat background metric upon which the system's locally curved spacetime is embedded. Examples might be a collapsing disk of 'dust' with or without radiation, or spin-orbit coupling between two gravitating masses. If that has not or cannot be done, what physical consequences are or can be definitely ascribed to the ambiguity mentioned in #31? If it only applies to say an expanding FRW universe, this gets back to which school is right, as per #30.

 

[TrickyDicky]

Noether's theorem is actually a bunch of different theorems, since it's been generalized in various ways. All the versions are specific mathematical results that make specific assumptions and derive specific conclusions from them. None of them leads to the result you claim.

Noether's theorem can be summarized as: for each symmetry of the Langrangian, there is a corresponding conserved quantity. In the case of time translation symmetry you get energy conservation. What part of this result(wich is the result I claim in what you quote) exactly do you find not derivable from Noether's theorem?

This is incorrect, because the integration step doesn't work the way you believe it does. The basic problem is that energy-momentum is a vector, and when you want to add up vector quantities that occur at different points in spacetime, you get an ambiguity due to the path-dependence of parallel transport.

What vector? I thought we were talking about the energy-momentum tensor. If you don't make any distinction between vectors and tensor fields with covariantly conserved energy, I don't think it leads to any productive discussion.

 

[PAllen]

This historic overview may be of interest to those reading this thread:

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

Note particularly, the following paragraph:

"In contemporary terminology the general theory of relativity is a gauge theory. The symmetry group of the theory, is a gauge group. It is the group of all continuous coordinate transformations with continuous derivatives, often called the group of general coordinate transformations. It is a Lie group that has a continuously infinite number of independent infinitesimal generators. In Noether's terminology such a group is an infinite continuous group. The symmetry group of special relativity, the Poincare' group4 is a Lie subgroup of the group of general coordinate transformations. It has a finite number (7) of independent infinitesimal generators. Noether refers to such a group as a finite continuous group. This distinction between a Lie group with a finite (or countably infinite) number of independent infinitesimal generators and an infinite continuous group is what distinguishes Noether's theorem I and theorem II in I.V.. Theorem I applies when one has a finite continuous group of symmetries, and theorem II when there is an infinite continuous group of symmetries. Field theories with a finite continuous symmetry group have what Hilbert called `proper energy theorems'. Physically in such theories one has a localized, conserved energy density; and one can prove that in any arbitrary volume the net outflow of energy across the boundary is equal to the time rate of decrease of energy within the volume. As will be shown below, this follows from the fact that the energy-momentum tensor of the theory is divergence free. In general relativity, on the other hand, it has no meaning to speak of a definite localization of energy. One may define a quantity which is divergence free analogous to the energy-momentum density tensor of special relativity, but it is gauge dependent: i.e., it is not covariant under general coordinate transformations. Consequently the fact that it is divergence free does not yield a meaningful law of local energy conservation. Thus one has, as Hilbert saw it, in such theories `improper energy theorems.'"

The remaining discussion is also relevant.

See also, the last paragraph (especially) of:

http://www.mathpages.com/home/kmath564/kmath564.htm

 

[bcrowell]

Back again. Asked earlier for any specific model of a finite time evolving gravitating system that can be shown to violate conservation of energy.

If by "violate conservation of energy" you mean a local violation, then that won't happen, because local conservation of energy is built into the Einstein field equations. If you mean a global violation, then this is covered in the FAQ: https://www.physicsforums.com/showthread.php?t=506985 The total energy cannot even be defined in a typical spacetime, so there is no way to discuss whether it changes over time.

This should be unambiguously determinable by referencing to coordinate measure - ie assume a flat background metric upon which the system's locally curved spacetime is embedded.

This sounds like you want to assume GR isn't valid in a discussion of GR. One of the most important and fundamental ideas of GR is that spacetime isn't naturally endowed with coordinates.

 

[Q-reeus]

If by "violate conservation of energy" you mean a local violation, then that won't happen, because local conservation of energy is built into the Einstein field equations. If you mean a global violation, then this is covered in the FAQ: https://www.physicsforums.com/showthread.php?t=506985 The total energy cannot even be defined in a typical spacetime, so there is no way to discuss whether it changes over time.

Thanks for the link. Well the first consideration is; how local does local have to be? If strictly speaking a point volume, not much use. My brief layman's translation is: In the 'global' context, GR is in general incapable of connecting all the internals of a system to give an unambiguous definition of overall energy-momentum. Hmm...could this fundamentally stem from that lack of gravitational field energy-momentum as source terms in the RHS of the EFE's? Or is it claimed any metric theory would be so burdened with ambiguity?

This sounds like you want to assume GR isn't valid in a discussion of GR. One of the most important and fundamental ideas of GR is that spacetime isn't naturally endowed with coordinates.

That wasn't what was meant. If the assumed global failure stems from this inability to connect the internals from within a curved spacetime, how about assessing a finite system 'from the outside'. Most or at least many cosmologists are happy with the idea of an overall flat universe, so that seems a natural backdrop (the coordinate reference frame) from which to assess the evolution of some given local system - the 'black box'. All I'm saying is, given some initial configuration, this should be able to be followed through to an arbitrary final configuration, and any difference in net energy-momentum unambiguously determined - all referenced to the coordinate values 'at infinity'. So has that been done? If even that is considered an ill-defined problem for GR, then yes I suppose I would be questioning validity of same.

 

[WannabeNewton]

You can't define extrinsic properties of a manifold in GR as it only concerns itself with intrinsic properties so I do not understand what you mean by "from the outside". The ambiguity results from the way parallel transport works on a curved manifold: transporting a vector around a closed loop in one direction will yield a vector with different components than if the vector was transported around the same loop the other way because remember covariant derivatives don't commute in curved space. Can you define the covariant derivative without a metric? It requires an affine connection but you can define an affine connection on a manifold without the use of a metric right?

 

[Q-reeus]

You can't define extrinsic properties of a manifold in GR as it only concerns itself with intrinsic properties so I do not understand what you mean by "from the outside". The ambiguity results from the way parallel transport works on a curved manifold: transporting a vector around a closed loop in one direction will yield a vector with different components than if the vector was transported around the same loop the other way because remember covariant derivatives don't commute in curved space.

I won't argue your comments about parallel transport issues - your '11th grade' education in GR is better than mine. However, let's try and get this into some sort of concrete setting. We have say a binary star system that spirals together and finishes up as a neutron star+exploding gas, light etc. Are you saying there is no way in principle we could have an unambiguous before and after accounting of the net energy-momentum, all referenced from some far vantage point where curvature of the system has negligible effect on the distant observer? If that's the case, this seems tantamount to saying it's impossible in GR to accurately specify any gravitating system to begin with.

 

[WannabeNewton]

Are you saying there is no way in principle we could have an unambiguous before and after accounting of the net energy-momentum, all referenced from some far vantage point where curvature of the system has negligible effect on the distant observer?

No I think in your scenario, where the observer is in an asymptotically flat - space time, the energy - momentum loss can be measured unambiguously through the angular momentum and energy flux carried away by gravitational waves. But I think the point is that in asymptotically flat space - times you can actually define global energy far away from the source but what about in general? You can't even define global energy in curved space - times in the absence of time - like killing fields.

 

[TrickyDicky]

Let's give some context to what is being discussed so we can avoid some gratuitous arguments, what I posted in #22 was in the context of the Noether theorem discussion about symmetries that I introduced previously, so it is obvious that it only works provided a timelike killing vector is introduced. The EFE by themselves tell us nothing about the existence of this Killing vector, that it doesn't exist is an assumption of the current cosmological model.

 

[Q-reeus]

No I think in your scenario, where the observer is in an asymptotically flat - space time, the energy - momentum loss can be measured unambiguously through the angular momentum and energy flux carried away by gravitational waves. But I think the point is that in asymptotically flat space - times you can actually define global energy far away from the source but what about in general? You can't even define global energy in curved space - times in the absence of time - like killing fields.

OK let's agree then that the evolution of a system's energy-momentum can be definitely established wrt asymptotically flat spacetime. I see two issues here. First, can we assume that conservative behavour is always then determined, regardless of the possibly violently energetic and highly curved system local parameters (eg high spinning binary neutron star inspiral)? If so, then shouldn't this logically be extrapolated to say that conservative behavour applies in any arbitrary curved spacetime setting, and it's just that we don't have an unambiguous means of determining it? Which is quite different to saying there can be an actual failure of coem. (hope this doesn't seem gratuitous TrickyDicky! )

 

[bcrowell]

First, can we assume that conservative behavour is always then determined, regardless of the possibly violently energetic and highly curved system local parameters (eg high spinning binary neutron star inspiral)?

In that situation, energy is being radiated away to infinity by gravitational waves.

If so, then shouldn't this logically be extrapolated to say that conservative behavour applies in any arbitrary curved spacetime setting,

No. There are conserved, global, scalar measures of mass-energy for static spacetimes and asymptotically flat spacetimes, but not for arbitrary spacetimes.

 

[Q-reeus]

In that situation, energy is being radiated away to infinity by gravitational waves.

You misunderstand - by conservative it was meant overall conservative, ie conservation of energy-momentum holds, not that there was no dissipative behavour.

No. There are conserved, global, scalar measures of mass-energy for static spacetimes and asymptotically flat spacetimes, but not for arbitrary spacetimes.

Well this is a sticking point for me. Do you agree that say the inspiralling NS scenario will be conservative (net!) wrt asymptotically flat spacetime measure? If not I desist but if you do then this creates a problem. The inspiralling NS system is an interaction of matter in a highly curved local spacetime, so if coem actually fails in GR it should be failing here and thus registered as such at 'infinity'. Conversely, if there is no failure by this reference measure, how can 'failure' in a generally curved background setting be other than a measurement problem, and not intrinsically 'real'?

 

[bcrowell]

Let's give some context to what is being discussed so we can avoid some gratuitous arguments, what I posted in #22 was in the context of the Noether theorem discussion about symmetries that I introduced previously, so it is obvious that it only works provided a timelike killing vector is introduced. The EFE by themselves tell us nothing about the existence of this Killing vector, that it doesn't exist is an assumption of the current cosmological model.

Your #22 was incorrect, for the reasons I gave in #31. Noether's theorem doesn't help, for the reasons given in #31.

Your logic about timelike Killing vectors is incorrect. What's relatively easy to show is that if there's a timelike Killing vector (i.e., the spacetime is stationary), then there is a conserved energy for test particles. That is completely different from being able to define a global, scaler measure of energy for the spacetime as a whole -- that was done by Komar in 1963, and it wasn't just some kind of trivial application of Noether's theorem.

I don't know if you will get around to admitting at some point that your #1 was incorrect. The FAQ gives references -- did you look at them?

 

[TrickyDicky]

Your #22 was incorrect, for the reasons I gave in #31. Noether's theorem doesn't help, for the reasons given in #31.

Your logic about timelike Killing vectors is incorrect. What's relatively easy to show is that if there's a timelike Killing vector (i.e., the spacetime is stationary), then there is a conserved energy for test particles. That is completely different from being able to define a global, scaler measure of energy for the spacetime as a whole

What do you find incorrect about this? It is the mathematical version of #22 and if you want citations you can find it in Penrose Road to reality, where the references to Noether theorem can also be seen:
We define a rate of energy transfer or flux with covector L:
L_a=T_{ab}\kappa^{b} with \kappa as the Killing vector
This will make the flux covector divergence vanish when \nabla^{a} T_{ab}=0
From this It is straight-forward that there is an integral conservation law:
\int_{V}L=0


Do you get it now?

 

[Ben Niehoff]

What do you find incorrect about this? It is the mathematical version of #22 and if you want citations you can find it in Penrose Road to reality, where the references to Noether theorem can also be seen:
We define a rate of energy transfer or flux with covector L:
L_a=T_{ab}\kappa^{b} with \kappa as the Killing vector
This will make the flux covector divergence vanish when \nabla^{a} T_{ab}=0
From this It is straight-forward that there is an integral conservation law:
\int_{V}L=0


Do you get it now?

Can you be more specific what you're integrating here? It looks like you're trying to integrate a 1-form over a 3-volume, which makes no sense.

 

[TrickyDicky]

I'd say L is a 3-form.

 

[bcrowell]

What do you find incorrect about this? It is the mathematical version of #22 and if you want citations you can find it in Penrose Road to reality, where the references to Noether theorem can also be seen:

What page in Road to Reality? Ben Niehoff has pointed out that your notation doesn't make sense, which makes me doubt that you're giving an accurate depiction of an argument by Penrose.

Actually I happen to have a relevant quote from the Penrose book handy:

For example, it is not at all a clear-cut matter to apply these ideas to obtain energy-momentum conservation in general relativity, and strictly speaking, the method does not work in this case. The apparent gravitational analogue [of EM gauge symmetry] is 'invariance under general coordinate transformations' [...] but the Noether theorem does not work in this situation, giving something of the nature '0=0'.

--Penrose, Road to Reality, p. 489

Have you still not bothered to look at the references given in the FAQ? Maybe you should write a stern letter to Misner, Thorne, and Wheeler telling them that they're "not even wrong."

 

[TrickyDicky]

What page in Road to Reality? Ben Niehoff has pointed out that your notation doesn't make sense, which makes me doubt that you're giving an accurate depiction of an argument by Penrose.

Actually I happen to have a relevant quote from the Penrose book handy:

--Penrose, Road to Reality, p. 489

Have you still not bothered to look at the references given in the FAQ? Maybe you should write a stern letter to Misner, Thorne, and Wheeler telling them that they're "not even wrong."

Take it easy man, that attitude is not productive.

 

[bcrowell]

Take it easy man, that attitude is not productive.

Still waiting for the page number from Penrose. Still waiting for any indication that you've looked at the references.