The first law of thermodynamics states that energy is conserved. It is one of the most fundamental laws of physics and not one that you would expect many physicists to challenge, so it comes as a surprise to find that a growing number of cosmologists and relativists are doing just that. Of course any law of physics is subject to experimental verification and as new realms of observation are opened up we should require that previous assumptions including conservation of energy are checked. But the subject under question is not new physics in this sense. It is the classical theory of general relativity. Whether general relativity is correct is not the issue, although it has withstood all experimental tests so far. The question concerns whether energy is conserved in the classical theory of general relativity with or without cosmological constant as given be Einstein nearly 100 years ago. This is a purely mathematical question.

It has indeed been said that too much ink has been spilt on this subject already, but the fact is that the wrong conclusions are still drawn. It does not matter how well-respected the cosmologists are or how many people have read their textbooks, the fact is that they are wrong. Energy *is* conserved in general relativity. There are no ifs or buts. The mathematics is clear and the errors in the thinking of those who think it is not conserved can also be traced. It is time to put the record straight.

Not all the cosmologists are so bold as to state directly that energy is not conserved, but some are. Here are some examples of the kind of things they do say:

“*there is not a general global energy conservation law in general relativity theory*” – Phillip Peebles in Principles of Physical Cosmology

“*In special cases, yes. In general — it depends on what you mean by ‘energy’, and what you mean by ‘conserved’*.” – John Baez and Michael Weiss in the Physics FAQ

“*The energy conservation law is an identity in general relativity*” – Felix Klein

“*the local conservation laws, integrated over a closed space [..] produce nothing of interest, only the trivial identity 0 = 0*“, John Wheeler in Geometrodynamics.

“*a local energy density is well-defined in GR only for spacetimes that
admit a timelike Killing vector*” – Steve Carlip on sci.physics.research

“*Energy is Not Conserved*” – Sean Carroll on Cosmic Variance

“*Energy is not conserved in cosmology. As is always the case
with confusing stuff in cosmology, this is covered well in Edward
Harrison’s COSMOLOGY textbook*.” – Phillip Helbig on usenet

Many other statements have been made to the effect that conservation of energy in general relativity is only approximate, quasi-local, trivial, non-covariant, ambiguous or only valid in special cases. They are all wrong. Energy is conserved in general relativity.

Discussions about conservation of energy in cosmology often arise when people write about redshift of the cosmic background radiation. Individual photons are not created or destroyed as they travel across space, so if they are redshifted they are losing energy. Where does it go? The answer is that it goes into the background gravitational field. The presence of the CMB affects slightly the rate at which the universe is expanding so there should be an energy term for the expansion rate. This is what happens for particles moving in other types of background such as electric and magnetic fields so it should work for gravity too.

Since the discovery of “dark energy” the level of confusion has become worse. People visualise dark energy as a constant density of energy that pervades space. If space expands then there should be more of it, so where does the energy come from? The anser is the same as for radiation. The dark energy, or cosmological constant as it used to be known, affects the expansion rate of the universe. The gravitational component of energy has a contribution from this exapnsion and the rate changes to conteract the amount of dark energy being added so that total energy is constant.

**Noether’s Theorem**

To make the case for energy conservation in general relativity sound, we need a valid mathematical formula for it in terms of the gravitational field (the metric tensor) and the matter fields. This problem was initially tackled as soon as general relativity was proposed by Einstein. The mathematician Emmy Noether was asked to look at the problem and she solved it eloquently by stating her theorem relating symmetry to conservation laws. Although the theorem is well-known to physicists it is not often appreciated that it was formulated to tackle this specific problem.

Noether’s theorem tells us that if a physical law derived from an action principle is invariant under time translations, then it has an energy conservation law. In fact the theorem provides a formula to derive an energy current whose divergence is zero. Such a current can always be integrated over a region space to provide a total energy whose rate of change is equal to the flux of the current from the surface bounding the region. This is exactly what we mean by conservation of energy.

For example if we take Maxwell’s equations in special relativity such invariance applies and we can derive a formula for the energy current. Of course in special relativity time is not absolute and there are different concepts of time dependent on an observers velocity. This means that we actually get an infinite number of energy conservation laws, one for each possible velocity. Conveniently this boils down to a single energy-momentum tensor that gives the energy current for any choice of the time coordinate in a reference frame. The same tensor can be used to provide momentum and angular momentum conservation laws. It is all very intuitive and nice!

**Energy-momentum pseudotensors**

What about the case of general relativity? Invariance under time translation still holds and general relativity is derived from the Hilbert action principle so Noether’s theorem can be applied to the gravitational field along with any matter fields to give a total conserved energy current, but there is a technical hitch. The Hilbert action includes second derivatives of the metric tensor as well as the first, and Noether’s theorem only deals with the case where there are first derivatives. The usual solution applied in the early days of relativity was to modify the Hilbert action in a way that removed the terms containing the second derivatives without affecting the dynamics of the Einstein equations derived from it. Noether’s theorem could then be applied. The only snag was that the procedure could not be made gauge invariant so the energy-momentum quantities derived did not form a covariant tensor as they did for special relativity. Sometimes they are called the energy-momentum pseudotensor. The solution works but some people just don’t like it. They complain that the pseudo-tensor can be made zero at any point in spacetime for example. It is not really a problem but people did not expect it so they complain about it.

The source of the problem (which is not really a problem) can be traced to the fact that the spacetime symmetry group in general relativity is bigger than it is in special relativity. Instead of just a choice of time coordinate for each velocity of an inertial reference frame, you have one for any choice of motion whether inertial or not. This gives a much larger set of conservation laws and with the extra choice you can always make the energy and momentum of the field zero at any given event in space and time.

The choice of time coordinate can be associated with a contravariant vector field that generates the time translation. We should expect the formula for our energy from Noether’s theorem to have a dependency on this field. Trying to express it as a tensor is not really appropriate and that is what causes the confusion.

**Modern Covariant Solution**

It turns out that there is a more general version of Noether’s theorem that can be used even when the action includes terms with second derivatives. This provides a more modern approach to the derivation of an energy current that has a dependency on the time translation vector field. Since it does not require any manipulations of the action the result is a covariant local expression. I am avoiding formulae here but you can look up, the answer in arXiv:gr-qc/9701028. This paper does not take into account the cosmological constant but that is not a problem. The conditions for Noether’s theorem still apply with the cosmological constant term in place and the derivation of this more general case is a straightforward exercise left for the reader.

So the outcome is that there is a local covariant expression for the energy current in general relativity after all. This is exactly the thing that many cosmologists claim does not exist, but it does, and energy conservation holds perfectly with no caveats.

To finish off let’s take a look at some of the specific things that cosmologists and relativists have been saying and debunk them one by one in the light of the solution we now understand.

**Energy Conservation in general relativity is approximate NOT**

It is sometimes claimed that energy conservation in general relativity is only approximate. On further examination of what is meant we find that the person who thinks this only knows of (or only accepts as valid) the extension of the covariant energy-momentum tensor from special relativity to the general theory. This tensor includes only contributions from the matter fields and not the gravitational field. Its covariant divergence is zero just as required for a conserved current vector, but unfortunately it is a symmetric tensor and you can not integrate a divergenceless symmetric tensor to get a conserved quantity in curved spacetime. That only works for vectors and anti-symmetric tensors. Because of this people say that the conservation is only approximate.

It should be clear now where the error in this argument lies. The energy-momentum tensor does not include contributions from the gravitational field and energy conservation cannot be formulated without it. Of course your energy conservation law is only going to be approximate if you neglect one of the fields that has energy.

The correction is to include the gravitational field either by using the pseudotensor method or by using the more modern derivation of the current as a function of the time translation vector field.

**Energy conservation only works in special cases in general relativity NOT**

The cause of this false claim is once again the use of the energy-momentum tensor. For some special cases the gravitational field has a killing vector that indicates that it is static in some specific reference frame. If you contract this killing vector with the energy-momentum tensor you get an expression for an energy current that is conserved. That’s very nice but nothing unusual. It is normal that you can get a conserved energy in a fixed background field which is static. The same happens for other fields such as the electromagnetic field. If the energy in the background field in not changing then the energy in the rest of the system can be conserved too without adding the energy from the background field.

Just because energy conservation is a bit simpler in special cases does not mean that it does not work in more general cases, which it does of course.

Another special case often cited is an asymptotically flat spacetime. You can work out the total energy and momentum and it takes the form of a familiar energy-momentum four vector in the asymptotic limit. Very nice, but again just a special case while the general case also works perfectly well.

**Energy conservation in general relativity is trivial NOT**

This particular version of the energy conservation “problem” in general relativity goes back to the early days when Noether, Einstein, Klein, Hilbert and others were investigating it. Klein claimed that the conservation law that Noether’s theorem gave was an identity, so there was no real physical content to the law. This claim has been echoed many times since, for example when Wheeler claimed that the law reduces to the trivial result 0 = 0 for closed spacetimes.

In addition to her well-known theorem, Noether had a second theorem that elaborated on what happens when there is a local gauge symmetry rather than just a global symmetry. In this case you can derive Bianchi type identities that provide formulae for currents that are conserved kinematically, even if the equations of motion are not. You can say that such a current is trivially conserved. The formula for the energy current derived from Noether’s theorem is not such a quantity, but it is the sum of two parts one of which is trivially conserved and the other of which is always zero when the field equations apply. For some people this is enough to make the claim that energy conservation is trivial in general relativity.

That this makes no sense is easily seen by considering any other gauge field and its conserved charges. For example, electromagnetism is a gauge theory that conserves electric charge. Because of Noethers second theorem the expression for the electric current can be written as the sum of a term depending only on the electromagnetic potential whose divergence is explicitly zero, plus a term which is obviously zero when Maxwell’s equations hold. This is exactly analogous to the case of energy conservation in general relativity. Nobody claims that this makes charge conservation trivial in the classical theory of electromagnetism so they should not make such a claim for energy conservation in general relativity.

**Conclusion**

I have debunked some of the major claims about energy conservation in general relativity that people use to justify the idea that there is something wrong with it. There are others but they are all just as shallow and easy to deal with. If you come across anyone making such claims, please just refer them to here and hopefully we can put an end to this nonsense.

Phil,

May I suggest “Energy-momentum is conserved” for the title for your posting?. This will remind everyone at a glance that space-time continuum is the cornerstone of Relativity.

Cheers,

Ervin

A related claim often appears in introductory treatments of quantum mechanics. Namely the implication that states that are short-lived such as virtual particles owe their existence to Heisenberg’s uncertainty principle, which lets them “borrow” energy. (From where they borrow this energy is never said.)

At the heart of it, this is just unfortunate terminology, any serious physicist understands the lie. Nevertheless I’ve run across graduate students who believe literally that you can violate energy conservation if you hurry!

Don’t be joking, Phil.

There’s no nonzero conserved “energy” in general situations described by general relativity.

If you calculate the Noether current, you will find out that it is only conserved if the background is guaranteed to be time-translationally invariant, which it’s mostly not in the relevant situations.

If you add the energy of the gravitational field, you will get a conserved quantity, but it’s identically zero. The stress-energy tensor describing its (and momentum’s) density and flow (current) is given by the difference of the left-hand side and right-hand side of Einstein’s equations – which is zero whenever the laws of Nature are satisfied.

One can only find a conserved quantity in GR in asymptotically AdS or Minkowski or other backgrounds whose physics can be understood as “finite perturbations” of a time-translationally invariant background. That’s the ADM energy.

Energy conservation explicitly holds in cosmology. For example, cosmological constant expands the volume of space exponentially, keeping the energy density constant (this density is called the cosmological constant). Guess what happens with the total energy. It goes up exponentially, just like the volume.

On the other hand, radiation’s energy goes down as 1/a where “a” is the linear dimension of the Universe. That’s why e.g. 1 photon stays one photon but its wavelength has to grow proportionally to the size of the Universe – imagine e.g. the fixed number of waves around the Universe which can’t change – and the energy of the photon is inversely proportional to the wavelength.

Only the dust – with no pressure – follows an energy-conserving evolution but the exact dust is measure zero among the energy types filling the space in GR.

Just think about these simple particular situations quantitatively to see that your opinion is completely wrong. In cosmology, the total energy is either defined so that one can prove it’s zero, or it is defined so that it is not conserved. There’s no other option.

Erratum: “Energy conservation explicitly holds in cosmology” should have been “fails”.

Lubos thanks, you have carefully provided versions of all the different claims that I have explicitly tried to refute in my post.😦 Obviously my points are not getting across but never mind, let’s just agree to disagree on this one, otherwise we will waste a lot more virtual ink.

Ervin, actually I am not too keen on the expression “energy-momentum” when talking about conservation in general relativity. It is nice that energy-momentum forms a 4 vector in special relativity and you can form an energy-momentum tensor as a combined current, but that does not work well in general relativity because the Poincare Group is not the right symmetry globally, except in special cases. You may not have been thinking about it in 4-vector those terms but some people do when they hear that term.

I’ll edit something about this into the post.

Dear Phil,

“Lubos thanks, you have carefully provided versions of all the different claims that I have explicitly tried to refute in my post.”

that’s because I am the reincarnation of all correct physics that exists as of 2010.😉 Before we find out that this disagreement of ours is lethal, could you please tell me what’s happening with energy carried by radiation and/or cosmological constant when the Universe is expanding? Instead of denying that your bold hypothesis can actually be tested (and refuted)?

Thanks a bunch

Lubos

The energy lost to the photons goes into the gravitational field affecting the rate of expansion.

The cosmological constant case is similar with energy going either way depending on sign.

Am I walking into a trap?

Dear Philip, I don’t know whether you are feeling nicely in the trap but you’ve been trapped since the beginning.

If the Universe gets K times bigger, a photon of wavelength L and energy E will become a much-lower photon of wavelength K.L and energy E/K, agreed? As the Universe gets 1 billion times bigger, the original energy of the photon will be almost completely gone and the Universe will look exactly as it would look like at some moment if it were without any photon.

How else could it look like? A near-zero-energy photon simply can’t make any difference.

So if you take this K=1 billion times bigger Universe and calculate its energy according to any formula you like, the result will clearly be identical in the cases with and without the near-zero-energy photon. So the original energy of the photon has been lost – or 99.999999.. percent of it.

I actually wanted you to do the calculation quantitatively. To say what the total energy is, and if you claim that the energy goes to XY, how to calculate the energy of XY. You say XY is the “gravitational field”. What you write makes no sense.

The answer is that we want the empty space to carry no energy, so all the energy is carried by the photon, and it goes down as 1/K i.e. E/K because its wavelength goes like K where K is the linear factor of the expansion of the Universe.

Of course, you may postulate a “God justice” version of energy that is kept by the Universe, compensates any violation of the conservation law😉, but can’t be measured or calculated from the configuration in any way. But if one uses this kind of “energy”, he is doing religion, not science. Energy as a conserved quantity only makes sense in science if its value is actually linked to the observable quantities, to the state of the physical system, to physical degrees of freedom.

There’s no such energy that would be both nonzero and conserved in cosmology.

Phil

Sorry, but I am with Lubos on this one.

One must be careful with properly defining the context where conserved quantities in GR are discussed.

A clear-cut presentation of this topic is included in Bernard Schutz’ classical text “A first course in General Relativity”. Here it is noted that “a general gravitational field will not be stationary in any frame, so no conserved energy can be defined”. It is straightforward to see that there is generally no coordinate system which makes a given metric time independent. The metric has 10 independent components (same as any 4 x 4 symmetric matrix), while a change of coordinates enables one to introduce only four degrees of freedom to change the components (these are the coordinate transformation functions x'(x)). It is a special metric indeed if all ten components can be made time independent this way.

Cheers,

Ervin

The following article by S.C. Tiwari may be of interest: he argues that if spacetime is approached from a thermodynamic standpoint and is regarded as fundamentally discrete, the use of unimodular relativity becomes appropriate, in which case the conservation of energy-momentum (sorry, Phil) is achieved in a simple, straightforward manner.

http://arxiv.org/abs/gr-qc/0612099

Lubos, I don’t have any problem with the energy current being zero in this case. The fact that it is zero depends on using the field equations so it is not trivially zero. We can write an expression for the energy current and equating it to zero tells us something about the dynamics.

I knew it was going to be zero because I can see it another way in the special case of a closed cosmology. The total of any conserved quantity in a closed universe is zero because when you integrate the current you are left with no boundary (you know this I am sure). In a homogeneous isotropic cosmology the current must have no spacial component in comoving coordinates. It follows that it must be zero everywhere. The contribution from gravity must be equal and oposite to the contribution from matter. Although this argument applies only to a closed universe it is true for an open one too.

In a more general spacetime this argument does not work. Total energy for a closed universe is still zero in general but the current vector does not have to be zero everywhere.

If you just wanted to show that the energy current is zero for a standard cosmology I agree with you and I am fine with that. It is an interesting feature of standard cosmology but it does not invalidate the law of conservation of energy.

You are right that I should illustrate this case with the equations to show that the result is still meaningful. However I am otherwsie occupied right now. I hope I can find time to do that as another post later.

ervin, I agree with the staement that “a general gravitational field will not be stationary in any frame” but this does not mean that no conserved energy can be defined unless you believe that only special cases with killing vectors work.

This situation is perfectly analogous to an electron moving in a dynamic electro-magnetic field. The electron has energy and the electromagnetic field has energy. These are not individually constant but their total is. The electromagentic field does not have to be static in some reference frame for this to work. Exactly the same principle applies in general relativity. You have to work out a contribution from the gravitational field and from the matter. The total energy is then constant.

Dear Philip, I expected somewhat more technical or “geeky” approach of yours to the homework I asked you to solve, you know, some mathematical expressions, instead of the vacuous, misleading, and wrong philosophical cliches you offered us.

It’s clear that if you say that the energy of the photon in an expanding Universe equals zero, then you will have to agree that the energy of anything is zero because pretty much any matter can be annihilated to photons, so if your energy is conserved, the energy of pretty much any matter must be equal to zero, too.😉

If you define your energy – or the stress-energy tensor – in such a way that it is zero whenever the equations of motion are satisfied, then the only thing that the vanishing can tell you is that the equations of motion are (probably) satisfied.😉 It’s a pure tautology and it tells you exactly nothing – the notion of energy doesn’t exist.

Noether’s theorem makes it very transparent why there’s no conserved energy in cosmology: because the background is not time-translationally symmetric. The only way to find something that is preserved is to admit that “everything” – including the would-be asymptotic metric tensor degrees of freedom – is dynamical. But if everything is dynamical in this way, and indeed, for a compact universe, everything is dynamical, you can create anything out of nothing, so the conservation law implies that the energy of anything must be equal to the energy of nothing, namely zero.

Phil,

Let me clarify my statements, to avoid any misinterpretation.

Of course, any isolated system containing a particle interacting with a field is conservative and the total energy of the system is constant.

But what I cautioned in my reply is that one must be careful with definition of conserved quantities in GR. This boils down to the fact that, in general, not ALL components of the metric tensor can be made time-independent through a suitable chosen coordinate transformation.

Consider the most general geodesic equation of motion describing the trajectory of a massive particle in a gravitational field. It is clearly a classical analogue of the case you mentioned, that is, an electron moving in an electromagnetic field. The four-momentum of the particle is conserved if and only if all components of the metric tensor are independent of space-time coordinates. This is clearly the case if one deals with a stationary gravitational field. But, this case is rather special since a general gravitational field is not stationary in any frame and no conserved energy can be defined. Only if you impose additional constraints and make the metric, say, axially symmetric, then coordinates can be found in which the metric is independent of the angle around the axis and the corresponding angular momentum is conserved.

Hope this clears up my previous reply.

Ervin

Hi Phil,

If I might politely interject on this discussion, and deal with things on a slightly more abstract (and possibly simpler) level, I might like to advance the opinion that you and Lubos may well be talking past each other.

My understanding of the subject matter involved is roughly as follows – general relativity is an optimal theory of physics grounded in the logic of 0-categories, ie the theory of sets (Riemannian geometry being the corresponding geometrisation). In this case I believe you are completely correct in stating that energy is conserved within the corresponding mathematical framework, that being the dynamics associated to requiring the Riemann-Cartan tensor be Ricci flat.

However general relativity is, at the end of the day, merely a model of physical reality and therefore an approximation. In particular it is my impression that it is

incomplete. Various observational evidence, such as the strange fractal structure of the distribution of galaxies at large scales, and the existence of so-calleddark energyetc are anomalies that support this assertion, I think, although experiment is not my primary area of expertise and I certainly would not mind being corrected on these points.The theory known as strings in my understanding is an attempt to address some of the incompleteness of general relativity by embedding the machinery within some as yet unmapped tensor theory based on the dynamics of function spaces as opposed to dynamics based on the theory of sets. So my take on it is that it is supposed to be a 1-categorical construction. Although once again, since I am self-trained in this area, and do not have formal grounding in the standard material on the subject, perhaps experts (Lubos?) could correct and/or clarify my remarks here.

Indeed the corresponding notion of “energy” for function spaces at the 1-categorical level we certainly would not expect to correspond to “energy” at the level of 0-categories in GR. In this way the “string” notion of energy is not conserved in GR, but as a mathematical framework the “energy” at the level of 0-categories is indeed conserved in GR.

It is my opinion furthermore that eventually when a tensor theory of strings is formulated it itself will also be incomplete and probably people will have this argument again, once more confusing different levels of foundational abstraction.

Hope this helps. I think I am probably guilty of ignoring most of the precise details of the discussion above but this is my rough take on what you are saying.

wman, Chris, Your comments are interesting asides. This post is really about classical general relativity and whether it has energy conservation. Of course the universe is quantum and general relativity may be replaced by something else in the real universe, but the question about classical general relativity remains as a mathematical question and it should be relevant to real physics as at least a very good approximation. In other words, we should not have to look beyond classical GR to resolve this.

Lubos, the mathematical details are in my old paper that I linked to at http://arxiv.org/abs/gr-qc/9701028 . It needs to be updated to include the cosmological constant and it would be nice to do an example to show how the standard expanding universe case works. I’ll do that if and when I get time. It is straightforward but I need to take time to get all the factors and signs right and code it in blogtex.

Yes, even when there is dust in addition to the photon background, the total energy current is zero in standard cosmology. I would not say “the energy of matter is zero”. I would say “the energy of matter plus the energy of the gravitational field is zero” This only happens in specific cases. In general the total energy current is not zero. Also there is nothing wrong with it being zero, it is not trivially zero.

In standard cosmology the conservation of energy is equivalent to the equation of motion being satisfied. This is not a tautology. The same thing happens in any basic maechanical system when there is only one degree of freedom. It happens in the standard cosmology because the condition of isotropy and homogeniety reduces the system to one remaining degree on freedom and therefore one ordinary differential equation to be solved. In the more general case energy conservation is implied by the equations of motion but is not equaivalent to it. This is the same principle that applies to energy conservation in any system.

I also agree that total energy in a closed system has to be zero, but this is just a nice feature of energy conservation. It is not a trivial 0 = 0 as Wheeler put it because it is only true when the equations of motion are satisfied. The same is true of any conserved quantity such as electric charge. In a basic dynamical systems the total energy can be zero in the same way. It is a perfectly OK for total energy in a system to be zero.

You know, everytime someone makes a claim that energy conservation is menaingless or wrong in GR I just compare their reasoning with an analogous non-gravitational conservation law, such as conservation of charge in electromagnetism, or even energy conservation in a basic mechanical system. I can always find such a system where the same reasoning can be used. Yet nobody thinks that these systems are trivial or pathological in any way. For some reason they think the reasoning makes the case of GR trivial or pathological. Obviously that is inconsistent.

Ervin, I agree with you until the point where you say “[for a nonstationary gravitational filed] no conserved energy can be defined.” Such a conserved energy can be defined using the pseudotensor methods, or the covariant methods in my paper http://arxiv.org/abs/gr-qc/9701028

Phil,

Point well taken. The pseudo-tensor method is indeed required to correctly formulate the four-momentum conservation for a general system including classical gravitational, electromagnetic and matter fields.

Phil,

This is an off-the-topic thought: at some point in the future, it maybe enlightening to discuss the (still controversial) topic of time asymmetry and intrinsic irreversibility in quantum theory. It is my view that proper resolution of this issue and correct understanding of non-equilibrium dynamics are critical for theoretical developments regarding physics beyond SM.

I include here a link to one of the many representative papers:

http://philsci-archive.pitt.edu/archive/00002595/01/TAQM-RII.pdf

Cheers,

Ervin

Quote: STAR data also suggest the local breaking of another form of symmetry, known as charge-parity, or CP, invariance. According to this fundamental physics principle, when energy is converted to mass or vice-versa according to Einstein’s famous E=mc2 equation, equal numbers of particles and oppositely charged antiparticles must be created or annihilated. http://www.bnl.gov/rhic/news2/news.asp?a=1073&t=pr

But this is not the case. Neutrino-antineutrino oscillations show a rather strong asymmetry. So the Einsteins conservation is not true?

Star showed that the strong force ev. could break also the parity, with aid of magnetism.

Lubos is quite right in that the conserved four-momentum vanishes identically in general relativity. If one has time translations as isometries, one can define conserved energy but this is of course only a special case. There would be also a problem related to the identification of the diffeomorphisms responsible for energy and momentum conservation unless isometries are in question. Second derivatives in the curvature scalar have nothing to do with the problem since Noether’s theorem generalizes to action containing also higher derivatives.

One can of course argue that gravitation is so weak that one can forget the fact that energy is not even well-defined except under additional assumptions.

This would mean local conservation laws but uncertainty principles forces to question this assumption.

The only way out of the problem that I see is that space-times are identified ass 4-D surfaces in some space of form M^4xS, such that the isometries of S give rise to additional conservation laws (color group, which implies S=CP_2). Weak symmetries correspond to holonomies of CP_2 and are broken from the beginning. Standard model quantum numbers including baryon and lepton conservation result automatically. Family replication has topological explanation. A further beautiful consequence is that if one identifies fundamental objects as light-like 3-surfaces (throats of wormhole contacts at which induced metric changes its signature), the conformal symmetries of string models generalize. This happens only for 4-D Minkowski space and 4-D space-time surface. If one requires that the identification of fundamental objects as space-like 3-surfaces is equivalent identification then effective 2-dimensionality follows: partonic 2-surfaces together with their 4-D tangent spaces are fundamental objects.

[…] Energy Is Conserved « viXra log […]

ervin, ulla, It is an interesting point about the implications of CPT violation for energy conservation. Nobody has formualted a verison of quantum field theory that respects Lorentz invariance but not CPT so it is hard to say what the answer would be. I’m going to stick with discussing just classical GR here otherwise everybody will be lost in confusion.

If matter and antimatter is oscillating all the time, about half of the matter is in the form of antimatter. If that portion is excluded the result is not good.

Is antimatter a time-question or a dark matter question or both?

Is a subtraction real, as GR and the inflanatory model says (annihilation- into what)? Then there is a vast mount of energy not included in Einsteins model. This energy must mean something (for gravity?).

This must be hold in mind.

I think in the end this question scales down to three major problems, the vacuum energy, the question what mass really is, and the question if light speed is constant.

Neutrinos are said to travel faster than light, but they have a small mass. It is not the mass that is important, but the interactions? The dissipation? Is this dissipation the same as gravity? Dissipation is entropic.

Antimatter is non-dissipative? Negentropy is non-dissipative?

So, energy conservation is true only for some situations?

Phil,

In reality, there are formulations of field theory that respect Lorentz invariance but violate well established symmetries such as the CPT invariance. The crucial ingredient here is the fundamental passage from equilibrium to non-equilibrium dynamics of underlying fields and from smooth topologies to fractal topologies (measurable metric sets with non-integer Hausdorff dimensions). This line of thinking has been pioneered by Prigogine and his Brussels-Austin school and continue to be the focus of research done by various groups involved in nonlinear dynamics and complexity theory.

But I completely agree with you that this is not the place and time to open up a discussion about these topics without running the risk of creating lots of confusion.

Ervin

Matti, I do not agree that conserved energy current is always zero in GR. This is the case for the standard cosmology because it is homogeneous and isotropic, but it is not true in general, nor is it a problem or a tautology that it is true in standard cosmology.

The same thing happens with electric charge. In a homogeneous spacetime the positive and negative electric charges must always balance exactly, otherwise a boundary surrounding a region would have a net flux from the electric field which would violate the symmetry. In the general case however, you can separate the negative and positive charges and have nonzero currents.

In GR it is the same for energy. The positive contribution from matter and radiation must be exactly balanced by the negative contribution from gravity in the standard cosmology at every point, but in general this is not the case.

When gravitational waves propogate from a binary pulsar there is a net flow of energy away from the source. This means there must be a non-zero energy current.

Another case where you can see it is not zero is when the spacetime is asymptotically flat. The energy current can be integrated to infinity to give

the ADM mass of the system. If the current were zero everywhere the ADM mass would be zero too, but it isn’t.

You also mention a “problem” relating to the ambiguity in choice of the diffeomorphism to define energy. I dont see this ambiguity as a problem. Even in special relativity there is a choice of reference frame so energy is not unique. In GR the choice is larger but that does not turn it into a problem.

Energy can be defined with respect to a parameterised set of diffeomorphisms generated by a contravariant vector field. The field must be timelike for it to correspond to an energy. It should also be normaliased on the boundary of the region in which you are summing up the energy. There are an infinite number of choices for this but that is just an aspect of the relative nature of energy in relativity. Energy currents in GR are well defined, they are just not unique. It is same story as in SR and was never considered a problem there.

Spacelike fields give you quantities corresponding to mementa and angular momenta. You cannot combine these into energy-momentum four vectors globally accept in the special case of asymptotically flat spacetime. Again this is not a problem. It just reflects the fact that Lorentz symmetry is just a local feature in GR. Globally you cant expect to form objects that are representations of the Lorentz group (although you can fake it to a certain extent using the pseudo-tensor methods.)

I wrote a few words (i.e. a few megabytes) about this simple issue here:

http://motls.blogspot.com/2010/08/why-and-how-energy-is-not-conserved-in.html

I almost understood it Very good, thanks.

The Susy is an implicit charachter in QM. But there are four dimensions that has ‘expanded’, also time. You forget to mention the mirror image of the four already expanded, making 8 totally making an extension in octets (bubbles)? Once you was very nasty with me on that point.

In what way do Susy, with extra (hidden) dimensions differ from theories with hidden dimensions? I think of Jerry Decker. Are the dimensions ‘hidden’ in a hidden phase? The gluon with its infinite end, the minimal black hole or monopole, the Higgs…?

“the energy stored e.g. in the cosmological constant will expand”

How does the expanding happen? A creation of ‘bubbles’, a phase extension? Do you realize that is ‘almost TGD’? Funny, isn’t it?

I prefer to discuss it here, with no aggression, if Phil agrees. I have bad experiencies on your blog.

And you must realize I am an novis.

I’ll post a more mathematical description of how energy conservation works tomorrow. Perhaps that will convince you

Phil, in general relativity the Noether current for diffeomorphism is

proportional to T- G which vanishes by Einstein’s equations. What this says physically is that general coordinate transformations are analogous to local gauge transformations.

It would be attractive to continue the analogy by saying that since constant gauge transformation gives rise to non-vanishing charge unlike non-constant gauge transformations then diffeomorpisms which do not approach identity asymptotically give rise to four-momentum and angular momentum. If space-time approaches asymptotically to Minkowski space, one can proceed in this manner. To me this is however very tricky and is the conserved currents do not follow from Noether theorem in this case which is essentially global. Asymptotic behavior of gravitational field could also identified as the carrier of information about Poincare charges.

In special relativity the symmetries are isometries and this distinguishes them. Energy and momentum current complex is unique but its representation depends on the coordinate frame, which could be arbitrary coordinate system. One can choose the rest system in many manners but this is not ambiguity and has a precise physical meaning: consider only effects such as time dilation.

My own approach is essentially the fusion of special and general relativities. Symmetries of special relativity and the description of gravitation in terms of space-time curvature from general relativity. Isometries do not move points of space-time surface but the entire space-time surface just like rotations move the entire rigid body. This leads also to a geometrization of gauge interactions at classical level since sub-manifold geometry is much richer than abstract manifold geometry: the shape of the space-time surface as seen from imbedding space perspective matters.

To be honest, I must make clear that in zero energy ontology causal diamonds (CDs) defined as intersections of future and past directed light-cones (times CP_2 of course) and containing space-time surfaces inside them lead to a potential problem. Translations and Lorentz boosts in general lead out of CD near its upper and lower boundaries (light-cone boundaries).

My proposal for the solution of problem is based on Kac-Moody symmetry generalizing imbedding space isometries to their local variants local with respect to space-time surface (their analogs are encountered also in string model). Since Kac-Moody generators corresponding to non-constant translation acting much like gauge transformations annihilate the physical states, the Kac -Moody generator corresponding to a constant translation is equivalent with a translation containing non-constant part chosen in such a manner that the net transformation does not lead out from the boundary of CD.

Matti says “in general relativity the Noether current for diffeomorphism is proportional to T- G which vanishes by Einstein’s equations.”

Sorry, but that’s not correct. The Noether current comes out to a term proportional to T-G plus another term known as the Komar Superpotential. It is only zero in special cases such as standard cosmology. I know this because I worked through the lengthy calculation myself in http://arxiv.org/abs/gr-qc/9701028

If what you said were true there would be no energy transmitted by gravitational waves as observed for binary pulsars. Furthermore, the ADM mass of any isolated system would be zero.

[…] Phil Gibbs is convinced that all relativists are wrong when they say that the energy conservation law is weakened, trivialized, corrected, or violated in general relativity in any way. […]

Conservation of energy in general relativity is only assured if there is a timelike Killing vector. This then defines an isometry which establishes K_t*E = const., for K_t a timelike Killing vector, to put it in cryptic terms. Cosmologies such as the FLRW metric of cosmology do not admit this. The time dependency of the metric terms prevent a K_t. As a result we can’t conclude that energy is conserved in cosmology!

Lawrence hi, the objection you mention is dealt with in the post under the heading “Energy conservation only works in special cases in general relativity NOT”

Most GR textbooks only deal with these special cases but a few such as Dirac, landau&Lifshitz and Weinberg deal with the general case using pseudotensors. For some more examples try this Google book search http://www.google.co.uk/search?tbs=bks%3A1&tbo=1&q=relativity+pseudotensor

Hi Phillip,

The contraction of a Killing vector with the energy-momentum tensor leads to an object , and the covariant derivative where the first is zero by the Killing equation and the second by continuity equation. The question is whether this is the same as conservation of energy. This would be the case if there is a and a , a universe with just dust and other special cases. The equivalency with energy conservation may not hold in general.

As for pseudo-tensors, I am not generally a big fan of those techniques.

Lawrence, the maths of the Killintg vector case is nice, but it is just a special case. In fact it is not even a real physical case because the background gravitational field is static while the matter field is dynamic. This can only occur as a limiting case where the matter fields are so weak their back reaction on the gravitational field can be neglected. there are not even many practical applications for this case. It just gets in the textbooks because the maths can be done in a few lines.

I also don’t like the pseudotensor methods either, even if I ackowledge that they are mathematically correct and have been applied to real situations such as gravitational wave energy radiation. That is why I developed my own covariant formulation which is described in more detail in the next post https://blog.vixra.org/2010/08/08/energy-is-conserved-the-maths/. It can do everything the pseudotensor methods can but without pseudotensors.

Phillip,

I looked at the maths here. It is interesting this quotes Sean Carroll, for a while back (2-3 years ago) I had a bit of an armwrestling match with him over this. He seemed more disposed to what you are saying. There are some questions I have here, for this construction is similar to OPEs in string theory. I gather this is TeX-able from what you did to my post.

I think Sean Carroll is busy on other things now so sadly he is not likely to enter the discussion. It would have been useful to hear his response

You can use latex between “$latex” and “$”. There is no preview or edit and it is buggy but I can probably fix errors.

Here is a trial “latex$”iH\psi~=~E\psi”$” I seem to remember on Carroll’s Cosmic Variance blog that he was not happy with the issue of Killing vectors and holonomies as the only case of energy conservation.