# Why is global conservation of energy not considered a tautology?

This question is in reference to my downvoted answer to this active physics.SE question. More than one user has indicated that it is simply wrong and I am having trouble understanding why. My point of view is that the fact that energy is conserved is essentially part of its definition. In particular I don't understand why it is necessary to invoke Noether's theorem, GR, or BB in order to "prove" the conservation of energy. I can't see how a universe without conservation of energy is even possible in principle.

A thought experiment (of a sociological nature) to illustrate my point: suppose a perpetual motion machine were discovered or invented that seems to create energy continuously from nothing in defiance of all known physical theories. So we create a new theory to explain it by saying that the energy comes from "somewhere else", i.e. a "different universe". But now we have expanded our concept of "universe" to include this other place; isn't it still fair to believe that energy is globally conserved within this newly-defined larger universe? I know that numerous discoveries have been made on the basis of missing energy, e.g. the existence of the neutrino, but seems like it is always preferable to violate Occam's razor by postulating new entities than it is to question the law of conservation of energy.

Can someone help clear this up for me? I am not so much interested in knowing the mathematics as I am in getting my scientific intuition straightened out. How is a universe without conservation of energy even possible in principle? If energy is not globally conserved as some users have suggested in response to the linked question, does this mean that some sort of cosmological-scale perpetual motion device is actually possible? If so, why not just define it away as I have done in my thought experiment?

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I like this question :-) And thanks for not getting too bitter about your previous answer getting downvoted, probably more than was strictly necessary. –  David Z May 26 '11 at 2:40
Your thought experiment is begging the question. Of course I can imagine a universe where energy isn't conserved, where it doesn't merely come from "somewhere else," but is in fact spontaneously generated or destroyed. It's not our universe, but that's the point, innit? –  wsc May 26 '11 at 2:50
wsc, the point is that there is no reason not to describe it as coming from "somewhere else". And there is at least one reason to describe it that way: we still have conservation of energy. I don't think the argument assumes its own conclusion as you suggest; it assumes that we have some pre-existing theory which cannot explain the extra energy, and constructs a new theory that can explain it. Whenever conservation of energy is violated we just fix the theory. GR and QM are falsifiable but how is conservation of energy falsifiable? –  Dan Brumleve May 26 '11 at 3:42

I will address the physical intuition, and I will accept as a premise that Neother's theorem is not necessary for a conserved quantity to exist, which is what you are saying. Sufficient but not necessary.

"Why is global conservation of mass not considered a tautology?"

Mass is conserved in our everyday life, and even in classical microsystems. It is only when we enter the realm of atomic that the anti-intuitive surprise of Quantum mechanical systems catches up with physics.

From the butcher to the baker/ to the candlestick maker, we knew mass was conserved. Weight is money and money is a serious question.

What happened with quantum mechanics? Mass became connected with momentum and energy in an intimate way, and our strong intuition that mass is conserved gave way to the observation, that this was not always true. And we fell back to mathematics to bring order out of chaos, hence the essential for logic Noether's theorem.

There is no simple way, going back to the intent of your real question, of simply adding up mass, (as we try to do when we measure the smoke from the candle and still find mass missing), so as to conserve it by enlarging the universe. We needed to develop an intuition for quantum mechanics and relativity. The underlying structure of the universe revealed that conservation of mass is a concept very useful for human life, but applicable in specific four dimensional coordinate systems only.

The same is true for conservation of baryon number (intimately connected with conservation of mass), as was discussed in another question on this site. When energies become large enough to turn everything to a quark gluon plasma, baryon number conservation becomes meaningless. We could define a quark number that would describe the plasma but it would have little meaning as most of the energy would be in the sea and the effective mass of the plasma would have little to do with the number.

Thus people were telling you, despite Feynman's interesting quote in your previous answer, that in a similar way, energy conservation can be well defined and very useful in various coordinate systems but cannot be a general law of the General Relativity universe.

Hope this helps on the intuition front.

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At one point we believed that mass and energy (according to the old definition of "energy" which excluded mass) were independently conserved. Then it was discovered otherwise, but the conservation law was recovered by redefining energy to include mass. Shouldn't energy defined as the sum of every time-invariant scalar quantity? Why not convert B to Joules and explain the redshift by saying that the photons create a neutron sometimes? Of course we demand evidence of this, and the epicycles continue, but why is one thing energy and not another just because no conversion process is known? –  Dan Brumleve May 27 '11 at 4:36
All of the answers are wonderful, and all point me to the wilderness instead of the path, but this one is the most helpful. The "conservation of mass" analogy is simple, it relates well to my existing intuition, and it suggests interesting questions. –  Dan Brumleve May 27 '11 at 7:54
Dan Brumleve , it is true that calling mass energy a new conservation law was found, but, and it is a big but, masses used to be additive, the conservation of total energy law becomes a non linear function of rest mass and momentum. We needed a new mathematical point of view, not a mass producing source. So in some frameworks of GR something might be defined that would include energy, but it would be a new functional form, not energy any longer,and not a source of "free energy" to balance the lack of conservation. –  anna v May 27 '11 at 10:40
I guess you are referring to KE = m c^2 gamma and that seems to refute my statement that it's just a linear combination. And it's not quite as simple as en.wikipedia.org/wiki/Grue_and_bleen because time itself is in question! Plenty to think about here. –  Dan Brumleve May 28 '11 at 5:14
Dear @Anna, Noether's theorem is a theorem, i.e. it is tautologically true, so it is surely not sufficient for the validity of any other statement - except for other tautologies - but it is a necessary condition for the validity of anything because if theorems and tautologies fail, everyone is screwed. ;-) You probably meant that a symmetry is not necessary for the existence of a conserved quantity. But in theories obeying mild assumptions of Noether's theorem, the symmetry is both necessary and sufficient. That's what can be established by those Noether techniques. –  Luboš Motl May 28 '11 at 8:01

Yes, we can imagine a universe without energy conservation. Just imagine that there's a dewar that's completely isolated from everything else, and there's an ideal gas inside. The gas suddenly gets hotter without any chemical change. Or imagine that all the electrons in the universe are gaining mass. Or imagine that there's an Energizer Bunny that never, ever stops.

More simply, imagine a universe with two states - one high energy and one low energy. If it ever changes states, then energy is not conserved.

These situations really do violate energy conservation. They would not be explained by bringing in some previously-unknown source of energy unless that source had some other, verifiable physical meaning.

Comparing a hypothetical perpetual motion machine to the neutrino is specious. The neutrino, a single particle, explained conservation of momentum, energy, and angular momentum in the reactions where it was involved - not energy alone. Still, people were indeed skeptical about neutrinos until they were directly detected. So the science behind neutrinos is not similar to a magical energy repository that exists simply to feed a perpetual motion machine.

If a perpetual motion machine were discovered, scientists would work very hard to find out where the energy is coming from. There's a good reason for that, which is that the laws behind everyday things are known, and those laws conserve energy, so most likely the perpetual motion machine really is drawing energy from some already-known source. If we ever discovered a true perpetual motion machine, though, we would not invent a new, unknown source of energy to add on to the universe to explain it. We would can the conservation of energy.

Something similar happened when it was discovered that in certain circumstances, the universe violates CP symmetry. Scientists did not invent otherwise-undetectable particles or in some other way invoke a deus ex machina to try to patch the situation up. It is simply accepted that the universe violates CP symmetry. Similarly, if we found energy is not conserved, we would have to acknowledge that time-translational symmetry is not perfectly true.

The point of energy conservation is to make useful predictions and help our actual understanding of the universe. Beginning with high school physics and continuing on through all the electromagnetism, quantum mechanics, analytical mechanics, relativity, and especially statistical mechanics that I've learned, energy conservation is an incredibly useful tool.

Specifically, it and other conservation laws provide constraints. If we know the state of a system now but don't know all the physics of that system, we might still at least calculate its energy. Then, either assuming the system is isolated or finding all the energy that goes in and out of it (all of which is transferred by known physical mechanisms), we know what the energy of the system will be in the future, even if we don't know exactly what the state will be. That's useful.

If energy could simply materialize in the system, then we could say "no worries; it just got some energy from the energy gods, who by the way are an established part of the scientific universe", but we'd be fooling ourselves because we'd have sacrificed the predictive power we used to have. I can't think of any case where energy is used in that way. Each time we claim energy enters of exits a system, it does it through a known mechanism that has a physical meaning. In the case of neutrinos, we could detect the neutrino leaving the system (with a certain probability). In the case of electromagnetic radiation leaving the system, we can detect that. In the case of the system losing thermal energy by conduction, we can detect the temperature change in the surrounding environment, etc.

In conclusion, it is not correct to say that we sacrifice Occam's razor to preserve energy conservation. In fact the opposite is true - positing the existence of the neutrino is a simple explanation for many different observed effects, not just the loss of energy when you ignore neutrinos.

I'm not knowledgeable enough to speak to your concerns with GR, so I'll leave that to the more experienced users.

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This is a helpful and intellectually honest answer. It will take me some time to finish studying it but I have one immediate concern: suppose the hypothetical free-energy machine is a singleton i.e. it has only one discovered instance. Fancifully let's say it is discovered buried on the moon like the monolith in the movie "2001". So it is not a thing which can be replicated by any known means. Would it still be the case that scientists would can the law of conservation of energy rather than ascribe its power to the energy gods? Why should they assume there are any more instances of it? –  Dan Brumleve May 26 '11 at 5:01
I hope that idea isn't too silly for this site but my question is essentially philosophical and I think the thought experiment of a perpetual motion machine requires some specific context for it to make any sense. I think an equivalent question is: does the falsification of energy conservation require the ability to independently and repeatedly replicate that falsification experiment? –  Dan Brumleve May 26 '11 at 5:10
@Dan In practice, when we find things like that (e.g. the pioneer anomaly en.wikipedia.org/wiki/Pioneer_anomaly) we tend to try to explain them, and if we can't, put it on hold and hope the answer will come out eventually. I don't what people would do if there were a big obvious one that defied explanation for a long time. I am not really expecting that to happen in physics, but if it did we'd have a problem. –  Mark Eichenlaub May 26 '11 at 5:17
Yet another way to phrase this objection: isn't it the case that both the assertion and denial of conservation of energy are philosophical and not physical statements? I find the former more palatable because it is true in everyday experience and were it to be demonstrated otherwise, "energy gods" is a short string compared to any long proof that conservation of energy is violated on the basis of some other unproven theory. So it is either conservation or energy gods by Occam's razor. –  Dan Brumleve May 26 '11 at 5:22
@Dan I can't answer because I don't know what all the "it"s are referring to in your comment. I also don't know what you mean by "a short string compared to any long proof..." Anyway, I don't see how energy is philosophical. Given a state of a system, we can calculate its energy. That calculation is based on things like temperature and mass. That's not philosophical. There are no energy gods. We don't need them and never have needed them to do physics. Ultimately, saying "energy is conserved" is both a summary of past observations and a prediction of future observations. It's physics. –  Mark Eichenlaub May 26 '11 at 5:35

The issue is finding that conserved energy. What Noether's theorem does is give us a nice, clean way to discover what the conserved energy is.

It turns out that for certain models in general relativity, there is no clear global notion of conserved energy. In particular, most of the "nice" ways that we would try and define energy are totally coordinate-dependent--both the components of the metric tensor and their first derivatives are right out as local components of a potential "energy of the gravitational field," since their value at a point can be set arbitrarily by a coordinate transformation, meaning that if I were to construct an energy density out of the metric and its first derivatives, its value would depend on whether I used (for example) spherical or rectangular coordinates in $\mathbb{R}^{3}$ There are some cases where we can be saved by Noether-type logic, though--if we either have a metric that has a global timelike translation symmetry, or at least a 3-surface that has a timelike or null translational symmetry along it, we can construct something that looks like a conserved energy for the dynamics in that spacetime. These different 'masses' go by different names, depending on the details of the problem that you're dealing with. For the sake of posterity, I'll refer to a pretty technical paper on this.

But, it turns out that there are a wide variety of spacetimes where none of this works out, and that expansionary cosmologies containing matter other than the cosmological constant are one of these type of spacetimes.${}^{1}$. In these cases, there is just no conserved energy or energy density that we can define. If the universe expands forever, and the photons just keep on redshifting, where is the energy of those redshifting photons going to, after all?

We just have to deal with the fact that the only energy conservation we have is in the local sense, where $T^{ab};{}_{b}=0$. Globally, energy is not conserved.

${}^{1}$In much the same way that there is no conserved energy for the Lagrangian $L=\frac{1}{2}m{\dot x}^{2} + a_{0}xe^{t}$. It's just that in this case, we know that it's because we have a time-dependent external potential driving the system, so we don't expect the energy to be conserved.

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This seems like saying that a statement isn't true unless it is provable, or that a number isn't real unless it is computable, but both are false assuming the consistency of the formal system. Why can't we see energy as something more primitive than the theories which attempt to provide methods to measure it? Just because global energy isn't defined in GR doesn't mean it doesn't exist, it just means GR is incomplete (which I guess we already know because it seems to have little to say about QM). –  Dan Brumleve May 26 '11 at 2:56
@Dan Brumleve: I can compute with GR, and I can do phenomenology with those predictions. What else can we hope to do, really? It seems to me that belief in the existence of something like a conserved energy/momentum/angular momentum should be a consequence of our observations, not one of our initial assumptions. –  Jerry Schirmer May 26 '11 at 3:01
Good answer, @Jerry. And despite some redundancy with previous answers, the data clearly indicate that there's still a lot of room for explaining these things. ;-) –  Luboš Motl May 26 '11 at 3:43

If you observe that energy is not conserved as in the perpetual motion example you give, you cannot just proclaim that there is somewhere else that the energy comes from. That would provide an empty concept of no value. You must find a dynamical system such as a new type of particle or wave and define energy as a function of the variables of the new dynamical entities in such a way that it accounts for any missing or excess energy.

To give an example, binary pulsars are observed to be losing energy when you add up the known forms of energy within the system. This is resolved by showing that the energy in radiated gravitational waves predicted by general relativity can precisely account for the missing energy. This is verified experimentally and will be confirmed more directly when gravitational waves are detected and found to transfer energy back to matter in accordance with the predictions. If they don't then energy conservation will be in trouble.

Noether's Theorem merely gives a way of showing that energy conservation laws of this sort exist when the dynamical laws are derived from a principle of least action that has time invariance. Einstein first formulated a correct law of energy conservation for gravity without using Noether's theorem and his result is still correct, although there was a long history of arguments before the matter was finally agreed. GR has time translation invariance as a subgroup of diffeomorphism invariance when you treat the gravitational field as a dynamical entity itself. Noether's theorem applies in this case and energy conservation can be derived in that way too.

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see also vixra.org/abs/1305.0034 for more discussion about energy conservation in GR –  Philip Gibbs - inactive May 6 '13 at 12:23

I'm sympathetic to this Question. A historical way to approach it is to note the mutations of Poincaré's conventionalism, from which I take the general point that can be extracted to be that the experimental evidence fits into some conceptual systems better than into others. If we take that better fit to mean something, we come to the usual way we do Physics.

You might consider the SEP entry on convention, Section 1.2 particularly, http://plato.stanford.edu/entries/convention/. You could certainly find other resources on Conventionalism, including http://philsci-archive.pitt.edu/ for preprints on Philosophy, if you don't mind risking never again getting an upvote here if you don't keep your head on straight.

I take the description of GR given by Jerry Schirmer partly to cede your point. On the cosmological and gravitational scales there is no global concept of energy because the models are not translation invariant. To make the models translation invariant would require us to introduce a flat background coordinate system, which I think we could do, but we would have to introduce endless epicycles to do so.

The cosmological and gravitational scales, however, are not the only component of your Question. You also introduce the somewhat epicyclic nature of the introduction of new particle types such as neutrinos to preserve translation and Lorentz invariance and hence energy and angular momentum conservation. Here I think the answer can only lie in detailed modeling. We might be able to introduce a model that is not translation and Lorentz invariant on meter or nanometer scales that justifies itself by a need for far fewer quantum field types, but that model has to be exhibited explicitly and its correspondence to the experimental results that are taken to support the current Standard Model of Particle Physics cashed out in detail. Remember the Correspondence Principle? For most working Physicists, put up or shut up is a reasonable response to this, because such conceptual musings will more often put you in a graveyard than not. As you've discovered, some ways of talking about these issues can get you bullied, your current Question worked better, but it's always nerve-wracking.

It's always difficult to know which concepts are getting in the way of progress and which we should regard as inviolate. That, FWIW, is my view of the Quine-Duhem thesis in a nutshell (somewhat curiously mixed up with other arguments in the SEP, http://plato.stanford.edu/entries/mathphil-indis/). People sometimes superficially behave as if their favorite concept to drop or modify is obviously the only alternative, although conversation with them will usually see a relatively more subtle appreciation of other people's ideas.

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This is a question of a metaphysical principle—"Nothing can give what he has not." (Nemo dat quod non habet)—bearing on a experimental law, the conservation of energy. Physics, whose theories are free classifications of experimental laws, is not subaltern to metaphysics. Cf. Pierre Duhem's "Physics & Metaphysics."

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