Why is global conservation of energy not considered a tautology?

Question

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?

Answer 1

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.

Answer 2

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.

Answer 3

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.

Answer 4

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.

I will remold your question and answer as:

"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.

Answer 5

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.

Answer 6

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."

Answer 7

Your question goes to the difference between intuitions and reasoning. When we seen something over and over again, it becomes and intuition for us. This intuition, however, is not a logical necessity. It is still an observed phenomenon, and could have been otherwise in principle. All laws in physics are based on observed regularities, they are not logical necessities. And Physics often finds itself discovering violations of "laws".

One example of a postulated exception for Conservation of energy is here: https://physicsworld.com/a/dark-energy-emerges-when-energy-conservation-is-violated/ If conservation of energy were a tautology, this proposal would violate the principles of logic. It does not, hence C of E is NOT a tautology.

Further, the violation of conservation of energy is not only possible in theory, it is actually predicted by Gauge Symmetry: https://www.pnas.org/content/93/25/14256

It has also been observed in Time Crystals: https://www.space.com/38100-the-significance-of-time-crystals.html

As other answers have noted, it is also not a coherent claim in the context of General Relativity.

So Conservation of Energy is not only not a tautology, it is also not true as a universal principle.