# How do we know that energy and momentum are conserved?

How do we know energy and momentum are conserved? Are these two facts taken as axioms or have they been proven by an experiment?

This question has been in part addressed here: Conservation of Momentum but I don't see how translational symmetry implies conservation of momentum. If the reasoning behind this could be explained that would be great.

Conservation of energy, like conservation of momentum, seems intuitive to me but similarly how do we know for certain that it is impossible to create or destroy energy? Is this taken as an axiom or has it been proved by an experiment?

I hope it is clear that I'm not trying to suggest that I don't trust these laws to be true but rather that I'd like to know how we know they are true.

Thanks for the help

• You can't prove anything by experiement. You can only verify postulates, or disprove them. In short: (Because the answers will explain much more in detail) Momentum conservation stems from translational symmetry (See: Noether Current). Energy conservation stems from translational symmetry in time. The branch of Physics that proves this (Yes, this is actually something that can be proved) is Analytical Mechanics. (Or Lagrangian Mechanics) – Omry Jul 4 '16 at 3:14
• – dmckee Jul 4 '16 at 3:37
• We don't know, we conclude. – yo' Jul 4 '16 at 20:14
• Physics doesn't have axioms. – OrangeDog Jul 5 '16 at 8:40

We know through experimental observation. That is the beginning and end of the subject of physics, at least the part of it the tells it apart from, say mathematics. Conservation of momentum is simply an inductively reasoned hypothesis to summarize certain patterns in experimental data.

You are alluding to the conservation of momentum's being "explained" through Noether's Theorem. As I discuss in my answer to the Physics SE question "What is Momentum, Really?" here, whenever the Lagrangian of a physical system is invariant with respect to co-ordinate translation, there is a vector conserved quantity. That fact is wholly mathematical result, that continuous symmetries of a Lagrangian always imply quantities conserved by system state evolution described by that Lagrangian, one for each "generator" of continuous symmetry (i.e. basis vector of the Lie algebra of the Lie group of the Lagrangian's symmetries).

Note carefully, however, that Noether's theorem is an "if" theorem: a one-way implication. It's far from being the only way that a conservation might arise. Experimentally, it has been found to be fruitful to act on the hunch that it is the explanation, in the following way. Since the conserved quantity in a Lagrangian formulation of Newtonian mechanics implied by co-ordinate translation invariance is Newtonian momentum, we hypothesize that the result is more general and therefore deliberately construct Lagrangians for other theories to have this symmetry so that they too will have conserved momentums (i.e. spatial co-ordinate translational invariance). When we make predictions with these theories, they turn out, again determined experimentally, to be sound.

We say that the symmetry "explains" conservation of momentum, but what we really mean that is that we have found a compelling translation of the conservation law: it translates conservation into symmetry terms.

It is nonetheless an important translation; in my opinion it makes physics much more "visceral". The statement of conservation laws as givens seems abstract and, from a 21st century standpoint, arbitrary and open to question. In stark contrast, a symmetry description is much more accessible to us: even tiny children begin to understand that the World's behavior is independent of the way we choose to describe it. Why should the laws of physics change simply because I decide to shift my co-ordinate origin to another place, or rotate my co-ordinate system (rotational invariance of a Lagrangian gives rise to conservation of angular momentum)? Unless, of course, there is a clear, outside, experimentally measurable agent breaking this independence (e.g. grain structure in a crystal making laws depend on their orientation relative to the grain).

... I would just add that we are now so confident in energy/momentum conservation that it can be used "in reverse" to your method in paragraph 3: if we saw events at the LHC with missing energy, this would be taken as evidence for dark matter, not evidence against conservation of energy! We would change our Lagrangian, nothing more.

I can't really add any clarifying comment to that statement.

• Awesome answer, I would just add that we are now so confident in energy/momentum conservation that it can be used "in reverse" to your method in paragraph 3: if we saw events at the LHC with missing energy, this would be taken as evidence for dark matter, not evidence against conservation of energy! We would change our Lagrangian, nothing more. – knzhou Jul 4 '16 at 3:32
• @knzhou A large fraction of events at a collider have missing energy and transverse momentum because they involve neutrinos. The signature of beyond the standard model physics in that regard is a bit more complex than just missing energy. – dmckee Jul 4 '16 at 3:39
• For reference, another example of missing energy was from binary pulsars orbiting each other. Over time their orbits around each other got smaller, meaning they lost energy. The lost energy was the first evidence of gravitational radiation. The pulsars were detected in 1971, the closer in orbits in 1970s and they got the Nobel prize (for the pulsars and the accurate timing they provided). Grav waves were directly detected in 2015. Noether's theorem had a great impact in relating symmetries to conservation. Btw, Newtons equations says work becomes kin. energy, and with V energy is conserved. – Bob Bee Jul 4 '16 at 4:11
• @WetSavannaAnimal -- They took a few years and ruled out everything they could think of. See the Nobel poster nobelprize.org/nobel_prizes/physics/laureates/1993/illpres/… – Bob Bee Jul 4 '16 at 4:21
• Of course, in general relativity energy is not generally conserved. No time symmetry. It is for asymptotically flat spacetime, in the large. From the pulsars to here it's a good enough approximation. – Bob Bee Jul 4 '16 at 4:32

The answer by WetSavanna... is complete but I want to particularly address the part

I hope it is clear that I'm not trying to suggest that I don't trust these laws to be true but rather that I'd like to know how we know they are true.

Physics theories are mathematical models that fit current observations/data and are predictive of new ones. Prediction is the lynch pin that separates a Physics theory from a map with mathematics of the data.

The mathematical part is rigorous and the variable phase space it can explore is much larger than the phase space that physical observations defined. Laws ( or postulates) are extra axioms to the mathematical axioms, to pick up the subset of the functions which are compatible with the physical data to be modeled. From observations it has been determined by data that momentum and energy are conserved and they become the physical axioms to pick up the relevant mathematics from the larger set.

Reading physics history one sees that these laws are upheld to the point of defining new matter. In the elementary particles, the neutrino was discovered by the missing energy and momentum in beta decay . Dark matter is postulated because energy and momentum would have to be violated for the rotational curves of galaxies which were observed.

Upholding the laws is a basic tenet when getting new data and observations. Nevertheless, there are questions even about energy and momentum when one goes to physics theories like General Relativity, where the law of conservation of energy is not mapped one to one to a procedure in GR mathematics, even the range of validity of these basic laws may be questioned.

Questioning in physics leads to new knowledge and should not be anathema. What one should take great care with, when questioning, is to be founded well in the existing physical theories, to be sure that a new proposal is not a "crank" one. It should fit existing data and predict new observations so as to be able to be falsified , again within a mathematical model of physics.

We know or reasonably assume that momentum and energy are conserved because of two reasons mainly:

• Mathematical plausibility: If we assume that nature follows mathematical descriptions, then e.g. Noether's theorem makes it a necessity that momentum is conserved. Otherwise, something with the mathematical description would be wrong. And so far, in the vast majority of cases the mathematical description in physics fits perfectly what me actually measure in the experiment.
• Pure observation: So far, there was not even one single experiment reported (in a serious journal) that shows that energy and momentum conservation can be violated. In fact, the conservation of momentum and energy is so accurately confirmed by experiments (e.g. in the particle accelerator CERN) that it is just reasonable to assume they are true.

Of course, no one can tell what physics will look like in the future and one single experiment can disprove what we say at the moment. If you can prove that the conservation of momentum and energy can in fact be violated, a Nobel prize is probably waiting for you! Until then, it makes more sense to say: As far as we know, energy and momentum are conserved in closed systems.

When exploring deep questions in physics, like you are, it is important to remember that nature appears to obey the laws of physics. Empirical studies such as science cannot prove ontologically that nature does obey the laws of physics, or obey laws at all. For an extreme test case, consider the concept of idealism. In idealism, one claims that there is no such thing as "matter," only a mental substance that forms the fabric of consciousness. Matter is an illusion, under idealism, brought forth by the shared experiences of those consciousnesses. It sounds a bit absurd, but if you dig into philosophy, you find it is remarkably difficult to find an empirical way to disprove it. People have been trying for centuries. And, obviously, if matter is only an illusion, so is the idea that matter obeys any laws at all!

Coming back from that extreme, even the idealists have to admit that the laws of physics are remarkably effective. As several other answers have mentioned, we know of no single experiment which ever demonstrated a violation of the conservation of energy nor momentum. In fact, there's only one point in all of spacetime where we even postulate that conservation of energy might not have been conserved: the big bang. Even in our study of black holes, we build our equations around the assumption that energy and momentum are conserved.

So does that mean the laws are true? Well, it doesn't prove it. We have gobs of evidence in support of the claim that it is true, but evidence does not mean proof. In philosophy, there is actually a tool to deal with this called abduction. We are all familiar with deduction, a. la. Sherlock Homes, and we are all familiar with induction. A third -duction is adbduction. In abduction, one states that the most likely hypothesis is indeed the true one. Based on this, it would be reasonable to use abductive reasoning to justify the claim that "the laws of conservation of energy and momentum are indeed true." However, abduction is a bit of a sticky wicket when it comes to philosophy. It turns out that trying to pen an exact definition of abduction leads to all sorts of strange behaviors. I believe this is why it takes a back seat to deduction and induction, which people are more comfortable with.

In this sense, we can treat the conservation of momentum and energy as hypotheses. Whether we accept them using abduction is up to us. However, no matter what approach we use, we must admit that these hypotheses have remarkable amounts of evidence behind them. So much evidence, in fact, that most people simply accept them as law.

And perhaps they are law. If one wants to assume that physics applies to "everything," one has to be careful with abduction. However, there is another approach which may be valid. One might define the physical world to be the part of "everything" which obeys the laws of physics, such as the conservation of energy and momentum. In such a case, we have now defined the physical world to be a subset of "everything." If indeed the physical world is "everything," and there is nothing beyond, then the physical world is equal to "everything" (which makes it a subset, though not a strict-subset). On the other hand, if there is "something more," then we know the physical world is a strict subset of "everything," and that physics does not specify the interactions between that subset and "something else."

We could see this in the form of dualism, the belief that there is both physical matter and some mental substance. Dualism is very popular, both because it describes our experiences very well and because it plays nicely with major religions. In dualism, the mental substance is not bound by the laws of physics. There is nothing in physics to suggest that this mental substance, unbound by physical laws, couldn't interact with the physical world, adding energy or momentum. In fact, a great deal of philosophy has gone into trying to find ways to describe that magical process!

So all of that shows the limits of our "laws of conservation." We cannot prove them. That being said, you will not find me relying on the existence of something that rejects those laws any time soon without a commensurate cause to spur me in that direction. The quantity of evidence is just so large. Hypothesizing the existence of something which refutes those laws is one thing; depending on said existence is another.

• Please tell [physics girl] your second paragraph! – JDługosz Jul 5 '16 at 12:42
• Great discussion. I agree. It's not proof, but there's nothing to replace the tools we rely on at this time. – Inquisitive Jul 22 '16 at 0:14

Way back in time, scientists studied the following equation intensely because it was so new:

$$F=m*a$$

By experiment, they discovered that if you apply a force to a mass and it's free to move, then it will accelerate according to that equation. Back then, it was considered a LAW. It was immutable. It was a "law of physics". It wasn't derivable.

Never in their wildest dreams would they have believed that it was completely wrong. Because, as a mass accelerates, its velocity increases...and so does its mass.

As we look at this very simple equation today, we can easily see that there's nothing about it OR the original experiments that can say with authority (prove) that mass would never change with velocity.

Their early instrumentation was unsophisticated and their logic was completely flawed. The point is, scientific belief, in totality, is always wrong. It simply becomes less wrong over time.

Proof is extremely difficult to achieve. I wouldn't bet my life on anyone's proof of anything.