For instance, conservation of momentum, does it take time to propagate between two or more objects?

If it does, then there would be some moment that the momentum is not conserved.

If it doesn't take any time at all, since the law itself is information, then doesn't it prove that information can travel faster than light?

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    $\begingroup$ Conservation laws are manifestations of the continuous symmetries present in our laws(read more about Noether's theorem). They're abstractions. They're not physical entities that have attributes like speed of propagation. You're making a categorical mistake in thinking this way. $\endgroup$ – Omar Nagib Aug 3 '15 at 13:36
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    $\begingroup$ It is worth noting that the way we treat collisions in an introductory course---as a black boxes which happen on very short time scales and the details of which are ignored---misses exactly those features of real collisions that "transmit" the conserved quantities in finite time. That could lead to the impression the momentum leaps instantly from one extended object to another, but this is not a feature of proper understanding of what is happening in even very simple collisions. $\endgroup$ – dmckee --- ex-moderator kitten Aug 3 '15 at 13:44
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    $\begingroup$ I think you have something fundamentally backwards: 'laws' are just expressions of the human understanding of how the universe works, so thinking that they 'propagate' is one of those "not even wrong" ideas. $\endgroup$ – jamesqf Aug 3 '15 at 16:39
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    $\begingroup$ I like to think of conserved quantities as being properties of the observer. It bears no need to mention that quantities like total energy are in fact relative to a frame of reference. But I think of it as a fundamental limit on what an observer is capable of seeing. An observer will never see the total amount of energy change, so if he sees there being more energy in one part of the system, he can immediately conclude there is less in the other. In that sense, they take no time to propagate, whether as a result of logical inference or its more extreme manifestation, quantum entanglement. $\endgroup$ – Aleksandr Dubinsky Aug 3 '15 at 22:00
  • $\begingroup$ An interesting consequence of certain potential modes of time travel is conservation laws hold locally but not globally. $\endgroup$ – Joshua Aug 4 '15 at 15:44

Conservation laws don't "propagate". They are inevitable consequences of symmetries of the dynamics by Noether's theorem, and the dynamics propagate with whatever finite speed they do.

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    $\begingroup$ In other words: Conservation laws hold everywhere, at all times. There is no need for them to propagate, they are a fundamental property of physics. $\endgroup$ – Neuneck Aug 3 '15 at 13:30
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    $\begingroup$ @Neuneck That's a wonderful concise rendering of the answer. $\endgroup$ – Selene Routley Aug 3 '15 at 13:40
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    $\begingroup$ To make it even more concise: conservation laws hold locally, so they involve no propagation. Note that local laws can usually be re-cast into global laws. For example, local charge conservation can be re-cast as a statement that the current flowing into a volume must be equal to the current flowing out of that volume plus the time rate of change of total current inside the volume. $\endgroup$ – DanielSank Aug 4 '15 at 16:53
  • $\begingroup$ Related : physics.stackexchange.com/q/103724/44080 $\endgroup$ – J... Aug 5 '15 at 17:31
  • $\begingroup$ Well you did not include the speed @ACuriousMind. It would have to be at the speed of light. $\endgroup$ – Roghan Arun May 16 '20 at 20:07

See https://en.wikipedia.org/wiki/Continuity_equation - I was just writing some text there that I think helps explain.

Continuity equations are a stronger, local form of conservation laws. For example, the law of conservation of energy states that energy can neither be created nor destroyed—i.e., the total amount of energy is fixed. But this statement does not immediately rule out the possibility that energy could disappear from a field in Canada while simultaneously appearing in a room in Indonesia. A stronger statement is that energy is locally conserved: Energy can neither be created nor destroyed, nor can it "teleport" from one place to another—it can only move by a continuous flow. A continuity equation is the mathematical way to express this kind of statement.

If conservation of energy only said that total energy is fixed, and energy actually could teleport, then your question would be a very important question. But actually, every conservation law in practice is the stronger type, a local conservation law. Energy can only move by a continuous flow, and same for momentum and anything else.

A flow of energy, just like a flow of any other physical quantity, moves at or below the speed of light.

  • $\begingroup$ Does continuity apply in the context of quantum mechanics? What about disentanglement and "action at a distance"? $\endgroup$ – Aleksandr Dubinsky Aug 3 '15 at 19:30
  • $\begingroup$ There is no such thing as just "continuity", there are a wide variety of "continuity equations" describing many different quantities. Energy and momentum are conserved in quantum mechanics, and yes, those conservation laws are written as continuity equations in quantum mechanics too. There is no conflict between entanglement (which is ubiquitous in QM) and these continuity equations. I'm not sure why you think there would be. Can you explain what you have in mind? $\endgroup$ – Steve Byrnes Aug 3 '15 at 19:44
  • $\begingroup$ I mean something like: two entangled particles are separated by a considerable distance. Is the measure of each particle's energy also in superposition? When the states of the particle(s) is measured, does it determine how much energy is in one location (and how much in the other)? $\endgroup$ – Aleksandr Dubinsky Aug 3 '15 at 20:18
  • $\begingroup$ Yes, you can have two distant entangled particles (called an EPR pair), where one is more energetic and one is less energetic, but it is not determined which is which until you measure them. You can look online or in an intro QM textbook for an explanation of why information does not flow faster than light when you measure an EPR pair. This information-flow issue is commonly discussed. Well, energy also does not flow faster than light when you measure an EPR pair ... and it's for the exact same reason. $\endgroup$ – Steve Byrnes Aug 3 '15 at 21:30
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    $\begingroup$ @AleksandrDubinsky: a common name may be teleportation, but there's a good reason that we use mathematics instead of prose in physics. $\endgroup$ – MSalters Aug 4 '15 at 8:12

Conservation laws do not propagate instantaneously (or really at all), it really means that some property of the system does not change over time.

In the case of momentum conservation, the information about the collision propagates (at most) at the speed of sound in the medium. This is why you see the cars continuing along their path during a collision:

enter image description here

If the information of the crash travelled instantaneously, then the rest of the car would not move forwards after the front of the car impacts the wall, which is not what we see here. So no faster than light information travelling here.

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    $\begingroup$ What a gruesome example! It reminds me of our physics professor, ranting about students overtaking him on the way to the lecture, who will probably fail the momentum questions later at the exam... $\endgroup$ – Pavel Aug 3 '15 at 20:07
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    $\begingroup$ Errr... correct me physics noob if I am wrong, but: The momentum conservation is what makes the dummy keep going forward while the car around him is decelerating, spending its momentum in deforming the front section. (Until the dummy hits something, like the windshield, and spends its momentum in deforming that, or itself.) There is no need for anything propagating anywhere, much less limited to the speed of sound in the medium. The dummy has momentum, and it is conserved until applied forces change that. I don't see how "information about the impact" is propagating anywhere. Poor example? $\endgroup$ – DevSolar Aug 4 '15 at 9:40
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    $\begingroup$ @DevSolar: That is the issue at hand. OP expects that it takes no time for the information to propagate, which would suggest that the driver would not move during the collision (car stops, so does the driver). Since it is clear he does, then it takes time for the information of the collision to pass to the driver for their reaction. $\endgroup$ – Kyle Kanos Aug 4 '15 at 17:13
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    $\begingroup$ @KyleKanos: I think you're making a central mistake in that example. The driver doesn't collide with the wall at all. He's colliding with the steering wheel, and the windshield. The "information" that the car collided with the wall is irrelevant. Just the deceleration of the car (making the steering wheel and windshield an obstacle) is relevant. $\endgroup$ – DevSolar Aug 4 '15 at 19:19
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    $\begingroup$ @KyleKanos The speed of the propagation of information tells only when the rear of the car starts possibly decelerating. The magnitude of deceleration, and thus the time it takes for the rear to stop, does not depend on the speed of information at all. You say "If the information of the crash travelled instantaneously, then the rest of the car would not move forwards after the front of the car impacts the wall" but that is not true. $\endgroup$ – JiK Aug 5 '15 at 14:38

Let's look at an example, electromagnetism.

The electromagnetic field (the combination of electric and magnetic fields) has momentum.

Charges have momentum.

The charge feels a force right where it is, a force based on the fields right where it is. This changes the momentum of the particle.

The field loses an equal and opposite amount of momentum and also does so right in the same place. (The field has momentum distributed in little bits all over space, so has momentum right there to give to the charge.) Technically there is just a flow of momentum from the fields to the charge because the charge is also just changing its momentum at a certain rate, nor all at once. So you can also think of it as the charges momentum being given to the field in a way where it starts to spread out through the field into larger and larger regions. Since momentum has a direction there is no objectivity about whether you lose $p_x$ or gain $p_{-x}.$

So momentum propagates through space, and can be stored in the electromagnetic field and flow through space via the flow through the electromagnetic fields (and yes, technically the fields have a momentum and a flow of momentum) in a conserved way up until it meets a charge at which point the momentum in the field is no longer conserved but the total momentum (field and charge) is conserved.

Momentum is conserved locally. And it can take time for momentum to get from one object (charge) to another object (charge). But when it does so, momentum it is still conserved in the time in between because in between, the fields have the momentum.

Same with energy.


Others have explained that they don't propagate. That's a real puzzler though, and the real question is how does the stuff (energy, charge, whatever) always add up when a transfer takes finite time?

The answer to figuring that out was a profound pillar of modern physics: the field contains momentum (etc.). So while an electron is handshaking with another charged particle to say "you go this way, I'll go that way", you have the further puzzle of relativity of time as there is no "simultaneous" in an absolute sense. One changes before the other, or vice versa, in different reference frames. The solution is that the electric field (electromagnetic, if you are using relativistic effects) contains the momentum that is not at the moment (in your frame of reference) ascribed to either particle.

I'm sure I butchered that, but you get the idea: fields are real things that carry these properties, not just a way of drawing maps.


All conservative phenomena (not laws), do indeed, take a finite time to propagate, therefore "information" can't travel faster than light. Also, just because phenomena take time to propagate, does not mean that the phenomena are not being conserved at any given instant of time. In fact, there is no moment, when the phenomena are not conserved!


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