# Feynman's layman proof of local charge conservation

And it ends at 16 minutes. I have a great love for Feynman's explanations but right now I seem to have failed to understand exactly how his example proved charge is conserved locally. The important quotes would be just before and just after the example is shown:

The total amount of charge in a box might stay the same in two ways. It may be that the charge moves from one place to another within the box. But another possibility is that the charge in one place disappears, and simultaneously charge arises in another place, instantaneously related, and in such a manner that the total charge is never changing. [This second possibility for the conservation is of a different kind from the first, in which if a charge disappears in one place and turns up in another something has to travel through the space inbetween.] The second form of charge conservation is called local charge conservation, and is far more detailed than the simple remark the total charge does not change.

I would now like to describe to you an argument, fundamentally due to Einstein, which indicates that if anything is conserved it must be conserved locally.

If we are unable, by any experiment, to see a difference in the physical laws whether we are moving or not, then if the conservation of charge were not local only a certain kind of man would see it work right, namely the guy who is standing still, in an absolute sense. But such a thing is impossible according to Einstein's relativity principle, and therefore it is impossible to have non-local conservation of charge.

I understand the process of the experiment, and how each person sees two different situations. My confusion is the definition of "local" conservation and the later mentioned antonym "non-local" conservation. Directly from his explanation in the first quote it seems that local conservation is defined as "if charge pops up somewhere it must have disappeared somewhere else". What, then, does it mean to have non-local conservation of charge?

I also don't understand how this example hasn't shown the opposite of what he was trying to demonstrate. If a charge can pop up in one place (at the back of the train) and disappear in another (at the front), i.e. local conservation, then the person in the top trailer sees a situation different than the one in the bottom, but this is what violates Einstein's relativity principle, isn't it? The ability to create a charge in one place and have it disappear in another?

Is my understanding of local conservation wrong?

I think you've confused global and local charge conservation. The quote you included actually skips parts of the video, and I think the video may actually have been edited as well, so it is not obvious from the text alone what Feynman is referring to when he says "the second form of charge conservation...". He's actually referring to the first sentence. The correct definitions are:

A quantity in a box is conserved globally if the amount of the quantity in the box never changes. This allows, in principle, for the stuff in the box to instantaneously zap from place to place in the box, so long as the total amount never changes.

A quantity is conserved locally if the only way it can leave a region is by moving with some velocity through the walls (no teleporting). We say there is a "flux" of material through the boundary, and the amount by which the stuff inside changes is exactly accounted for by the flux entering or leaving the boundary.

So your statement "if charge pops up somewhere it must have disappeared somewhere else" is global or non-local conservation, not local conservation.

I think this should clarify the argument. If a quantity is globally conserved but not locally conserved, then some observer with a relative velocity will see it violate global conservation as well. The lesson is that given relativity, local conservation is necessary to have global conservation in all frames.

• I missed a line in my quote which I've edited in, in block quotes. But yeah, I misunderstood which form of conservation the second form was, and your answer was very clear so thank you. – AntiElephant Jun 23 '15 at 22:51
• Does this another way of saying that a local symmetry is actually a gauge, not actually a symmetry? – an offer can't refuse Jun 24 '15 at 14:57

I'm not sure whether everything's clear from your description and from Geoff's Answer.

If you haven't already worked it out, the essential problem that Feynman and Einstein are getting at is the relativity of simultaneity.

In the second situation, which Geoff's answer explains well, a relatively moving observer will not see the disappearance of charge and its reappearance elsewhere at the same time. There will be some observers for whom there is a nonzero time interval when there is neither the disappeared nor the reappeared charge present, and other relatively moving observers for whom both are present at the same time over a nonzero time interval. For all relatively moving observers, the charge becomes time varying.

I struggled with this myself. The exact comments by Feynman are in the book " The Character of Physical Law , Chapter 3, page 59. After much study, I agree with the other commenter that Feynman refers to the "first sentence". Local conservation is when the charge has to move from one place to another, not when it instantly pops up in a different place. If it pops up in a different place then one observer might see no charge at all for a certain time interval. Thus charge would not be conserved for him, but conserved for the other observer, and thus violate special relativity which says the physics should be the same in all reference frames.