What if everything in the universe was actually
a bit to the right of where it is now? Or if this orbiting planet was actually half
a rotation ahead? What changes? More importantly what stays the same? These seem like fun but useless thought experiments
until Emmy Noether discovered, what I think, is the most profound and far-reaching idea
in physics. Knowing what happens to a system under these
imaginary transformations, gives us insight into the systems real behaviour. The usual summary is: symmetries imply conservation
laws. In this video, Iâll explain what that means. Weâll start with symmetry. Normally we use the word symmetry to mean
that if we took the mirror image along some line, a symmetric object looks the same. Mirror symmetries are pretty, but we can make
the word symmetry mean so much more. For example rotational symmetry: when you
can rotate an object a certain amount and it looks just the same as before, or another
example is translational symmetry. In fact mathemations took the idea of symmetry
and generalised it completely. a symmetry is anything where you have some
sort of object and apply some sort of transformation to it, and you canât tell the difference-
in some sense. This might seem like theyâve taken a good
descriptive word and then generalized it till itâs meaningless. But actually this idea is very useful. These abstract symmetries are a constantly
reoccurring theme in mathematics - in fact, the study symmetry helped motivate a one of
the most important fields of modern mathematics called abstract algebra. Emmy Noether was an expert in symmetry, developing
foundational concepts in abstract algebra. It was during a small pause from her extremely
influential mathematical career that she thought about physics. She wondered if she could apply the idea of
symmetry to the world, and thatâs what lead to her beautiful theorem. This is the symmetry that she considered. The object is some system, a part of the universe. It could be a thing someone is throwing. Or a particle in a void. Or maybe some binary stars. Or if you want, the whole universe. Then you transform it. For example, you could rotate it by some angle
lambda. Or shift it up or down by lambda, or stretch
all the distances by lambda. Now weâre interested in if the system is
âthe sameâ in some sense. Noether decided the interesting thing to check
is if the total energy of the objects would be the same. So we say that a system has a symmetry under
a transform if the total energy of the objects didnât change. For example, if I had a particle all by itself
and then compared it to a shifted version, clearly the energy is the same. So this system is translationally symmetric. On the other hand, say there was a big planet
near by. A particle that is closer has got more gravitational
potential energy, so this isnât translationally symmetric. Or consider this object orbiting in a circle,
and compare it to a rotated version. Both objects are an equal distance from the
planet and so both ways, they have the same gravitational potential energy. So this system is rotationally symmetric. So thatâs the symmetry part of Noetherâs
theorem. Now letâs look at conservations. If youâve ever studied physics, for example
at school, youâll know how important these things called conservation laws are. It means that if you have a bunch of things
and you counted up their total momentum letâs say, then you let them go for any amount of
time and counted the momentum again, it would be the same number. Technically, you can do physics without ever
needing to use these conservation laws. But. Often theyâll give you some insane problem
that looks like you shouldnât be able to solve- at least not easily... But if you invoke the magical conservation
laws your answer just falls out. Conservations laws arenât just useful for
classical physics either, they help out in quantum mechanics and really all of modern
physics. I used to not like using conservation laws
because they can make it seem too easy. As in, Iâd get the solution with so little
work that it really feels like magic and so I didnât feel like I understood why it worked. After all, I didnât understand why energy
is conserved or why momentum is conserved, so if I used one of those to solve a problem
then clearly I didnât understand the solution Noetherâs theorem is powerful because it
explains where conservations come from. Let me go back to an example. I said that momentum is conserved. But this, is kind of not true not always true. If I choose my system to be a ball rolling
on the ground, we all know that eventually it stops. Or if I dropped something, it gets faster
and faster. Sure, if you take everything as your system
momentum is always conserved, but how can I know whether a particular systemâs momentum
wonât change. Noetherâs theorem gives us a simple way
to know, regardless of whether the system is one particle or the whole universe. She proved that you only get conservations
if the system has the right symmetries. Again, letâs look at examples. If you have translational symmetry, the theorem
says you have conservation of momentum. We know that a particle thatâs on its own
has this symmetry, so itâs momentum is conserved. Thatâs true, it will continue on at the
same speed in the same direction forever. If we instead had a bunch of particles by
themselves as our system, this system is also translationally symmetric-if they all over
there instead, that doesnât change their energy. So again, Noether tells us we have conservation
of their total momentum, which wouldnât be that obvious otherwise. In fact, if we consider a shifted version
of the universe, no one would be able to tell the difference and so thereâs no difference
in the energy. Hence the momentum of the universe is conserved. When isnât momentum conserved for a system? What about this object that gains speed as
it falls? Noetherâs theorem says that this system
canât have translational symmetry, so letâs check. What if this object was nearer to the ground? It would have had less gravitational potential
energy- Good! It isnât symmetric. How about rotational symmetry? Like we said, this object could have been
rotated here and the energy wouldnât change, so it has rotational symmetry around this
axis. We also know it has angular momentum in this
direction, and that it goes at the same speed the whole way, so its angular momentum is
conserved. And this is what noetherâs theorem predicts,
if you have rotational symmetry around one axis, then the angular momentum in that direction
is conserved. One last example, this one is a weird one. Weâve talked about translating in space
and in angle, but what about translating in time? In otherwords, you have a system doing something
at the moment and you compare it to the same system some time later. If it has the same energy then it is time
translation symmetric. What does Noether say is conserved then? Itâs energy. I know, thatâs a bit circular here, but
it is more important when we come to quantum mechanics- so I had to mention it. Noether didnât just come up with these three
examples. Instead, she gave us a mathematical way to
turn any symmetry into a conservation and vis versa. See these conserved quantities are called
the generators of these transformations and you can calculate what the generator is for
any transformation you come up with. If I encountered some exotic system and noticed
it is symmetric under a transformation, there is a mathematical way for me to calculate
whatâs conserved. Thereâs also the converse. Say I notice
noticed that some mysterious new quantity is conserved. Noetherâs theorem says that conservation
is from some symmetry, and the conserved quantity is the generator of the transformation, so
I can calculate which transformation it is. Thatâs very powerful, but the theorem is
amazing because it is just as beautiful and it is useful. Symmetries appeal to us, and seem natural. We think it makes sense that if the universe
was shifted, or rotated that nothing should change, thereâs no difference between here
and there. So showing that symmetry and conservation
laws are equivalent shows that conservation laws must be just as natural. Homework
Let me know what you think of this idea. Have you heard of it before? Maybe youâve heard about things like super
symmetry in physics- try find out how thatâs related. The version of Noetherâs theorem I talked
about here is the one for classical physics (including GR), only its much less powerful
version of the theorem than she created (but I donât understand that one so...). If you know some calculus and classical physics,
try and find a proof of this theorem. And this is a fun activity, try come up with
strange systems with strange symmetries- then see if you can figure out whatâs conserved.
As a personal note, while some would think that as a supporter of the EMdrive, I also deride CoM. Let me be clear, I believe beyond a shadow of a doubt that CoM is practical, useful, and very pertinent to understanding the world around us. What I do believe with the EMdrive is that there is something new to be discovered if force is happening. Sorta like when chemistry had the idea of conservation of mass before the discovery of radioactive materials. Back then, radioactive materials would have been said to be a violation of the consevation of mass, until it was determined that the mass was being changed into energy. An update or correction had to be established to make it true. Thus the equations for consevation of mass for a radioactive material looks entirely different then for any other element, and should rightly be so.
Getting back to the point, CoM is elegant and very much a useable law, but can it be said to be based on everything that can possibly happen in the universe? No, it can't. Mostly because we as humans don't have access to all known systems of the universe just yet. Does that mean we might find reasons to modify? Yes. Does it mean we might have to understand a system better before we list there is a conflict with CoM? Yes. At present we lack enough information to state either way, which most detractors use as their reason to side aginst it and for some reason conclude that they don't have to do any experiments because of said lack of info. Which to me is like a snake that eats itself, but hey I guess that is why they call it circular logic. Does the statement of any CoM immediately make me doubt or state the EMdrive is automatically wrong? No, it doesn't, and won't until we have the validated experimental evidence to support or contradict it.
Richard P. Feynman: "LOL. It doesn't matter how beautiful your idea is, it doesn't matter how smart you are. If it doesn't agree with experiment, it's wrong :-)"
Great post, don't know why it hasn't been posted before. Everyone should know this. It's a shot through the heart of the emdrive and one of the most important results in the history of physics.
This is a good video that I've been pointing newbies at to cut-to-the-chase about the emdrive's foolishness.
It seems to be a good strategy. If people really understand the video then they will laugh at the emdrive of their own accord. It is very satisfying to save souls this way.
The Noether theorem is based on classical Newton laws (conservation of momentum) - therefore it shouldn't suprise us, that the EMDrive would violate it too, at least seemingly.
Noether theorem gets violated all the time