Insights I Know the Math Says so, but Is It Really True?

[PeterDonis]

Einstein and his collaborator (Rosen?)

Marcel Grossmann.

 

[fresh_42]

I agree 100%. IMO it is the single most important theorem in all of physics.

For anyone who is interested in the original paper (which can be translated if you use google's chrome, right-click + "Translate to English"):
https://de.wikisource.org/wiki/Invariante_Variationsprobleme

and here are the facsimile (pages 37 ff. and 235 ff.)
https://gdz.sub.uni-goettingen.de/id/PPN252457811_1918

 

[strangerep]

IMO [Noether's theorem] is the single most important theorem in all of physics.

Surely that can't be true.

(Try deriving Kepler's 3rd law using Noetherian symmetry/conservation techniques... )

 

[strangerep]

Marcel Grossmann.

Hmm. I thought it was just Einstein+Grossman who put together GR using Riemannian geometry (but I haven't read a full history). What exactly did they need help with from Gottingen?

 

[PeterDonis]

I thought it was just Einstein+Grossman who put together GR using Riemannian geometry

Einstein came up with the field equation (at least by one possible route--see below for another route that Hilbert took). Grossmann helped him to learn Riemannian geometry.

What exactly did they need help with from Gottingen?

Einstein was stuck regarding a particular aspect of the field equation. (It's been a while since I read up about this so I can't say off the top of my head exactly what aspect it was.) I don't know that his primary purpose in visiting Hilbert in Gottingen was to see if Hilbert could help him get unstuck, but it probably was at least in the back of his mind. I also don't know that the talks with Hilbert were the primary thing that got Einstein unstuck, although they might well have helped.

However, the greater impact of the Gottingen visit was not on Einstein but on Hilbert. After mulling over his talks with Einstein, Hilbert came up with a simple, quick route to the field equation for gravity using the principle of least action. He came up with what is now called the Einstein-Hilbert Lagrangian based on obvious and simple considerations, and then it was a simple matter to derive the Euler-Lagrange Equation for it, and boom! he had the field equation. As MTW say in their discussion of six different routes to the field equation, "no route to the field equation is quicker".

 

[Dale]

Surely that can't be true. (Try deriving Kepler's 3rd law using Noetherian symmetry/conservation techniques... )

I am not sure why you think that is a necessary feature for a theorem to be classified as the single most important theorem in physics. I stand my my opinion, but being an opinion you are welcome to your own.

 

[fresh_42]

Try deriving Kepler's 3rd law using Noetherian symmetry/conservation techniques...

Is that really hard? Kepler's third is basically Newton's gravitation approximated, and Newton's gravitation is energy conservation in an $r^{-2}$ potential.

This would be at least my plan to approach it. Where is my mistake?

 

[PeterDonis]

Newton's gravitation is energy conservation in an $r^{-2}$ potential.

I think you mean an $r^{-1}$ potential.

 

[fresh_42]

I think you mean an $r^{-1}$ potential.

Sure. But is the plan ok?

 

[PeterDonis]

is the plan ok?

Newtonian gravitation is not the only theory that has energy conservation in an $r^{-1}$ potential (the obvious other such theory is electromagnetism). So that alone is not enough to get you to Newtonian gravitation.

 

[fresh_42]

I have read a comment on stackexchange that the only reason that Newton's mechanics can't be completely derived from Noether would be its incompleteness. I'm not sure how to take this since my physics knowledge is far too basic.

 

[fresh_42]

Newtonian gravitation is not the only theory that has energy conservation in an $r^{-1}$ potential (the obvious other such theory is electromagnetism). So that alone is not enough to get you to Newtonian gravitation.

As far as I understood the challenge, uniqueness wasn't required. The gravitational potential for masses can be measured and from there on only math is necessary.

 

[strangerep]

Kepler's third is basically Newton's gravitation approximated, [...]

Huh? Kepler's third law is an exact result from the theory of Newtonian gravitation. (A symmetry underlies it, but that symmetry is not associated with a conserved quantity.)

[Btw, I leave it to Mentors to decide if/when to fork this into a new thread. ]

 

[strangerep]

I have read a comment on stackexchange that the only reason that Newton's mechanics can't be completely derived from Noether would be its incompleteness.

? The Kepler problem is overcomplete in that it has more first integrals (constants of the motion) than are needed to solve the equations of motion, i.e., energy, angular momentum, and the LRL vector.

But none of these yield Kepler's 3rd law.

 

[bhobba]

I have read a comment on stackexchange that the only reason that Newton's mechanics can't be completely derived from Noether would be its incompleteness. I'm not sure how to take this since my physics knowledge is far too basic.

Well, one reason is you need to assume the Principle of Least Action (PLA). That is derivable from QM. So at rock bottom, classical mechanics is a limiting case of QM. Even assuming the PLA and Noether, a bit more is required as detailed in Landau - Mechanics eg why does mass appear in the free particle Langrangian.

Thanks
Bill

 

[strangerep]

To tie off this little detour about Kepler's 3rd law (K3L hereafter), a short answer is that K3L can be derived from a property of the Lagrangian known as Mechanical Similarity. (L&L vol1 p22 has more detail.)

In essence, if some of the variables in a Lagrangian are homogeneous (meaning that, e.g., $L(\alpha r) = a^k L(r)$ for some constant $k$ (where $k$ might be different for different variables), there are cases where the homogeneities in the different variables can combine to result in merely multiplying the Lagrangian by a constant factor. This doesn't change the equations of motion, but does correspond to the existence of similarly shaped orbits of different size and energy. This symmetry doesn't commute with the Hamiltonian, hence is not associated with a conserved quantity.

There are other cases where mechanical similarity is useful. L&L give examples.

 

[droogiefret]

I think what happens is that lay people (like myself) have these flashes of awe and there is this need to try to comprehend. Then we talk to physicists and everyone gets frustrated..lol!

My most recent example is magnetism. I'm reading that one of the objections to Newtonian explanations of gravity is that it is force acting at a distance - and I realize that that is exactly what magnetism is. So then I'm googling and Youtubing in a fairly haphazard way.

And I get to the point of realising that there is very little discussion of what a magnetic field could be made of. How can something exist that is not made of anything? And, understandably I guess, the physicists and mathematicians tend not to be particularly long suffering. It's not a scientific question. Or, if you studied the maths and understood it you'd gradually find you're more comfortable with the existence of non-physical fields. Or, the maths works if we assume it's made of photons, and although it's not really photons, it's virtual photons, because that's how the maths works.

I get it that physics and maths need to be practical. Measurement and prediction are real - wanting to intuit directly some kind of basic reality is a kind of wrong headedness. But still ... trying to comprehend a magnetic field is irresistible.

 

[fresh_42]

My most recent example is magnetism. I'm reading that one of the objections to Newtonian explanations of gravity is that it is force acting at a distance - and I realize that that is exactly what magnetism is. So then I'm googling and Youtubing in a fairly haphazard way.

Here is a nice interview with Richard Feynman about such questions in general, and magnetism in particular:

How can something exist that is not made of anything?

How can it fail to exist if we can measure it and see what it does?

 

[droogiefret]

Thank you - yes that sums up the problem nicely. Feynman is being as patient as he can be. Yet I'm sure the interviewer is thinking that Feynman has avoided the core of his question. I guess it's a language problem.

 

[phinds]

I guess it's a language problem.

Depends on what you mean. It is NOT a problem w/ the English language, it is only a "language" problem in the sense that math is not English.

 

[phinds]

@droogiefret perhaps you are not aware of it but there ARE things that simply cannot be accurately described w/ human languages, but only with math. **

The best example is Hawking Radiation. EVERYWHERE in popsci presentations you will read/hear/see that this is explained as virtual particles near the Event Horizon. That explanation is, at best, a heuristic that sort of describes kind of what's more or less happening. It was stated by Hawking and he clarified it by saying that it wasn't really what's happening but it was as close as he could come using English and not math.

** Well, that not exactly true since you COULD use English to describe the math but to those who don't already understand the math, it would require reading several advanced math textbooks plus a physics book or three and it would require that the audience understand everything that you read.

 

[fresh_42]

** Well, that not exactly true since you COULD use English to describe the math but to those who don't already understand the math, it would require reading several advanced math textbooks plus a physics book or three and it would require that the audience understand everything that you read.

The other way around is much easier. You can write the math without any spoken language. And it surprisingly translates into any other spoken language. Same as music. Math and music are the only universal languages. Well, maybe bureaucracy, too, according to Bones.

 

[PeterDonis]

I'm reading that one of the objections to Newtonian explanations of gravity is that it is force acting at a distance - and I realize that that is exactly what magnetism is.

No, magnetism is not "force acting at a distance" in the sense of Newtonian gravity. Magnetism is propagated at the speed of light, in accordance with relativity.

I get to the point of realising that there is very little discussion of what a magnetic field could be made of.

Magnetic fields are part of the electromagnetic field. The electromagnetic field itself is what magnetic fields are "made of". There is no need for them to be "made of" anything else.

 

[Dale]

How can something exist that is not made of anything?

I guess it's a language problem.

I don’t think it is a language problem. I think it is an erroneous philosophical concept that you are trying to push onto the world. You are assuming that something that exists must be made of something and are getting frustrated because the best description of the world doesn’t seem to conform to your philosophy. The best solution is to discard the erroneous philosophical concept.

 

[phinds]

The other way around is much easier. You can write the math without any spoken language. And it surprisingly translates into any other spoken language. Same as music. Math and music are the only universal languages. Well, maybe bureaucracy, too, according to Bones.

Sure, but you're talking about just reading off the symbols and perhaps even the meaning of each ("force" instead of "F") but you're not talking about translating the meaning of a whole equation.

 

[droogiefret]

I don’t think it is a language problem. I think it is an erroneous philosophical concept that you are trying to push onto the world. You are assuming that something that exists must be made of something and are getting frustrated because the best description of the world doesn’t seem to conform to your philosophy. The best solution is to discard the erroneous philosophical concept.

Exactly this.

But I wonder what the consequence is of accepting that particles (solar wind) can be deflected by something that has no particles (earth's magnetic field). And magnetic fields store energy do they not? So does that mean (philosophically) that energy exists as a separate entity to matter - rather than a property of it?

See how frustrating I become? And it's not deliberate - it's how lay people think.

 

[fresh_42]

See how frustrating I become? And it's not deliberate - it's how lay people think.

In order to answer a why question, you first need a framework that defines accepted answers.

So if this ...

Magnetic fields are part of the electromagnetic field. The electromagnetic field itself is what magnetic fields are "made of". There is no need for them to be "made of" anything else.

... isn't in your frame, what is?

 

[droogiefret]

@droogiefret perhaps you are not aware of it but there ARE things that simply cannot be accurately described w/ human languages, but only with math. **

The best example is Hawking Radiation. EVERYWHERE in popsci presentations you will read/hear/see that this is explained as virtual particles near the Event Horizon. That explanation is, at best, a heuristic that sort of describes kind of what's more or less happening. It was stated by Hawking and he clarified it by saying that it wasn't really what's happening but it was as close as he could come using English and not math.

** Well, that not exactly true since you COULD use English to describe the math but to those who don't already understand the math, it would require reading several advanced math textbooks plus a physics book or three and it would require that the audience understand everything that you read.

I am certainly aware of mathematics being regarded as the only true way to describe nature. And certainly I do feel that it is possible to intuit 'truths' about reality from equations and theorems (as opposed to simply using the equations and theorems for practical purposes). You present me with a huge undertaking though.

 

[Dale]

I wonder what the consequence is of accepting that particles (solar wind) can be deflected by something that has no particles (earth's magnetic field)

Why would there be any particular consequences?

All the mental knots you are tied in are just your own philosophical assumptions. Stop telling the world that it needs to conform to your philosophy. Let it show you how it actually behaves instead.

 

[droogiefret]

Why would there be any particular consequences?

All the mental knots you are tied in are just your own philosophical assumptions. Stop telling the world that it needs to conform to your philosophy. Let it show you how it actually behaves instead.

I shall try. I suppose the idea that everything that exists is made up of particles is, for whatever reason, ingrained. I shall try to let it go. Thank you.

 

[benorin]

Tyvm for this insightful insight @PeterDonis I feel like I could almost follow it.

 

[Dale]

I suppose the idea that everything that exists is made up of particles is, for whatever reason, ingrained. I shall try to let it go.

Well, maybe this will help:

You are made up of particles, and those particles are made up of particles, and those particles are made up of particles. But eventually you get to the last particles, the fundamental ones that we describe as fields. They cannot be made of anything, because then they wouldn’t be fundamental. They are just themselves. And why should they be anything else than themselves?

 

[phinds]

it's how lay people think.

But ... the universe is under no obligation whatsoever to act the way lay people think it acts.

Or, put another way, if the universe does not meet your expectations, that is not the universe's problem.

 

[bhobba]

It boils down to what is meant by 'exist'. That is a deep philosophical issue by forum rules, not something we discuss. But physicists are a rather literal lot. If something has effects that can be measured, mathematically modeled etc, then it exists. In modern times we know everything is a quantum field, matter, EM, etc - everything. Yes, even gravity:
https://blogs.umass.edu/donoghue/research/quantum-gravity-and-effective-field-theory/

It is generally thought all our QFT theories are just effective and break down at about the Plank Scale. The next stage is to find a theory for below the Plank Scale and experimentally test it. Could such be the fabled Theory Of Everything? Maybe. It could also be turtles all the way down. We simply do not know, but it is the quest of science to find out.

This brings up the old conundrum of renormalisation. Thanks to Ken Wilson, we know a lot more about it these days. But a discussion of that requires a whole new thread.

Thanks
Bill

 

[collinsmark]

But I wonder what the consequence is of accepting that particles (solar wind) can be deflected by something that has no particles (earth's magnetic field).

But the Earth's magnetic field is created by moving, charged particles in the Earth's core via a dynamo effect. So it's really a question of how the charged particles in the Earth's core interact with the charged particles in the solar wind.

Also note that the charged particles in the solar wind of interest are primarily electric monopoles; i.e., particles that have a net charge, unlike electric dipoles or electric quadrapoles, etc. The range of the electric fields associated with electric monopoles is much greater than that of electrically net neural particles such as dipoles, etc. The range can extend really, really far. Also, the trajectory of electric monopoles can change quite significantly when moving through a magnetic field. Contrast that with electric dipoles that might feel a torque, but their trajectories don't change much, comparatively. Doing the physics, complete with the mathematics, will show why this is.

(And as @PeterDonis mentioned, electric and magnetic fields don't come into being instantaneously over a distance; they propogate at speeds less than or equal to the speed of light in a vacuum, each one being proportional to [a function of] the rate of change of the other.)

Now, you might be saying, "sure, but the charges in the Earth and the charges in the solar wind don't touch each other." Well, don't they? Not by most definitions, no. But how exactly do you define "touch" in this scenario?

The reason your feet don't fall through the floor is ultimately the interaction of your feet and the floor's electromagnetic forces together with the Pauli Exclusion Principle as applied to particles subject to the electromagnetic force. It's the net balance of attractive and repulsive forces that define what "touching" is. We might define the interaction between the Earth and solar wind in a similar fashion, except that since we're dealing with electric monopoles, the interactions can easily propagate out to much greater distances. [Edit: Of course, the Pauli Exclusion Principle plays no role at all at such distances. I'm just saying that it's still about interacting forces and their associated fields.]

 

[droogiefret]

Thank you all very much for taking the time to try to explain to me. It's helped me a lot.

 

[zdcyclops]

Continue reading...

Some things cannot be said in "normal language" and can only be understood if one learns the language of science which IS mathematics. On can make statements that are mathematically correct but have no relationship with the universe at large. As to black holes direct observations of what occurs inside the event horizon are speculations in that they cannot be observed. They are reasonable and logical but cannot be proved by direct observation. Come back in a couple thousand of years for an update.

 

[Dale]

direct observations of what occurs inside the event horizon are speculations in that they cannot be observed

No. They can be observed. Just not published in any journal outside the horizon.

 

[swampwiz]

Here is a nice interview with Richard Feynman about such questions in general, and magnetism in particular:

Feynman here does a great job of explaining that the understanding of the forces of nature is as per the level of abstraction that interested party is satisfied with. While he did a good job in discussing abstraction in general, he could have dug into electromagnetism at enough of a level that an intelligent person (e.g., the reporter) could grasp.

An intelligent person should know that there are electrons that are buzzing around in an atom; from there, magnetism could be described as a force whose field is generated by electrons moving in general, with a concentration of magnetic field being generated by electrons moving in a circle. Iron is a type of material where it is possible for these electron circles to not cancel out the other electrons, if as a solid (i.e., a liquid or gas would not be able to hold the atoms in place), and so if molten iron solidifies while under a magnetic field (such as with the Earth's), it would a non-random distribution of these circles, and thus a net "circle". The magnetic field generates a force on electrons that are similarly in motion, and also concentrated if the electrons in the material have a non-random distribution of circles. If these circles were in a totally random set of orientations, there would be no net circle and thus no magnetic force.