Relativity, speed of light and stuff

[atyy]

it isn't clear to me that the particles in a solid which is transmitting a sound would necessarily be moving at the speed of sound. one can increase the speed of sound simply by increasing the stiffness of the material and decrease the motion of the particles by simply decreasing the amplitude of the sound. or at least, I guess you can. I'm not an expert on sound. or anything else for that matter.

Yes, hence the second thoughts in my above post. So I guess the question can be split in 2:
1) What transformations are consistent with 2 invariant speeds (speed of light and something else). I suppose this us related to doubly special relativity.
2) Can light (and gravity) be usefully modeled as a medium? For light, it appears the answer is yes. For gravity, the answer is unknown, but there are several intriguing leads (http://arxiv.org/abs/0712.0427)

 

[granpa]

2 invariant speeds?

it gets confusing since we are talking about sound but what we are really talking about is light. I never meant to say anything implying 2 invariant speeds.

 

[atyy]

Well, if we did have a good "theoretical explanation", it would belong in a physics textbook, not a literature textbook. So I would consider it physics. The fact that we don't have such a theory doesn't mean that such a theory is not within the scope of physics.

Al

I repeat my question: but when you reach the level of the most fundamental laws, what exactly would it mean to have a "theoretical explanation" of these laws? I can't imagine what a theoretical explanation of fundamental laws would even look like (aside from boiling them down to some minimal set of axioms), so it's not clear what you're imagining here, but it sounds like you're talking about something totally unprecedented in the history of science.

There's an interesting discussion by Wen in his Quantum Field Theory book (OUP 2004, p 12):
Chinese philosophers theorized that the division could be continued indefinitely, and hence that there were no elementary particles. Greek philosophers assumed that the division could not be continued indefinitely ... Those ultimate particles were called "atomos".

He quotes the Dao De Jing (p11): The Dao that can be stated cannot be eternal Dao. The Name that can be named cannot be eternal Name. The Nameless is the origin of the universe. The Named is the mother of all matter.

Which he mischievously translates as (footnote, p11): The physical theory that can be formulated cannot be the final ultimate theory. The classification that can be implemented cannot classify everything. The unformulable ultimate theory does exist and governs the creation of the universe. The formulated theories describe the matter we see everyday.

Preface (pviii): we still know so little about the richness of nature. However, instead of being disappointed, I hope the readers are excited by our incomplete understanding. ... The human imagination is also boundless. ... I wonder which will come out as a 'winner', the richness of nature or the boundlessness of the human imagination.

 

[atyy]

2 invariant speeds?

it gets confusing since we are talking about sound but what we are really talking about is light. I never meant to say anything implying 2 invariant speeds.

Ah, I see, the discussion was just on the second point then. Another interesting quote from Wen's QFT book, this particular one is quite uncontroversial, but he has nice imagery:
Our vacuum is more like an ocean which is not empty. Light and fermions are collective excitations that correspond to certain patterns of 'water' motion.

 

[Fredrik]

I repeat my question: but when you reach the level of the most fundamental laws, what exactly would it mean to have a "theoretical explanation" of these laws? I can't imagine what a theoretical explanation of fundamental laws would even look like (aside from boiling them down to some minimal set of axioms), so it's not clear what you're imagining here, but it sounds like you're talking about something totally unprecedented in the history of science.

I don't know what his answer is, but my answer would be that a deeper, more fundamental theory, can be considered a theoretical explanation of the fundamental laws in your theory. For example, general relativity is a theoretical explanation of Newton's law of gravity (the inverse square law). Newton's theory of gravity can't explain the inverse square law because it's a fundamental law of the theory, but GR can explain it because it's just one of many results that can be derived from the fundamental laws of that theory.

 

[JesseM]

I don't know what his answer is, but my answer would be that a deeper, more fundamental theory, can be considered a theoretical explanation of the fundamental laws in your theory. For example, general relativity is a theoretical explanation of Newton's law of gravity (the inverse square law). Newton's theory of gravity can't explain the inverse square law because it's a fundamental law of the theory, but GR can explain it because it's just one of many results that can be derived from the fundamental laws of that theory.

My question was, 'but when you reach the level of the most fundamental laws, what exactly would it mean to have a "theoretical explanation" of these laws?' If a given theory turns out to be an approximation of some more fundamental theory, like Newtonian gravity is understood as an approximation of GR, that shows that the first theory (Newtonian gravity) wasn't really one of the "most fundamental laws". Of course I'm assuming here that there are some final, most fundamental laws out there waiting to be discovered; as atyy brought up, it's conceivable that it's just wheels within wheels forever, that every particle is really a composite entity made up of even smaller particles, etc.

 

[MeJennifer]

INewton's theory of gravity can't explain the inverse square law because it's a fundamental law of the theory, but GR can explain it because it's just one of many results that can be derived from the fundamental laws of that theory.

I think you are mistaken. In fact if you look how Einstein derived GR you will see he simply included the Newtonian limit as a given. Technically GR is simply Newtonian gravity plus relativistic effects. GR does not explain anything, it is simply a more accurate theory.

If you think I am mistaken, please demonstrate how it explains the inverse square law. Or an even simpler question: How does GR get to the Newtonian limit.

 

[rbj]

I think you are mistaken. In fact if you look how Einstein derived GR you will see he simply included the Newtonian limit as a given. Technically GR is simply Newtonian gravity plus relativistic effects. GR does not explain anything, it is simply a more accurate theory.

If you think I am mistaken, please demonstrate how it explains the inverse square law. Or an even simpler question: How does GR get to the Newtonian limit.

does John Baez do that here?

or maybe Sean Carroll does that http://preposterousuniverse.com/grnotes/grtinypdf.pdf ?

i think they can derive the inverse-square relationship (or maybe it's a 1/r relationship for potential energy) for the flat space-time limit. the constant of proportionality in the Einstein equation (8 \pi G) does come about to be compatible with Newtonian gravitation.

i can't actually do the math myself (i am ashamed to confess i never figured out tensors), but it appears that this is what they do.

 

[yuiop]

I tend to agree with Jennifer here. I can not see how a physicist isolated in a small spacestation that had never experienced gravity or even heard of it, would conclude from a knowledge of Special Relativity alone, that two particles would have to move towards each other, let alone that they accelerate towards each other with an acceleration inversely proportional to the distance separating them.

As far as I can tell General Relativity started with a knowledge that we experience "Newtonian gravity" and extrapolated or reverse engineered that knowledge to more extreme conditions than we normally experience. It is hardly surprising that Newtonian gravity is recovered from GR in the weak field limit because GR started with that assumption. lease do not get me wrong here. I am not saying there is anything wrong with GR, I am just saying that it does not fundementally explain or predict gravity and just provides a pretty good mathematical description of what we observe.

Put it another way. In multiverse theories where there are any number of possible universes each with their own laws of nature, would a universe that obeys the laws of Special Relativity have to have an inverse square law of gravity in the weak field limit or come to that, any gravity at all?

 

[JesseM]

I tend to agree with Jennifer here. I can not see how a physicist isolated in a small spacestation that had never experienced gravity or even heard of it, would conclude from a knowledge of Special Relativity alone, that two particles would have to move towards each other, let alone that they accelerate towards each other with an acceleration inversely proportional to the distance separating them.

When did Fredrik say anything like that? He didn't say you could discover the inverse-square law from pure thought, he just said that if you already know the equations of GR you can get the inverse-square law as a derived consequence. Of course you could say the same thing about the equations of Newtonian gravity in some sense, so I'm not sure this is a totally clear distinction, but at least in Newtonian gravity it's obvious from the fundamental equations whereas in GR it's not.

As far as I can tell General Relativity started with a knowledge that we experience "Newtonian gravity" and extrapolated or reverse engineered that knowledge to more extreme conditions than we normally experience.

I don't know whether or not that's true of Einstein's original derivation as a historical matter, but it is at least true that GR can be derived from assumptions that have nothing to do with Newtonian gravity--on this page Steve Carlip writes:

If you want to derive the Einstein field equations from scratch, you can do so without making very many assumptions. You must assume that

1. the geometry of spacetime is dynamical;
2. there are no extra fixed, nondynamical "background structures" that influence the geometry;
3. special relativity becomes a good approximation when gravitational fields are weak;
4. the field equations can be derived from a Lagrangian, or an action principle; and
5. the field equations involve no more than second derivatives; that is, they determine "accelerations" rather than requiring accelerations as initial data.

These assumptions lead almost uniquely to a set of field equations with two undetermined constants. One of these is Newton's constant, which determines the strength of the gravitatonal interaction. The other is the cosmological constant, Lambda.

I think it is also true that you can come up with theories that are identical to Newtonian gravity in every respect except for the fact that the strength of the force is inversely proportional to some other real power like r^2.05, whereas in GR you don't have this sort of wiggle room, trying to make it no longer obey an inverse-square law would give a very different theory (presumably it would require violating one of Carlip's basic assumptions above).

 

[granpa]

the electric field follows an inverse square law because space is 3 dimensional. aether theory explains this very well.

 

[Fredrik]

My question was, 'but when you reach the level of the most fundamental laws, what exactly would it mean to have a "theoretical explanation" of these laws?'

To me, the idea of a law being fundamental or not only makes sense within the framework of a specific theory. What you call "fundamental laws" seems to be what I would describe as "a final theory". If such a theory is found, it won't be possible to explain its postulates. This isn't a very deep statement. It's just the definition of what I mean by "final".

When did Fredrik say anything like that? He didn't say you could discover the inverse-square law from pure thought, he just said that if you already know the equations of GR you can get the inverse-square law as a derived consequence.

Thanks. If I hadn't been asleep I would have said something very similar.

To Kev and MeJennifer, I would like to add a couple of things:

I don't consider the way Einstein discovered SR and GR to be "derivations" of those theories. In both cases he wrote down a somewhat ill-defined list of properties that he wanted the theory to have, and then searched for a theory that had those properties. The reason why I can't consider this method a "derivation" is that the "list of properties" was ill-defined to begin with, and later made well-defined by the theory that was found. (E.g. we need Minkowski space to properly define the inertial frames in which the speed of light is supposed to be a constant).

I understand that your opinion is that the fact that GR was found by looking only for theories that could reproduce the Newtonian limit means that GR can't be said to explain the inverse square law. That is a valid opinion (about the meaning of the word "explain") but I don't agree with it. There is no deeper form of understanding than having a theory that agrees with experiment, so if derivation from a theory that agrees with experiment can't be considered an explanation, nothing can. It makes no difference to me (at all) how the theory was found. All that matters to me is what range of phenomena it's capable of describing and how well it agrees with experiment.

 

[rbj]

the electric field follows an inverse square law because space is 3 dimensional.

if you bring into this an additional concept of flux, which is conserved and makes Gauss's Law possible. the concept of conserved flux seems natural and satisfying, but it wouldn't have to necessarily be the case. the inverse-square law of gravitation would require the same hypothesis; a gravitational flux emitted by quantities of mass, unless, like we're discussing here, the Newtonian inverse-square law is derived from some other more fundamental principle (like GR).

now, inverse-square laws regarding radiant intensity (E&M or acoustic) do necessarily follow from a combination hypotheses of conservation of energy and 3-dim space (both reasonable). the radiant energy (or power) comprises a natural form of "flux", which is conserved.

BTW, it is because of this concept of flux in inverse-square laws that make me wish that Planck units had originally normalized 4 \pi G and \epsilon_0 rather than normalizing G and 4 \pi \epsilon_0 as was done. i believe these rationalized Planck units are a little more natural (yielding simpler field equations) than the existing definitions. with any extraneous constants removed from the field equations, i think that might lead to insight to what might be behind such. we know that Nature isn't really performing a multiplication in her head to convert a particle wave frequency to its energy. that multiplication is necessary only because of the anthropocentric units we arbitrarily chose to use. and Nature doesn't give a rat's as$ what units humans (or some alien race) chose to use.

aether theory explains this very well.

i don't see a hypothetical aether having anything to do with the inverse-square relationship.

As far as I can tell General Relativity started with a knowledge that we experience "Newtonian gravity" and extrapolated or reverse engineered that knowledge to more extreme conditions than we normally experience. It is hardly surprising that Newtonian gravity is recovered from GR in the weak field limit because GR started with that assumption.

it's not surprising because of the correspondence principle. any newer, more advanced, theory must degenerate to the old theory in the context where the old theory was known to be valid. even though Einstein knew that his new GR theory would need to do that, i don't think that Newtonian gravity was where he started and extrapolated from. i think it was those classic elevator and spaceship thought experiments.

When did Fredrik say anything like that? He didn't say you could discover the inverse-square law from pure thought, he just said that if you already know the equations of GR you can get the inverse-square law as a derived consequence.

I don't know whether or not that's true of Einstein's original derivation as a historical matter, but it is at least true that GR can be derived from assumptions that have nothing to do with Newtonian gravity--

but, because of a concept of flux (which can be cooked up from pure thought) and knowledge of the mathematical fact that a sphere in 3-dimensional space has a surface area of 4 \pi r^2 can lead one to predict or hypothesize an inverse-square law for some quantity. doesn't mean, of course, that the hypothesis need not be tested in reality.

i think that Einstein first, from pure thought experiments with just a few really reasonable postulates (like the laws of physics are invariant for every inertial observer and that a free-falling observer cannot differentiate his or her state from being inertial - the equivalence principle), came up with SR, and with a little mathematical help from folks like Mercel Grossman, the GR. there is no evidence that Einstein ever drew on or referred to the Michaelson-Morley experiment and the null result, and i am convinced that it made little difference to him ("as if God had any choice in the matter"). assuming he knew of the experiment and result, Einstein was likely utterly not surprized. it's amazing what you can cook up from a very few extremely reasonable postulates, thought experiments, and math (all from pure thought). that is, if your brain is the size of a small planet and you have truly historical levels of insight. such persons are rare in history.

 

[JesseM]

there is no evidence that Einstein ever drew on or referred to the Michaelson-Morley experiment and the null result, and i am convinced that it made little difference to him ("as if God had any choice in the matter").

Well, in his original 1905 paper, in his first paragraph he discussed some theoretical reasons to suspect that electromagnetism doesn't have a preferred frame, but then in his second paragraph he said:

Examples of this sort, together with the unsuccessful attempts to discover any motion of the Earth relatively to the "light medium,'' suggest that the phenomena of electrodynamics as well as of mechanics possesses no properties corresponding to the idea of absolute rest.

And the Einstein quote you're referring to is "What really interests me is whether God had any choice in the creation of the world"--this was not a positive assertion that he was confident God had no choice about special relativity as you made it sound, it seems like more of a general philosophical question about the laws of physics as a whole (if it's an accurate quote at all, there are a lot of fake Einstein quotes that have circulated around, you can really only trust the ones attributed to some published source).

 

[rbj]

thanks for the reference to the original 1905 paper. i stand corrected about that. he clearly indicates he knew of the MM experiment and result (but he should have cited it).

And the Einstein quote you're referring to is "What really interests me is whether God had any choice in the creation of the world"--this was not a positive assertion that he was confident God had no choice about special relativity as you made it sound, it seems like more of a general philosophical question about the laws of physics as a whole.

i disagree with you about his position about this (and i assume the quote is for real). i really think that Einstein is questioning whether the form of reality could possibly be different. of the fundamental (dimensionless) constants that go into the description of reality, that's different, but the functional form, i think that Einstein was wondering, even challenging, if they could possibly be different.

 

[JesseM]

disagree with you about his position about this (and i assume the quote is for real). i really think that Einstein is questioning whether the form of reality could possibly be different. of the fundamental (dimensionless) constants that go into the description of reality, that's different, but the functional form, i think that Einstein was wondering, even challenging, if they could possibly be different.

How is that disagreeing with me, though? That's just what I said, it was a philosophical question about the laws of physics as a whole.

edit: also, note that the quote is listed in the "misattributed" section of http://www.billionquotes.com/index.php/Albert_Einstein#Misattributed claims that he said it to his assistant, Ernst Straus, but doesn't give a reference.

 

[granpa]

if light is thought of as a KIND OF sound wave in the aether then I believe it follows naturally that electric fields must follow an inverse square law. there is no difference between the electric field in a light wave and the electric field from an electron.

 

[Dale]

A wave which propagates through a medium has a propagation velocity that depends on the medium and is relative to that medium. If, by some coincidence, the propagation velocity of a wave in some medium were equal to the invariant speed then all observers would measure the propagation velocity to be the invariant speed regardless of what they measure the velocity of the medium to be.

However, a wave that does require a medium must propagate at the invariant speed. Since light does not require a medium it propagates at the invariant speed, which is how we originally discovered the invariant speed and its implications for the geometry of spacetime.

Since the aether is otherwise undetectable, and since it would be an enormous coincidence if the propagation of light through the aether were equal to the invariant speed, and since the speed of light is more simply explained by assuming it does not require a medium, what is the value of the concept of aether?

 

[granpa]

your second paragraph is unclear.

it might be an enormous coincidence or it might indicate the existence of an underlying symmetry that we haven't been smart enough to figure out yet.

why should light alone of all known waves not require a medium? it is much simpler to just take its wave nature as evidence of the existence of such a medium. in any event, relativity doesn't entirely eliminate the aether. it just renames it 'space'. according to relativity even empty space has properties.

 

[Dale]

If a wave does not propagate in a medium then what other speed could it possibly propagate at besides the invariant speed?

 

[granpa]

zero

 

[Dale]

Then it wouldn't be a wave

 

[granpa]

exactly

 

[atyy]

Since the aether is otherwise undetectable, and since it would be an enormous coincidence if the propagation of light through the aether were equal to the invariant speed, and since the speed of light is more simply explained by assuming it does not require a medium, what is the value of the concept of aether?

Not much within SR (only pedagogically for explaining the significance of the Michelson-Morley null result). But apparently it's useful within condensed matter physics (artificial "light" and "electrons"), and the interplay between condensed matter theories, quantum field theories and the search for a quantum theory of gravity. In these theories, the "aether" is typically not so much embedded in space, but spacetime and matter emerge from the "aether".

 

[atyy]

I understand that your opinion is that the fact that GR was found by looking only for theories that could reproduce the Newtonian limit means that GR can't be said to explain the inverse square law. That is a valid opinion (about the meaning of the word "explain") but I don't agree with it. There is no deeper form of understanding than having a theory that agrees with experiment, so if derivation from a theory that agrees with experiment can't be considered an explanation, nothing can. It makes no difference to me (at all) how the theory was found. All that matters to me is what range of phenomena it's capable of describing and how well it agrees with experiment.

Would it help to say that the inverse square law cannot be derived from GR, only an approximate inverse square law. So GR explains why we were deceived that it is an inverse square law? (If the inverse square law were exactly derivable, we wouldn't have perihelion precession)

 

[Dale]

granpa, you are not making any sense

 

[granpa]

it does not follow that a wave without a medium would travel at c. it follows that a wave without a medium would not travel at all and if it did it wouldn't be a wave.

 

[Dale]

granpa, first, even a propagation velocity of 0 wouldn't work, because 0 in one frame is non-zero in another frame. Second, what you are really saying is that it is not possible for any wave to propagate without a medium, do you have any logical reason to think that?

None of the four fundamental forces have a medium in which they propagate.

 

[granpa]

none of the forces have a medium? what do you think the aether is?

do you have any reason to think that a wave can propagate without a medium? I've never seen one do so.

 

[Dale]

Yes, I have a reason to think that a wave can propagate without a medium, and if you have never seen one do so then you must be blind.

\nabla \cdot \mathbf{E} = 0

\nabla \cdot \mathbf{B} = 0

\nabla \times \mathbf{E} = - \frac{\partial\mathbf{B}} {\partial t}

\nabla \times \mathbf{B} = \mu_0\varepsilon_0 \ \ \frac{\partial \mathbf{E}} {\partial t}

So according to you, all four fundamental forces require a medium in which to propagate, they all share the same medium, it is completely undetectable, it just happens that all four forces have the same propagation speed in this medium, and that propagation speed also happens to be the invariant speed.

Have you even one piece of evidence to support this rather long list of coincidences?