joigus

Einstein said: 1) The laws of physics are the same in every inertial reference system 2) The speed of light is the same in all inertial frames of reference This gives a group of transformations that, in turn, result in a transformation of velocities that's called Einstein's transformation of velocities, which is consistent with it, as couldn't be otherwise. For systems moving in the x direction, the Einstein transformation of velocities gives, \[ V_{x}'=\frac{V_{x}-v}{1-vV_{x}/c^{2}} \] Now, if you feed into it a light ray moving parallel to this direction (x, called the 'boost' direction), you get, \[ c'=\frac{c-v}{1-vc/c^{2}}=\frac{c-v}{1-v/c}=c \] So indeed the speed of light doesn't change, consistent with Einstein's principle of relativity. Now, what you are suggesting is a different principle of relativity. Let's call it Killtech's principle of relativity, which says, 1) The laws of physics are the same in every inertial reference system 2) The speed of sound is the same in all inertial frames of reference That is a principle of relativity that's perfectly logically consistent. Same with the speed of 1 mile per hour, same with any other speed that you choose. That's not how Nature behaves though, so the speed of sound must change according to Einstein's transformation of velocities, and consequently it changes from reference frame to reference frame. The observation that one could have one different principle of relativity for every which velocity that one choses, and still be logically consistent is trivial, and obviously doesn't lead anywhere useful. Is that what all this is about? Do you have some reason to suspect that sound plays a pivotal role in it all? The speed of light is fundamental for reasons that other members have told you about. You can do all the maths that you want, and you can rephrase it any way you want. You still got it wrong.

"Classical" is an adjective that people use to mean "as opposed to quantum". I don't think you want to say "classical" there, but "non-relativistic". Namely: referring to a spacetime where time is observer-independent. I don't understand your formulas, so I'l tell you what I know. For a wave that is monochromatic (has just one frequency), the wavelength divided by the period is obviously the speed. What other speed is there? So, \[ v_{p}=\frac{\omega}{k} \] Is both the phase velocity and the group velocity. More general waves have many frequencies. They're not monochromatic. When one such wave is concentrated around a particular region of space, ie, it has certain spatial "lumpiness", it is possible to prove it must be made out of a range of frequencies. It's not obvious, but not too difficult to see either, that the centre of such "lumps" move at a speed that's given by the derivative, \[ v_{g}=\frac{d\omega}{dk} \] In quantum mechanics, you must remember that the \( E \), \( p \) mechanical properties are related to the \( \omega \), \( k \) wave properties (frequency and wave number) by, \[ E=\hbar\omega \] \[ p=\hbar k \] If you use the non-relativistic expression for kinetic energy, \[ E=\frac{p^{2}}{2m} \] you get the dispersion relation, \[ \omega\left(k\right)=\frac{\hbar k^{2}}{2m} \] which produces a phase and group velocities, \[ v_{p}=\frac{\omega}{k}=\frac{\hbar k}{2m} \] \[ v_{g}=\frac{d\omega}{dk}=\frac{\hbar k}{m} \] Now, for light waves (which are always relativistic, there's no non-relativistic approximation for photons), you have, \( c\omega=k \), which gives, \[ v_{p}=\frac{\omega}{k}=c^{-1} \] \[ v_{g}=\frac{d\omega}{dk}=c^{-1} \] For relativistic matter waves, on the other hand, if you repeat these calculations, you get, \[ v_{p}=\frac{\omega}{k}=c\sqrt{1+\frac{m^{2}c^{2}}{k^{2}\hbar^{2}}}\geq c \] \[ v_{g}=\frac{d\omega}{dk}=\frac{1}{2\sqrt{k^{2}+\frac{m^{2}c^{2}}{\hbar^{2}}}}\left(2k\right)=\frac{c}{\sqrt{1+\frac{m^{2}c^{2}}{k^{2}\hbar^{2}}}}\leq c \] So the phase velocity is generally greater than c, while the group velocity is less than c. That's what I meant when I said that the relativistic approach is apparently paradoxical. This is very far from a rigorous analysis in terms of quantum fields, but it gives you an idea that the relativistic formalism implies some superluminal modes, which later we learn are non-observable, and the group velocity is the one that seems to correspond to the measurable degrees of freedom. Quantum field have these virtual or non-observable degrees of freedom. They always do.

Ok. It's not that transversal waves must have a dispersion relation consistent with E=pc. Rather, massless waves, those that have a dispersion relation E=pc, must satisfy a transversality in the context of a gauge theory. You can prove this from Maxwell's equations in terms of the vector potential, if I remember correctly.

Here I explained how that is not the case in terms of frequency (inverse period and proportional to energy) and wave number (inverse wavelength and proportional to momentum): Different waves have different dispersion relations. Relativistic wave equations, eg, have still another different dispersion relation that's apparently paradoxical (if one tries to interpret it in terms of a 1-particle theory).

So is it clear now that the sound equation is not Lorentz invariant, as vs is not a universal constant, nor is it a Lorentz scalar, or are we still discussing that?

Sorry. A c4 should be c2 inside the square root of what I wrote there. No. Each wave has a different dispersion relation. E=pc gives you a photon D.R. Matter waves are a different matter. 😬

No, it's not \( E=\left|\boldsymbol{p}\right|c\) for massive particles. As @MigL said, it's, for massive particles. So \( E=\left|\boldsymbol{p}\right|c\sqrt{1+\frac{m²c^{4}}{\left|\boldsymbol{p}\right|^{2}}} \). With this factorisation maybe it's clearer how and why it's not the same for massive particles?

Sorry I didn't read carefully. You actually mentioned the eikonal equation. What is it exactly that makes it non-wavy? The problem with it is when the wave finds inhomogeneities of size the order of the wavelength, then it no longer is a good approximation. But otherwise it's quite wavy isn't it? I mean, if you solve for the amplitude and the direction of the 'rays' you're home free I suppose.

No. It's similar as when you define mass as m=F/a, while you define F=ma. Things like F=-Gmm'/r² get you out of the tautology. In the case of electromagnetism, you have many other relations that get you out of the tautology, like E=hx(frequency), or E(nu,k) is a solution of Maxwell's equations in free space. Etc. It all works out, and you never look like a dog chasing its own tail. Believe me.

\[ \left|\boldsymbol{p}\right|=E/c \] with, \[ E=h\nu \] with h Planck's constant, and \( |nu \) being the photon's frequency in Hertzs. Direction of momentum given by, \[ \frac{\boldsymbol{p}}{\left|\boldsymbol{p}\right|}=\boldsymbol{k} \]

https://en.wikipedia.org/wiki/Eikonal_equation

No reason to expect it does. In order to function, fiction only needs to create enough sense of plausibility for you to temporarily suspend your critical thinking under the implicit assumption that the narrative will be entertaining: https://en.wikipedia.org/wiki/Suspension_of_disbelief

Or... It may sound weird and be a false assumption as well. Example: Seals are grey because polar bears are white. It sounds weird and it is false. "Photons don't carry mass because they always entail a very small disturbance of the vacuum" is as false a statement as can be. X rays are an example. Extremely-high-energy photons are as massless as extremely-low-energy photons. x-posted with @exchemist, who said more or less what I said.

It's just an attempt to focus the discussion on physical determinism, which is what you meant, I think. Is it not? And we've been talking about free will for quite a while now. The words "elbow room" kind of gave it away. PS: Edited.

One thing is physical determinism, and quite a different thing is behavioural, psicological --or what may have you-- determinism. Ie. Suppose some kind of physical determinism has been established and we all agree on it being the basis for the physical world. There would still be a long way to go in order to prove or convince anybody that this has any bearing on the question of free will, as @Eise has argued somewhere else, if I'm not mistaken. Those are two different things.

Might this not be just a tad glass-is-half-full as to determinism and/or predictability? I think we who are trained in science tend to overestimate the scope of determinism. Determinism suggests itself very strongly when you first fall in love with science. Then it hardly ever occurs in practice.

It's a very interesting question. Of course, I don't know the answer, but here are some ideas. What seems to be a fairly simple question turns out not to be so easy when you start asking relevant questions on the basic definitions, primitive concepts on which it rests, hidden assumtions, and so on. So, for example (taken from Wikipedia): What is an event? How do you characterise it? What is a cause? What does it mean to be determined? So, for further example, suppose we accept the model of state of a system, state variables, one-dimensional time, law of motion, etc. Is it even possible to factor the state of the universe into environment + system under scrutiny? If so, is it always possible? Can prediction be extended indefinitely in principle without appealing to cosmic events? Could any non-predictability be attributed to unknown --and what's worse, presumably unknowable-- circumstances of past states of the universe such that, were those to be known, a mathematician could carry out the prediction successfully? What's clear to me is that QM dealt a very heavy blow to any intentions of formulating any naive determinism. What's not so clear to me is that there is no chance at all of elaborating on the concept of state variables, evolution law, etc, so as to explain why the universe looks deterministic at some simple level (free fall, penduli, 2-body problem, and the like), inherits this property from some nearly unfathomable cosmic condition which later reappears here and there, but for the most part is lost in the middle ground, which constitutes most of what we see.

No. You want to make it look like we've reached some kind of a stalemate, "we agree to disagree", when the truth is that you've been awash in facts, valid criticism and analysis, and your arguments don't stand any ground. You have no theory. You have no valid criticism. You have nothing and you must go back, either to the drawing board or, rather, to the books on basic physics and learn it all. Learn why we believe SR is right to zeroth, first, second, and third order --in a manner of speaking-- and, if proven wrong one day, it will be in a much more subtle way than what you're suggesting here. Along with the explanation of why it looks so damn right. As I said, the whole thing is ludicrous.

Then I suggest you stop ignoring the facts. The facts say special relativity is correct.

Not just that, but so much more. 4-dim formulation of SR is essential to understand theory of scattering (the so-called Mandelstam variables are relativistic invariants in 4-D language that make many other symmetries obvious: crossing symmetries, CPT, and the like). It provides the necessary preamble for the theory of gravitation (which-point-to-point has a local version of special relativity in it). It also makes many aspects of electromagnetism obvious. 4-dim SR explains magnetism. Tells you exactly how it comes about when charges move. It also indirectly explains spin 1/2 when you try to make the quantum theory every bit as 4-dim relativistic as the classical theory. I'm only talking about the 4-dimensional aspect of SR. It really, really lets you see behind corners that otherwise stay in the dark. So yes. A theory gives you a 'big picture' perspective. It affords you a tool to predict new range of phenomena. So @Bufofrog was spot on, I think, when saying you don't really have a theory. People who aren't trained in physics really do not understand the full import of what 'beautiful' means to physicists. It's not about looking pretty on a blackboard, with fancy Greek symbols. Rather, it's about empowering you to understand, it's about economy of thought. It's about being able to solve complicated problems immediately. Also about relating apparently unrelated phenomena. It's about understanding the language in which the laws of physics are written.

Sorry, but I couldn't disagree more. People like Minkowski, Wheeler, and others understood aspects of relativity that didn't occur to Einstein himself. Example: What Einstein discovered is equivalent to space and time being a 4-dimensional continuum, and relations from one inertial frame to another being equivalent to hyperbolic rotations in that space. This idea is not in Einstein's writing prior to 1908, and for a while he was reluctant to wholeheartedly accept it. Other people helped him --and everybody else-- understand his own ideas much better.

Oh, you said "sine of a degree", but what you meant was probably "sine of an angle expressed in degrees", right? So, naturally, I understood "sine of 1 degree". Anyway, that's not x-x³/3!+... Neither is it (180⁰/π)(x-x³/3!+...) Rather, it's, \[ \left(\frac{\pi}{180}x\right)-\left(\frac{\pi}{180}x\right)^{3}/3!+\left(\frac{\pi}{180}x\right)^{5}/5!-\cdots \] Is it not? Sorry. I was confusing because I was confused.

Wrong again. It's not about re-scaling sin(x). It's about re-scaling x. Repeat.

LOL. So it is possible to start an argument with a non sequitur.

No, I mean, I see what you're doing here: The regular crackpot routine. See what happens when the voice in your head speaks louder than the world outside?