Constancy of the speed of light

[Ziang]

Nothing is wiggling though. At every point in space there is an electrical field, and at every moment that field points in some direction. If the direction is always the same (so only the amplitude of the field is changing with time - written as a vector we have $\vec{E}=(A\sin{\omega}t)\hat{x}$ where $\hat{x}$ is a fixed unit vector in the direction of polarization), then we say that the wave is linearly polarized.

May you tell me how can the direction of the field vector be changed when the observer is moving as same direction as the vector?

 

[Ibix]

May you tell me how can the direction of the field vector be changed when the observer is moving as same direction as the vector?

Because it's not an electric wave, it's an electromagnetic wave. You need to remember both the electric and magnetic waves and how they transform. What one frame regards as the electric part of the wave contributes to the electric and magnetic parts as measured in the other frame. Similarly the magnetic part. Putting them together gives you a transverse wave, as I said before.

 

[Dale]

May you tell me how can the direction of the field vector be changed when the observer is moving as same direction as the vector?

See:

https://en.m.wikipedia.org/wiki/Classical_electromagnetism_and_special_relativity

And Section 6 of http://hermes.ffn.ub.es/luisnavarro/nuevo_maletin/Einstein_1905_relativity.pdf

 

[Ibix]

May you tell me how can the direction of the field vector be changed when the observer is moving as same direction as the vector?

I'll do this explicitly. If you can't follow this then you have some learning to do - maybe read Dale's links.

The electromagnetic field tensor is, in general,$$F^{\mu\nu}=\pmatrix{
0&-E_x/c&-E_y/c&-E_z/c\cr
E_x/c&0&-B_z&B_y\cr
E_y/c&B_z&0&-B_x\cr
E_z/c&-B_y&B_x&0\cr}$$It's easy to see how you can split that up into a sum of two tensors, one that describes the electric field and one that describes the magnetic field:
$$\pmatrix{
0&-E_x/c&-E_y/c&-E_z/c\cr
E_x/c&0&0&0\cr
E_y/c&0&0&0\cr
E_z/c&0&0&0\cr}
+\pmatrix{
0&0&0&0\cr
0&0&-B_z&B_y\cr
0&B_z&0&-B_x\cr
0&-B_y&B_x&0\cr}$$
For the case of an electromagnetic wave heading in the z-direction with its E-field in the x direction, the electromagnetic field tensor is $$F^{\mu\nu}=\pmatrix{
0&-B\sin \left(kz-\omega t\right)&0&0\cr
B\sin \left(kz-\omega t\right)&0&0&B\sin \left(kz-\omega t\right)\cr
0&0&0&0\cr
0&-B\sin \left(kz-\omega t\right)&0&0\cr }$$where I have used that the amplitude $E$ of the E-field is c times the amplitude $B$ of the B field. Transforming it into another frame is straightforward - $F^{\mu'\nu'}=\Lambda^{\mu'}{}_{\mu} F^{\mu\nu}\Lambda^{\nu'}{}_\nu$, where $$\Lambda^{\mu'}{}_{\mu}=
\pmatrix{
\gamma&-{{v}\over{c}}\gamma&0&0\cr
-{{v}\over{c}}\gamma&\gamma&0&0\cr
0&0&1&0\cr
0&0&0&1\cr }$$is the Lorentz transform (or, in matrix notation, $F'=\Lambda F \Lambda^T$). Decomposing our tensor into the electric and magnetic components before transforming, we find that the electric component becomes$$\pmatrix{0&-B\sin \left(k z-\omega t\right)&0&0\cr B\sin \left(k
z-\omega t\right) &0&0&0\cr 0&0&0&0\cr 0&0&0&0\cr }$$and the magnetic component becomes
$$\pmatrix{0&0&0&-B{{v \sin \left(k z-\omega t\right) }\over{
\sqrt{c^2-v^2}}}\cr 0&0&0&B{{c \sin \left(k z-\omega t\right)
}\over{\sqrt{c^2-v^2}}}\cr 0&0&0&0\cr B{{v \sin \left(k z-\omega t
\right) }\over{\sqrt{c^2-v^2}}}&-B{{c \sin \left(k z-\omega t
\right) }\over{\sqrt{c^2-v^2}}}&0&0\cr }$$That is, what the first frame calls an electric field, the other frame sees the same (a special case because the E field is parallel to the velocity - generally a B field component will appear). But what the first frame calls a magnetic field, the other frame sees as both a magnetic field and an additional electric field. Adding them together, we see that the electric field in the primed frame is parallel to $(1,0,\gamma v/c)$.

It's straightforward to transform the direction of travel of the light to get $(-\gamma v/c,0,1)$ and hence see that the electric field is perpendicular to the direction of the light since the dot product is zero. The magnetic field is also perpendicular, trivially so since it only includes a y component in either frame. It's also straightforward to show that the magnitudes of both the electric and magnetic components have changed by a factor of $\gamma$.

So, in short, the transformed electromagnetic wave is an electromagnetic wave.

Edit: Note that I was lazy and didn't bother transforming z and t into z' and t', which is why there doesn't appear to have been any Doppler effect. If you do carry out that substitution the Doppler effect (concealed behind my laziness) will become apparent. It has no effect on the argument above, since all it does is change $kz-\omega t$ into $\vec{k'}\cdot\vec{x'}-\omega't'$ and then promptly cancels out of all the direction vectors I was interested in.

 

[Sorcerer]

Special relativity is the Lorentz transforms. Everything follows from them.

But if you want to regard relativity as a consequence of electromagnetism then you should also argue that it is (independently) a consequence of the strong force, the weak force, and gravity, since all of those are also relativistic theories.

Well, what I meant was that I am fairly confident that Maxwell's equations already had special relativity in them when they were put together, way back in the mid to late 1800s.

But would you be willing to elaborate? I know (obviously) that SR is compatible with EM (since it basically came from it) and with gravity (because of GR). I also know with QED that SR is compatible with quantum mechanics. What theories have SR compatible with the strong and weak forces? I know next to nothing about QED so if it's that, then I wouldn't know.

 

[Sorcerer]

May you tell me how can the direction of the field vector be changed when the observer is moving as same direction as the vector?

Because it's not an electric wave, it's an electromagnetic wave. You need to remember both the electric and magnetic waves and how they transform. What one frame regards as the electric part of the wave contributes to the electric and magnetic parts as measured in the other frame. Similarly the magnetic part. Putting them together gives you a transverse wave, as I said before.

I like this link on discussing this topic, if you're interested Ziang. I believe I posted it in our conversation.

http://www.mathpages.com/rr/s2-02/2-02.htm

It really is an excellent read, and explains the answers many, many of the questions you have.
I especially love how it shows how the invariant E² - B² is analogous to the invariant X² - T² of spacetime intervals (you have to get to the very end to see this in the article). It, to me, is a very attractive symmetry, which I'm noticing a lot of in special relativity as I learn it.

 

[PeterDonis]

I am fairly confident that Maxwell's equations already had special relativity in them when they were put together

They "had special relativity in them" in the sense that they are Lorentz invariant. But special relativity doesn't just say that EM is Lorentz invariant; it says that everything is Lorentz invariant, including ordinary mechanics, the sort of thing that Maxwell and other physicists of his time believed was governed by Newtonian mechanics and was therefore Galilean invariant, not Lorentz invariant.

What theories have SR compatible with the strong and weak forces?

The Standard Model of particle physics, which is, very schematically, a souped-up version of QED with enough fields in it to cover all of the known particles and (non-gravitational) interactions.

 

[Sorcerer]

They "had special relativity in them" in the sense that they are Lorentz invariant. But special relativity doesn't just say that EM is Lorentz invariant; it says that everything is Lorentz invariant, including ordinary mechanics, the sort of thing that Maxwell and other physicists of his time believed was governed by Newtonian mechanics and was therefore Galilean invariant, not Lorentz invariant.

So Einstein was the key because he believed that the divide between mechanics and Maxwell was an artificial one, correct?

Is that the origin of Lorentz' idea that matter was squished when moving fast, rather than space and time itself being what is relative? That he wanted to keep the rules of mechanics separate from the Lorentz transformation, and devised artificial ways of explaining experimental results which pointed to his transformation laws being universal? (I ask because if matter comprises chemicals, which are held together by electrons and their attraction to protons, due to electrical force, then presumably Maxwell's equations would apply to the very material objects thought to be in the domain of Newtonian/Galilean mechanics.)

The Standard Model of particle physics, which is, very schematically, a souped-up version of QED with enough fields in it to cover all of the known particles and (non-gravitational) interactions.

Thank you for the information. I don't know a lot about the Standard Model or QED. But by this do you mean that QED is embedded within the Standard Model? And does that mean that SR, QM, the strong, weak, and EM forces are all already united in a unified theory?

 

[Sorcerer]

What then is the meaning of a polarized wave? It is created by placing a slit in the light beam so that only waves which are aligned with the slit pass through.

I don't see how the equations help one to visualize this.

A polarizer usually does not have slits, at least not at optical wavelengths. Typically it is simply a material whose electrical properties are anisotropic.

The equations are not about visualizing anything. They are simply about refuting his absurd claim.

I know this is off topic, but guys like the now apparently banned Julian would do well to attempt to learn the math presented as refutation to a claim made against currently accepted physics by respected and well educated members who have spent years studying this topic. Physics is hard. You can't get beyond lower division undergraduate physics without doing much more than dipping your toes in the ocean of math. You have to jump in, because the subjects become far too abstract and to use simple "every man" models to explain them 100% accurately.

Even Newton's law of gravitation is much more complicated than you might think. Most physics newbies such as myself think it's this:

$F \ = \ G \ \frac{m_1 \ m_2}{r^2}$

When in reality, that's a vast oversimplification of a complicated vector integral that pushes the limits of undergraduate physics, used because spatially extended bodies are a lot harder to model than magical point masses.
More info on that here: https://physics.info/gravitation-extended/

Point being, physics is a lot more complicated than it seems when you start out looking at it, so don't make assumptions based on your intuition. Particularly when experiment has proven time and time again that our intuition is based on a limited perspective and often leaves us with incorrect assumptions.

 

[PeterDonis]

So Einstein was the key because he believed that the divide between mechanics and Maxwell was an artificial one, correct?

Basically, yes.

Is that the origin of Lorentz' idea that matter was squished when moving fast, rather than space and time itself being what is relative? That he wanted to keep the rules of mechanics separate from the Lorentz transformation, and devised artificial ways of explaining experimental results which pointed to his transformation laws being universal?

I'm not sure. Lorentz contraction is part of SR, so it's not like Lorentz was wrong about it. Also, from the viewpoint of the frame in which the object is moving, Lorentz's reasoning--that the way EM fields work when the source is moving causes the object to contract--is perfectly valid. (John Bell wrote an excellent essay about this--it's the one in which he introduces the Bell spaceship paradox, in fact.) The difference between Lorentz and Einstein was that Lorentz believed (at least if the historical claims I've read about him are correct) that there was a particular "privileged" inertial frame, and he intended his argument about EM field behavior causing Lorentz contraction to apply in that particular frame. Whereas Einstein realized that, no, if Lorentz invariance is correct then every inertial frame is the same, none of them are "privileged".

if matter comprises chemicals, which are held together by electrons and their attraction to protons, due to electrical force, then presumably Maxwell's equations would apply to the very material objects thought to be in the domain of Newtonian/Galilean mechanics

Yes, that's correct. And Lorentz understood that. See above.

But by this do you mean that QED is embedded within the Standard Model?

Yes. QED is what you get if you ignore all the fields in the Standard Model except the electron and photon.

does that mean that SR, QM, the strong, weak, and EM forces are all already united in a unified theory?

Yes.

 

[Ibix]

But would you be willing to elaborate? I know (obviously) that SR is compatible with EM (since it basically came from it) and with gravity (because of GR). I also know with QED that SR is compatible with quantum mechanics. What theories have SR compatible with the strong and weak forces? I know next to nothing about QED so if it's that, then I wouldn't know.

Of the so-called "four forces" we've only ever had a non-relativistic theory for one of them - gravity. EM was developed before relativity but its theory turned out to be compatible with SR and, historically, it was the mismatch between relativistic EM theory and the rest of physics that led to the discovery of SR. And the realisation that Newtonian gravity is incompatible with SR, no matter how hard we try to wedge it in there, led to the development of GR, a classical relativistic theory of gravity.

We then realized that Maxwell's EM was only an approximation to a relativistic quantum theory of EM, which is QED. Further thinking along those lines led to relativistic quantum field theories for the strong and weak forces - so far as I'm aware we never had a non-quantum nor non-relativistic version of those.

Similar lines of thought led to a plausible-looking candidate for a relativistic quantum theory of gravity, but the maths doesn't work and we've been kind of stalled there ever since. Despite ever more creative efforts.

 

[stevendaryl]

Of the so-called "four forces" we've only ever had a non-relativistic theory for one of them - gravity.

It seems to me that Coulomb's law makes for a consistent nonrelativistic theory of charged particles. Why doesn't it qualify?

I'm not sure whether there is a consistent nonrelativistic theory of the electric field + magnetic field.

 

[Sorcerer]

It seems to me that Coulomb's law makes for a consistent nonrelativistic theory of charged particles. Why doesn't it qualify?

I'm not sure whether there is a consistent nonrelativistic theory of the electric field + magnetic field.

I don’t think Coulomb’s law fits the bill as a theory encompassing the totality of one of the so-called four interactions/forces. That would be like saying Newton’s gravitational law encompasses all of classical mechanics (in fact the two look very similar).

That is, Coulomb’s law is a subset theory of the larger electromagnetic theory, and Ibix was referring to the big four, not any subset of them alone.

 

[Ibix]

It seems to me that Coulomb's law makes for a consistent nonrelativistic theory of charged particles. Why doesn't it qualify?

Fair point. I hadn't thought of it like that.

However, Coulomb's law leads simply to Gauss' law, which is one of Maxwell's equations. So is it a non-relativistic approximation or just an incomplete version of Maxwell?