Exploring the Constancy of Light in Vacuum: Einstein vs Maxwell's Equations

[harrylin]

So, as a newbie, is this saying Lorenz transforms form a group? [..]

I did not understand the part that I left out, but yes, Poincare (who was first of all a mathematician) emphasized that these transformations form a group.
- https://en.wikisource.org/wiki/On_the_Dynamics_of_the_Electron_%28June%29

 

[Devils]

"It is, in fact, possible to derive the Lorentz transformations from the principle of relativity alone and obtain the constancy of the speed of light as a consequence."
http://en.wikipedia.org/wiki/Principle_of_relativity

This is interesting. Can somebody point me to a proof?

 

[bcrowell]

"It is, in fact, possible to derive the Lorentz transformations from the principle of relativity alone and obtain the constancy of the speed of light as a consequence."
http://en.wikipedia.org/wiki/Principle_of_relativity

This is interesting. Can somebody point me to a proof?

The statement in WP is footnoted to a 2004 paper by Friedman, which I don't have access to. But I think they are probably referring to an argument that, in various forms, dates back to 1911:

W.v.Ignatowsky, Phys. Zeits. 11 (1911) 972
Rindler, Essential Relativity: Special, General, and Cosmological, 1979, p. 51
Morin, Introduction to Classical Mechanics, Cambridge, 1st ed., 2008, Appendix I
Palash B. Pal, "Nothing but Relativity," http://arxiv.org/abs/physics/0302045v1
http://www.lightandmatter.com/html_books/0sn/ch07/ch07.html (my own presentation)

 

[Devils]

OK I can see that there has to be a universal speed limit, but why does this have to be the speed of light (or other electromagnetic waves)? What implies that light must travel at a constant speed anyway (waves in water can have various speeds)?

 

[bcrowell]

OK I can see that there has to be a universal speed limit, but why does this have to be the speed of light (or other electromagnetic waves)? What implies that light must travel at a constant speed anyway (waves in water can have various speeds)?

Water waves have a speed relative to the water. The speed of light in a vacuum can't be relative to anything, because there isn't any medium for it to be relative to.

 

[harrylin]

"It is, in fact, possible to derive the Lorentz transformations from the principle of relativity alone and obtain the constancy of the speed of light as a consequence."
http://en.wikipedia.org/wiki/Principle_of_relativity

This is interesting. Can somebody point me to a proof?

Instead I can point to a counter claim in that same encyclopedia:
http://en.wikipedia.org/wiki/Histor...rentz_transformation_without_second_postulate
In fact, that should be obvious: Classical mechanics has the PoR but with the Galilean transformations.

 

[harrylin]

OK I can see that there has to be a universal speed limit, but why does this have to be the speed of light (or other electromagnetic waves)? What implies that light must travel at a constant speed anyway (waves in water can have various speeds)?

That was based on observation combined with Maxwell's theory which models light as a wave with constant speed, similar to the speed of sound in a homogeneous medium.

 

[Devils]

Water waves have a speed relative to the water. The speed of light in a vacuum can't be relative to anything, because there isn't any medium for it to be relative to.

Instead I can point to a counter claim in that same encyclopedia:
http://en.wikipedia.org/wiki/Histor...rentz_transformation_without_second_postulate
In fact, that should be obvious: Classical mechanics has the PoR but with the Galilean transformations.

So the implication here is that if we have 'something' that doesn't travel relative to anything, this 'something' has to travel at the 'universal speed limit' (this is probably provable). Light in a vacuum is an example of this 'something'; are there any others (gravitons?)

Also the speed of light in a vacuum is an axiom, rather than being able to be derivable; even Pauli thought so.

 

[harrylin]

So the implication here is that if we have 'something' that doesn't travel relative to anything [..]

Instead, and sticking with the topic, the model that is used is that of Maxwell, according to which light propagates at speed c relative to a "stationary" frame, or "space":

"light is always propagated in empty space with a definite velocity c which is independent of the state of motion of the emitting body"
"Any ray of light moves in the “stationary” system of co-ordinates with the determined velocity c, whether the ray be emitted by a stationary or by a moving body."
- Einstein 1905

"We [..] assume that the clocks can be adjusted in such a way that the propagation velocity of every light ray in vacuum - measured by means of these clocks - becomes everywhere equal to a universal constant c, provided that the coordinate system is not accelerated.
- Einstein 1907

Einstein explained it as follows in 1907:

" It is by no means self-evident that the assumption made here, which we will call the "principle of the constancy of the velocity of light", is actually realized in nature, but - at least for a coordinate system in a certain state of motion - it is made plausible by the confirmation of the Lorentz theory [1895], which is based on the assumption of an ether that is absolutely at rest, through experiment". [footnote refers to Fizeau's experiment]

And with the PoR this model can be used for any inertial frame:
"the phenomena of electrodynamics as well as of mechanics possesses no properties corresponding to the idea of absolute rest"
- Einstein 1905

 

[bcrowell]

Also the speed of light in a vacuum is an axiom, rather than being able to be derivable; even Pauli thought so.

Whether it's a postulate or a theorem depends on what system of axioms you pick. We have a FAQ about this: https://www.physicsforums.com/showthread.php?t=534862

 

[Devils]

(Sorry to derail this thread).

So to summarise
- we start with a Euclidean space
- we introduce a "speed limit", which is light, which leads to hyperbolic functions (and hyperbolic identities), which leads to the Lorenz transform
- normally distance = speed * time, but the speed limit distorts this as well, to dilate time

So special relativity is a dressed-up version of hyperbolic algebra. I think it really is that simple, I don't know why people want to make it so complicated.