A question on why Special Relativity holds

[thenava]

I was once told that the reason that the following two questions are fundamentally the same:

1) Are laws of Electricity and Magnetism are the same for all inertial reference frames?
2) The speed of light is 3.0*10^8 with respect to what?

I can fathom how Einstein could have shown the first question to be "Yes", but a quick explanation couldn't hurt =), but my real question is how are the two questions fundamentally asking the same thing? Thanks very much!

 

[James R]

Maxwell's equations for electromagnetism predict the speed of electromagnetic waves (light) to be $3.0 \times 10^8$ m/s. The problem is, though, that they appear to predict the same speed regardless of a change in inertial reference frame.

There are two possible solutions to this apparent problem:

1. Maxwell's equations are wrong, and the speed of light is actually different in different frames.

OR

2. Maxwell's equations are correct, and the speed of light is actually the same in all inertial frames.

Obviously, the accepted answer today is answer (2).

 

[thenava]

Thank you very much.

 

[JesseM]

Maxwell's equations for electromagnetism predict the speed of electromagnetic waves (light) to be $3.0 \times 10^8$ m/s. The problem is, though, that they appear to predict the same speed regardless of a change in inertial reference frame.

That's not really right, Maxwell's equations are just equations, there's no label attached to them that says they must hold in every frame. In fact, prior to Einstein virtually all physicists (including Maxwell) believed the equations would only hold exactly in the rest frame of a substance called the "luminiferous aether" which was supposed to fill all of space, and which light waves were imagined to be vibrations in much like sound waves are vibrations in the air (note that in the rest frame of the air, all sound waves move at the same speed regardless of the speed of the source of the sound waves). In a frame moving at some speed v relative to the aether, it was imagined that light waves would be measured to move at c + v in one direction and c - v in the opposite direction. But attempts to experimentally find differences in light's speed at different points in the Earth's orbit (when it would have different velocities relative to the aether), like the Michelson-Morley experiment, failed, and this was part of what lead Einstein to explore the postulate that perhaps the laws of physics might allow the speed of light to be the same in every form. But he didn't "prove" this, he just showed that the postulate leads to some predictions about the way the laws of physics work, predictions which have held true in experiment after experiment.

 

[James R]

That's not really right, Maxwell's equations are just equations, there's no label attached to them that says they must hold in every frame.

I'm sorry, but it is "really right". You have just said what I said in a slightly different way.

If you try to transform Maxwell's equations using a Galilean transformation, then they no longer allow for the existence of electromagnetic waves in the new frame. This is a real problem. The only possible solutions are: (a) assume that there is a single "preferred frame" which is the only one in which the equations hold; or (b) replace the Galilean transformations with transformations that preserve the invariance of Maxwell's equations.

Obviously, the Lorentz transformations of relativity satisfy option (b).

But attempts to experimentally find differences in light's speed at different points in the Earth's orbit (when it would have different velocities relative to the aether), like the Michelson-Morley experiment, failed, and this was part of what lead Einstein to explore the postulate that perhaps the laws of physics might allow the speed of light to be the same in every form. But he didn't "prove" this, he just showed that the postulate leads to some predictions about the way the laws of physics work, predictions which have held true in experiment after experiment.

I'm not sure what you mean by saying that Einstein didn't "prove" this. If I recall correctly, his famous 1905 paper was titled "On the electrodynamics of moving bodies", and he explicitly showed in it how Maxwell's equations transformed under the Lorentz transformations, thus holding in all inertial frames.

 

[JesseM]

If you try to transform Maxwell's equations using a Galilean transformation, then they no longer allow for the existence of electromagnetic waves in the new frame.

They no longer allow for the existence of electromagnetic waves which obey Maxwell's equations, but as I said, pre-Einstein, no one thought that Maxwell's equations would be precisely obeyed in any frame but the aether frame. In other frame's you'd have to modify them with a Galilei transform, and the modified equations would of course allow for waves that move at c+v in one direction and c-v in the other, where v is the frame's speed relative to the aether frame.

This is a real problem. The only possible solutions are: (a) assume that there is a single "preferred frame" which is the only one in which the equations hold;

Yes, and this was the option pretty much all physicists expected would be true before Einstein (or at least before Michelson-Morley). Nothing about Maxwell's equations themselves predicts that they will hold in all frames, which is why I objected to your statement "The problem is, though, that they appear to predict the same speed regardless of a change in inertial reference frame"...perhaps this was just poorly-worded though.

I'm not sure what you mean by saying that Einstein didn't "prove" this. If I recall correctly, his famous 1905 paper was titled "On the electrodynamics of moving bodies", and he explicitly showed in it how Maxwell's equations transformed under the Lorentz transformations, thus holding in all inertial frames.

I meant he didn't prove all the fundamental laws of physics of the real universe actually have this property of Lorentz-invariance, it was just a postulate he made. If all laws of physics (not just Maxwell's laws) weren't Lorentz-invariant, then it could then be that there would be types of physical rulers that didn't shrink and physical clocks that didn't slow down (as measured by other rulers/clocks which they are moving relative to), in which case inertial coordinate systems constructed from such rulers/clocks wouldn't be related by the Lorentz transformation, and there would be a unique physically-constructed frame where Maxwell's laws would hold. We must rely on experiments to show that all the fundamental laws of physics are in fact Lorentz-invariant, there's no theoretical argument that proves it.

 

[Heisenberg.]

A thing to note - before light was given a set speed - theory said time was a straight line equal for everyone. My time here on this planet would then be the same as the time anywhere else in the universe. This was replaced with the idea that time is not the same for everyone but rather speed is. The fact that time is not the same for everyone has been proven true through experiment. A famous example explaining this is imagine I have a twin. We each have our own clocks. My twin starts to travel at or near the speed of light (again this is hypothetical), I would age, but my twin comes back from wherever she traveled to at the speed of light and looks exactly the same. Just a fact that might help you out in some way - or at least I hope so.

 

[James R]

They no longer allow for the existence of electromagnetic waves which obey Maxwell's equations, but as I said, pre-Einstein, no one thought that Maxwell's equations would be precisely obeyed in any frame but the aether frame. In other frame's you'd have to modify them with a Galilei transform, and the modified equations would of course allow for waves that move at c+v in one direction and c-v in the other, where v is the frame's speed relative to the aether frame.

I have a sneaking suspicion that you are wrong. I think I recall seeing a proof that if you take Maxwell's equations for electric and magnetic fields and apply a Galilean transformation to them, the fields you obtain no longer satisfy the wave equation. But, maybe I'm wrong. I might check on this myself when I get a spare minute.

Nothing about Maxwell's equations themselves predicts that they will hold in all frames...

I didn't say that they do predict that. And my wording was not "poor"; it was quite clear. What I said was that electromagnetic waves in any inertial frame are predicted by Maxwell's equations to travel at the speed c.

I meant he didn't prove all the fundamental laws of physics of the real universe actually have this property of Lorentz-invariance, it was just a postulate he made.

Well, obviously he didn't go out and test all the laws of physics one by one. That would be impossible. In that sense, all of physics is just a bunch of postulates (or, more accurately, theories).

We must rely on experiments to show that all the fundamental laws of physics are in fact Lorentz-invariant, there's no theoretical argument that proves it.

There's no argument in theoretical physics that proves anything in the kind of mathematical sense you seem to be thinking of.