C as the speed of light and the speed of sound in Michelson-Morley.

[GerryB]

So we agree. Then please ask your question in terms of the meanings rather than the symbols. So far, it seems as though you are asking about the ink on paper rather than the physics.

Yes we agree on the importance of the meanings of words, but we seem to disagree on the meaning of invariance as it applies to sound, which is a key aspect of my question. So it seems pretty unresolvable.

 

[Dale]

To my knowledge "invariance" has only one meaning in special relativity. It means that a quantity or an equation does not change under the Lorentz transform. Is that not your understanding also?

 

[GerryB]

To my knowledge "invariance" has only one meaning in special relativity. It means that a quantity or an equation does not change under the Lorentz transform. Is that not your understanding also?

Well I agree with your definition of invariance in STR, but I think invariance has meanings beyond that. Since gallean transformations are contained within STR, but whose effects are negligible at the speeds of an average train, we can still use the old formulas. In classical galilean transformations, acceleration, distance intervals, and time intervals are also invariant (unchanging) quantities. So I am speculating that the observers in two reference frames moving relative to one another will measure the same velocity c for the sound wave, then I speculate that the equation I presented in my first post is valid. But it is only speculation, which is why I have asked the question.

Will these two observers use the same formula: L = ct + vt? This formula describes the idea that as the sound wave (velocity, c) travels rearward, it meets the caboose (velocity, v) traveling forward during the sane time. Each begins at the endpoints of the distance, L. This formula can be rearranged to the MM form: L / (c + v) = t; v = [L / t] - c, to find the velocity of the train relative to the earth.

 

[Dale]

Well I agree with your definition of invariance in STR, but I think invariance has meanings beyond that. Since gallean transformations are contained within STR, but are negligible at the speeds of an average train, we can still use the old formulas. In classical galilean transformations, acceleration, distance intervals, and time intervals are also invariant quantities. So I am speculating that the observers in two reference frames moving relative to one another will measure the same velocity c for the sound wave, then I speculate that the equation I presented in my first post is valid. But it is only speculation, which is why I have asked the question.

Even in Galilean relativity it is clear that the speed of sound is frame variant. In fact, in Galilean relativity ALL finite speeds are frame variant.

Let $v=dx/dt$ be any velocity. Then by the Galilean transform:

$t'=t$
$x'=x+ut$
$y'=y$
$z'=z$

then

$v=dx/dt=d(x'-ut)/dt=d(x'-ut')/dt'=dx'/dt'-u=v'-u \ne v'$

 

[Dale]

Closed pending moderation.

Edit: after discussion with the other mentors, the thread will remain closed. The idea that the speed of sound is frame variant has been succinctly proven here (for Galilean relativity) and is not a controversial topic (for either Galilean or special relativity) in the professional literature.