Spin Time Questions

From what I can gather, Lorentz was trying to show why an ether wasn’t detectable with the Michelson–Morley thingy. Lorentz believed that things [matter] contracted in the direction of motion, due to the modification of molecular forces through an ether. Relativity shows the same contraction without the need for a hypothesis about the structure of matter, but it’s still open as to whether the length contraction should be regarded as real or apparent, i.e., measurable [real] but frame dependent [apparent]. I think the muon thing is regarding time dilation, not length contraction.

Thanks for the info, C2!

In order to get a consistent picture in both the Earth and muon frames, there must be a physically real length contraction operative in the muon frame.
Just as there must be a physically real time dilation operative in the Earth frame.
But to see that requires more than just tossing ideas and povs about like salad.
And what the hell would motivate thinking there is 'a more complex Lorentz transformation than..Wikipedia knows about'?!

(Dec 4, 2023 12:57 AM)Kornee Wrote: In order to get a consistent picture in both the Earth and muon frames, there must be a physically real length contraction operative in the muon frame.
Just as there must be a physically real time dilation operative in the Earth frame.
But to see that requires more than just tossing ideas and povs about like salad.
And what the hell would motivate thinking there is 'a more complex Lorentz transformation than..Wikipedia knows about'?!

Yeah, I guess you’re right. You just flip the switch and from the muon’s perspective, the earth’s atmosphere is rushing past it at 99.99% of the speed of light.

https://en.wikipedia.org/wiki/Length_con...ontraction

"Another famous paradox is the Ehrenfest paradox, which proves that the concept of rigid bodies is not compatible with relativity, reducing the applicability of Born rigidity, and showing that for a co-rotating observer the geometry is in fact non-Euclidean."

Is it true that the concept of rigid bodies aren't compatible with relativity?

Hypothetically though, would a spinning disc look like a bowl in non-Euclidean geometry?

Kornee Wrote:And what the hell would motivate thinking there is 'a more complex Lorentz transformation than..Wikipedia knows about'?!

If you use four vectors the Lorentz Transform can be expressed as a matrix which I regard as more complicated than the wiki page I was looking at. I have in the past encountered the view that the four vector version is The Laplace Transform. I take it you were unaware of the alternative notation so please explore further - hopefully my motivation for mentioning "a more complex version" will become clear.

Kornee Wrote:In order to get a consistent picture in both the Earth and muon frames, there must be a physically real length contraction operative in the muon frame.
Just as there must be a physically real time dilation operative in the Earth frame.
But to see that requires more than just tossing ideas and povs about like salad.

I would be delighted to see it done rigorously .. the floor is yours.

(Dec 4, 2023 02:18 AM)Secular Sanity Wrote: Yeah, I guess you’re right. You just flip the switch and from the muon’s perspective, the earth’s atmosphere is rushing past it at 99.99% of the speed of light.

https://en.wikipedia.org/wiki/Length_con...ontraction

"Another famous paradox is the Ehrenfest paradox, which proves that the concept of rigid bodies is not compatible with relativity, reducing the applicability of Born rigidity, and showing that for a co-rotating observer the geometry is in fact non-Euclidean."

Is it true that the concept of rigid bodies aren't compatible with relativity? 

Hypothetically though, would a spinning disc look like a bowl in non-Euclidean geometry?

The Ehrenfest Paradox so-called is only a paradox in the extremely restrictive sense of an idealized absolutely rigid disk material. Any rotation speed greater than zero would over stress the rim in tension and cause shattering/cracking. In the real world where finite elasticity applies, there is merely an imperceptible strain adjustment. Many orders of magnitude lower than that extant for ordinary Newtonian mechanics centrifugal forces operating.

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(Dec 4, 2023 02:46 AM)confused2 Wrote:

Kornee Wrote:And what the hell would motivate thinking there is 'a more complex Lorentz transformation than..Wikipedia knows about'?!

If you use four vectors the Lorentz Transform can be expressed as a matrix which I regard as more complicated than the wiki page I was looking at.  I have in the past encountered the view that the four vector version is The Laplace Transform. I take it you were unaware of the alternative notation so please explore further - hopefully my motivation for mentioning "a more complex version" will become clear.

Kornee Wrote:In order to get a consistent picture in both the Earth and muon frames, there must be a physically real length contraction operative in the muon frame.
Just as there must be a physically real time dilation operative in the Earth frame.
But to see that requires more than just tossing ideas and povs about like salad.

I would be delighted to see it done rigorously .. the floor is yours.

Let's go to the relevant Wikipedia page.... https://en.wikipedia.org/wiki/Lorentz_transformation
That's about as comprehensive and exhaustive a treatment one can readily find. Looks at it from many angles. You want matrix notation? Just check out the relevant sub-headings
'Mathematical formulation' and further on again 'Tensor formulation'.
But none of that represents a more complex LT - just various ways of formulating the very same thing.

Recall the restriction that applies to LT, given in 2nd or 3rd (depending on how one divides it up) para:

"Frames of reference can be divided into two groups: inertial (relative motion with constant velocity) and non-inertial (accelerating, moving in curved paths, rotational motion with constant angular velocity, etc.). The term "Lorentz transformations" only refers to transformations between inertial frames, usually in the context of special relativity."

Regarding Wikipedia not knowing..

Kornee Wrote:Let's go to the relevant Wikipedia page.... https://en.wikipedia.org/wiki/Lorentz_transformation

You're right .. my raspberry wasn't loading the whole page .. after a crash and reset it loads all of it - my mistake.
If you're happy with tensor notation then all formulations are the same - I'm not so they aren't.

Muons are very much a special relativity thing so we're good to go with the LT.

I (still) think the crux of length contraction in the muon experiment is reasonably well expressed by:

I Wrote:At (close to) the speed of light the muons take 34us to travel 10km in the earth frame but in the (stationary) muon frame only 6.8us passes so either the muons think the the Earth is rushing past them at nearly 5 times the speed of light or 'something else'. The 'something else' is (has to be?) length contraction.

I'm tempted to have another go at it starting with invariant spacetime interval unless you have something better..

Edit.. The LT gave a good prediction for muon time dilation (it matched the experimental result) but this doesn't 'prove' length contraction. The invariant spacetime interval is a much more physical thing and gives the same result.

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Starting with
s²=x²-c²t²

In the muon frame elapsed time =t we have
s²=-c²t²
In the Earth frame t' and x' (where x' ~10km) we have
s²=x'²-c²t'²
since s² is the same in both frames
-c²t²=x'²-c²t'²
we know x' = vt' from Newton and the definition of velocity
so
-c²t²=(vt')²-c²t'²
c²t²=c²t'²-(vt')²
t= t'√(1-v²/c²)
so t=t_muon is less than t'=t_Earthclock
experiment shows t_Earthclock=34us and t_muon=6.8us

and the length contraction is .. ?
If we say the speed of the Earth in the muon frame is the same as the speed of the muon in the Earth frame then..
we know x' in the Earth frame was 10km in 34us so in 6.6us we get x is 2km at the same speed. Seems ok (ish)

I like it the lazy way. Just drill down via:
https://en.wikipedia.org/wiki/Lorentz_tr...ntz_boosts -> Physical implications -> Length Contraction

Suppose there is a rod at rest in F aligned along the x axis, with length Δx. In F′, the rod moves with velocity -v, so its length must be measured by taking two simultaneous (Δt′ = 0) measurements at opposite ends. Under these conditions, the inverse Lorentz transform shows that Δx = γΔx′. In F the two measurements are no longer simultaneous, but this does not matter because the rod is at rest in F. So each observer measures the distance between the end points of a moving rod to be shorter by a factor 1/γ than the end points of an identical rod at rest in his own frame. Length contraction affects any geometric quantity related to lengths, so from the perspective of a moving observer, areas and volumes will also appear to shrink along the direction of motion.

And two above that is one dealing with nonsimultaneity. Which is needed to resolve an apparent paradox.
How to explain that in the muon frame, the Earth frame measured length interval is from above contracted by factor 1/γ, as also is the clock rate wrt Earth frame rate.
Implying the muon measured clock interval will be less by (1/γ)*(1/γ) = (1/γ)^2.
Yet when the muon passes the final clock (placed at the lower point of elapsed trajectory), the reading must agree with the Earth frame elapsed interval of ~ 34ms. Nonsimultaneity to the rescue.

I'm not interested in pursuing this topic further but continue to do so if you wish.

In above post 34ms should have been 34us (phonetic thinking error).

Is it an either/or situation? Evidence for one, is evidence for the other? If so, the sleight of hand [frame-dependent] bit takes matter off the table…hmm, matter doesn’t matter. Am I right?

"This is an example of the Lorentz-Fitzgerald contraction, in which a moving length is contracted by a factor of the square root of 1 minus V squared over C squared along the direction of its motion when it’s moving with a speed V relative to an observer. This is just the same factor as in the Einstein time dilatation. The experiment that we’ve done can be interrupted as showing that moving clocks run slow or that moving lengths are contracted."

https://youtu.be/tbsdrHlLfVQ?si=99q_1wdgJbHdnaEQ&t=2069

Said I was out. OK one more post. This 'paradox' thought through may help clear up some confusion:
https://www.youtube.com/watch?v=0TU1tKTOIj4