So here's a trick question for you Victor. A factor of 10 time dilation for the muon allows it to barely cover the distance from the upper atmosphere to the surface of the earth. But relativity also says the muon will see that distance contract by a factor of 10. So why does this not mean the muon can easily cover 10 atmospheres of distance before it decays?

Well that is simple because it is because time dilation and length contraction are the same effect that would be like measuring the same effect twice, if it covered 10 atmospheres, it would be like trying to take 2 * 4 = 8, but instead you are taking it twice so it is 2 * 4 * 4 = 32. Length Contraction and Time dilation are different views of the same effect on matter. If you did take the same effect twice it wouldn't be 2x but rather x^{2 }anyways ralfcis but having the same root cause you only take the effect of relativistic motion once.

This equation is why

That's right. So in every example of relativity you can either use time dilation or length contraction but not both. Correct? So why would you need to consider length contraction if time dilation is equivalent and can stand in for it? But the constancy of the speed of light depends on both time dilation and length contraction occurring concurrently. How is that different from every other example? I've shown length contraction is never needed for the same reason you gave even when you consider c constancy. So why does Einstein's relativity need it?

It is still a process that needs to be measured so you will know the exact length and time but they happen simultaneously It could be said why would you need to know relativistic mass increase either but it is just so you know all the properties effected by relativistic motion.

So, again, why does a muon get the choice of either applying time dilation or length contraction but the constancy of the speed of light needs both concurrently?

I have already answered this question for you, but apparently you have me on ignore, which is great!

It's unbelievable you would even ask this.

Sorry for meddling into this problem. I posted a thread about muons recently:

http://www.sciencefo...on-and-history/

but nobody cared about it.

As Lorentz transforms are symmetrical, you can choose frame K(x,t) being at rest (sign -) or

frame K'(x',t') being at rest (sign +).

Measuring time elapsed or distances requires differences, which are represented by **d (DELTA)**.

Also, **Y **is the **Gamma factor**):

Being K(x,t) at rest referenced to K'(x',t'):

dx' = Y (dx-vdt)

dt' = Y (dt-vdx/c^{2})

Also can be used this way, being K'(x',t') at rest referenced to K(x,t):

dx = Y (dx'+vdt')

dt = Y (dt'+vdx'/c^{2})

Now is when relativity became **NASTY**. This is the way science explain muon's time dilation and length contraction:

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1) For **time dilation**, K'(x',t') is selected for Earth observer and K(x,t) for the observer travelling with the muon.

**dt' = Y.dt**

dx'=0 because **the observer at Earth doesn't move when measuring dt'. **It means that, simultaneously:

**dx = v.dt**

Then, for dt=2.2 microsec (muon's decay time), and for v/c = 0.99587 then Y =10.

dt' = 22 microsec (measured at ground level)

dx = 660 meters (measured at the muon's reference frame).

This values are concurrent, **BUT **are only valid for TIME DILATION (Length contraction **can't be used **simultaneously).

2) For **length contraction**, I **switch references **and K(x,t) is selected for Earth observer, and K'(x',t') for the observer travelling with the muon.

dx' = Y (dx-vdt) : distance measured from muon's RF.

dt' = 0 because both distances in dx' **are measured at the same time t' in the muon's RF. **

dt = vdx/c^{2} : time lapse at Earth.

so, by replacing in dx', it gives:

**dx' = Y.dx (1-v**^{2}/c^{2}) = dx/Y : distance measured at muon's RF (330 mt). It's contracted with reference to Earth.

so

**dx = Y.dx' **= 10 x 660 mt = 6,600 mt (distance measured from Earth's RF.)

dt = vdx/c^{2} is a mathematical identity without physical meaning, and can't be used for anything more than substitutions.

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This is how tricky relativity is applied to the muon's phenomena by "stablished physics".

The theory behind this can be consulted at HyperPhysics link, here:

Muon Experiment
http://hyperphysics....ase/hframe.html