Time Dilation: Relative to Medium or Vacuum?

[jk22]

If a medium with speed of light $c_1$ is considered, shall the Lorentz transformation be considered relative to it or to speed of light in the vacuum ?

I don't know if we could send particles like muons through water for example, to check this with their life time.

 

[phinds]

If a medium with speed of light #c_1# is considered

There is no such thing

Time dilation is always relative to the observer; it has nothing to do with the speed of light and there's no medium that has anything to do with it, unless I am misunderstanding your question

 

[Ibix]

Are you asking if the $c$ in the Lorentz transform is ever anything other than the speed of light in vacuum? If so, no. It's always $c$.

 

[Dale]

If a medium with speed of light $c_1$ is considered, shall the Lorentz transformation be considered relative to it or to speed of light in the vacuum ?

I don't know if we could send particles like muons through water for example, to check this with their life time.

The $c$ in the Lorentz transform is always the speed of light in vacuum, not in a medium. It is perfectly possible for an object to have a velocity $c_1<v<c$. This causes Cherenkov radiation, but does not violate physics in any way.

 

[jartsa]

I don't know if we could send particles like muons through water for example, to check this with their life time.

What particle let's water flow through itself?

Or, what does it matter if there is a water flow outside of the particle?

A submarine fleet would experience some effects, like slowed down communication by light signals.

 

[PeterDonis]

I don't know if we could send particles like muons through water for example, to check this with their life time.

The muons created by cosmic rays in the Earth's upper atmosphere, and detected at the Earth's surface, travel through air, not vacuum; the speed of light in air is slower than that in vacuum (not by as much as in water, true, but measurably slower), but their lifetimes are still the ones calculated by using the standard Lorentz transformation, using $c$ in vacuum.

 

[Ibix]

I don't know if we could send particles like muons through water for example, to check this with their life time.

In this vein, Fizeau's experiments with measuring the speed of light included measuring its speed through flowing water. The results are consistent with the Lorentz transforms, and hence time dilation.

 

[vanhees71]

I'm a bit puzzled by the question in #1. You must carefully specify which time dilation you want to calculate, or more clearly formulated, which physics case you want to discuss. Is it about the Fizeau experiment?

 

[jk22]

It was meant measuring the lifetime of muons traveling through water which has a refractive index to deduce the value of the time dilation. But I don't know if muons are absorbed by water.

 

[Ibix]

It was meant measuring the lifetime of muons traveling through water which has a refractive index to deduce the value of the time dilation. But I don't know if muons are absorbed by water.

The constant $c$ in the Lorentz transforms is a scale factor between natural units of time and distance. It's more or less irrelevant what speed light travels at, except that the fact it travels at $c$ makes a lot of thought experiments easier.

We have plenty of evidence that it is always $c$, not $c/n$, that is the invariant speed. Fizeau's experiments and the existence of Cerenkov radiation have already been pointed out.

 

[Dale]

It was meant measuring the lifetime of muons traveling through water which has a refractive index to deduce the value of the time dilation. But I don't know if muons are absorbed by water.

This has been done with high energy neutrons from nuclear reactions. They go through water at $c_1 < v < c$. If the time dilation were based on $c_1$ then they would have infinite energy and the Cherenkov radiation would vaporize everything on Earth instead of just emitting a pretty blue glow.

 

[vanhees71]

It was meant measuring the lifetime of muons traveling through water which has a refractive index to deduce the value of the time dilation. But I don't know if muons are absorbed by water.

What should the lifetime of muons have to do with the phase velocity of em. waves in water?