Postulates of Special Relativity: Speed of Light in Inertial Frames

[Q1111]

TL;DR Summary

About the second postulate

Would the second postulate (The speed of light in free space has the same value c in all inertial reference frames.)be also true if it was in some medium instead of in free space? I know the value won't be c anymore but I want to know whether the speed of light in that medium would be the same in all inertial reference frames? Please tell how do we know.

 

[Dale]

No. The speed of light in a transparent medium is not invariant. There is only one invariant speed and that is the speed of light in vacuum.

 

[PeroK]

Summary:: About the second postulate

Would the second postulate (The speed of light in free space has the same value c in all inertial reference frames.)be also true if it was in some medium instead of in free space? I know the value won't be c anymore but I want to know whether the speed of light in that medium would be the same in all inertial reference frames? Please tell how do we know.

The velocity transformation law applies in all cases. If $u$ is the velocity of an object in one reference frame (e.g. the speed of light in a medium); and $v$ of relative velocity of the reference frame to another reference frame; then the velocity of the object in the second reference frame, $u'$ is given by:
$$u' = \frac{u + v}{ 1 + \frac{uv}{c^2}}$$ Some examples:

1) If $u = c$, i.e. the speed of light in vacuum in one reference frame, then: $$u' = \frac{c + v}{ 1 + \frac{cv}{c^2}}= c\frac{1 + \frac v c}{1 + \frac v c} = c$$ And we find that the invariance of the speed of light is maintained by our velocity addition formula. Note that was independent of $v$ the relative velocity between the reference frames.

2) If $u = 0.9c$ and $v = 0.9c$, then $$u' = \frac{1.8c}{ 1 + \frac{0.81c^2}{c^2}} = \frac{1.8c}{1.81} < c$$ and we see that the speed of the object is till less than $c$. I.e. this formula never results in a speed greater than $c$.

3) The second example applies to your case of light moving at less than $c$ in some medium - e.g. moving at $0.9c$ in a medium with a refractive index of $1.1$. If that medium is moving at velocity $v$ in some reference frame, then the velocity transformation rule applies.

 

[vanhees71]

Also see

https://en.wikipedia.org/wiki/Aether_drag_hypothesis
https://en.wikipedia.org/wiki/Fizeau_experiment

 

[Q1111]

No. The speed of light in a transparent medium is not invariant. There is only one invariant speed and that is the speed of light in vacuum.

Thank you.

 

[Q1111]

The velocity transformation law applies in all cases. If $u$ is the velocity of an object in one reference frame (e.g. the speed of light in a medium); and $v$ of relative velocity of the reference frame to another reference frame; then the velocity of the object in the second reference frame, $u'$ is given by:
$$u' = \frac{u + v}{ 1 + \frac{uv}{c^2}}$$ Some examples:

1) If $u = c$, i.e. the speed of light in vacuum in one reference frame, then: $$u' = \frac{c + v}{ 1 + \frac{cv}{c^2}}= c\frac{1 + \frac v c}{1 + \frac v c} = c$$ And we find that the invariance of the speed of light is maintained by our velocity addition formula. Note that was independent of $v$ the relative velocity between the reference frames.

2) If $u = 0.9c$ and $v = 0.9c$, then $$u' = \frac{1.8c}{ 1 + \frac{0.81c^2}{c^2}} = \frac{1.8c}{1.81} < c$$ and we see that the speed of the object is till less than $c$. I.e. this formula never results in a speed greater than $c$.

3) The second example applies to your case of light moving at less than $c$ in some medium - e.g. moving at $0.9c$ in a medium with a refractive index of $1.1$. If that medium is moving at velocity $v$ in some reference frame, then the velocity transformation rule applies.

Thanks for explanation. I understood.

 

[Q1111]

Also see

https://en.wikipedia.org/wiki/Aether_drag_hypothesis
https://en.wikipedia.org/wiki/Fizeau_experiment

OK thanks I will read them.

 

[Mister T]

An invariant speed must also be the maximum speed. It seems that it's just a coincidence that light travels at this speed. Perhaps there's a deeper underlying explanation as to why they're the same. On the other hand, it may be that they are not the same. Note that if they are not the same then no modifications of relativity would need to be made. The speed $c$ would simply be called the invariant speed instead of the speed of light.

 

[vanhees71]

At the present status of the most fundamental theory describing matter and radiation (the Standard Model of elementary particle physics) there's indeed no fundamental reason that the photon must be exactly massless. An Abelian gauge boson can be massive without using the Higgs mechanism and without violating gauge symmetry (in contradistinction to non-Abelian gauge bosons which can only get massive using the Higgs mechanism without violating gauge symmetry and then making the entire model inconsistent). In this sense the photon mass has to be determined empirically as all of the masses of the elementary particles in the Standard model. The accepted upper limit is $m_{\gamma}<10^{-18} \; \text{eV}$:

https://pdg.lbl.gov/2020/listings/rpp2020-list-photon.pdf

 

[Q1111]

An invariant speed must also be the maximum speed. It seems that it's just a coincidence that light travels at this speed. Perhaps there's a deeper underlying explanation as to why they're the same. On the other hand, it may be that they are not the same. Note that if they are not the same then no modifications of relativity would need to be made. The speed $c$ would simply be called the invariant speed instead of the speed of light.

Thanks for mentioning this.