Modern Physics/Principle of Relativity Applied

Special Relativity
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An application of relativity to waves

Returning to the phase of a wave, we immediately see that

$\phi =\mathbf {k} \cdot \mathbf {x} -\omega t=\mathbf {k} \cdot \mathbf {x} -(\omega /c)(ct)={\underline {k}}\cdot {\underline {x}}.$

Thus, a compact way to write a wave is $A({\underline {x}})=\sin({\underline {k}}\cdot {\underline {x}}).$(6.8)

Since ${\underline {x}}$ is known to be a four-vector and since the phase of a wave is known to be a scalar independent of reference frame, it follows that ${\underline {k}}$ is indeed a four-vector rather than just a set of numbers. Thus, the square of the length of the wave four-vector must also be a scalar independent of reference frame

${\underline {k}}\cdot {\underline {k}}=\mathbf {k} \cdot \mathbf {k} -\omega ^{2}/c^{2}=const.$

Resolution of a four-vector into components in two different reference frames.

Let us review precisely what this means. As this figure shows, we can resolve a position four-vector x into components in two different reference frames, e.g (X,T) and (X′,T′), but these are just different ways of writing the same vector.

This is exactly the same as the way a three-vector has different components in a rotated frame.

Similarly, just as a three-vector has the same magnititude in all frames, so does a 4-vector; i.e,

$X^{2}-c^{2}T^{2}=X^{\prime 2}-c^{2}T^{\prime 2}$

Applying this to the wave four-vector, we infer that

$k^{2}-c^{-2}\omega ^{2}=k^{\prime 2}-c^{-2}\omega ^{\prime 2}$

where the unprimed and primed values of k and ω refer to the components of the wave four-vector in two different reference frames.

Up to now, this argument applies to any wave. However, waves can be divided into two categories, those for which a special reference frame exists, and those for which there is no such special frame.

As an example of the former, sound waves look simplest in the reference frame in which the gas carrying the sound is stationary. The same is true of light propagating through a material medium with an index of refraction not equal to unity. In both cases the speed of the wave is the same in all directions only in the frame in which the material medium is stationary.

If there is no material medium, then there is no unambiguous way of finding a special frame so the waves must fall into the second category. This includes all waves in a vacuum, such as light.

In this case the following argument can be made. An observer moving with respect to waves of frequency ω and wave number k sees waves of frequency ω′ and wave number k′. If the observer can tell in any way that they come from a source moving with respect to them, then they can use this to identify a special frame for those waves, so the waves must look just like ones from a stationary source of frequency ω

This forces us to conclude that for such waves

$\omega ^{2}=c^{2}k^{2}+\mu ^{2}\,$

where μ is a constant. All waves in a vacuum must have this form, a much more restricted choice than in classical physics.