Modern Physics/Waves in Spacetime

Special Relativity
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Applications of Special Relativity

In this chapter we continue the study of special relativity by applying the ideas developed in the previous chapter to the study of waves.

First, we shall show how to describe waves in the context of spacetime. We then see how waves which have no preferred reference frame (such as that of a medium supporting them) are constrained by special relativity to have a dispersion relation of a particular form. This dispersion relation turns out to be that of the relativistic matter waves of quantum mechanics.

Second, we shall investigate the Doppler shift phenomenon, in which the frequency of a wave takes on different values in different coordinate systems.

Third, we shall show how to add velocities in a relativistically consistent manner. This will also prove useful when we come to discuss particle behaviour in special relativity.

A new mathematical idea will be presented in the context of relativistic waves, namely the spacetime vector or four-vector. Writing the laws of physics totally in terms of relativistic scalars and four-vectors ensures that they will be valid in all inertial reference frames.

Waves in Spacetime

Waves in Spacetime

We now look at the characteristics of waves in spacetime. Recall that a wave in one space dimension can be represented by

$A(x,t)=A_{0}\sin(kx-\omega t)\,$

where $A_{0}$ is the (constant) amplitude of the wave, $k$ is the wavenumber, and $\omega$ is the angular frequency, and that the quantity $\phi =kx-\omega t$ is called the phase of the wave. For a wave in three space dimensions, the wave is represented in a similar way,

$A(\mathbf {x} ,t)=A_{0}\sin(\mathbf {k} \cdot \mathbf {x} -\omega t)$

where $\mathbf {x}$ is now the position vector and $\mathbf {k}$ is the wave vector. The magnitude of the wave vector, $|\mathbf {k} |=k$ is just the wavenumber of the wave and the direction of this vector indicates the direction the wave is moving. The phase of the wave in this case is $\phi =\mathbf {k} \cdot \mathbf {x} -\omega t$.

Figure 5.1: Sketch of wave fronts for a wave in spacetime. The large arrow is the associated wave four-vector, which has slope $\omega /ck$. The slope of the wave fronts is the inverse, $ck/\omega$.

In the one-dimensional case $\phi =kx-\omega t$. A wave front has constant phase $\phi$, so solving this equation for $t$ and multiplying by $c$, the speed of light in a vacuum, gives us an equation for the world line of a wave front:

$ct={\frac {ckx}{\omega }}-{\frac {c\phi }{\omega }}={\frac {ck}{u_{p}}}-{\frac {c\phi }{\omega }}\quad {\mbox{(wave front).}}$

The slope of the world line in a spacetime diagram is the coefficient of $x$, or $c/u_{p}$, where $u_{p}=\omega /k$ is the phase speed.