5.1: Waves in Spacetime

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

$$A(x, t)=A_{0} \sin (k x-\omega t)\label{5.1}$$

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

$$A(\mathbf{x}, t)=A_{0} \sin (\mathbf{k} \cdot \mathbf{x}-\omega t)\label{5.2}$$

where $k$ is now the position vector and $k$ is the wave vector. The magnitude of the wave vector, $|\mathbf{k}|=\mathrm{k}$ 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 $\PageIndex{1}$:: Sketch of wave fronts for a wave in spacetime. The large arrow is the associated wave four-vector, which has slope $\omega / \mathrm{ck}$. The slope of the wave fronts is the inverse, $\mathrm{ck} / \omega \text { . }$. The phase speed of the wave is greater than c in this example. (Can you tell why?)

In the case of a one-dimensional wave moving to the right $\phi=k x-\omega t$. 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:

$$c t=\frac{c k x}{\omega}-\frac{c \phi}{\omega}=\frac{c x}{u_{p}}-\frac{c \phi}{\omega} \quad(\text { wave front })\label{5.3}$$

The slope of the world line in a spacetime diagram is the coefficient of x, or $c / \mathrm{u}_{\mathrm{p}}, \text { where } \mathrm{u}_{\mathrm{p}}=\omega / \mathrm{k}$ is the phase speed. The world lines of the wave fronts of a wave are illustrated in Figure $\PageIndex{1}$:.