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

$$A(x,t) = A_0 \sin ( kx - \omega t ) ,$$ (6.1)

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,

$$A( \mbox{\bf x} , t) = A_0 \sin ( \mbox{\bf k} \cdot \mbox{\bf x} - \omega t ) ,$$ (6.2)

where $\mbox{\bf x}$ is now the position vector and $\mbox{\bf k}$ is the wave vector. The magnitude of the wave vector, $\vert \mbox{\bf k} \vert = k$ 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 $\phi = \mbox{\bf k} \cdot \mbox{\bf x} - \omega t$.

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, $ck/ \omega$. The phase speed of the wave is greater than $c$ in this example. (Can you tell why?)

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:

$$ct = \frac{ckx}{\omega} - \frac{c \phi}{\omega} = \frac{cx}{u_p} - \frac{c \phi}{\omega} ~~~ \mbox{(wave front)} .$$ (6.3)

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.