5.1: Waves in Spacetime
( \newcommand{\kernel}{\mathrm{null}\,}\)
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)=A0sin(kx−ωt)
where A0 is the (constant) amplitude of the wave, k is the wavenumber, and ω is the angular frequency, and that the quantity ϕ=kx−ωt is called the phase of the wave. For a plane wave in three space dimensions, the wave is represented in a similar way,
A(x,t)=A0sin(k⋅x−ωt)
where k is now the position vector and k is the wave vector. The magnitude of the wave vector, |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 ϕ=k⋅x−ωt.

In the case of a one-dimensional wave moving to the right ϕ=kx−ωt. A wave front has constant phase ϕ, 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=ckxω−cϕω=cxup−cϕω( wave front )
The slope of the world line in a spacetime diagram is the coefficient of x, or c/up, where up=ω/k is the phase speed. The world lines of the wave fronts of a wave are illustrated in Figure 5.1.1:.