I am solving (numerically) a wave equation for a general relativity problem given by:
$$ \frac{\partial^2 u}{\partial t^2} - c(t)^2 \frac{\partial^2 u}{\partial x^2} = 0.$$
I have already written a solution using RK4 for the Fourier transformed solution, $ u(x,t) = \sum_k u_k(t) e^{-ikx} $, which gives (for each $k$):
$$ \frac{\partial^2 u}{\partial t^2} + c(t)^2 k^2 u = 0. $$
Since the Fourier transform ignores $t$, I don't see any problem with this. However all of the sources that I look at switch to conformal time $t \rightarrow \eta$, which makes the equation more or less time independent:
$$ c(\eta)^2 \left( \frac{\partial^2 \tilde{u}}{\partial t^2} - \frac{\partial^2 \tilde{u}}{\partial x^2} \right) = 0, $$
before performing a Fourier transform on the equation.
Is this just convention? I haven't studied enough Fourier analysis to see why this might be a problem, and a cursory glance at the first equation listed tells me that the equation is separable to begin with. However, this might just be naive of me.