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I am having trouble understanding parametrizing a path in proper time. So my understanding is that using proper time to parametrize a path corresponds to the rest frame of the particle whereby it only has a time coordinate of $c \tau$ and spatial components of zero. Will there ever exist a case where a particles motion is in proper time and it has non-zero spatial components?

The reason why I ask is because of the geodesic equation. If one uses a proper time parametrization, we have:

$$ \frac{\mathrm d^2x^{\mu}}{\mathrm d\tau^2} + \Gamma^{\mu}_{\alpha\beta}\frac{\mathrm dx^{\alpha}}{\mathrm d\tau}\frac{\mathrm dx^{\beta}}{\mathrm d\tau} = 0$$

But in proper time, the motion will always boil down to zero spatial components and a time component of $c\tau$. I just solved this equation for a particle in the presence of a gravitational wave. It was very complicated, but at the end of the day, the solutions became exactly what I described here (zero spatial components and $c\tau$ time component given initial conditions of zero velocity and initial position of 0 for all components. There was no effect of the perturbed metric on the solutions.

So then I thought, "OK, let me change coordinates to an observer that is located in a different spatial location," but I got confused on how to do that. Which led me to post a question about it the other day.

Am I misunderstanding proper time parametrization? A lot of concepts in this topic I seem to know just enough to be dangerous and would appreciate filling in the gaps.

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user41178 wrote: "So my understanding is that using proper time to parametrize a path corresponds to the rest frame of the particle whereby it only has a time coordinate of cτ and spatial components of zero."

The spatial components of the test particle's frame don't change in the test particle's frame, so $\rm d s/d \tau=1$ (or $0$ if it's a photon), therefore we can set that to $1$ or $0$ in the line element to solve for the other components.

The $\rm x^{\mu}$ are the temporal and spatial components of the selected coordinate bookkeeper, which may be a stationary, free falling or accelerated observer, depending on the coordinates. The spatial components of $\rm x^{\mu}$ are not generally $0$ since they are the coordinates of the test particle in the bookeeper's coordinates, not in his own frame.

Only $\tau$, which is not part of $\rm x^{\mu}$, is a coordinate of the test particle's frame, so the $\rm d x^{\mu}/d \tau$ is generally neither $0$ nor $1$. If the spatial components of $\rm d x^{\mu}/d \tau$ are $0$ you might have chosen comoving coordinates and a comoving testparticle, then constant spatial coordinates might go hand in hand with a time dependend metric coefficient and the distance would in fact change (like in the comoving FLRW coordinates, where in the spatial coordinates of the galaxies don't change despite their distance growing due to the expansion) or oscillate (in the case of gravitational waves).

But since you didn't really show what you did in your gravitational wave example it is impossible to tell, and I also wouldn't start with gravitational waves if I had trouble with the concept, but with something easier like Schwarzschild where the method is the same, but the equations are much shorter. The gravitational wave coordinates I know are comoving, and the changing distances between free floating beads are not reflected in changing spatial coordinates, but their time dependend metric coefficients (just like the comoving galaxies in comoving FLRW coordinates).

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