Suppose I have a space ship that can travel at $0.9c$, and I'm going to a star located at 20 light years or so from the Sun.
From a practical point of view, if I keep pointing the nose of my space ship towards the destination star, am I following the shortest distance path to that star?
The reasoning follows this vague intuition: light itself is telling us the shortest path through spacetime (is that true?), so instead of solving a geodesic based on the shape of the gravitational field (probably nasty because the stars in between move (but maybe not by much for only 20 years)), the path of the light ray is the solution you are looking for? - but there might be a hitch because $0.9c \neq c$ (i.e. all of a sudden, the best path for the light is no longer the best path for the spaceship)? note: this "vague intuition" is begging to be deconstructed and falsified.
[Update: I did some numerical simulations, and it looks like under some circumstances (I was varying the speed of the star, of the spaceship..., it is not possible to not even converge, i.e. I see the spaceship get into a neat trajectory that "follows" the star, without ever getting closer! - of course, it's possible there are bugs in my simulation, but it does raise the question of whether the method even "converges", i.e. reaches the star...]