If we describe spacetime with a Lorentzian manifold, it is always possible to choose a coordinate system such that at any particular point $x^\alpha$, the components of the metric are: $$ g_{\mu\nu}(x^\alpha) = \left( \begin{array}{cccc} 1 & 0 & 0 & 0 \\ 0 &-1 & 0 & 0 \\ 0 & 0 &-1 & 0 \\ 0 & 0 & 0 &-1 \end{array} \right) $$
But our freedom is much greater than this. In curved spacetime, the equivalence principle suggests we can choose coordinates such that the metric is of that form for every point along a chosen time-like geodesic. And in flat spacetime, we can see an explicit example that it doesn't even need to be a geodesic, for the Rindler metric has that form on every point of a particular worldline with constant proper acceleration. I have a feeling this is possible for any time-like worldine.
So my question is:
Given a coordinate system and metric for a Lorentzian manifold and a time-like worldline on this spacetime, is it always possible to find a coordinate transformation such that for every point on the world line the components of the metric in this coordinate system are just (1,-1,-1,-1) on the diagonal?
I realize that even for simplified cases (say a geodesic on the Schwarzschild background), such a coordinate system could be incredibly complicated. So if someone creates an incredibly complicated explicit construction, please also show that the solution to the equations you setup exist with some kind of existence proof.
This originally started from trying to find such a local coordinate chart for a free falling observer towards a blackhole, but realized I didn't know a good mathematical way to represent such coordinate freedom to get me started. Eventually I ended up pondering this current question. So even if you can't give a full answer, but can suggest some mathematical tools or where to read up on them, any help would be appreciated.