Question about principle of equivalence Set up: Suppose we have a worldline and I have a system of coordinates that can label all points that belong to the worldline (in some macroscopic region of spacetime). The principle of equivalence says that at each point P there exists a change in coordinates that will make the metric tensor at P Lorentzian (this is how I learned it).
Let us call this “inertial” system of coordinates at P: $\xi_{\mu}=\Lambda_{\mu} ^ \nu X_{\nu}$ and those at another point Q as $\xi^{'} _{\mu}=\Lambda^{' \nu} _{\mu} X _{\nu}$ where the $\Lambda$’s are the change of coordinate matrix and $X_{\mu}$ are the original coordinates.
Question: The principle of equivalence guarantees the existence of $\xi_{\mu}$ and $\xi^{'} _{\mu}$ separately, but does it also guarantee that we can find the inertial coordinates to be
$\xi_{\mu}=\xi^{'} _{\mu}$?
If $\xi_{\mu}\neq \xi^{'} _{\mu}$, then wouldn't we see test particles accelerate even at a local level inside a free falling lab?
 A: 
The principle of equivalence says that at each point P there exists a change in coordinates that will make the metric tensor at P Lorentzian

It is actually a little stronger than that. The metric at P is Lorentzian and the metric in the neighborhood of P is Lorentzian to first order.

The principle of equivalence guarantees the existence of ξμ and ξ′μ separately, but does it also guarantee that we can find the inertial coordinates to be ξμ=ξ′μ?

What you are interested in is Fermi coordinates.
https://en.m.wikipedia.org/wiki/Fermi_coordinates
With Fermi coordinates if you start with the inertial coordinate basis vectors $\mathbf{e_t}$, $\mathbf{e_x}$, $\mathbf{e_y}$, and $\mathbf{e_z}$ where the $\mathbf{e_i}$ are orthonormal and $\mathbf{e_t}$ is tangent to the worldline at $P$, then you can parallel transport the $\mathbf{e_i}$ along the worldline to $Q$ and they will still be orthonormal.
Additionally, if the worldline is a geodesic then $\mathbf{e_t}$ will be tangent at $Q$ also because geodesics parallel transport their tangent vector. Such parallel transported basis vectors can define a locally (to first order) inertial frame at $Q$.
