Locally Inertial Coordinates: Spacetime Points & Freely Falling Particles

[Ratzinger]

Locally inertial coordintates in GR refer to spacetime, not only to space, right?
Are they only defined for events, for spacetime points, so that when a particle is freely falling it passes through infinitely many such locally lorentz frames?
But if so, how can we say that the particle runs straight lines in each locally inertial frame?

?
thank you

 

[dicerandom]

Locally Lorentzian coordinates refer to both space and time, it basically means that if you consider a small enough patch of spacetime then you should be able to ignore the effects of the curvature and the metric tends to $\eta_{\mu \nu}$, i.e. flat Minkowsky space. It's an analagous situation to a tangent plane on a curved surface.

That does not, however, mean that the effects are gone; it simply means that for a short enough time and a short enough distance you can ignore them and only be off from the true results by some small ammount.