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I've seen the phrase "as measured in a local Lorentz frame" tagged on the end of so many sentences.

What does it mean precisely?

To give an explanation with an example, consider the context of measuring the speed of an object, if this applies.

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You consider processes where you have no geodesic deviation over the whole area, in the appropriate time frame.

Notice that in every point $p$ of your pseudo-Riemannian manifold, you can find a coordinate system such that $g(x)=\eta+\mathcal{O}(x^2)$. Mathematically, in any neightborhood $U_p$ around $p$, you can't turn off the graitational effects completely, but if you speak of a "local inertial frame", or "Lorentz frame", then they are supposed to be negligible for the other physics you're interested in.

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I think it's safe to say that this phrase means the same thing as "local inertial frame." Assuming that this is actually is a safe assumption, I'll describe, physically, what a local inertial frame is. I can give a lot of mathematical detail in an addendum if you wish.

In special relativity, we assume that spacetime is flat, and in this case, any observer moving with a constant three-velocity below $c$ is called an inertial observer. The laws of physics look the same to all such observers.

In general relativity, the spacetime is not necessarily flat, but in small regions of spacetime, the spacetime looks approximately flat, and freely falling massive observers, which move along timelike geodesics, when making measurements in small regions of spacetime around them, will see physics that is approximately the same as that seen by an inertial observer in flat spacetime as in special relativity.

One then says that such observers are making observations from reference frames that are locally inertial.

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