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Following up on the questions raise here and trying to get a little more clarity on what it means to be 'local' and 'flat'. In the first chapter of Gravitation, the author(s) state:

The geometry of spacetime is locally Lorentzian everywhere

I can visualize this. If I'm 8 light minutes from a large star, I can obviously see curvature in the path of a planet around that star, but if I take smaller and smaller measurements of shorter duration, so that for all intents and purposes I'm measuring a point on a geodesic around this star, then any experiment I do with test particles will meet the predictions of a Lorentzian geometry.

But I don't follow this logic if I do the experiment at the center of this mass. If I do my experiment at the center of this star, then no matter how small I make my laboratory, even making it small enough to be considered a point, I'm still going to see curvature all around me.

How is the geometry of spacetime is locally flat at the exact center of a exceedingly large mass?

Edit: In trying to understand this statement from Gravitation, I'm trying to understand how a local geometry might not be flat. An image of the geodesics around a singularity came to mind as a place on the manifold that wasn't differentiable. This is the context for the question.

Quark Soup
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The phrase "then no matter how small I make my laboratory, even making it small enough to be considered a point, I'm still going to see curvature all around me" is always true, no matter how far you are away from any star.

Spacetime is locally Lorentzian in a way similar to that a parabola can be locally approximated by a line. In no finite region will you find that the parabola is congruent to the line (unless it is a degenerate parabola of course). But you are guaranteed that you can find a line that concides with the parabola in one point and with the same slope.

Similarly you are guaranteed to find a Lorentzian metric and a connection that conincide in one point with the gravitational manifold.

oliver
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