# How is spacetime locally Lorentzian at the center of a large mass?

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.

• Commented Feb 11, 2021 at 18:18
• The phrase “center of curvature” does not have any meaning in GR. In general, spacetime is locally flat at every point and has curvature at every point. There is no contradiction because, as has been explained multiple times in response to your past questions, “locally flat” is very different from “flat”. Commented Feb 11, 2021 at 18:21
• I hope you can step back and appreciate how contradictory that statement appears to someone trying to understand this subject. Commented Feb 11, 2021 at 18:24
• Spacetime as a manifold is locally flat at the centre of a large mass because also there exists a tangent space with a Lorentzian metric. Commented Feb 11, 2021 at 18:24
• It should be intuitively obvious that the surface of the Earth is “locally flat” but not “flat”. Spacetime works in the same way, but in four dimensions instead of two. Commented Feb 11, 2021 at 18:31

• @GluonSoup: as to 3): take the simple hyperpola $f(x)=1/x$ as a basic metaphor: you can't approximate it by a line at $x_0=0$ because that would require $f(x_0)$ and $f^\prime(x_0)$ to be be defined. By contrast, $f(x)=x^2/x$ is also singular at $x_0=0$, but this singularity is removable (similar to a coordinate singularity) and then you can approximate it by $x$ at $x_0=0$ even if it is not defined there. Locality in $locally Lorentzian$ is defined by the metric and it's derivatives/connection. If they are not defined in the point of interest, you're in trouble with applying that definition. Commented Feb 14, 2021 at 16:52