# Locally flat - what does it mean?

What does it mean in general relativity that space is locally flat. It means that in neighbourhood of each point we can chose such coordinates that the metric is flat or we can do it only in a point (not in neighbourhood - open set).

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Ben Niehoff
Gold Member
It means that given any point, we may choose a set of coordinates in which the metric is flat up to first order (i.e., the metric is flat at the point, and its first derivatives all vanish). The second derivatives cannot be made to vanish in general (since the second derivatives are basically the same thing as curvature).

Stated another way, it means that given any positive epsilon, we may find a neighborhood around the point where the deviation from flatness is smaller than epsilon.

George Jones
Staff Emeritus
Gold Member
What does it mean in general relativity that space is locally flat. It means that in neighbourhood of each point we can chose such coordinates that the metric is flat or we can do it only in a point (not in neighbourhood - open set).
Unfortunately, the term "locally flat" is not used consistently, and you have identified the two main uses of the term. The first definition is used by mathematically careful references, but the second definition is used by many physicists.

An example ot the former, the book Modern Mathematical Physics: Groups, Hilbert Space and Differential Geometry by Peter Szekeres, writes:

"A manifold M with affine connection is said to be locally flat if for every point p in M there is a chart (U; x^i) with such that all the components of the connection vanish throughout U. This implies of course both torsion tensor and curvature tensor vanish throughout U, ..."

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George Jones
Staff Emeritus
Gold Member
An example ot the former, the book Modern Mathematical Physics: Groups, Hilbert Space and Differential Geometry by Peter Szekeres, writes:

"A manifold M with affine connection is said to be locally flat if for every point p in M there is a chart (U; x^i) with such that all the components of the connection vanish throughout U. This implies of course both torsion tensor and curvature tensor vanish throughout U, ..."
Other mathematically careful reference that have similar definition for local flatness include: Analysis, Manifolds and Physics; Tensor Analysis on Manifolds.

Ben Niehoff
Gold Member
"A manifold M with affine connection is said to be locally flat if for every point p in M there is a chart (U; x^i) with such that all the components of the connection vanish throughout U. This implies of course both torsion tensor and curvature tensor vanish throughout U, ..."
How in the world does this even make sense? Wouldn't such a manifold automatically be globally flat (though with possibly non-trivial topology)?

George Jones
Staff Emeritus
Gold Member
How in the world does this even make sense? Wouldn't such a manifold automatically be globally flat (though with possibly non-trivial topology)?
And if the manifold has non-trivial topology, then there isn't necessarily a global coordinate system (in which the components of the connection are zero), hence the concept is formulated in terms of coordinate neighbourhoods in manner in which "local" is used for other mathematical concepts, for example, "locally compact".

Now, let me ask you a question. What restriction does your definition place on (semi)Riemannian manifolds? None; every (semi)Riemannian manifold satisfies your definition. Each definition is, in some sense, trivial.

What does it mean in general relativity that space is locally flat. It means that in neighbourhood of each point we can chose such coordinates that the metric is flat or we can do it only in a point (not in neighbourhood - open set).
A quote from me explains what LF means in general:

Completely true! You don't need to even think of "local flatness" in terms of Christoffel symbols when in GR. One just needs to set the first derivatives of metric tensor equal to zero which fits within the definition of "geodesic coordinates" which is not actually an actual coordinate system (read the point 1 below)!! But remember that there is a strongly local coordinates in the sense that just at a point you can make the values of metric equal those of Minkowski metric. Here two points must be made:

1- In general, we can hardly determine a "coordinate transformation" $$x^{\mu}\rightarrow {\bar{x}}^{\mu}$$ by which at a point P, one has $$g^{\mu\nu}(P)={\eta}^{\mu\nu}(P)$$. If the metric is diagonal, the number of equations of its transformation would be much less than the case when metric is considered to be symmetric. This is because, for instance, if we count the number of equations involved in a symmetric metric trans., that is,

$$n(n+1)/2$$,

then you must fit at least this number of arbitrarily-chosen contants within the coordinates transformation at any given point to form a set of equations with the same number of unknowns and equations. But in the diagonal case, this number reduces to n, so the system of equations gets much simpler to be solved.

2- Geodesic coordinates just account for the first derivatives of metric being all equal to zero, so this way of leading to the local flatness at some point is another alternative.

3- Following 1, in the neighborhood of P the spacetime is "nearly" flat within a range that the equivalence principle issues. This is because we don't know what is meant by "neighborhood" in GR and this is only evaluated\estimated by EP. This might not be applicable for the case 2.

AB

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