Local Geometry of General Relativity Theory

[mikeeey]

Hello guys .
Through all the analysis of theory of general relativity we used what so called Manifolds
Manifolds as we know are topological spaces that resemble ( look like) euclidean space locally at tiny portion of space
And an euclidean space is the pair ( real coordinate space R^n , dot product ),
And any euclidean space is flat space,
So manifolds locally are flat , do not have curvature locally
But solutions of GR's equations show the manifolds are not flat locally even they are locally look like euclidean space .
My question is that if the space is curved , Then the Curvature tensor does not depend on the chosen local real coordinate space ( system ) , is it ?!

Thanks .

 

[Nugatory]

But solutions of GR's equations show the manifolds are not flat locally even they are locally look like euclidean space.

Minkowski, not Euclidean - you can find coordinates in which the metric tensor is diagonal with components arbitrarily close to (-1,1,1,1) but not (1,1,1,1) which would be Euclidean.

The curvature tensor does not depend on the chosen local real coordinate space (system), does it?

All tensors are coordinate-independent objects - their value does not depend on the coordinate system. The values of the components of a tensor do change with the coordinate system, but that's just a result of using different coordinate systems to represent the same object - a physics problem may look very different (and be much easier or harder to solve) when it's written in one coordinate system instead of another, but it's still the same problem.

 

[mikeeey]

My question , let's enter to 2-manifolds( surfaces ) if the surface of the shpere(e.g.) is locally look like euclidean space ( made up from gluing small planes together )that mean its curvature locally flat ! , but its not in fact

 

[Nugatory]

My question , let's enter to 2-manifolds( surfaces ) if the surface of the shpere(e.g.) is locally look like euclidean space ( made up from gluing small planes together )that mean its curvature locally flat ! , but its not in fact

That is correct. Somewhere in whatever text you're using you'll find a proper definition of what "locally flat" means. It will be something along the lines of: the difference between the metric tensor and the flat-space metric tensor can be made arbitrarily small by considering a small enough region.

 

[mikeeey]

Last question , how does the curvature tensor of the surface of the sphere is not zero and still locally flat ?! Thanks

 

[Nugatory]

Last question , how does the curvature tensor of the surface of the sphere is not zero and still locally flat ?

The curvature tensor is not zero, but a zero curvature tensor is not a requirement for local flatness. Local flatness means that a sufficiently small region can be approximated as flat, and the smaller you make the region the better the approximation is. Whatever text you're using should have a proper formal definition - keep looking until you find it and understand it.

 

[mikeeey]

Thank you