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## Main Question or Discussion Point

Many of the formulas of physics assume a background of flat spacetime. But it's natural to think that this is just a special case of a more general formulation in curved spacetimes. So I'd like to develop a general procedure to transform flat space formulas and integrals, etc, into their curved space counterparts.

It seems that the Jacobian matrix can be used to transform volume elements of curvilinear coordinates to the equivalent volume element in flat space. This is done by transforming differential moves in the curved space to differential moves in flat space. The Jacobian matrix does not seem to care what kind of space (curved or flat) is being transformed into flat space. One just does the differentiation of the curvilinear coordinates with respect to the flat space coordinates. So I have to wonder if the Jacobian matrix also applies when transforming from spaces with curvature and torsion to flat spaces. If it does, then does that mean that every coordinate system whatsoever is locally flat? I assume every coordinate system has coordinate lines that are locally linear independent. And using the Gram-Schmidt procedure this can be made into an orthogonal coordinate system. So is every coordinate system locally flat?

But more to the point, I wonder how to transform from flat to curved spaces. I assume that the inverse Jacobian can be used to transform volume elements in flat space to volume elements in curved space, right? And does this mean that it is just a matter of transforming differential moves in flat space to differential moves in curved space?

Any insight that anyone can give would be appreciated. Thanks.

It seems that the Jacobian matrix can be used to transform volume elements of curvilinear coordinates to the equivalent volume element in flat space. This is done by transforming differential moves in the curved space to differential moves in flat space. The Jacobian matrix does not seem to care what kind of space (curved or flat) is being transformed into flat space. One just does the differentiation of the curvilinear coordinates with respect to the flat space coordinates. So I have to wonder if the Jacobian matrix also applies when transforming from spaces with curvature and torsion to flat spaces. If it does, then does that mean that every coordinate system whatsoever is locally flat? I assume every coordinate system has coordinate lines that are locally linear independent. And using the Gram-Schmidt procedure this can be made into an orthogonal coordinate system. So is every coordinate system locally flat?

But more to the point, I wonder how to transform from flat to curved spaces. I assume that the inverse Jacobian can be used to transform volume elements in flat space to volume elements in curved space, right? And does this mean that it is just a matter of transforming differential moves in flat space to differential moves in curved space?

Any insight that anyone can give would be appreciated. Thanks.