# Transformations in curved spacetime?

#### wpan

I know that the spacetime in special relativity is not curved and that the axis can be transformed via the lorentz transformations.

I was wondering if the curved spacetime in general relativity can be transformed in such a way, and if so, how?

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#### dauto

Yes of course. The Lorentz transformations are the set of transformations that preserve the space-time interval $$ds^2 = dt^2 - dx^2 - dy^2 - dz^2$$. A similar concept of space-time interval also applies to GR.

#### Naty1

Depends on just what you mean, but generally, 'no'.

Wikipedia puts it this way: [Minkowski space is the the flat space-time of SR]
http://en.wikipedia.org/wiki/Lorentz_transformation

In Minkowski space, the Lorentz transformations preserve the spacetime interval between any two events. They describe only the transformations in which the spacetime event at the origin is left fixed, so they can be considered as a hyperbolic rotation of Minkowski space....

In the flat spacetime of SR the space-time separation between two events or two observers is the integral of ds along a straight line from one event to the other. [There is only one such straight line.] The analogous measure in curved spacetime would be the integral of ds along a geodesic [free fall path], generally a curved worldine. But in general in curved spacetime there are multiple geodesics connecting the events so you won't see much talk of “space-time separation” in GR because there are many such paths.

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