Undergrad Shwartzschild coordinate system

[omidj]

TL;DR

Transformation equations from Schwarzschild coordinate and metric to Minkowski space time

Is there any direct transformation equations from schwarzschild coordinate components of time and distance to a flat Minkowsky space time?
These basis vectors seem to be capable of producing the Shwarzschild metric:
et=√(1-rs/r)cosh(ct) eT +√(1-rs/r)sinh(ct) eR
er=(1/√(1-rs/r))sinh(ct) eT +(1/√(1-rs/r))cosh(ct) eR
(et&er for Schwarzschild and eT&eR for Minkowsky) But when Jacobian matrix is derived, the problem emerges ...

 

[Dale]

They are different manifolds so there is not a coordinate transform between them. Of course locally you can always find coordinates that approximate Minkowski to first order.

 

[Nugatory]

Is there any direct transformation equations from schwarzschild coordinate components of time and distance to a flat Minkowsky space time?

It cannot be done, for about the same reason that we cannot find a transformation between latitude and longitude on the curved surface of the earth and cartesian x-y coordinates on a euclidean plane.

We can of course ignore the curvature completely and use ordinary Minkowski polar coordinates as an approximation valid over sufficiently small regions of spacetime, analogous to the way that we treat the surface of the earth as flat across a sufficiently small area.

 

[JimWhoKnew]

Is there any direct transformation equations from schwarzschild coordinate components of time and distance to a flat Minkowsky space time?

Please use LaTeX.

As it was already said, such a transformation is impossible. Each spacetime comes with its inherent quantities that are invariant under coordinate transformations. Geodesics in Minkowski spacetime (straight lines) can intersect each other once at most. In Schwarzschild spacetime, they may do so many times (for example: circular orbits in opposite directions). No coordinate transformation can change that.
The question of equivalence up to a coordinate transformation can be a complex challenge in the general case. Sometime it is sufficient to consider a few curvature scalars. Scalar fields are invariant under coordinate transformations, like the Ricci scalar $R=R^\mu{}_\mu$ . For the example given by @Nugatory in post #3, the 2D Euclidean plane has $R=0$ , while the unit sphere has $R=2$ , which proves the claim that they are fundamentally distinct.

Edit: corrected error

 

[PeterDonis]

the Ricci scalar

Note that in Schwarzschild spacetime, which is a vacuum solution, this scalar is zero. The simplest nonzero curvature scalar in Schwarzschild spacetime is the Kretschmann scalar:

https://en.wikipedia.org/wiki/Kretschmann_scalar

 

[JimWhoKnew]

Note that in Schwarzschild spacetime, which is a vacuum solution, this scalar is zero. The simplest nonzero curvature scalar in Schwarzschild spacetime is the Kretschmann scalar:

https://en.wikipedia.org/wiki/Kretschmann_scalar

Yes. I should have mentioned it in #4.