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If you are given a spacetime embedded with a particular metric tensor that satisfies the vacuum field equations of general relativity, how do you confirm that you aren’t simply dealing with a Minkowski spacetime that has been changed via some exotic coordinate transformation?

Or, more specifically if I have an exact solution to the vacuum field equations that has a Kretschmann scalar value of $0$. Is this just the Minkowski spacetime in a different coordinate system?

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    $\begingroup$ Not necessarily. Take any subset of Minkowski space or quotients of Minkowski space as counterexamples. $\endgroup$ – Slereah Feb 21 '18 at 9:59
  • $\begingroup$ Thanks @Slereah. Can you think of an example in hard language. For example, the following line element satisfies the field equations $(ds)^2= ...$ and has a K scalar of $0$ but is not equivalent to Minkowski space. I am an Applied mathematician and have studied GR entirely via tensor calc and have avoided diff geom which can often be my downfall in these situations. $\endgroup$ – Rumplestillskin Feb 21 '18 at 22:14
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A zero Kretschmann scalar does not mean the Riemann tensor is zero. For example in the Kerr metric the sign of $K$ can change as we move towards the black hole, and that means it necessarily passes through zero. For details of this see this paper on the Arxiv.

There is some related discussion in Interpreting the Kretschmann scalar.

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  • $\begingroup$ In my particular case I have a zero Riemann tensor, zero Kretschmann scalar and also a metric tensor that identically satisfies the vacuum field equations. $\endgroup$ – Rumplestillskin Feb 21 '18 at 21:44
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    $\begingroup$ @Rumplestillskin if the Riemann tensor is zero the the spacetime is Minkowski $\endgroup$ – John Rennie Feb 22 '18 at 4:47
  • $\begingroup$ so would you describe the Riemann tensor, among other things, as a device to determine whether or not you may simply be describing Minkowski space that may just be expressed in exotic coordinates? $\endgroup$ – Rumplestillskin Feb 22 '18 at 5:03
  • $\begingroup$ @Rumplestillskin yes, calculating the Riemann tensor is the only sure guide. No matter what your coordinate system the Riemann tensor will always be zero for Minkowski spacetime. $\endgroup$ – John Rennie Feb 22 '18 at 5:24
  • $\begingroup$ Back to the drawing board it would appear $\endgroup$ – Rumplestillskin Feb 22 '18 at 5:32

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