# Significance of Kretschmann scalar to flat spaces?

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?

• Not necessarily. Take any subset of Minkowski space or quotients of Minkowski space as counterexamples. – Slereah Feb 21 '18 at 9:59
• 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. – Rumplestillskin Feb 21 '18 at 22:14

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.