Kretschmann Scalar: Flat Spacetime & Singularities

[Cusp]

The Kretschmann scalar (the full contraction of the Reimann tensor K = R_abcd R^abcd) is often used to identify singularities - i.e. for a Schwarzschild black hole, K \propto 1/r^6, so we have a singularity at r=0, but not at the Schwarzschild horizon).

Clearly, as r->\infinity, K->0. Is K=0 a measure of flat spacetime in general? Is there a reference that shows this?

Cheers

 

[George Jones]

The Kretschmann scalar (the full contraction of the Reimann tensor K = R_abcd R^abcd) is often used to identify singularities - i.e. for a Schwarzschild black hole, K \propto 1/r^6, so we have a singularity at r=0, but not at the Schwarzschild horizon).

Clearly, as r->\infinity, K->0. Is K=0 a measure of flat spacetime in general? Is there a reference that shows this?

Cheers

The person best able to answer this has stopped posting, but, from previous posts of his, the answer is "No." There are pp-waves that have curvature singularities, and that have K = 0. I suspect that this somewhere in Exact Solutions of Einstein's Field Equations by Hans Stephani, Dietrich Kramer, Malcolm MacCallum, and Cornelius Hoenselaers.

See 4. in
https://www.physicsforums.com/showthread.php?p=1351759#post1351759

1. in
https://www.physicsforums.com/showthread.php?p=1124707#post1124707

and the last paragraph of (the first post)
https://www.physicsforums.com/showthread.php?p=1176876#post1176876

 

[George Jones]

According to Hawking and Ellis, page 260, it was pointed out by Penrose that curvature can be non-zero even when stuff like K is zero.

 

[Cusp]

Thanks George - will check it out.