Kretschmann Scalar

**Cusp** · Jun11-09, 07:46 PM

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** · Jun11-09, 08:47 PM

Originally Posted by 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

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
http://www.physicsforums.com/showthr...59#post1351759

1. in
http://www.physicsforums.com/showthr...07#post1124707

and the last paragraph of (the first post)
http://www.physicsforums.com/showthr...76#post1176876

**George Jones** · Jun12-09, 09:01 AM

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** · Jun12-09, 05:32 PM

Thanks George - will check it out.