According to wiki under http://en.wikipedia.org/wiki/Kretschmann_scalar" [Broken]-(adsbygoogle = window.adsbygoogle || []).push({});

While Riemann curvature tensor is proportional to tidal forces [itex](\Delta g=2Gm/r^3)[/itex], in some models of rotating (and charged) black holes, K is considered to diverge at the Cauchy horizon while tidal forces remain finite- ..The Kretschmann invariant is

[tex]K=R_{abcd}R^{abcd}[/tex]

where [itex]R_{abcd}[/itex] is the Riemann curvature tensor. Because it is a sum of squares of tensor components, this is a quadratic invariant..

source- http://relativity.livingreviews.org/open?pubNo=lrr-2002-1&page=node5.html" [Broken] ..the mass function (qualitatively [itex]R_{\alpha\beta\gamma\delta} \propto M/r^3[/itex]) diverges at the Cauchy horizon (mass inflation). However, Ori showed both for RN and Kerr that the metric perturbations are finite (even though [itex]R_{\mu\upsilon\rho\sigma}R^{\mu\upsilon\rho\sigma}[/itex] diverges) so that an observer would not be destroyed by tidal forces (the tidal distortion would be finite) and could survive passage through the CH..

For static black holes-

[tex]\tag{1}C=\frac{48 M^2}{r^6}[/tex]

which remains finite at 2M and can be loosely translated for rotating black holes to-

[tex]\tag{2}C(r,a)=\frac{12 (M+\sqrt{M^2-a^2})^2}{r^6}[/tex]

another source has-

[tex]\tag{3}R_{abcd}R^{abcd}=\frac{48M^2(r^2-a^2cos^2\theta)[(r^2+a^2cos^2\theta)^2-16r^2a^2cos^2\theta]}{(r^2+a^2cos^2\theta)^6}[/tex]

yet neither seem relative to the Cauchy horizon (i.e. none seem to diverge at the CH while tidal forces remain finite). Is there an equation for K that takes into the account that the Cauchy horizon is a null singularity and that M/r^3 remains finite even though [itex]R_{abcd}R^{abcd}[/itex] diverges? Also, what are the units for K? are they simply geometric or can they be multiplied by anything (such as c^2 or G/c^2) and recognisable as SI units?

Steve

(1)http://members.tripod.com/~albert51/cool.html

(2)http://members.tripod.com/~albert51/bhole.htm

(3)'The Kerr spacetime: A brief introduction' http://arxiv.org/abs/0706.0622v3 page 7

________________________________

UPDATE-

I did find this paper which shows a curvature scalar equation by A. Ori relative to rotating black holes-

http://arxiv.org/PS_cache/gr-qc/pdf/0304/0304052v2.pdf page 7

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# Curvature invariants

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