TL;DR

Any known references where ##\sqrt{K}## rather than ##K## is used as a geometric quantity?

##K = R_{abcd}R^{abcd} = \frac{48M^2}{r^6}##

I notice that ##\sqrt{K}## has dimensions of ##[\text{length}]^{-2}## and a clean ##1/r^3## profile.

Has ##\sqrt{K}## (as opposed to ##K## itself) appeared in any published work as a physically meaningful quantity? I'm aware of ##K## being used for singularity detection and invariant classification, but
I'm curious whether the square root has ever been studied separately.

Thanks for any references.

Having the dimensions of an energy density does not make something an energy density, any more than having dimensions of acceleration makes something an observable acceleration. Soooo, it all feels a bit like numerology to me and less like physics.

I found this: https://link.springer.com/article/10.1140/epjc/s10052-024-13204-8

Having the dimensions of an energy density does not make something an energy density, any more than having dimensions of acceleration makes something an observable acceleration. Soooo, it all feels a bit like numerology to me and less like physics.

I'm not an astrophysicist, but this seems to do that:

What does a measurement of mass and/or radius of a neutron star constrain: Equation of state or gravity?
Kazim Yavuz Ekşi, Can Güngör, Murat Metehan Türkoğlu
Phys. Rev. D 89, 063003 – Published 6 March, 2014
https://doi.org/10.1103/PhysRevD.89.063003
https://arxiv.org/abs/1402.0488

I'm not an astrophysicist, but this seems to do that:

What does a measurement of mass and/or radius of a neutron star constrain: Equation of state or gravity?
Kazim Yavuz Ekşi, Can Güngör, Murat Metehan Türkoğlu
Phys. Rev. D 89, 063003 – Published 6 March, 2014
https://doi.org/10.1103/PhysRevD.89.063003
https://arxiv.org/abs/1402.0488

I'm not sure why it matters whether we use the full contraction itself or its square root though?

In fact, my original curiosity came from the observation that, in Schwarzschild vacuum,

## \mathcal{K} \equiv \sqrt{R^{\mu\nu\alpha\beta}R_{\mu\nu\alpha\beta}}
=
\frac{4\sqrt{3}\,GM}{c^2 r^3}. ##

I was therefore interested in whether this quantity had ever been considered independently of the Kretschmann scalar itself.

The references posted above are very helpful because they show that the square root of the Kretschmann scalar has indeed been used as a physically meaningful measure of spacetime curvature, gravitational field strength, or effective gravitational energy density.

That is exactly the kind of prior literature I was hoping to find, so thank you.

• https://en.wikipedia.org/wiki/Kretschmann_scalar
• Kretschmann Scalar for a Kerr-Newman Black Hole
  Richard C. Henry , 2000 ApJ 535 350 ( https://iopscience.iop.org/article/10.1086/308819, https://arxiv.org/abs/astro-ph/9912320v1 )
• (Mathematica) Determining curvature invariants for the Kerr-Newman black hole https://github.com/Nico2046/Kerr-newman-black-hole
• Second Order Scalar Invariants of the Riemann Tensor: Applications to Black Hole Spacetimes
  C. Cherubini, D. Bini, S. Capozziello, R. Ruffini, Int.J.Mod.Phys.D11:827-841,2002 ( https://doi.org/10.1142/S0218271802002037 , https://arxiv.org/abs/gr-qc/0302095 )
• https://en.wikipedia.org/wiki/Carminati–McLenaghan_invariants
• Algebraic invariants of the Riemann tensor in a four‐dimensional Lorentzian space J. Carminati; R. G. McLenaghan
  J. Math. Phys. 32, 3135–3140 (1991)
  https://doi.org/10.1063/1.529470
• (old thread) https://www.physicsforums.com/threads/fun-with-carminati-mclenaghan-invariants.844556/
• A Complete Set of Riemann Invariants
  E. Zakhary & C. B. G. Mcintosh General Relativity and Gravitation, Volume 29, pages 539–581 (1997) https://link.springer.com/article/10.1023/A:1018851201784
• (Maple) https://www.maplesoft.com/support/h...DifferentialGeometry/Tensor/RiemannInvariants