Square root of Kretschmann scalar: any known uses?

[Roberto Pavani]

TL;DR

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

In Schwarzschild vacuum, the Kretschmann scalar is

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

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.

 

[robphy]

From a search
https://www.google.com/search?q="square+root+of+the+Kretschmann"
https://scholar.google.com/scholar?&q="square+root+of+the+Kretschmann"
point to an application as a measure of "effective gravitational energy density", among others...

 

[QuarkyMeson]

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.

 

[Markus Hanke]

Has K (as opposed to K itself) appeared in any published work as a physically meaningful quantity?

Can’t comment on whether or not it appears in the literature, but just looking at its form, it’s basically a type of ‘norm’ of the Riemann tensor, so it measures the overall strength of tidal gravity. As such, otherwise different spacetimes can have the same square-rooted Kretschmann scalar, so I suspect there is no more specific physical meaning behind it.
Just my own thoughts.

 

[robphy]

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.

While dimensional analysis is merely a guide,
it obviously has to be followed up with seeing if
it (obtained from a scalar formed from the riemann tensor) helps characterize quantities of physical interest.

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

 

[Matterwave]

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

It seems this paper is exactly what @Roberto Pavani asked for -- a publish paper which uses the square root of the full contraction of the Riemann tensor:

...one can employ the square root of the full contraction of the Riemann tensor (the Kretschmann scalar) $\mathcal{K}\equiv\sqrt{\mathcal{R}^{\mu\nu\alpha\beta}\mathcal{R}_{\mu\nu\alpha\beta}}=4\sqrt{3}GM_*/c^2r^3$, as a measure of the curvature in vacuum.

Indeed, it seems they use the square root to define the Kretschmann scalar in the first place.

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

 

[Roberto Pavani]

Thank you all for the references.

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.

 

[robphy]

postscript (other curvature invariants that could be given a similar treatment):

• 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