QuarkyMeson said:

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

robphy said:

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?

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