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If the curvature inside the horizon of a black-hole is not infinite in some quantum gravity theories (as in Loop quantum gravity), then what is the expression of the maximum value of the curvature that is reached at the former "singularity"?

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    $\begingroup$ Could you add a reference/expand in which quantum gravity theories it is not infinite? $\endgroup$ – ACuriousMind Jul 28 '14 at 17:58
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    $\begingroup$ Could you expand this question a little bit? It reads to me like "If a is true, then is it possible that a is valid?" $\endgroup$ – Jerry Schirmer Jul 28 '14 at 18:10
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There is no (widely accepted) theory that describes the structure of spacetime down to the quantum scale. You mention loop quantum gravity, but as far as I know the removal of singularities has been addressed only in the simplified form of loop quantum cosmology.

However as far back as the 60s there have been suggestions that quantum effects would cause the Schwarzschild geometry to become de Sitter near the singularity of a black hole, and this would mean there was a maximum radius of curvature. As it happens the recent question How does the friedmon solution to Einstein's equations resolve paradox of bounded infinities? covered some of this ground. You might also want to look at the paper Implementing Markov's Limiting Curvature Hypothesis for some more background information.

These ideas are not based on any fundamental theory of quantum gravity, because no such theory exists, but rather they argue from general principles. So you won't be surprised to learn that there is no precise prediction for the maximum curvature, but rather that it is expected to be around the Planck length. More precisely the principal curvature in any dimension cannot exceed the inverse of the Planck length.

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