A The uncertainty principle in quantum gravity

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The main role in quantum gravity can be played by the uncertainty principle
{\displaystyle \Delta r_{s}\Delta r\geq \ell _{P}^{2}}
, where
r_{s}
is the gravitational radius,
r
is the radial coordinate,
\ell _{P}
is the Planck length. This uncertainty principle is another form of Heisenberg's uncertainty principle between momentum and coordinate as applied to the Planck scale. Indeed, this ratio can be written as follows:
{\displaystyle \Delta (2Gm/c^{2})\Delta r\geq G\hbar /c^{3}}
, where
G
is the gravitational constant,
m
is body mass,
c
is the speed of light,
\hbar
is the reduced Planck constant. Reducing identical constants from two sides, we get the Heisenberg's uncertainty principle
{\displaystyle \Delta (mc)\Delta r\geq \hbar /2}
. Uncertainty principle
{\displaystyle \Delta r_{s}\Delta r\geq \ell _{P}^{2}}
predicts the appearance of virtual black holes and wormholes (quantum foam) on the Planck scale.
Is such a form of Heisenberg's uncertainty principle possible?
 
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This uncertainty principle is another form of Heisenberg's uncertainty principle between momentum and coordinate as applied to the Planck scale.
I don't see how since ##r_s## is not a momentum.

It's a common speculation that the Planck length will play some fundamental role in quantum gravity theory, but right now that's all it is, a speculation.
 

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