Two papers, one by Sergio Doplicher, Klaus Fredenhagen and John Roberts[5] and the other by D. V. Ahluwalia,[6] set out another motivation for the possible noncommutativity of space-time. The arguments go as follows: According to general relativity, when the energy density grows sufficiently large, a black hole is formed. On the other hand, according to the Heisenberg uncertainty principle, a measurement of a space-time separation causes an uncertainty in momentum inversely proportional to the extent of the separation. Thus energy whose scale corresponds to the uncertainty in momentum is localized in the system within a region corresponding to the uncertainty in position. When the separation is small enough, the Schwarzschild radius of the system is reached and a black hole is formed, which prevents any information from escaping the system. Thus there is a lower bound for the measurement of length. A sufficient condition for preventing gravitational collapse can be expressed as an uncertainty relation for the coordinates. This relation can in turn be derived from a commutation relation for the coordinates. "
@Voidisyinyang Voidisyinyang do you have links for those papers you're talking about? I'd like to look over them more, when I have a little time for some brain stretching. 😉
@Mary Ann Bittle Thanks for the question. The idea of micro black holes from noncommutative spacetime is something I first encountered from Professor Shahn Majid - he mentions it in one of his youtube talks. He told me though that noncommutative spacetime has not been proven. I'm not sure that is true considering the "weak measurement" experiments of Yakir Aharonov's research group. The quotation I gave is from wiki and so the research papers can just be googled "Quantum Measurement, Gravitation, and Locality" by
D. V. Ahluwalia and The quantum structure of spacetime at the Planck scale and quantum fields
Sergio Doplicher, Klaus Fredenhagen, John E. Roberts - of course there should be follow up papers and discussions that we can obtain by entering those titles into googlescholar to get the citation links.
The proposal extends to substantially
relate NC to the essential quantum fluctuations of
gravitational field [35] by showing that classical gravity
is indeed unique “shadow” in the commutative limit
of the noncommutative “Fuzzy Spacetime” [36–40].
Almost at the same time, Virtual Black Holes (VBH)
were introduced to appear/disappear due to quantum
fluctuations of spacetime too [41–43] as consequence of
relating uncertainty principle to Einstein equations of
gravity such that VBH would gravitationally resemble
particle-antiparticle pairing in vacuum state of QFT
[44]. In light of this proposal, a VBH is to carry a
mass ∼MP and to share features of Wheeler’s quantum
foam [45, 46].
Hence from the previous discussion, we would expect
that the spacetime at the micro-scale to consist of a sea
of VBH [41, 79]. However, in all the models studying
VBH, the noncommutative structure of spacetime has
not been taken into an account, a priori, although the
motivation is the same for both phenomena. VBH could
have a measurable effect in particle physics, permitting
events/decays that are forbidden within the realm of the
standard model. The most important decay that could
be caused by VBH is the proton decay [80]. Noncom-
mutative spacetime models also predict phenomenolog-
ical aspects on particle physics, but they seem to be
rather ill-defined or even unjustified.
The LHC will not generate black holes in the cosmological sense.
However, some theories suggest that the formation of tiny 'quantum'
black holes may be possible. The observation of such an event would be
thrilling in terms of our understanding of the Universe; and would be
perfectly safe.
Although we are interested in a
matrix version of surfaces primarily as a model of an eventual noncommutative
theory of gravity they have a certain interest in other, closely related, domain
of physics. We have seen, for example, that without the differential calculus
the fuzzy sphere is basically just an approximation to a classical spin r by a
quantum spin r with It in lieu of A. It has been extended in various directions
under various names and for various reasons [17, 58, 105, 22]. In order to
explain the finite entropy of a black hole it has been conjectured, for example
by 't Hooft [207], that the horizon has a structure of a fuzzy 2-sphere since the
latter has a finite number of 'points' and yet has an S03-invariant geometry.
The horizon of a black hole might be a unique situation in which one can
actually 'see' the cellular structure of space.
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