### From quantum black hole "atoms" of gravitation to quanta of spectral noncommutative spacetime / Des trous noirs quantiques, "atomes" de la gravitation, aux quanta d'espacetemps spectral non commutatif

**Celebrating 40 years of quantum black hole physics... / Célébrons le quarantième anniversaire de la physique quantique des trous noirs...**

Expressions are derived for the mass of a stationary axisymmetric solution of the Einstein equations containing a black hole surrounded by matterand for the difference in mass between two neighboring such solutions. Two of the quantities which appear in these expressions, namelythe area A of the event horizon and the "surface gravity"; of the black hole, have a close analogy with entropy and temperature respectively. This analogy suggests the formulation of four laws of black hole mechanics which corre- spond to and in some ways transcend the four laws of thermodynamics.

J. M. Bardeen, B. Carter and S. W. Hawking24 Jan, 1973

Using well understood arguments from quantum theory we discuss the nature of the quantum spectra of the mass, charge, and angular momentum of the Kerr black hole. We argue that the mass spectrum is discrete, and infer a for the allowed mass levels by firstpointing out the analogy between the squared irreducible mass of the Kerr hole and the action integral of mechanics, then quantizing the squared irreducible mass by the Bohr-Sommerfeld quantization rule... The result is consistent with the correspondence principle...

Jacob D. Bekenstein1974

QUANTUM gravitational effects are usually ignored in calculations of the formation and evolution of black holes. The justification for this is that ... the energy density of particles created by the gravitational field is small compared to the space-time curvature.Even though quantum effects may be small locally, they may still, however, add up to produce a significant effect over the lifetime of the Universe ≈ 10. As a black hole emits this thermal radiation one would expect it to lose mass. This in turn would increase the surface gravity and so increase the rate of emission. The black hole would therefore have a finite life of the order of 10^{17}s which is very long compared to the Planck time ≈ 10^{−43}s. The purpose of this letter is to show that this indeed may be the case: it seems that any black hole will create and emit particles such as neutrinos or photons at just the rate that one would expect if the black hole was a body with a temperature of (κ/2π) (ħ/2k) ≈ 10^{−6}(M/M)K where κ is the surface gravity of the black hole^{71}(M/M)^{−3}s. For a black hole of solar mass this is much longer than the age of the Universe. There might, however, be much smaller black holes which were formed by fluctuations in the early Universe. Any such black hole of mass less than 10^{15}g would have evaporated by now. Near the end of its life the rate of emission would be very high and about 10^{30}erg would be released in the last 0.1s. This is a fairly small explosion by astronomical standards but it is equivalent to about 1 million 1 Mton hydrogen bombs

S. W. Hawking,01 March 1974

**... the spectral noncommutative way! / ... à la mode spectrale non commutative !**

In the construction of spectral manifolds in noncommutative geometry, a higher degree Heisenberg commutation relation involving the Dirac operator and the Feynman slash of real scalar ﬁelds naturally appears and implies, by equality with the index formula,the quantization of the volume. We ﬁrst show thatthis condition implies that the manifold decomposes into disconnected spheres which will represent quanta of geometry.We then reﬁne the conditionby involving the real structure and two types of geometric quanta, and show that connected manifolds with large quantized volume are then obtained as solutions. When this condition is adopted in the gravitational actionit leads to the quantization of the four volume with the cosmological constant obtained as an integration constant. Restricting the condition to a three dimensional hypersurface implies quantization of the three volume and the possible appearance of mimetic dark matter.When restricting to a two dimensional hypersurface,under appropriate boundary conditions,this results in the quantization of area and has many interesting applications to black hole physics.

Ali H. Chamseddine, Alain Connes, and Viatcheslav Mukhanov,Mon, 8 Sep. 2014

//retouches éditoriales le 13/09/14

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