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Giammarchi, M.; Ricci, F. Black Holes and Quantum Mechanics. Encyclopedia. Available online: https://encyclopedia.pub/entry/32307 (accessed on 23 June 2024).

Giammarchi M, Ricci F. Black Holes and Quantum Mechanics. Encyclopedia. Available at: https://encyclopedia.pub/entry/32307. Accessed June 23, 2024.

Giammarchi, Marco, Fulvio Ricci. "Black Holes and Quantum Mechanics" *Encyclopedia*, https://encyclopedia.pub/entry/32307 (accessed June 23, 2024).

Giammarchi, M., & Ricci, F. (2022, November 01). Black Holes and Quantum Mechanics. In *Encyclopedia*. https://encyclopedia.pub/entry/32307

Giammarchi, Marco and Fulvio Ricci. "Black Holes and Quantum Mechanics." *Encyclopedia*. Web. 01 November, 2022.

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Mass and spin distributions of stellar mass black holes (BH) are important sources of information on the formation mechanism and the evolution of galaxies. The birth of a stellar-mass BH, ranging in the interval ~5–150 M_{⊙}, is due to the spectacular phase of a massive star’s core collapse, an event involving the emission of multi-messenger signals such as neutrinos, GW’s and electromagnetic radiation in several bands.

black holes
gravitational waves
event horizon

General relativity (GR) is the modern theory of gravitation, a geometric theory proposed by A. Einstein in 1915 ^{[1]}, based on the equivalence principle that has been, as of time at writing, verified at the level of 10^{−15} ^{[2]}. Among the most important predictions of the theory was the existence of gravitational waves (GW) ^{[3]} and of black holes (BH) ^{[4]}. Black holes are vacuum solutions of general relativity (GR) that, at equilibrium, are fully characterized by their mass M, charge Q and spin J (No hair theorem). When a BH results from the collapse of an isolated massive star with zero charge and zero angular momentum, the resulting BH is the simplest classical body, defined only by its mass M. It is a pure static geometrical object and the spacetime around it has spherical symmetry and is asymptotically flat: it is the GR solution of the gravitational field generated by a static mass derived by Karl Schwarzschild in 1916 ^{[4]}. In addition, according to the Birkhoff theorem ^{[5]}^{[6]} this is the only possible solution for this physical scenario. The mass of the Schwarzschild BH defines the event horizon, the spherical surface of radius R_{S}
where G is the gravitational constant and c the speed of light. Following the classical vision of GR, the horizon allows the entrance but not the escape of matter and radiation from its interior and hides a singularity inside it: this surface is the barrier beyond which any information is lost. The hypothesis that BHs could exist lacked observational evidence for a long time, so that the possibility of their observation was considered unlikely. Theoretical indications in favor of their existence had become more robust already at the end of the 30s’ of the past siècle, but the physics community was still reluctant to accept this GR prediction. It had to wait at least thirty more years for the prevailing atmosphere of skepticism to be reversed. At the end of the 60s’ J. A. Wheeler actually invented the term “black hole” for the physical state of this GR solution. Then, the first very strong black hole candidate was discovered: Cygnus X-1, located within the Milky Way in the constellation of Cygnus, the Swan. The astronomers detected X rays emitted from a bright blue star orbiting a strange dark object. In 1971 it was suggested that the detected X-rays were a result of stellar material being stripped away from the bright star and “swallowed” by a dark spot, the black hole. While indirect, this evidence was considered rather convincing ^{[7]}.

R_{S} = 2 G M/c^{2 }(1)

Other evidence of the existence of BHs followed: in the case of potential supermassive black holes (SMBH) located at the center of some nearby galaxies, evidence was obtained by carefully tracing the motion of stars near the BHs or thanks to the very energetic emission of accretion disks. The disks consist of gas molecules swirling around the BH so fast as to emit electromagnetic radiation that is detected on Earth.

In parallel, theoretical studies progressed and, in 1963, the GR solution for the rotating BH, the Kerr solution, was found ^{[8]}. Rotating black holes (actually, the vast majority if not the totality) were called Kerr black holes or (if charged) Kerr–Newman. They feature an exterior region, outside the event horizon, in which any reference frame rotates alongside it at the speed of light, the so-called frame-dragging effect. This region, named the ergosphere has, as an inner border the event horizon, while the outer is an approximate spheroid having an oblateness proportional to the BH angular momentum.

Mass and spin distributions of stellar mass BHs are important sources of information on the formation mechanism and the evolution of galaxies. The birth of a stellar-mass BH, ranging in the interval ~5–150 M_{⊙}, is due to the spectacular phase of a massive star’s core collapse, an event involving the emission of multi-messenger signals such as neutrinos, GW’s and electromagnetic radiation in several bands.

For main sequence stars, the collapse proceeds in a rather complicated way: once the nuclear pressure is unable to sustain the overall stellar equilibrium, the core becomes unstable and gravitationally collapses inward upon itself. With the onset of contraction, increasing density and electron chemical potential, electron captures by nuclei speed up and accelerates the implosion. Then, the collapse stops abruptly when nuclear densities (>2.7 × 10^{14} g cm^{−3}) are reached; the overshooting inner core rebounds and the following shock wave leads to the disruption of the star in the supernova explosion. However, the process seems to be much more complex: simulations indicate that the shock wave generated after the bounce can stall because of the opacity of matter surrounding the core. In order to produce the supernova explosion, a mechanism is therefore needed to revive the shock. The most efficient way for this to happen seems to be neutrino re-heating, which should determine, some hundreds of milliseconds later, the mantle ejection. The neutrino heating is a consequence of the contraction of the collapsing core of the star associated with the compactification of its surroundings during the post-bounce accretion phase. This contraction leads to the increase of the neutrino temperatures and therefore the increase of the average energy of the radiated neutrinos.

This hypothetic scenario is still uncertain; a significant effort both in three-dimensional simulations and theoretical modeling is needed to have a complete understanding of the collapse, a process giving rise to neutron stars, pulsars, magnetars and stellar-mass black holes, such as those detected by LIGO and Virgo.

Core-collapse supernova is not the only mechanism that can end the life of a massive star. When the helium core of a star grows to ≥60 M_{⊙} and the central temperature reaches ∼10^{9} K, electron and positron pairs are produced at an efficient rate. The star then undergoes electron–positron pair instability, where oxygen, neon, and silicon are burned explosively and the entire star is disrupted. This leaves no remnant, unless the star’s helium core is ≥130 M_{⊙}.

The combination of all the predictions concerning the core-collapse mechanisms come together to define the BH mass spectrum. In particular, the electron–positron pair instability should determine a gap in the mass interval between ~50(−10, +20) M_{⊙} and 100–130 M_{⊙}. The uncertainty around this mass gap is mainly connected with poorly known nuclear reaction rates in the collapse of the residual hydrogen envelope and with the role of stellar rotation.

The gravitational wave signals detected by LIGO and Virgo are generated by binary systems with short orbital separation so that the GW emission can guarantee the coalescence in a time lower than Hubble time (the initial separation of the two masses should be less than a few tens of solar radii). Thus, here is the challenge of clarifying the formation mechanism of a binary black hole (BBH) system. In the present literature two main scenarios are confronted: (a) the isolated binary evolution scenario, (b) the formation of the binary BH starting in a star cluster, the densest place of the universe in terms of stars (>10^{3} stars pc^{−}^{3}), where the star orbits are constantly perturbed by the dynamic interaction with other objects in the cluster.

A binary black hole system (BBH) formation in isolation (first scenario) is complicated by several different processes occurring during the evolution of the system, primarily the mass transfer and the role of the gas surrounding both binary stars, usually called in astronomy the common envelope. This gigantic cloud can be formed when one of the stars expands rapidly and does not rotate generally with the binary system, leading to a drag effect which tends to reduce the orbital separation between a BH and the massive companion star. Then, if the common envelope is ejected and the core of the companion star collapses in a BH without receiving a strong kick, then a BBH is formed. For massive BHs the most likely spin configuration of the BBH system features spins well aligned to the orbital angular momentum and nearly zero orbital eccentricity.

The dynamical formation of BBHs in dense stellar environments (second scenario) is based on the hypothesis that an original binary system interacts with a third body, which replaces one of the binary components. The process tends to reduce the major axis of the orbit by absorbing momentum thereby assisting the formation of the couple. The relevant dynamical exchange involved in this process is compatible with a random distribution of the spin directions.

In 1974, S. Hawking and J. Bekenstein theorized that black holes were more than simple geometrical objects. Indeed, in pure general relativity, black holes do not emit any radiation, so they should be regarded as bodies at absolute zero temperature. If quantum mechanics (QM) is included in the BH description, the compact object then acquires some temperature. Hawking evaluated the particle creation effects for a body that collapses into a black hole and discovered that a distant observer would see a thermal distribution of particles emitted at finite temperature (Hawking radiation ^{[9]}). From this perspective, BHs are macroscopic thermodynamic objects with an entropy S given by the Bekenstein–Hawking Formula ^{[10]}^{[11]}^{[12]}:
where k_{B} is the Boltzmann constant. S is proportional to the area A of the BH horizon and depends on χ = [c J/(G M^{2})], the dimensionless spin parameter of a rotating black hole of mass M. In Equation (2) the quantities l_{P} = (h G/2 π c^{3})^{1/2} and m_{P} = (h c/2 π G)^{1/2} are the Planck length and mass, respectively. According to this formula, a black hole has a huge entropy, much larger than that of a star of the same mass, which seems reasonable since a BH is the final possible stage of gravitational evolution.

S (M, χ) = k_{B} A/(4 l_{P}^{2}) = 2 π k_{B} (M/m_{P})^{2} [1+(1 − χ)^{1/2}] (2)

Entropy and internal energy of the BH concur to define its temperature (often known as Hawking temperature) which are recalled here in the case of a Schwarzschild BH:
leading to the conclusion that BHs are probably the coldest objects in the whole universe (and of course especially the supermassive ones).

T = (h c^{3})/(16 π^{3} k_{B} G M) (3)

Thus, the search within BH physics is interlaced with one of the most intriguing problems of modern physics: how to harmonize general relativity and quantum mechanics. Following the Boltzmann approach the laws of thermodynamics should emerge as a macroscopic description of an ensemble of many microscopic states corresponding to the different possible ways of forming the same macroscopic situation. Enumerating these microstates leads to the entropy S as indeed—using statistical mechanics—the laws of thermodynamics could be drived from the kinetic theory of gases. Similarly, the laws of black hole thermodynamics are properties of gravity: BH entropy and temperature, while intrinsically quantum in nature, must be related to macroscopic quantities such as horizon area and surface gravity (as provided by GR). Therefore, it should be possible to derive black hole thermodynamics starting from a fundamental theory of quantum gravity and taking some appropriate average limit.

For a given quantum system described through its density matrix ρ the fine-grain Von-Neumann entropy:
is the variable quantifying author's ignorance about the precise quantum state of the system (if it is equal to zero, as in the case of a pure state, it certifies author's full knowledge of the quantum state). The classical concept of entropy is more related to the semi-classical notion of coarse-grained entropy, i.e., for a given density matrix that was measured just few observables of the system, which in ordinary thermodynamics, for example, can be energy and volume. Numerous different theoretical approaches have been proposed. However, up to now, it could be concluded that Equation (2) is robust when it is challenged following the statistical approach, i.e., to derive the entropy starting from the computation of the microstates of the microcanonical ensemble behaving as the BH: N ~ e ^{S (M, χ)}.

S = Tr|ρ log ρ| (4)

Nowadays, after the first GW detection of 2015 and the more recent Event Horizon Telescope (EHT) observations, a new phase is opening where the theoretical effort to challenge the GR picture of a black hole can be supported by experimental observations. In the following sections, the first attempts would be shortly reviewed to use the GW signals emitted by a collapsing black hole to provide tests of the microcanonical ensemble of the Bekenstein–Hawking entropy (the coarse-grained entropy). More generally, the study of GW signals and of the event horizon of supermassive black holes will provide decisive physical insight at the edge between quantum mechanics and general relativity.

- Einstein, A. Die Feldgleichungen der Gravitation; Sitzungsbericht; Königlich Preussische Akademie der Wissenschaften: Berlin, Germay, 1915; pp. 844–847.
- Touboul, P.; Métris, G.; Rodrigues, M.; André, Y.; Baghi, Q.; Bergé, J.; Boulanger, D.; Bremer, S.; Chhun, R.; Christophe, B.; et al. The MICROSCOPE mission: First results of a space test of the equivalence principle. Phys. Rev. Lett. 2017, 119, 231101.
- Einstein, A. Gravitationswellen (On Gravitational Waves); Erster Halbband; Königlich Preussische Akademie der Wissenschaften: Berlin, Germany, 1918; pp. 154–167.
- Schwarzschild, K. Über das Gravitationsfeld einer Kugel aus inkompressibler Flüssigkeit; Reimer, Berlin 1916, S. 424-434 (Sitzungsberichte der Königlich-Preussischen Akademie der Wissenschaften; 1916)—On the Gravitational Field of a Sphere of Incompressible Liquid, According to Einstein’s Theory. Abraham Zelmanov J. 2008, 1, 20–32.
- Israel, W. Event Horizons in Static Vacuum Space-Times. Phys. Rev. 1967, 164, 1776.
- Jebsen, J.T. Uber Die Allgemeinen Kugelsymmetrschen Losungen Der Einstei’Schen GavitationsgleiChungen Im Vakuum. Ark. Mat. Astron. Fys. 1921, 15, 1.
- Shipman, H.L. The implausible history of triple star models for Cygnus X-1 Evidence for a black hole. Astrophys. Lett. 1975, 16, 9.
- Kerr, R.P. Gravitational Field of a Spinning Mass as an Example of Algebraically Special Metrics. Phys. Rev. Lett. 1963, 11, 237.
- Visser, M. Hawking Radiation without Black Hole Entropy. Phys. Rev. Lett. 1998, 80, 3436.
- Hawking, S.W. Black holes and thermodynamics. Phys. Rev. 1976, D13, 191.
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