Topology-induced quantum transition in multiparticle systems in vicinity of a black hole

Let us consider a folded space-time described by Schwarzschild metric [23] being the solution of Einstein general relativity equations for a spherical non-rotating and uncharged body with the mass M and radius R, for r > R,

$$\begin{align} \begin{aligned} -c^2d\tau^2 = -\left(1-\frac{r_s}{r}\right) c^2dt^2+\left(1-\frac{r_s}{r}\right)^{-1}dr^2 +r^2\left(d\theta^2+\sin^2\theta d\phi^2\right),\\ \end{aligned} \end{align} \tag{ 1 }$$

the coordinates $t,r,\phi,\theta$ are time-space rigid coordinates (in spherical variables) the same as for a remote observer, whereas τ denotes the proper time, $r_s = \frac{2GM}{c^2}$ is the Schwarzschild radius, which defines the event horizon (G is the gravitational constant, c is the light velocity in the vacuum). The metric (1) in the limit $R\rightarrow 0$ can be addressed to a black hole with gravitational singularity at r = 0, which is surrounded by the event horizon sphere with the radius r_s , from which neither matter nor light can escape. Besides the singularity in r = 0, another one occurs in (1) on the event horizon at $r = r_{s}$. This singularity is apparent (called as coordinate singularity) and disappears if one changes to other coordinates [11, 24, 25]. A choice of a distinct coordinate system displays a different slicing of the same folded space-time onto its spatial and temporal parts, which can be done in a variety manners allowing various representations of particle motion without changing, however, the trajectory homotopy governed by invariant space-time curvature. In particular, in the metric (1) a remote observer can notice matter falling onto the event horizon of the black hole infinitely long, never crossing the event horizon. Changing, however, to the proper time instead of t, the matter smoothly passes the event horizon within a finite period of the proper time and next after of also finite period of the proper time any motion terminates in the central singularity [11, 24]. The metric (1) expressed in stationary (time independent spatial coordinates) is convenient to analyze the topology (homotopy) of trajectories of particles when they approach the event horizon. Impossibility to cross the event horizon in finite time t follows from the simple property of general relativity singularity, that it does not exist a rigid metric describing simultaneously inner and outer region with respect to the event horizon [26], though it can be done in other coordinates, e.g. in metrics by Novikov, Peinlevé–Gullstrand, Eddington–Finkelstein or Kruskal–Szekeres [11, 24–28].

The homotopy of trajectories of noninteracting identical indistinguishable particles (with mass m vanishing small in comparison to central mass M) in the upper neighborhood of the Schwarzschild event horizon is displayed by geodesics in the metric (1). These geodesics must lie in planes (because of spherical symmetry) and it is enough to consider the motion in a plane $\theta = \frac{\pi}{2}$. The Hamilton–Jacobi equation for geodesics,

$$\begin{align} g^{ik}\frac{\partial S}{\partial x^i}\frac{\partial S}{\partial x^k}-m^2c^2 = 0, \end{align} \tag{ 2 }$$

attains for Schwarzschild black hole the following form (with g^ik metric tensor components corresponding to metric (1) [26])

$$\begin{align} \begin{aligned} & \left(1-\frac{r_s}{r}\right)^{-1}\left( \frac{\partial S}{c\partial t}\right)^2 -\left(1-\frac{r_s}{r}\right)\left(\frac{\partial S}{\partial r}\right)^2 -\frac{1}{r^2}\left(\frac{\partial S}{\partial \phi}\right)^2-m^2c^2 = 0,\\ \end{aligned} \end{align} \tag{ 3 }$$

with the function S in the form,

$$\begin{align} S = -{\cal{E}}_0t+{\cal{L}}\phi+S_r\left(r\right), \end{align} \tag{ 4 }$$

where ${\cal{E}}_0$ and ${\cal{L}}$ are the particle energy and its angular momentum, respectively (${\cal{E}}_0$ and ${\cal{L}}$ are constants of motion). Substituting equation (4) into equation (3) one can find,

$$\begin{align} \begin{aligned} S_r & = \int dr\left[ \frac{{\cal{E}}_0^2}{c^2}\left(1-\frac{r_s}{r}\right)^{-2} -\left(m^2c^2 +\frac{{\cal{L}}^2}{r^2}\right)\left(1-\frac{r_s}{r}\right)^{-1}\right]^{1/2}. \end{aligned} \end{align} \tag{ 5 }$$

The radial dependence for a trajectory $r = r(t)$ is given by the condition $\frac{\partial S}{\partial{\cal{E}}_0} = \textrm{const}.$, whereas the angular dependence $\phi = \phi(t)$ by the condition $\frac{\partial S}{\partial{\cal{L}}} = \textrm{const}.$, which have an explicit form, for radius,

$$\begin{align} ct = \frac{{\cal{E}}_0}{mc^2}\int \frac{dr}{\left(1-\frac{r_s}{r}\right)\sqrt{\left(\frac{{\cal{E}}_0}{mc^2}\right)^2-\left(1+\frac{{\cal{L}}^2}{m^2c^2r^2}\right)\left(1-\frac{r_s}{r}\right)}}, \end{align} \tag{ 6 }$$

and for azimuth angle,

$$\begin{align} \phi = \int dr\frac{{\cal{L}}}{r^2}\left[\frac{{\cal{E}}_0^2}{c^2}-\left(m^2c^2+\frac{{\cal{L}}^2}{r^2}\right) \left(1-\frac{r_s}{r}\right)\right]^{-1/2}, \end{align} \tag{ 7 }$$

respectively

If one rewrites equation (6) in a differential form,

$$\begin{align} \frac{1}{1-r_s/r}\frac{dr}{cdt} = \frac{1}{{\cal{E}}_0}\left[{\cal{E}}_0^2-U^2\left(r\right)\right]^{1/2}, \end{align} \tag{ 8 }$$

the effective potential can be introduced, dependent on angular momentum ${\cal{L}}$ of the particle.

$$\begin{align} U\left(r\right) = mc^2\left[\left(1-\frac{r_s}{r}\right)\left(1+\frac{{\cal{L}}^2}{m^2c^2r^2}\right)\right]^{1/2}. \end{align} \tag{ 9 }$$

An accessible region for the motion is given by ${\cal{E}}_0\unicode{x2A7E} U(r)$. Additionally, the condition ${\cal{E}}_0 = U(r)$ defines circular orbits. The standard analysis [26] of extreme properties of U(r) allows determination of stable circular orbits (corresponding to minima of U(r)) and unstable ones (related to maxima of U(r)). The conditions $U(r) = {\cal{E}}_0$ and $\frac{\partial U(r)}{\partial r} = 0$ (for extreme) have the explicit form,

$$\begin{align} \begin{aligned} {\cal{E}}_0 &= {\cal{L}}c\sqrt\frac{2}{rr_s}\left(1-\frac{r_s}{r}\right),\\ \frac{r}{r_s} &= \frac{{\cal{L}}^2}{m^2c^2r_s^2}\left[1\pm \sqrt{1-\frac{3m^2c^2r_s^2}{{\cal{L}}^2}}\right], \end{aligned} \end{align} \tag{ 10 }$$

with + and − in the second equation corresponding to stable and unstable orbits, respectively. The solution of equation (10) can be illustrated in figure 1, where the circular orbit radius r versus angular momentum ${\cal{L}}$ of the particle is plotted. The upper curve (red one in this figure) gives positions of stable circular orbits and the lower curve (blue one) gives positions of unstable circular orbits. The stable circular orbits terminate at $r = 3r_s$ (point P in figure 1 corresponding to particle angular momentum ${\cal{L}} = \sqrt{3}mcr_s$ and particle energy ${\cal{E}}_0 = \sqrt{\frac{8}{9}}mc^2$). Below the innermost stable circular orbit there are possible unstable ones, but they also terminate at $r = 1.5r_s$ (for infinite ${\cal{L}}$). Below the innermost unstable circular orbit $r = 1.5r_s$ particle trajectories defined by equations (6) and (7) for $r\in(r_s,1.5r_s)$ have the form of unidirectional short spirals exemplified in figure 2. This is different in comparison to Newton gravitation center, for which conic section trajectories including circular orbits are available arbitrarily close to the center. The specific and transparently noticeable behavior of trajectories in metric (1) close to the event horizon of a black hole has pronounced consequences, because the homotopy class of classical trajectories of particles decides on quantum statistics in multiparticle systems.

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Quantum statistics of identical indistinguishable particles is not a general their property, but can change in response to topological properties of trajectories for particles in some local manifold on which all particles are placed (as commented in appendices A and B). This has been demonstrated (experimentally in agreement with the theory) for quantum Hall systems with dynamics of electrons confined to 2D local manifolds [29, 30]. In the case of a black hole vicinity, the qualitative change of particle trajectory homotopy also causes a local modification in quantum statistics in multiparticle systems of indistinguishable particles falling onto the event horizon.

The qualitative change of the trajectory homotopy takes place at the innermost unstable circular orbit for any massive particle at $r = 1.5r_s$. This orbit is also an unstable smaller orbit for photons [26] and the region ranged by the radius $r = 1.5r_s$ is called as photon sphere. Below the photon sphere upper rim no circular orbits for photons exist.

Any massive particle unavoidable spirals onto the event horizon if it passes the photon sphere rim in inward direction. This concerns a macroscopic body as well as particles of the microscopic its structure. In particular, if a multiparticle system passes the photon sphere rim inward all particles must fall toward the event horizon along one-way spirals. For free particles these spirals are given by equations (6) and (7) for $r\in(r_s,1.5r_s)$. Considering free particle systems is sufficient to define local quantum statistics in systems of indistinguishable identical particles. The photon sphere rim separates two regions with distinct types of available particle trajectories. Beyond this rim any direction of particle movement is admitted along conic section-like trajectories, whereas for particles passing inward the photon sphere rim, trajectories are only of spiral form irrevocably directed toward the event horizon regardless of initial conditions. This is a qualitative homotopy change of trajectories for free particles, which can be addressed to local multiparticle systems of identical indistinguishable particles considered to define quantum statistics.

Below the innermost unstable circular orbit (photon sphere rim in Schwarzschild geometry) no closed small local loops are admissible including also small loops built of pieces of one-way spiral-type particle trajectories. This is different in comparison to Newton gravitation center, for which all conic section trajectories are accessible for particles arbitrarily close to the center. Despite that for large distances from the gravitation center the Schwarzschild trajectories are similar to conic sections-type orbits (modified by some additional precession of elliptical orbits, like that observed for Mercury in Sun gravitation), conic section-like particle trajectories completely disappear below the photon sphere rim. Conic section-type trajectories can form closed arbitrarily small local loops from their pieces, what is easy to be noticed, as conic sections can cross in two arbitrarily close points (like circle with ellipse, hyperbole or parabola), as illustrated schematically in figure 2. For example, two inversely oriented trajectory pieces of conic sections can create together local arbitrarily small loops for mutual particle exchanges (particles can be located in crossing points). This is impossible below the innermost unstable circular orbit (the photon sphere rim), where unidirectional spirals cannot form local closed loops using even an arbitrary large number of small fragments of these spirals. It is impossible to built here local loops for particle exchanges—local loops built of pieces of spirals given by equations (6) and (7) and $r\in(r_s,1.5r_s)$ cannot be closed because of one-way their directions for particles passing the photon sphere inward. Hence, the innermost unstable circular orbit separates two space regions with different classes of homotopy of trajectories for massive particles in multiparticle systems passing these limiting sphere inward.

The existence of real classical trajectories for particle exchanges is essential for the definition of quantum statistics for these particles. When N identical indistinguishable particles are located on some spatial manifold ${\cal{M}}$, then to these particles can be assigned a quantum statistics in compliance with one dimensional unitary representation of the braid group [31, 32], which describes all possible exchanges of particles on ${\cal{M}}$. The braid group is the first homotopy group $\pi_1(F_N)$, where $F_N = ({\cal{M}}^N-\Delta)/S_N$ is the multiparticle configuration classical space of N identical indistinguishable particles with dynamics confined to the manifold ${\cal{M}}$. In this definition ${\cal{M}}^N = {\cal{M}}\times{\cal{M}}\times\dots\times{\cal{M}}$ is N-fold product of the manifold ${\cal{M}}$ to equally account for all particles. Δ is the diagonal subset of ${\cal{M}}^N$ (collecting points with coinciding coordinates of at least two particles) and subtracted from the coordination space to assure particle number conservation. Division by the permutation group S_N (the quotient structure of space F_N ) introduces indistinguishability of particles, i.e. all distributions of particles, which differ only in their numbering, are unified into a single point in F_N . Hence, braids from the first homotopy group $\pi_1(F_N)$, which are closed loops in F_N , display mutual exchanges of particles positions (braid loops link in F_N differently numbered particle distributions) and their scalar unitary representations defines quantum statistics of these N particles.

Quantum statistics is of fundamental significance and imposes severe restrictions on general quantum collective properties. In Schrödinger picture any multiparticle wave function must transform according to scalar unitary representation of a particular braid, which describes interchanges of arguments of this function (arguments are classical positions of particles on ${\cal{M}}$). The same concerns quantum fields of second quantization with restrictions imposed on their commutation algebra. For 3D manifolds (in unfolded space) the braid groups are always permutation groups S_N [31, 33], which have only two different scalar unitary representations, defined on generators of S_N as $\sigma_i \rightarrow e^{i\pi} = -1$ or $\sigma_i \rightarrow e^{i 0} = 1$ (where σ_i , $i = 1,\dots, N-1$, are elementary braids—exchanges of ith particle with $(i+1)$th one, when other particles stay a rest; numeration of particles is arbitrary but fixed; σ_i are generators of any braid group [33], in particular of S_N ). Two different scalar unitary representations of S_N give rise to statistics of fermions and bosons, respectively. In 3D $\sigma_i^2 = \varepsilon$ (neutral element in the braid group), but in 2D $\sigma_i^2\neq \varepsilon$ and braid groups in planar systems of the same classical particles are different than S_N (instead of finite permutation group, the braid group for N particles on a plane, i.e. for ${\cal{M}} = R^2$, is infinite Artin group B_N [33, 34], with infinite number of different scalar unitary representations, related to so-called anyons with fractional quantum statistics [30]).

The definition of fermionic or bosonic quantum statistics in 3D requires, however, the existence of classical trajectories in ${\cal{M}}$ for particle exchanges σ_i . Otherwise the braid group generators (here the permutation group) cannot be defined and quantum statistics cannot be assigned (a precise mathematical formulation of the linkage of trajectory homotopy class and quantum statistics can be demonstrated in the framework of path integration quantization for systems of identical indistinguishable particlesis, as summarized in appendix A).

For the existence of σ_i generators of the braid group for a particular system, the mutual exchanges for particle pairs on the manifold ${\cal{M}}$ must be admitted, i.e. there must exist closed loops in ${\cal{M}}$ built of pieces of individual particle trajectories in ${\cal{M}}$. It is unreachable to form such closed loops in local ${\cal{M}}$ from pieces of one-way spirals defined by equations (6) and (7) for $r\in(r_s,1.5r_s)$, in contrast to conic-section-like trajectories beyond the photon sphere. This precludes the definition of the braid group generated by σ_i in the local multiparticle configuration space beneath the photon sphere rim and related to ${\cal{M}}$ defined by a macroscopic system passing this rim. Hence, for $r\unicode{x2A7E} 1.5r_s$ the fermionic or bosonic quantum statistics can be assigned [31, 33] to particles in local manifolds in contrary to the region $r\in(r_s,1.5r_s)$, where none such quantum statistics is defined.

Below the innermost unstable circular orbit the dominating term in the effective potential (equations (8) and (9)) is ${\sim} -\frac{{\cal{L}}^2}{r^3}$, which causes an unavoidable one-way spiral movement toward the event horizon of any particle passing this orbit inward despite its energy and angular momentum and local mutual interaction between particles. The phase shift for such spirals is all the more limited the larger ${\cal{E}}_0$ for given ${\cal{L}}$ (as illustrated in figure 3). From pieces of these short unidirectional spirals it is impossible to form any closed local loop required for interchanges of particles and braid group implementation. This is in contrary to conic section-like trajectories admissible beyond the photon sphere rim, which can form arbitrary small closed loops.

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Any local multiparticle system which passes the photon sphere rim inward loses its quantum statistics defined on braid group generators σ_i because these generators cannot be constructed in related F_N space beneath this rim. The space F_N is thus simply-connected here and its first homotopy group is trivial $\pi_1(F_N) = \{\varepsilon\}$ (scalar unitary representation of neutral group element ε is 1, as $\varepsilon\cdot \varepsilon = \varepsilon$, which does not display any statistics because ε is not particle exchange).

The foremost quantum property of fermions is expressed as the so-called exclusion principle by Pauli, which asserts that fermionic quantum particles cannot share any common single-particle quantum state. Localized quantum states preclude thus approaching by fermions a space region already occupied by another identical fermion, which results in fermion repulsion, beyond fundamental interactions. This so-called quantum degeneracy repulsion can exert a giant pressure in a dense system and actually stops the collapse of white dwarfs [35] or neutron stars [18, 19, 36]. In white dwarfs electrons halt a collapse, whereas in neutron stars neutrons are of significance.

In sufficiently dense fermionic systems and at temperatures T lower than the chemical potential, i.e. when $k_BT\ll \mu$ (k_B is the Boltzmann constant, µ is the chemical potential—the energy increase caused by the addition of a single particle to the system), fermions form the so-called Fermi sphere, corresponding to one by one filled single-particle stationary states by all fermions respecting Pauli exclusion principle. Even though the Fermi sphere structure is transparent for noninteracting particles, inclusion of inter-particle interaction does not change the notion of the Fermi sphere [37] and its radius (in momentum space) is invariant with respect to interaction according to Luttinger theorem [38]. This radius equals to [39, 40],

$$\begin{align} p_F = \hbar\left(3\pi^2 \rho\right)^{1/3}, \end{align} \tag{ 11 }$$

and is the function of fermion concentration only, $\rho = \frac{N}{V}$, where N is the total number of particles, V is the spatial volume of the system, $\hbar = \frac{h}{2\pi}$ is reduced Planck constant. The proof of Luttinger theorem resolves itself to the observation that in the phase space $V\frac{4}{3}\pi p_F^3$, there exist $2V\frac{4}{3}\pi p_F^3/h^3$ different single-particle quantum states (due to Bohr–Sommerfeld rule for quantization, factor 2 is caused by double spin degeneracy for $\frac{1}{2}$ spin of fermions), and if they are filled by N particles, the relation (11) is obtained. Quasiclassical Bohr–Sommerfeld rule [41] is immune to interaction, thus p_F defined by equation (11) is universal for any interacting fermionic system. The energy of fermions on a Fermi surface is called as Fermi energy, $\varepsilon_F = \sqrt{p_F^2c^2+m^2c^4}-mc^2$ (in relativistic case), which equals to the chemical potential at zero temperature T = 0 K, which is familiar from the Fermi–Dirac distribution function for noninteracting fermions,

$$\begin{align} f\left(\varepsilon\left(\mathbf{p}\right)\right) = \frac{1}{e^{\frac{\varepsilon\left(\mathbf{p}\right)-\mu}{k_BT}}+1} \rightarrow_{T\to 0} 1-\theta\left(\varepsilon\left(\mathbf{p}\right)-\varepsilon_F\right), \end{align} \tag{ 12 }$$

where θ is the Heaviside step function, µ is the chemical potential, which at T = 0 K equals to Fermi energy ε_F and weakly changes with the temperature increase [37].

The whole Fermi sphere can store a high energy, which can be assessed per spatial volume V (neglecting interaction) as follows [39, 40],

$$\begin{align} \begin{aligned} E & = \sum_{\mathbf{p}} \varepsilon\left(\mathbf{p}\right) f\left(\varepsilon\left(\mathbf{p}\right)\right)\\ & = \frac{V}{\left(2\pi \hbar\right)^3}\int d^3\mathbf{p}\varepsilon\left(\mathbf{p}\right)f\left(\varepsilon\left(\mathbf{p}\right)\right)\\ & = \int_0^{p_F}dp \int_0^{\pi} d \theta\int_0^{2\pi} d\phi p^2 \sin\theta \varepsilon\left(\mathbf{p}\right) \frac{V}{\left(2\pi \hbar\right)^3}\\ & = \frac{V}{2 \pi^2 \hbar^3}\int_0^{p_F} dp p^2 \varepsilon\left(p\right),\\ \end{aligned} \end{align} \tag{ 13 }$$

the sum runs over occupied states p in momentum space, which form an isotropic Fermi sphere $|\mathbf{p}|\unicode{x2A7D} p_F$ with radius p_F ; $p,\theta, \phi$ are spherical variables in momentum space and $\varepsilon(\mathbf{p}) = \sqrt{p^2c^2+m^2c^4}-mc^2$ (the kinetical energy of a single particle in relativistic case), $\frac{V}{(2\pi \hbar)^3}$ is a number of single-particle quantum states in the element of the phase space $Vd^3\mathbf{p}$ (the density of states according to Bohr–Sommerfeld rule). The energy estimation (13) concerns T = 0 K case, but it very well approximates energy of fermions in Fermi sphere also for nonzero temperatures, provided that $k_BT\ll\mu\simeq\varepsilon_F$.

To demonstrate how large is the energy accumulated in a Fermi sphere one can estimate it via equations (11) and (13) in exemplary systems. In a metal (like Fe) in normal conditions free electrons (with concentration of order of Avogadro number per cm³) form the Fermi sphere accumulating ${\sim} 10^{10}$ J m⁻³. In a neutron star merger at TOV stability limit (of ca. 2.3 Sun masses compressed to the sphere with radius lower than 10 km) the Fermi sphere of neutrons has energy ${\sim} 10^{47}$ J. The energy collected in Fermi spheres is blocked by Pauli exclusion principle, but can be released when this principle is locally waived off, as at passing the photon sphere rim of a black hole. In both mentioned above examples we deal with quantumly degenerated systems, as the Fermi energy in metal is of order of 8 eV (i.e. ${\simeq} 10^5$ K in temperature units, when $k_B = 1$) and highly exceeds even the melting temperature of a metal, and for a neutron star merger $\varepsilon_F \simeq 0.3$ GeV (i.e. ${\simeq} 10^{12}$ K in temperature units, cf table 1 for detail) and also highly exceeds the supposed temperature of a merger (10⁶ K), so the formula (13) holds with high accuracy.

Table 1. Fermi momentum p_F (acc. to equation (11)), Fermi energy $\varepsilon_F = \varepsilon(p_F)$ and total energy of the Fermi sphere E released at the collapse of the Fermi sphere, for neutron star (acc. to equation (13)), for quasar luminosity—the energy per second; ξ—density of compressed electron-hadron matter, n—total number of neutrons (ne) in a neutron star; for a quasar of electrons (el) or protons (pr) per second.

pF (kg m s−1)	ξ (kg m$^{-3})$	n	εF (GeV) ((K) if $k_B = 1$)	$E\,$(J)/luminosity (W)
Unstable neutron star merger of 2.3 $\text{M}_{\odot}$ with radius ca. 6 km
$4.67 \times 10^{-19}$	$5\times10^{18}$	$2.4 \times 10^{57}$	ne 0.34 ($3 \times 10^{12}$)	$4.7 \times 10^{46}$ J
Quasar with BH of 109 $\text{M}_{\odot}$ consuming 0.06 Earth mass per second ($5.6~\text{M}_{\odot}$ per year)
$5.4 \times 10^{-19}$	$9\times 10^{18}$	$2.14 \times 10^{50}$	el$\,1\,(10^{13})$, pr $0.4\,(4\times10^{12})$	1040 W

At collapse of neutron star merger into a black hole the photon radiation is supposed to be released, though the mechanism of the photon burst with energy equivalent to half of Sun mass converted into e-m radiation within subsecond time has not yet been provided. Neutron star attains the density stability limit when its mass crosses ca. 2.3 Sun masses (the TOV limit [18, 19]), what happens if smaller neutron stars merge together and due to the gravitational compression, the merger attains the radius lower than 10 km (for M = 2.3 Sun masses $r_s\simeq 6.8$ km). When all the mass is contained inside its own photon sphere, then the local waiving off of Pauli principle takes place and due to the relief in quantum pressure of neutrons the star collapses. Simultaneously, the Fermi sphere of neutrons decays and its energy is released [42]. Neutrons are stabilized in a neutron star by Pauli exclusion principle and at the quantum statistics transition they decay into protons and electrons, which are charged particles interacting with e-m field, thus able to radiate photons according to Fermi golden rule. The energy of neutrons in their Fermi sphere in a neutron star merger at TOV limit, being of order of 10⁴⁷ J, is thus emitted at suddenly admitted quantum jumps of protons and electrons from their positions in momentum space onto ground state. A neutron with momentum p decays into a proton and electron and antineutrino (along β⁻ transition scheme with time rate strongly accelerated by forced emission of avalanche type in the system of ca. 10⁵⁷ neutrons in a star merger). Liberated charged particles lose their kinetic energy at suddenly admitted quantum transitions to lower states. The spectrum and timing of these transitions for electrons and protons can be assessed by the Fermi golden rule. Assuming the same concentration of electrons and protons, their initial Fermi spheres have the same Fermi momentum (11).

According to Fermi golden rule, the probability of a quantum transition per time unit due to interaction of a charged quantum particle with e-m field is as follows [41],

$$\begin{align} w_{1,2} = \frac{2\pi}{\hbar} \vert\langle 1 \vert\hat{V}\left(\mathbf{r}\right)\vert 2 \rangle\vert^2\delta\left(E_1-E_2-\alpha\hbar\omega\right), \end{align} \tag{ 14 }$$

where $\langle 1\vert \hat{V}(\mathbf{r})\vert 2 \rangle$ is a matrix element of an operator $\hat{V}(\mathbf{r})$ describing coupling of a charged particle to e-m field taken between an initial $\vert 1\rangle$ quantum stationary state of a particle with energy E₁, and a final one $\vert 2\rangle$ with energy E₂. $\alpha\hbar\omega$ is the energy of emitted photon at the particle transition $1\rightarrow 2$, including gravitational redshift close to the black hole horizon, $\alpha = (1-\frac{r_s}{r})^{1/2}\simeq0.57$ for $r = 1.5 r_s$. For relativistic electrons or protons the kinetic energy single-particle Hamiltonian has the form,

$$\begin{align} \hat{H} = \sqrt{\left(\hat{\mathbf{p}}\mp e\mathbf{A}\left(\mathbf{r},t\right)\right)^2c^2+m_{e\left(p\right)}^2c^4}-m_{e\left(p\right)}c^2, \end{align} \tag{ 15 }$$

where $\mathbf{A}(\mathbf{r},t) = \mathbf{A}_0e^{i(\mathbf{q}\cdot\mathbf{r}- cq t)/\hbar}$ is the vector potential of the e-m field for a plane wave (at gauge $\mathrm{div}{\mathbf{A}} = 0$), $\hat{\mathbf{p}} = -i\hbar\nabla$. As the Fermi golden rule is of first order perturbation accuracy [41], thus, the form of operator $\hat{V}$ is defined by the term in single-particle Hamiltonian linear with respect to A,

$$\begin{align} \hat{V}\left(\mathbf{r},t\right) = \mp\frac{e c^2 \mathbf{A}\left({\mathbf{r}},t\right) \cdot\hat{\mathbf{p}}}{\sqrt{\hat{\mathbf{p}}^2c^2+m_{e\left(p\right)}^2c^4}}. \end{align} \tag{ 16 }$$

Initial and final states $\vert1(2)\rangle$ can be assumed in the form $\vert 1(2)\rangle = \vert \mathbf{p}_1 (\mathbf{p}_2)\rangle = \frac{1}{(2\pi\hbar)^{3/2}} e^{i (\mathbf{p}_{1(2)}\cdot \mathbf{r}-E_{p_{1(2)}}t)/\hbar}$ representing states in the isotropic Fermi sphere for $p_{1(2)}\unicode{x2A7D} p_F$, $p_1\gt p_2$. One can calculate the matrix element in equation (14) for $\vert 1(2)\rangle = \vert \mathbf{p}_1 (\mathbf{p}_2)\rangle$, which gives,

$$\begin{align} \langle \mathbf{p}_1 \vert \hat{V}\left(\mathbf{r}\right)\vert \mathbf{p_2}\rangle = \mp \delta\left(\mathbf{p}_1-\mathbf{p}_2-\mathbf{q}\right) \frac{ec^2\mathbf{A_0}\cdot \mathbf{p}_2}{\sqrt{p_2^2c^2+m_{e\left(p\right)}^2c^4}}. \end{align} \tag{ 17 }$$

Substituting it into equation (14), we obtain,

$$\begin{align} \begin{aligned} w_{1,2} & = \frac{{\cal{V}}}{\left(2 \pi\right)^2\hbar^4}e^2c^2A_0^2cos^2\theta f\left(p_2\right)\delta\left(\mathbf{p}_1-\mathbf{p}_2-\mathbf{q}\right) \delta\left(E_{p_1}-E_{p_2}-\alpha cq\right), \end{aligned} \end{align} \tag{ 18 }$$

where $f(p_2) = \frac{p_2^2c^2}{p_2^2c^2+m_{e(p)}^2c^4}\lt1$ and for $\delta^2(\mathbf{p}) = \delta(\mathbf{p} = 0)\delta(\mathbf{p}) = \frac{\cal{V}}{(2\pi\hbar)^3}\delta(\mathbf{p})$ with ${\cal{V}}\rightarrow \infty$ (because $\hbar^3\delta(\mathbf{p}) = \frac{1}{(2\pi)^3}\int e^{i\mathbf{p}\cdot \mathbf{r}/\hbar}d^3r$ and $\delta(\mathbf{p} = 0) = \frac{1}{(2\pi \hbar)^3}\int e^{i\mathbf{p}\cdot \mathbf{r}/\hbar}d^3r\vert_{\mathbf{p} = 0}\simeq \frac{{\cal{V}}}{(2\pi\hbar)^3}\vert_{{\cal{V}}\rightarrow \infty}$, ${\cal{V}} = \int d^{3}r1$), $E_{p_{1,2}} = \sqrt{p^2_{1(2)}c^2+m_{e(p)}^2c^4}-m_{e(p)}c^2$, $\cos\theta = \mathbf{A}_0\cdot \mathbf{p}_2/A_0p_2$ (${\cal{V}}$ is the volume of a local multi-particle system). The probability $w_{1,2 }$ tends to zero for $\mathbf{p}_2\rightarrow 0$, but for $p_2^2 c^2\gg m_{e(p)}^2c^4$ the function $f(p_2)$ practically does not depend on p₂ and in this limit $f(p_2)\simeq 1$. For electrons (protons) this inequality is satisfied for $p_2\gt5 \times 10^{-22}$ ($5 \times 10^{-19}$) kg m s⁻¹. Higher energies of photons dominate as the density of states in isotropic Fermi sphere grows as p².

Integration of equation (18) over all occupied states in the Fermi sphere allows the estimation of time-span for entire Fermi sphere collapse. The integration over p₁ and p₂ gives from equation (18),

$$\begin{align} \begin{aligned} w & = \int_{p_F}d^3p_1\int_{p_F}d^3p_2 w_{1,2} = \left(N+1\right)\gamma,\\ \gamma & = \frac{m_{e\left(p\right)} c e^2}{3 \pi \varepsilon_0 \hbar^2 y} \int_{-1}^1 dz\int_0^{p_F/m_{e\left(p\right)}c}dx \frac{x^4}{x^2+1} \delta\left(\sqrt{x^2+y^2+2 x y z+1}-\sqrt{x^2+1}-0.57 y\right), \end{aligned} \end{align} \tag{ 19 }$$

where $x = p_2/m_{e(p)}c$, $y = q/m_{e(p)}c = \hbar \omega/m_{e(p)}c^2$, ε₀ is the dielectric constant and $N = \varepsilon_0 \frac{{\cal{V}}E_0^2}{2 \hbar\omega}$ is the number of photons $\hbar \omega$ in the volume ${\cal{V}}$, $A_0 = E_0/\omega$ (where E₀ and A₀ are amplitudes of electric field and of vector potential of e-m radiation, respectively) and the density of energy of e-m field is $\varepsilon_0 E_0^2 /2$. Non-zero value of w for N = 0 reveals the spontaneous emission (as in optical transitions the only non-zero matrix element of the creation operator for photons is out of diagonal and equals to $\sqrt{N+1}$ for the quantized vector potential in equation (16)). (A technical note: in (19) the integrals over p₁ and p₂ should be accompanied with the density of states, $\frac{{\cal{V}}}{(2\pi \hbar)^3}$, but the states $\vert 1(2)\rangle$ are normalized on the Dirac delta and to restore their probability sense, the matrix element (17) must be divided by $\frac{{\cal{V}}}{(2\pi \hbar)^3}$ to change the normalization to volume ${\cal{V}}$; since the matrix element (17) enters into (14) in the square, then the state density coefficients in (19) cancel. Moreover, the integration over arbitrary orientation of the vector A₀ with respect to the vector p₂ (assumed along z-axis) has been performed in (19), resulting in $\int \frac{d\Omega}{4\pi} \cos^2\theta = 1/3$, $d\Omega = \sin\theta d\theta d\phi$.) The above derivation of equation (19) has been performed for wave functions $\vert\mathbf{p}_1 (\mathbf{p}_2)\rangle = \frac{1}{(2\pi\hbar)^{3/2}} e^{i (\mathbf{p}_{1(2)}\cdot \mathbf{r}-E_{p_{1(2)}}t)/\hbar}$ normalized on the Dirac delta. One can repeat calculations also for wave functions normalized on the finite volume ${\cal{V}}$, as shown in detail in C (and leading to the same equation (19)).

The number of photons satisfies the dynamic equation,

$$\begin{align} dN = w dt = \left(N+1\right)\gamma dt, \end{align} \tag{ 20 }$$

hence, the solution of (20) attains the form $\ln N = \gamma t$, or $\text{ln} N_0 = \gamma\Delta t$, where $\Delta t$ is the time of total Fermi sphere decay for N₀ of order of particle number in the system. $\Delta t = \frac{\ln N_0}{\gamma(y)}$ depends on photon frequency $\hbar\omega = y m_{e(p)}c^2$ via function $\gamma(\hbar \omega)$ given by equation (19).

For a neutron star merger with 2.3 Sun masses, one can estimate the time-span $\Delta t$ of the decay of Fermi spheres of electrons (assuming in (19) $m_e = 9.1 \times 10^{-31}$ kg) and protons (ca. 1837 times more massive) for particular energy of emitted photons—as shown in table 2, which displays the rapid process of photon emission. In this assessment we did not take into account the time of the decay of neutrons into electrons and protons (and antineutrinos), which slows down the photon burst. The β⁻ decay of an isolated neutron takes ca. 880 s mostly due to quark transitions, the conversion of the negatively charged $-\frac{1}{3}e$ down quark to the positively charged $\frac{2}{3}e$ up quark by emission of a W⁻ boson (the W⁻ boson subsequently decays, almost instantly, into an electron and an electron antineutrino). In a neutron star neutrons are stabilized by Pauli exclusion principle. If the latter is locally waived off, then neutrons decay in a collective manner also according to Fermi golden rule (similar as in equation (20) with specific $\gamma^{^{\prime}}$ for spontaneous decay of a neutron). If for $\gamma^{^{\prime}}$ one assumes $\frac{1}{880}$ 1/s, the decay of all neutrons would take ca 30 hours. Nevertheless, fermionic quarks also lose their quantum statistics at passing the photon sphere rim of a black hole, and the decay of unstable isolated neutron would be here much faster (neutron has larger rest mass than proton, which causes the decay not limited by fermionic nature of quarks and conversion $d\rightarrow u$ quarks will not be hampered by the presence of second u quark in a proton). This would accelerate the decay by few orders—to be consistent with time span of short giant gamma-ray burst associated with the neutron star collapse, to the order of $\gamma^{^{\prime}}\sim10^3$ 1/s, which gives the liberation of electrons and protons in the whole merger of mass 2.3 Sun masses within ca. $\textrm{ln}(2.53 \times 10^{57})/\gamma^{^{\prime}}\simeq 0.1$ s. The following release of both antineutrinos (with time scale 10⁻²⁷ s for decay of bosons W⁻) and photons (cf table 2) is almost instant. Such timing agrees with observations of some kind of short gamma-ray bursts (on the other hand, assuming such a scenario, one can infer about coupling for β⁻ transition, which should be the same regardless of the change in quantum statistics).

Table 2. The Fermi golden rule estimation of the time $\Delta t$ of complete decay of the Fermi sphere for neutron star merger with the mass 2.3 M$_{\odot}$, as in table 1, for different photon energies $\hbar\omega$ and for electron and proton contributions.

	Electrons		Protons
$\hbar\omega$ (eV)	$\gamma(\hbar\omega)$ (1/s)	$\Delta t$ (s)	$\gamma(\hbar\omega)$ (1/s)	$\Delta t$ (s)
0.1 GeV	$2.5 \times 10^{22}$	$5\times 10^{-21}$	$2.9 \times 10^{21}$	$4.5 \times 10^{-20}$
1 MeV	$2.4\times 10^{26}$	$5 \times 10^{-25}$	$2.4 \times 10^{24}$	$5.5\times 10^{-23}$
1 keV	$2.6 \times 10^{28}$	$5 \times 10^{-27}$	$2\times 10^{30}$	$6 \times 10^{-29}$

Protons give a bit longer burst in the range of high photon energy (though also almost instant, cf table 2), but their Fermi sphere total energy is only ${\sim} 0.7$ of the Fermi sphere energy of electrons (at the same Fermi momentum of both and due to larger rest mass, cf estimation in next section 4.2).

According to the presented above scheme of decay of Fermi sphere of neutrons at collapse of unstable neutron star merger, the emission of e-m radiation should be associated by preceding burst of electron-antineutrinos emitted at collective β⁻ decomposition of neutrons. These antineutrino bursts have not been observed as of yet, what can be linked to large distances of potential candidates for collapsing neutron star mergers and almost infinitesimal interaction of antineutrinos with matter in counters.

Conventional models of quasar luminosity consider accretion disc of plasma upon classical hydrodynamic approach [10, 11, 13, 15–17]. Though such an approach reproduces many observed features of quasar radiation, especially well for micro-quasars [13], the super-luminous quasars are still not fully described [14–17]. Thermal radiation from the accretion disc [12] does not cover the observed spectrum and intensity of quasar radiation and the source of x-ray part of this radiation is conventionally assumed [13] due to Comptonization of soft photons in small region of accretion disc with extremely hot plasma. In Shapiro at al. model of microquasar Cygnus X-1 [13] the inverse Compton effect needs 10⁹ K for electrons and 10¹¹ K for ions in the hot bulb in accretion disc at the distance of ca. $6 r_s$ from the center of black hole (of 15 Sun masses in this case) to match observed x-ray component of this microquasar radiation. Additionally, the sufficiently large supply of soft photons to such a small region is unclear. In the case of super-luminous quasars powered by $10^{8-9}$ Sun mass black holes the extension of the Shapiro model is problematic, as an extremely hot region in this case would have to be 10⁸ times larger, what is unlikely to be so hot.

The decay of Fermi spheres of electrons and protons in the accretion disc of quasars at passing the photon sphere rim supplies, however, a very effective channel for conversion of gravitational energy into e-m radiation, which can solve above mentioned problems of quasar luminosity. Matter falling onto the black hole event horizon is compressed to high density, which can achieve the uppermost its stable value when the supply of the surrounding gas to the accretion disc is unlimited. Assuming an accretion disc formation from the neutral hydrogen surrounding a black hole, a plasma is electrically balanced, thus the Fermi momentum (11) for electrons and protons is the same. Using equation (13) we estimate the energy accumulated in Fermi spheres of electrons and protons at the photon sphere rim for an uppermost fermion concentration (similar as in neutron star at TOW limit) for quasar with the black hole of ${\sim} 10^9$ Sun mass consuming in a steady way 5.6 Sun mass per year, i.e. ca. 0.06 Earth mass per second [43]. Because of the steadiness of the process the continuous in time decay of Fermi spheres of electrons and protons can be accounted for per single second resulting in additional contribution to quasar luminosity expressed in W = J s⁻¹. Total energy released from Fermi sphere of electrons and protons at passing the photon sphere rim (acc. to equation (13)) is just 10⁴⁰ J per second with prevailing contribution of x-ray radiation, which agrees with observable luminosity of such quasars. This energy had been accumulated in Fermi spheres of electrons and protons on the cost of gravitation energy of a black hole, which compressed the matter falling onto the event horizon. At approaching the event horizon the spatial volume in proper coordinates diverse, however. Thus, for calculation of the Fermi momentum (11) the local proper concentration should be taken, $\frac{dN}{d{\cal{V}}}$, instead of $\frac{dN}{dV}$ with $d{\cal{V}} = \left(1-\frac{r_s}{r}\right)^{-1/2} dV$ ($dV = drr^2 \sin\theta d\theta d\phi$ in rigid coordinates of a remote observer). For $r = 1.5 r_r$ the renormalization factor in fermion concentration is ${\simeq} 1.7$, which causes the reduction of p_F via equation (11) by factor $(1.7)^{-1/3}\simeq 0.84$. This factor is still of order of unity, thus, general relativistic corrections do not change the order of energy estimation via equation (13).

The conversion of gravitational energy at passing the photon sphere rim due to decay of Fermi spheres of electrons and protons achieves, in the exemplary considered here case, the efficiency of 30% mass to radiation energy conversion rate. It means that the compressed plasma at photon sphere rim is by 30% more massive than initial distanced hydrogen captured to the accretion disc, where this increase of the mass is the energy accumulated in Fermi spheres of electrons and protons approaching the photon sphere rim of a black hole (divided by c²), just before their release due to the decay of both Fermi spheres. The release of high energy radiation at the photon sphere rim does not interfere with radiation from more distant regions of an accretion disc and conveniently contributes to the total quasar luminosity filling a loophole in understanding of high energy radiation of super-luminous quasars [14–17].

Comparing the total Fermi sphere energies of electrons and protons with the same Fermi momentum (for the same concentration as in electrically balanced plasma in accretion disc) one can find via equation (13) that $\frac{E_{el}(r^*)}{E_{pr}(r^*)}\simeq 1.4$ (for $r^* = 1.5 r_s$), due to the difference in rest mass between these particles. The Fermi energy of electrons with Fermi momentum $p_F(r^*) = 5.48 \times 10^{-19}$ kg m s⁻¹ (cf table 1) equals to $\varepsilon_F = 1$ GeV (this gives the limit for uppermost possible energy of emitted photons). In the thermal scale (in units at $k_B = 1$) this energy is of order of 10¹³ K, which indicates that the electron liquid in compressed plasma at photon sphere rim of super-luminous quasar is quantumly degenerated at lower temperatures (quasars are not source of thermal gamma radiation, thus their actual temperatures are much lower). Similarly for protons with Fermi momentum $p_F(r^*) = 5.48 \times 10^{-19}$ kg m s⁻¹, for which Fermi energy equals to $\varepsilon_F = 0.4$ GeV (in thermal scale is of order of $4 \times 10^{12}$ K)—for lower temperatures of plasma, protons also form the quantum degenerate Fermi liquid at photon sphere rim of a quasar.

In many observable events of photon radiation from close vicinity of black holes, the rate of matter consumption is much lower than for super-luminous quasars because of limited matter supply from the black hole surroundings. To such effects belong transients of closer AGN, microquasars, and flares associated to tidal disruption events (TDEs). When the concentration of electrons and protons at the photon sphere rim does not reach its uppermost stable value, then local Fermi momentum (11) is smaller and the energy accumulated in Fermi spheres (13) is also lower. Nevertheless, the local waving off of Pauli exclusion principle still causes the Fermi sphere decay, though the efficiency of mass to radiation energy conversion is lower than 30%. In each such episode, large part of photons released according to Fermi golden rule exceeds in individual energies the thermal radiation in conventional classical models of AGN or TDEs. We refer below to the following examples of recent observations, which could be treated as a confirmation of the proposed model of Fermi sphere decay.

4.3.1. Transient AGN 1ES 1927+654.

4.3.2. TDE flare AT 2020neh.

4.3.3. Transient AT 2021lwx.

The problem of the fate of information (related to entropy) encoded in a matter falling to a black hole, in view of the second law of thermodynamics, leads to so-called information paradox [47–49]. It is still unsolved, because the Hawking radiation of black holes is fully stochastic [50, 51] and cannot take away entropy/information in a unitary manner. The concept of creation of particle-antiparticle pairs on the event horizon and escape of a particle associated with capturing by a black hole of its antimatter partner did not solve the problem, because of quantum entanglement property called monogamy. Particles and antiparticles are entangled and Hawking radiation must be also entangled in time, which forces the instant breaking of the entanglement at event horizon leading to the concept of a firewall [49] (immediately broken entanglement between the infalling particle and the outgoing particle would release large energy, creating in this way a searing firewall at the event horizon). Neither Hawking radiation nor firewall have been observed as of yet (some evidences of fuzzy black hole horizon in gravitational wave signature [52] have been eventually excluded [53]).

According to theoretical field models of gravitation singularity, including holographic formulation [54], the unitarity at matter consumption by black holes should be maintained, which challenges, however, Einstein's equivalence principle or existing quantum field theory [49].

The quantum statistics transition at passing the photon sphere rim, presented in this paper, seems to contribute to the debate on information paradox and related concept of a black hole firewall, updating the related premises, to some extent. The decay of quantum statics in multiparticle systems at passing the photon sphere rim removes the quantum statistics origin entanglement, as antisymmetrized wave functions for fermions or symmetrized ones for bosons were entangled nonseparable functions in tensor productive Hilbert space of all particles. The related release of energy (including that released due to local waiving off of Pauli exclusion principle) forms some kind of firewall, shifted to the photon sphere rim and visible to any distant observers, substituting a hypothetical firewall on the event horizon [49] unobservable for distant observers. Although the entropy behavior at passing the photon sphere rim needs a more thorough analysis, it is seen that energy along with entropy escapes into outer space from the the photon sphere rim in a unitary manner at quantum statistics transition, and quantum fields lose their statistics assignment beneath this rim (which should affect local quantum field theory formulation).