Black Holes as Gravitational Dirac Spinors

1. Introduction

Starting from an initial suggestion by Hawking [1], that the final result of the gravitational collapse should not be an event horizon, but an apparent horizon instead, Vaz derived a quantum gravitational model of dust collapse that allowed him to win the Second Prize in the 2014 Gravity Research Foundation Essay Competition [2]. Vaz’s approach has shown that continued collapse to a singularity can only be obtained by combining two independent and complete solutions of the Wheeler-DeWitt equation. In that case, the physical interpretation of the BH information paradox [3] is to simply forbid such a combination. The final result of quantum collapse is that the collapsing matter condenses on the Schwarzschild surface, which is an apparent horizon rather than a real one. This enabled the emergence of an entirely new BH framework [2] that was also consistent with Einstein’s idea in 1939 of the localization of the collapsing particles within a thin spherical shell [4]. It would be an extremely compact “dark star,” that is, a sort of BH “gravitational atom” being held up not by any degeneracy pressure but by quantum gravity in the same way that ordinary atoms are sustained by quantum mechanics. From now on, Planck units, i.e. $G=c=k_B=\hslash ={\displaystyle \frac{1}{4\pi {ϵ}_0}}=1$, will be used. Vaz’s quantum approach concerns the spherical collapse of inhomogeneous dust in Anti-de Sitter spacetime of dimension $d=n+2$ [2], which is described by the Lemaître-Tolman-Bondi (LTB) family of metrics [5,6,7]. Dirac quantization of the constraints leading to a Wheeler-DeWitt equation permits finding two independent solutions in terms of shell wave functions supported everywhere in spacetime [2]

$${\psi }_i={\psi }_i^{\left(1\right)}+A_i{\psi }_i^{\left(2\right)},$$

(1)

where ${\psi }_i^{\left(1\right)}$ represents dust shells condensing to the apparent horizon on both sides of it and ${\psi }_i^{\left(2\right)}$ represents dust shells moving away from the apparent horizon on either side of it where the exterior, outgoing wave is suppressed by the Boltzmann factor at the Hawking temperature for the shell [2,8]. The key point is that there is nothing within the theory that suggests a value for $A_i$ [2,8]. In addition, further input is needed to determine these amplitudes [2,8]. If $0<\left|A_i\right|\le 1,$ then the dust will ultimately pass through the horizon via continued collapse, arriving at a central singularity [2,8]. Consequently, an event horizon will form, with emission of thermal radiation in the exterior [2,8]. To avoid the existence of horizons and singularities, $\left|A_i\right|$ must vanish [2,8]. Then, ${\psi }_i^{\left(1\right)}$ alone results in the complete description of the quantum collapse [2,8]. The meaning of ${\psi }_i^{\left(1\right)}$ is that each shell will condense to the apparent horizon, by stopping the gravitational collapse [2,8]. Thus, there should be no tunneling into the exterior, and no horizons, no singularities, and no information paradox [2,8]. Each shell converges to the apparent horizon, and a thin “dark star” forms. So far, Vaz’s important achievement in 2014 [2], similar to Einstein’s one in 1939 [4], has been summarized. Using the Newtonian approximation, the mass and energy spectra of this BH “gravitational atom” have been found via a Schrödinger-like approach in [8], while special relativity corrections have been provided in [9], showing that these BHs obey the Klein-Gordon equation. Now, the results in [8,9] will be further generalized using the correct form of the potential energy in general relativity for these BHs and replacing the Klein-Gordon theory with the Dirac theory.

2. The Gravitational Energy of a Black Hole as “Gravitational Atom” in General Relativity

Let’s try to clarify Vaz’s approach better. In classical theory, inside the event horizon of a Schwarzschild BH, given by the condition

$$r_g\equiv 2M,$$

(2)

the radial space coordinate, $r,$ and the time coordinate, $t,$ swap roles due to the change of sign in the coefficients of the spacetime metric. Beyond what is a real event horizon, that is, $r<r_g,$ the term associated with the radial coordinate becomes negative (behaving like a temporal coordinate), while the temporal one becomes positive. It follows that the singularity at $r=0$ no longer represents a fixed position in space that can be avoided, but an unavoidable moment in the future. In other words, just as one cannot stop the flow of time to avoid reaching tomorrow, inside the BH one cannot move in the opposite direction to avoid the temporal singularity. So, falling into the singularity doesn’t mean going to the center, but running out of time. The singularity is the end of time for any unfortunate observer who happens to be inside the horizon. Vaz’s approach [2] eliminates the temporal singularity and coordinate inversion simply because it prevents the event horizon and classical BH from forming at all, replacing them with a stable quantum object. Vaz applies the Wheeler-DeWitt equation to a collapsing dust cloud. His analysis undermines the classical view of gravitational collapse through precise physical steps. In quantum physics, the continuous collapse of matter as described above would require combining two independent, integral wave solutions of the Wheeler-DeWitt equation. Vaz prohibits the combination of these two solutions. Consequently, the quantum state of the system does not allow matter to continue its collapse until it runs out of time. Once the collapse towards the end of time is prevented, what does matter do? During quantum collapse, matter condenses, forming a thin spherical shell centered on its apparent horizon. All the mass of the dying star stops and is distributed across the outer surface, forming a structure that strongly resembles an idea originally proposed by Albert Einstein in 1939 [4]. The inversion between the radial space coordinate, $r,$ and the time coordinate, $t,$ occurs, in classical theory, only and exclusively when the matter crosses the event horizon and enters the Schwarzschild radius. In Vaz’s approach, matter does not end inside. Since all matter stops and condenses on the surface of the horizon, which is therefore apparent, the empty internal region does not have an inverted metric. The object does not possess an interior in which space becomes time. The radial spatial coordinate retains its classical mathematical sign and remains a spatial coordinate in all respects also for $r<r_g$. In fact, outside the apparent horizon spacetime is described by the classical, empty Schwarzschild metric. This is the standard geometry one knows, stable and static. All the matter in the dying star does not fall beyond the gravitational radius, but condenses to form an ultra-compact spherical quantum object (a thin shell) located exactly where the event horizon should have formed. Since matter has settled on the surface, the interior region is completely empty of matter. According to Birkhoff’s Theorem (applied to the boundary conditions of the model), the interior spacetime experiences no residual gravity and is a simple flat Lorentz-Minkowski spacetime [10]. No observer or particle ever crosses the surface of the quantum shell into a zone of no return; the metric never experiences that geometric breaking point. The radial coordinate r outside is exactly the Schwarzschild radial coordinate; beneath the massive shell, it becomes the radial coordinate of flat Lorentz-Minkowski spacetime and measures the distance between you and the geometric center (which is a flat, empty point in space, not a singularity).

Now, to study the quantum behavior of any physical system, the history of physics teaches one to start by determining the classical energy configuration of the system itself, and then to reason in quantum terms. One must remember, however, that, due to Einstein’s equivalence principle, in classical general relativity, the energy of the gravitational field cannot be localized [10]. The sole exception concerns the gravitational energy of a spherical star [10], which is exactly the case analyzed in this paper. In this case, the gravitational energy is indeed localized not by mathematical conventions, but by the circumstance that the transfer of energy is detectable by local measures [10]. In the case of a spherical star, using Schwarzschild coordinates, the gravitational energy is given by formula (7) of Box 23.1 of the reference [10], i.e.

$$U=-{\int }_0^r\rho \left[{\left(1-{\displaystyle \frac{r_g}{r}}\right)}^{-{\displaystyle \frac{1}{2}}}-1\right]4\pi r^{2}dr,$$

(3)

where $\rho$ and M are the density and mass of the spherical star, respectively. If the star turns out to be, as in the present case, a spherical shell having mass M, then it will be subjected to the potential

$$U=-{\int }_{r_i}^{r_e}\rho \left[{\left(\left|1-{\displaystyle \frac{r_g}{r}}\right|\right)}^{-{\displaystyle \frac{1}{2}}}-1\right]4\pi r^{2}dr,$$

(4)

where $r_i$ and $r_e$ are the inner and outer radii (in the sense of the Schwarzschild radial coordinate) of the thin spherical shell, which is centered around the apparent horizon $r_g=2M.$ The absolute value in Eq. (4) was included because the proper volume in a Riemannian or pseudo-Riemannian manifold is calculated from the determinant of the space’s metric. For a purely spatial coordinate, the metric coefficient $g_{rr}$ must be positive [10]. Assuming that the spherical shell is extremely thin, one can consider the density constant and write

$$U=-{\displaystyle \frac{M}{V_{schell}}}{\int }_{r_i}^{r_e}\left[{\left(\left|1-{\displaystyle \frac{r_g}{r}}\right|\right)}^{-{\displaystyle \frac{1}{2}}}-1\right]4\pi r^{2}dr,$$

(5)

where it was defined $\rho \equiv {\displaystyle \frac{M}{V_{schell}}}$ and the proper volume of the shell is indicated with $V_{schell}$. Eq. (5) can be rewritten as

$$U=-{\displaystyle \frac{M}{V_{schell}}}{\int }_{r_i}^{r_e}4\pi r^2{\left(\left|1-{\displaystyle \frac{r_g}{r}}\right|\right)}^{-{\displaystyle \frac{1}{2}}}dr+{\displaystyle \frac{M}{V_{schell}}}{\int }_{r_i}^{r_e}4\pi r^{2}dr.$$

(6)

But the integral ${\int }_{r_i}^{r_e}4\pi r^2{\left(\left|1-{\displaystyle \frac{r_g}{r}}\right|\right)}^{-{\displaystyle \frac{1}{2}}}dr$ is precisely $V_{schell}$, while

$${\displaystyle \frac{M}{V_{schell}}}{\int }_{r_i}^{r_e}4\pi r^{2}dr={\displaystyle \frac{M}{V_{schell}}}{\displaystyle \frac{4\pi }{3}}\left(r_e^3-r_i^3\right).$$

(7)

Thus, one gets

$$\begin{array}{c}U=-{\displaystyle \frac{M}{V_{schell}}}{V}_{schell}+{\displaystyle \frac{M}{V_{schell}}}{\displaystyle \frac{4\pi }{3}}\left(r_e^3-r_i^3\right)=\\ \\ =-M+{\displaystyle \frac{M}{V_{schell}}}{\displaystyle \frac{4\pi }{3}}\left(r_e^3-r_i^3\right).\end{array}$$

(8)

One can perform the exact calculation of $V_{schell}$ using traditional mathematical methods. The three-dimensional volume element is defined uniformly with respect to standard spatial coordinates as

$$d{V}_{schell}\equiv 4\pi r^2\sqrt{\left|g_{rr}\right|}=4\pi r^2{\left(\left|1-{\displaystyle \frac{r_g}{r}}\right|\right)}^{-{\displaystyle \frac{1}{2}}}dr.$$

(9)

Let $2\epsilon$ be the thickness of the spherical shell. Then, the calculation must be divided into two perfectly symmetrical parts straddling the apparent horizon:

1.

Portion above the apparent horizon $(r_g\le r\le r_g+\epsilon )$: For $r>r_g$, the absolute value is superfluous since the term is positive:

$$\begin{array}{c}V{\left(ext\right)}_{schell}\equiv {\int }_{r_g}^{r_g+\epsilon }4\pi r^2{\left(1-{\displaystyle \frac{r_g}{r}}\right)}^{-{\displaystyle \frac{1}{2}}}dr=\\ \\ ={\int }_{r_g}^{r_g+\epsilon }4\pi r^{5/2}{\left(r-r_g\right)}^{-{\displaystyle \frac{1}{2}}}.\end{array}$$

(10)

For a very thin shell $(\epsilon \to 0)$, $4\pi r^{5/2}$ is approximated by setting $r\simeq r_g$. One gets

$$\begin{array}{c}V{\left(ext\right)}_{schell}\simeq 4\pi r_g^{5/2}{\int }_{r_g}^{r_g+\epsilon }{\left(r-r_g\right)}^{-{\displaystyle \frac{1}{2}}}dr=\\ \\ =4\pi r_g^{5/2}{\left[\sqrt{r-r_g}\right]}_{r_g}^{r_g+\epsilon }=8\pi r_g^{5/2}\sqrt{\epsilon }.\end{array}$$

(11)

2.

Portion below the horizon ($r_g-\epsilon \le r\le r_g$): For $r<r_g$, By applying the modulus, the argument of ${\left(1-{\displaystyle \frac{r_g}{r}}\right)}^{-{\displaystyle \frac{1}{2}}}$ is inverted. Then, one gets

$$\begin{array}{c}V{\left(int\right)}_{schell}\equiv {\int }_{r_g-\epsilon }^{r_g}4\pi r^2{\left({\displaystyle \frac{r_g}{r}}-1\right)}^{-{\displaystyle \frac{1}{2}}}dr=\\ \\ ={\int }_{r_g-\epsilon }^{r_g}4\pi r^{5/2}{\left(r_g-r\right)}^{-{\displaystyle \frac{1}{2}}}.\end{array}$$

(12)

By adopting the same approximation for $4\pi r^{5/2},$that is $r\simeq r_g,$ and performing the integration with respect to r, one finds

$$\begin{array}{c}V{\left(int\right)}_{schell}\simeq 4\pi r_g^{5/2}{\int }_{r_g-\epsilon }^{r_g}{\left(r_g-r\right)}^{-{\displaystyle \frac{1}{2}}}dr=\\ \\ =4\pi r_g^{5/2}{\left[\sqrt{r_g-r}\right]}_{r_g-\epsilon }^{r_g}=8\pi r_g^{5/2}\sqrt{\epsilon }.\end{array}$$

(13)

By adding the two portions, the two contributions are mathematically identical and add symmetrically. Therefore, the total proper volume of the thin spherical shell centered on the apparent horizon is

$$V_{schell}\equiv {\int }_{r_i}^{r_e}4\pi r^2{\left(\left|1-{\displaystyle \frac{r_g}{r}}\right|\right)}^{-{\displaystyle \frac{1}{2}}}dr\simeq 16\pi r_g^{5/2}\sqrt{\epsilon }.$$

(14)

Thus, despite the presence of the singularity in the coordinates at $r=r_g,$ the proper volume of the thin spherical shell remains finite and proportional to the square root of the shell’s thickness. For thin shells, i.e. when $r_e$ approaches $r_i,$ the quantity ${\displaystyle \frac{M}{V_{schell}}}{\displaystyle \frac{4\pi }{3}}\left(r_e^3-r_i^3\right)$ in Eq. (8) can therefore be approximated as follows:

$${\displaystyle \frac{M}{V_{schell}}}{\displaystyle \frac{4\pi }{3}}\left(r_e^3-r_i^3\right)\simeq {\displaystyle \frac{M}{12\pi r_g^{5/2}\sqrt{\epsilon }}}\left(r_e^3-r_i^3\right)={\displaystyle \frac{r_g^{1/2}}{12\pi }}\sqrt{\epsilon },$$

(15)

where in the last step the difference of cubes was developed and it was taken into account that $r_e-r_i=2\epsilon .$ Hence, for a very thin shell $(\epsilon \to 0)$ the quantity ${\displaystyle \frac{M}{V_{schell}}}{\displaystyle \frac{4\pi }{3}}\left(r_e^3-r_i^3\right)$ in Eq. (8) tends to zero because the proper volume $V_{schell}$, which scales like $\sqrt{\epsilon }$, tends to be increasingly larger than the Schwarzschild volume ${\displaystyle \frac{4\pi }{3}}\left(r_e^3-r_i^3\right)$, which scales like $\epsilon .$ Then, from Eq. (8) one gets

$$U=-M.$$

(16)

Using the gravitational radius $r_g=2M$, Eq. (16) becomes

$$U=-{\displaystyle \frac{2M^2}{r_g}}.$$

(17)

If one now sets

$$r_{\alpha }\equiv r_g/2,$$

(18)

one obtains

$$U=-{\displaystyle \frac{M^2}{r_{\alpha }}},$$

(19)

which is formally equal to the Coulomb potential energy of the hydrogen atom if the the BH mass is replaced by electron charge

$$V\left(r\right)=-{\displaystyle \frac{e^2}{r}}.$$

(20)

One notices that a further important correction is necessary. In fact, although the two Eqs. (19) and (20) are formally identical, there is an important difference between them. In the case of Eq. (20), the electron charge is constant for all energy levels of the hydrogen atom. Instead, in the case of Eq. (19), the BH mass changes during jumps from one energy level to another due to the emission of BH radiation or the absorption of external particles. In fact, this BH mass decreases for emissions and increases for absorptions. Therefore, it is also necessary to consider this BH dynamical behavior. A good way to account for the BH dynamical behavior is to introduce the BH dynamical state [8,9,11,12]. This consists of introducing some dynamical quantities. Considering the initial BH mass before an energy transition (an emission of a quantum or an absorption of an external particle), $M,$ and the final BH mass after the transition, $M\pm \omega ,$ where $\omega$ is the mass-energy of the particle involved in the transition (the + or - sign corresponds to an absorption or an emission respectively), one introduces the BH dynamical mass and the BH dynamical horizon as [8,9,11,12]

$$M_D\equiv M\pm {\displaystyle \frac{\omega }{2}},r_D\equiv 2M_D,$$

(21)

respectively. The dynamical quantities in Eq. (21) represent average quantities. The variable $r_D$ is in fact the average of the initial and final horizons while $M_D$ is the average of the initial and final masses [8,9,11,12]. They represent the mass and the horizon of the BH during the contraction (expansion) of the BH, i.e. during the emission (absorption) of the particle. Hence, on the one hand, the introduction of the dynamical quantities in Eq. (21) into the BH equations is very intuitive. On the other hand, such introduction is rigorously justified through the Gibbons-Hawking periodicity argument [48]. Let us analyze the matter in detail. A traditional Schwarzschild BH having mass M emits radiation at the Hawking temperature [49]

$$T_H={\displaystyle \frac{1}{8\pi M}},$$

(22)

with a corresponding probability of emission

$$\Gamma \sim \exp (-{\displaystyle \frac{\omega }{T_H}}),$$

(23)

where $\omega$ is the energy-frequency of the outgoing particle. Hawking’s derivation was based on a previous idea by Bekenstein [50] who associated the BH with an entropy given by

$$S_{BH}={\displaystyle \frac{A}{4}}=\pi r_g^2,$$

(24)

where A is the horizon area and is the Schwarzschild radius in Eq. (2). These initial insights by Bekenstein and Hawking form the basis of the fascinating theoretical framework of BH thermodynamics, which is still actively studied today by many theoretical physicists around the world. Hawking later demonstrated that this approach led to a non-trivial problem, subsequently known as the BH information loss paradox [3]. Hawking radiation is, in fact, generated by the extreme gravitational force present in the vacuum surrounding the BH event horizon. However, it appears that this radiation, produced by the quantum vacuum, is incapable of carrying any physical information. Consequently, even if one resolves the issue and demonstrates that the BH actually possesses a different physical structure—namely, that of an ordinary body emitting radiation from its surface, just like any other object—one would still need to explain why such a body exhibits the thermodynamic properties typical of traditional BHs. This problem was resolved in [51]. The authors of [51] demonstrated that any extremely compact object (ECO) with a radius sufficiently close to the Schwarzschild radius possesses the same thermodynamic properties as the semiclassical BH analyzed by Bekenstein and Hawking. However, the analysis conducted in [51] starts from the assumption that the Hawking radiation spectrum must be strictly thermal. Yet, solid arguments based on energy conservation and BH back-reaction suggest that the spectrum of radiation emitted by BHs cannot be exactly thermal [52]. Indeed, energy conservation implies that the BH contracts during particle emission, resulting in a change in its geometry [52]. In this way, an important corrective term emerges [52]

$$\Gamma \sim \exp [-{\displaystyle \frac{\omega }{T_H}}(1-{\displaystyle \frac{\omega }{2M}})].$$

(25)

The additional term ${\displaystyle \frac{\omega }{2M}}$ appears in Eq. (25). One introduces the dynamical (effective) temperature[8,9,11,12]

$$T_D\left(\omega \right)\equiv {\displaystyle \frac{2M}{2M-\omega }}{T}_H={\displaystyle \frac{1}{4\pi (2M-\omega )}},$$

(26)

and rewrites Eq. (25) in a Boltzmann-like form similar to Eq. (23)

$$\Gamma \sim \exp [-{\beta }_D\left(\omega \right)\omega ]=\exp (-{\displaystyle \frac{\omega }{T_D\left(\omega \right)}}).$$

(27)

Here $\exp [-{\beta }_D\left(\omega \right)\omega ]$ is the dynamical (effective) Boltzmann factor, where

$${\beta }_D\left(\omega \right)={\displaystyle \frac{1}{T_D\left(\omega \right)}}={\beta }_H\left(1-{\displaystyle \frac{\omega }{2M}}\right),$$

(28)

with ${\beta }_H={\displaystyle \frac{1}{T_H}}.$ Therefore, to account for the BH dynamic state—determined by the variation of its geometry during particle emission—it is necessary to replace $T_H$ with $T_D\left(\omega \right)$ in the expression for the emission probability. $T_D\left(\omega \right)$ is a function of the energy-frequency of the emitted radiation. One can therefore quantify the deviation of the radiation spectrum from strict thermality via the ratio ${\displaystyle \frac{T_D\left(\omega \right)}{T_H}}={\displaystyle \frac{2M}{2M-\omega }}=1+{\displaystyle \frac{\omega }{2M}}+\mathcal{O}\left({\left|{\displaystyle \frac{\omega }{2M}}\right|}^2\right)$. In [11,12] it was shown that this type of thermodynamic analysis applies not only to traditional BHs possessing horizons and singularities, but also to ECOs with a radius sufficiently close to the Schwarzschild radius, and, in particular, to the thin spherical shells analyzed in the present paper. Indeed, in this case, the ECO has dynamical temperature $T_{D\left(ECO\right)}$ that matches the BH dynamical temperature in Eq. (26). Following [11,12,53], the standard Schwarzschild line-element is

$$d{s}^2=-\left(1-{\displaystyle \frac{2M}{r}}\right)d{t}^2+r^2\left({\sin }^2\left(\theta \right)d{\phi }^2+d{\theta }^2\right)+{\displaystyle \frac{d{r}^2}{1-\frac{2M}{r}}}.$$

(29)

Setting $\tau \equiv it$ and $x\equiv 4M\sqrt{1-{\displaystyle \frac{2M}{r}}}$ [54], the Gibbons-Hawking argument [48] allows one to write down the Euclidean form of the metric in Eq. (29) as

$$d{s}_e^2=x^2{\left[{\displaystyle \frac{d\tau }{4M\left(1-\frac{\omega }{2M}\right)}}\right]}^2+{\left({\displaystyle \frac{r}{r_D}}\right)}^{2}d{x}^2+r^2\left({\sin }^2\left(\theta \right)d{\phi }^2+d{\theta }^2\right).$$

(30)

In [53,54], corrections to the standard semiclassical Hawking temperature arising from quantum back-reaction are estimated using the Hamilton-Jacobi method. Specifically, the (complex) semiclassical wave function of a massless particle in spacetime—satisfying the Klein-Gordon equation—is expressed in terms of a (real) Hamilton-Jacobi function (proportional to the phase of the complex wave function) expanded in powers of ℏ. Terms of order $O(\hslash )$ and higher are then treated as quantum corrections to the semiclassical value of the Hamilton-Jacobi function. In this framework, the corrected expression for the BH temperature is $T_h=T_H{\left[1+{\sum }_i{\beta }_i{\displaystyle \frac{{\hslash }^i}{M^{2i}}}\right]}^{-1},$ where $T_H$ is the Hawking temperature. Note that the ${\beta }_i$ are dimensionless constant parameters and that the quantity ${\beta }_i{\displaystyle \frac{{\hslash }^i}{M^{2i}}}$ is also dimensionless, given that the reduced Planck constant has dimensions equal to the square of a mass. This is a consequence of the fact that units $G=c=k_B=1$ are used in [53,54]. The Euclidean form of the metric in Eq. (30) is regular at $x=0$ and $r=r_D$, while $\tau$ is considered to be an angular variable with period ${\beta }_D\left(\omega \right)$ [11,12,53,54] . Now, if one replaces the quantity ${\sum }_i{\beta }_i{\displaystyle \frac{{\hslash }^i}{M^{2i}}}$ in [53] with the quantity $-{\displaystyle \frac{\omega }{2M}}$ as in Eq. (25), and follows step by step the analysis in [53] one obtains the dynamical Schwarzschild metric as

$$d{s}^2=-\left(1-{\displaystyle \frac{2M_D}{r}}\right)d{t}^2+r^2\left({\sin }^2\left(\theta \right)d{\phi }^2+d{\theta }^2\right)+{\displaystyle \frac{d{r}^2}{1-\frac{2M_D}{r}}},$$

(31)

with the dynamical surface gravityequal to ${\displaystyle \frac{1}{4M_D}}.$ Using Eq. (21) one can also rewrite Eq. (31) as

$$d{s}^2=-\left[1-{\displaystyle \frac{2\left(M-\frac{\omega }{2}\right)}{r}}\right]d{t}^2+r^2\left({\sin }^2\left(\theta \right)d{\phi }^2+d{\theta }^2\right)+{\displaystyle \frac{d{r}^2}{1-\frac{2\left(M-\frac{\omega }{2}\right)}{r}}}.$$

(32)

Thus, to take the BH dynamic behavior into account, one must replace the BH mass M with the BH dynamical mass $M_D$ in Eq. (21). Then, Eq. (19) becomes

$$U=-{\displaystyle \frac{M_D^2}{r_{\alpha }}}.$$

(33)

Now, in the quantum analysis conducted, one wants to obtain the energy eigenvalues as absorptions starting from the BH formation, that is, from the massless BH, where the massless BH refers to the gravitational collapse situation before the formation of the first apparent horizon. This implies that one must replace $M\to 0and\omega \to M$ in Eqs. (21) and one must take the + sign (absorptions) in the same equations. Thus, one obtains

$$M_D={\displaystyle \frac{M}{2}},r_D\equiv 2M_D=M.$$

(34)

Thus, Eq. (33) becomes

$$U=-{\displaystyle \frac{M^2}{4r_{\alpha }}},$$

(35)

and, using Eq. (18) one has

$$U=-{\displaystyle \frac{M^2}{2r_g}}.$$

(36)

Thus, an interesting result was obtained. In the search for BH energy eigenvalues, the spherical shell potential to be inserted into the quantum equations is identical to its Newtonian counterpart [13]

$$U=-{\displaystyle \frac{M^2}{2r}},$$

(37)

where the Newtonian distance r is replaced by the gravitational radius $r_g.$ Setting

$$R\equiv 2r_g,$$

(38)

one gets

$$U=-{\displaystyle \frac{M^2}{R}},$$

(39)

which, again, is formally equal to the Coulomb potential energy of the hydrogen atom in Eq. (20) if the BH mass M is replaced by electron charge e. In the next sections, the potential energy in Eq. (39) will be used to discuss the generalized Schrödinger theory and the Dirac theory of BHs.

3. Generalized Schrödinger Theory of Black Holes

In a classical context, the potential energy in Eq. (39) might suggest a static configuration for the extremely massive thin spherical shell. However, if one wants to consider this BH in terms of a gravitational atom,” one must think of it as a highly excited quantum system, in accordance with a general belief. Thus, one assumes that the thin spherical shell of collapsing matter which has condensed onto the apparent horizon is subject to quantum oscillations due to its self-interacting gravitational potential energy. Moreover, it is well known that the Schwarzschild radius in a quantum framework is subject to quantum fluctuations due to potential absorptions of external particles or emissions of radiation [14,15]. Via Eq. (39), one can think of the BH gravitational atom found by Vaz [2] as equivalent to a fictitious quantum system formed by a fictitious particle, the electron, rotating around a fictitious central field, the nucleus, with a gravitational fine-structure constant

$${\alpha }_g\equiv M^2.$$

(40)

In other words, the mass M is interpreted not only as the BH mass, but also as the gravitational charge of the “electron.” Thus, twice the oscillating gravitational radius, denoted by the new dynamic variable R in Eq. (38), is considered to be the variable distance associated with the electron. Clearly, the real BH quantum physical state remains that of a thin oscillating spherical shell, while the equivalence with a fictitious system formed by an electron and a central field is a mathematical artifice that will, however, allow one to perform the calculations necessary to thoroughly analyze the physical properties of the system. The radius R is no longer a rigid classical geometric parameter, but becomes a quantum operator (or a dynamic coordinate) describing the intrinsic fluctuations of the shell itself under the influence of its own self-interacting gravitational potential energy. In a quantum context, to complete the analogy with the hydrogen atom, one must also assume that the thin, massive shell of condensed matter on the apparent horizon, is subject to rotations and, consequently, to deformations away from perfect spherical symmetry. In other words, one must consider the spherical shell in terms of probabilistic quantum orbitals, which are precisely subject to quantum oscillations and rotations. Following [16], with the replacement $e\to M,$ since both the mass of the central field and that of the “electron” are equal to $M,$ one can write the Schrödinger equation for the gravitational atom as

$$-{\displaystyle \frac{1}{2M}}{\nabla }^2\Psi \left(\mathbf{R}\right)-{\displaystyle \frac{M^2}{R}}\Psi \left(\mathbf{R}\right)=E\Psi \left(\mathbf{R}\right),$$

(41)

where ${\nabla }^2$ is the Laplacian operator and $\Psi \left(\mathbf{R}\right)$ is the wave function. Using traditional methods [16], the total wave function $\Psi (R,\theta ,\varphi )$ is broken down into two parts using the separation of variables technique:

$$\Psi (R,\theta ,\varphi )=Y_{\ell }^m(\theta ,\varphi ){\displaystyle \frac{y_l\left(R\right)}{R}},$$

(42)

where $y_l\left(R\right)$ is solution of the radial equation

$${\displaystyle \frac{d^2{y}_l\left(R\right)}{d{R}^2}}+\left[2M\left(E+{\displaystyle \frac{M^2}{R}}\right)-{\displaystyle \frac{l(l+1)}{R^2}}\right]y_l\left(R\right)=0,$$

(43)

and $Y_{\ell }^m(\theta ,\varphi )$ is an angular function known as spherical harmonics. The angular component describes the spatial geometry and shape of the orbitals. It depends on the angular momentum quantum numbers, while the radial component describes how the “electron’s” probability density changes as one moves away from the nucleus; see [16] for details. Eq. (41) generalizes the equation found in [8], which concerned only the s states ($l=0)$ of this gravitational hydrogen atom. Here, instead, the radial operator includes a centrifugal force term

$$V_{centrifugal}={\displaystyle \frac{l(l+1)}{2M{R}^2}}.$$

(44)

So, for $l>0,$ the centrifugal term repels the “electron” away from the nucleus. One recalls that the s states (with azimuthal quantum number $l=0)$ represent orbitals with perfect spherical symmetry, since their wave function depends only on the radial coordinate R and not on the polar angles $\theta$ and $\varphi .$ One can therefore identify them as the quantum counterpart of the classical Schwarzschild BH. The quantum number l defines the shape of the “atomic orbital” and determines the orbital angular momentum of the electron. Thus, one can identify states with non-zero l as rotating Kerr BHs, although a better description of the latter, which seems in agreement with astrophysical observations, will be given via Dirac theory in the next section.

The energy spectrum for the bound states ($E<0$) depends only on the principal quantum number $n=1,2,3,\dots$ [16] and is found as

$$E_n=-{\displaystyle \frac{M^5}{2n^2}}.$$

(45)

Hence, in order to completely solve the problem, also finding the BH mass spectrum, one must find the relationship between the mass and the total energy of the quantum shell. To find the relation between the BH mass and the BH total energy, the Oppenheimer and Snyder (OS) gravitational collapse [17] will be used instead of the LTB one. Following [8], one recalls that, in the case of the OS gravitational collapse, the internal solution is given by the well-known Friedmann-Lemaître-Robertson-Walker line element, which, using spherical coordinates and comoving time, turns out to be [8,10]

$$d{s}^2=d{\tau }^2-a^2\left(\tau \right)\left({\displaystyle \frac{d{r}^2}{1-r^2}}+r^{2}d{\theta }^2+r^2{\sin }^2\theta d{\phi }^2\right).$$

(46)

Now, one considers the Einstein field equation [8,10]

$$G_{\mu \nu }=-8\pi T_{\mu \nu },$$

(47)

and, assuming that the pressure is zero, as in the traditional OS collapse, one obtains the following relations [8]

$$\begin{array}{c}{\dot{a}}^2={\displaystyle \frac{8}{3}}\pi a^2\rho -1\\ \\ \ddot{a}=-{\displaystyle \frac{4}{3}}\pi a\rho ,\end{array}$$

(48)

where $\dot{a}={\displaystyle \frac{da}{d\tau }}$. To be consistent, one gets [8,18]

$${\displaystyle \frac{d\rho }{da}}=-{\displaystyle \frac{3\rho }{a}}.$$

(49)

By integrating Eq. (49), one gets

$$\rho ={\displaystyle \frac{C}{a^3}}.$$

(50)

The integration constant C can be obtained using the initial conditions as [8,10]

$$C={\displaystyle \frac{3a_0}{8\pi }}.$$

(51)

One rewrites Eq. (50) as

$$\rho ={\displaystyle \frac{3a_0}{8\pi a^3}}.$$

(52)

Now, one considers the standard Einstein-Hilbert Lagrangian [8,10]

$$L_{EH}={\displaystyle \frac{\sqrt{-g}R}{16\pi }}+L_M,$$

(53)

where, as usual, $L_M$ represents the matter fields. For the collapsing dust star, $L_M=\rho$ [8,10]. Using the Friedmann-Lemaître-Robertson-Walker line element in Eq. (46) one obtains

$$L_{FLRW}={\dot{a}}^2+{\displaystyle \frac{8}{3}}\pi a^2\rho ,$$

(54)

which can be rescaled as [8]

$$L={\displaystyle \frac{M{\dot{a}}^2}{2}}+{\displaystyle \frac{4}{3}}M\pi a^2\rho .$$

(55)

Eq. (55) can be rewritten as [8]

$$L={\displaystyle \frac{M{\dot{a}}^2}{2}}-V\left(a\right),$$

(56)

defining [8]

$$V\left(a\right)\equiv -{\displaystyle \frac{4}{3}}M\pi a^2\rho .$$

(57)

Inserting Eq. (52) into Eq. (57), one gets [8]

$$V\left(a\right)=-{\displaystyle \frac{M{a}_0}{2a}}.$$

(58)

The energy function associated with the Lagrangian is [8]

$$E={\displaystyle \frac{\partial L}{\partial \dot{a}}}\dot{a}-L.$$

(59)

Finally, by inserting Eq. (55) into Eq. (59) and using the first of Eq. (48) one finds the sought relation

$$E=-{\displaystyle \frac{M}{2}}.$$

(60)

Even in this case however, to take the BH dynamic behavior into account, one must replace the BH mass M with the BH dynamical mass $M_D={\displaystyle \frac{M}{2}}.$ So the previous equation becomes

$$E=-{\displaystyle \frac{M}{4}}.$$

(61)

Although the relationship found between total energy and mass was obtained in the particular case of the OS collapse, which is completely radial and therefore without rotations, if one assumes the absence of dissipation, it will have a general character even in the case of rotations, since the BH mass and total energy must still be conserved. To better understand this concept, let’s introduce the Christodoulou irreducible mass $M_{irr}$, which allows for the separation of the static mass-energy contribution from that due to rotation as [55]

$$M^2=M_{irr}^2+{\displaystyle \frac{J^2}{4M_{irr}^2}}.$$

(62)

Clearly, in the absence of dissipation (i.e., considering the BH as an isolated system), the initial mass and energy must be conserved, regardless of whether or not ($J=0$) rotation is present.

Now, by inserting Eq. (61) into Eq. (45), a bit of algebra permits obtaining the mass spectrum

$$M_n=2^{-{\displaystyle \frac{1}{4}}}\sqrt{n}.$$

(63)

To find the BH energy spectrum one still uses Eq. (61), yielding

$$E_n=-2^{-{\displaystyle \frac{9}{4}}}\sqrt{n}.$$

(64)

The mass spectrum in Eq. (63) is consistent with the BH mass spectrum found by Bekenstein in 1974 [18]. Bekenstein in fact obtained $M_n=\sqrt{{\displaystyle \frac{n}{2}}}$ using the Bohr-Sommerfeld quantization condition and assuming that the Schwarzschild BH behaves as an adiabatic invariant. This cannot be a coincidence; rather, it is a strong indication that the path one is following is the right one. In fact, it has been demonstrated—in research that received an Honorable Mention in the Gravity Research Foundation’s 2025 Essay Competition [12], and summarized in the previous section of this paper —that the thermodynamics of these compact spherical shells are identical to those of traditional BHs, which possess horizons and singularities [11,12].

It should be emphasized that in the classical framework there are considerable differences in the geometry of evolving trapped regions for OS and LTB. In the former, the boundary of the star gets trapped much before the singularity, while for the latter there are scenarios where trapping occurs extremely close to the central singularity and thus allowing outgoing null geodesics from a very dense region. On the other hand, the situation changes in the quantum framework. In fact, two different approaches to the quantum OS collapse have been recently developed [19,20,21]. The first one [19,20] concerns the application of a quantization procedure originally developed in a cosmological framework by Einstein’s historical collaborator Nathan Rosen [22]. The latter [21] concerns the use of Feynman’s path integral approach [23]. In both the cases, the same Schrödinger equation has been obtained. This Schrödinger equation was a special case of Eq. (41), which concerned only the s states ($l=0$) of the gravitational hydrogen atom. This means that the two different gravitational collapses, i.e., the OS and the LTB, generate the same quantum object. Thus, in this paper the results in [19,20,21] are generalized.

For the BH ground state ($n=1$), from Eq. (63) one obtains the mass as

$$M_1=2^{-{\displaystyle \frac{1}{4}}}\simeq 0.84,$$

(65)

in Planck units. Then, in standard units one gets $M_1=2^{-{\displaystyle \frac{1}{4}}}{m}_P$, where $m_P$ is the Planck mass, $m_P\equiv 2.17645*{10}^{-8}Kg.$ A negative total energy from Eq. (64) and a (fixed) gravitational radius are associated with the mass in Eq. (65)

$$E_1=-2^{-{\displaystyle \frac{9}{4}}}\simeq -0.21,$$

(66)

$$r_{g1}=2^{{\displaystyle \frac{3}{4}}}\simeq 1.68.$$

(67)

This is the state with minimum mass and minimum energy (the energy of this state is minimum in absolute value; in its real value, being negative, it is maximum). It represents the smallest possible BH. In the case of the semi-classical Bohr model of the hydrogen atom, the Bohr radius, which represents the classical radius of the electron in the ground state, is [16]

$$Bohrradius={\displaystyle \frac{1}{m{e}^2}},$$

(68)

where m is the mass of the electron. To obtain the corresponding Bohr radius for the gravitational hydrogen atom, it is necessary to replace both m and e in Eq. (68) with the mass $M_1$ that represents also the “gravitational charge.” Thus, the Bohr radius now becomes

$$b_1={\displaystyle \frac{1}{M_1^3}}=2^{{\displaystyle \frac{3}{4}}}\simeq 1.68,$$

(69)

which in standard units is read as $b_1={\displaystyle \frac{1}{M_1^3}}=2^{{\displaystyle \frac{3}{4}}}{l}_P$, where $l_P\equiv 1.61625*{10}^{-35}m$ is the Planck length. However, one must be careful here. In quantum mechanics, the Bohr radius is interpreted as the most probable distance at which the fictitious electron can be found when in its ground state. In this case, one can observe from Eq. (38) that the radial coordinate of the fictitious “electron” is twice the radial coordinate of the spherical shell, which is the real BH. Consequently, the most probable distance at which the spherical shell can be found is, in fact, equal to $2^{-{\displaystyle \frac{3}{4}}},$which corresponds exactly to the half of the BH classical (fixed) gravitational radius in Eq. (67):

$$b_1\left(BH\right)={\displaystyle \frac{1}{2}}{b}_1=2^{-{\displaystyle \frac{1}{4}}}\simeq 0.84.$$

(70)

Following [16], the wave function associated with the BH ground state is

$${\Psi }_1={\pi }^{-{\displaystyle \frac{1}{2}}}{b}_1^{-{\displaystyle \frac{3}{2}}}R\exp -\left({\displaystyle \frac{R}{b_1}}\right),$$

(71)

where ${\Psi }_1$ is normalized as ${\int }_0^{\infty }{\Psi }_1^{2}dR=1.$ Thus, the size of the fictitious quantum system is

$$\overline{R_1}={\int }_0^{\infty }{\Psi }_1^{2}RdR={\displaystyle \frac{3}{2}}{b}_1=1.5*2^{{\displaystyle \frac{3}{4}}}\simeq 2.52.$$

(72)

It should be remembered that in quantum mechanics the Bohr radius in Eq. (69) gives the radius with the maximum radial probability density rather than its expected radial distance (the size) [24]. The latter is in fact $3/2$ times the Bohr radius [24]. This depends on the long tail of the radial wavefunction [24] and is therefore given by Eq. (72). It is emphasized once again that the radial coordinate of the fictitious “electron” is twice the radial coordinate of the spherical shell. Therefore, the expected radial distance, i.e., the size of the real BH in its ground state, is, in fact, equal to $3*2^{-{\displaystyle \frac{5}{4}}}$ Planck lengths

$$\overline{R_1}\left(BH\right)={\displaystyle \frac{3}{2}}{b}_1\left(BH\right)=3*2^{-{\displaystyle \frac{5}{4}}}\simeq 1.26,$$

(73)

which turns out to be exactly three-quarters of its (fixed) classical gravitational radius, given by Eq. (67). So, a very interesting result has been found here: the size of the quantum BH is exactly three-quarters that of its classical counterpart. Upon reflection, one might have expected something of this sort. In fact, this is likely the price that must be paid for Vaz’s mechanism [2] to eliminate both the event horizon and the singularity: on average, the distance of the oscillating spherical shell from its center must be smaller than the fixed gravitational radius of its classical counterpart, due to the extremely strong self-interacting gravitational potential energy generated by the spherical shell itself. Thus, an interesting quantum mechanical representation of the Schwarzschild BH ground state at the Planck scale has been obtained. This Schwarzschild BH ground state represents the minimum energy level of a BH, compatible with the generalized uncertainty principle (GUP) [25]. The GUP in fact prevents a BH from completely evaporating by halting the Hawking evaporation process in exactly the same way that the Heisenberg uncertainty principle prevents the hydrogen atom from completely collapsing [25].

Let us now look at the situation at a generic energy level $n,$ still with $l=0$ (quantum counterpart of the classical Schwarzschild BH). One recalls that, in the hydrogen atom, the Bohr radius of the electron excited at the n level is [16],

$$Bohrradiusleveln={\displaystyle \frac{n^2}{m{e}^2}}.$$

(74)

In the case under examination, however, it is not enough to multiply $b_1$ by $n^2$, because the mass is varying; see also Rosen’s reasoning for the similar problem in [22]. Instead, one will need to replace m and e in Eq. (74) with the BH mass when the BH is excited at level n. Thus, by using Eq. (63) one finds

$$b_n={\displaystyle \frac{n^2}{M_n^3}}=2^{{\displaystyle \frac{3}{4}}}\sqrt{n}\simeq 1.68\sqrt{n},$$

(75)

with an associated expected radial distance (size)

$$\overline{R_n}={\displaystyle \frac{3}{2}}{b}_n=1.5*2^{{\displaystyle \frac{3}{4}}}\sqrt{n}\simeq 2.52\sqrt{n}.$$

(76)

From Eq. (63) one finds that the gravitational radius of the Schwarzschild BH of the $n-$th level is

$$r_{g\left(n\right)}=2M_n=2^{{\displaystyle \frac{3}{4}}}\sqrt{n}\simeq 1.68\sqrt{n}.$$

(77)

Recalling once again that the radial coordinate of the fictitious “electron” is twice the radial coordinate of the quantum spherical shell (the real BH), one finds that the “Bohr radius” for the spherical shell in an excited state n is equal to the half of what its classical (fixed) gravitational radius would be, that is

$$b_n\left(BH\right)={\displaystyle \frac{1}{2}}{b}_n=2^{-{\displaystyle \frac{1}{4}}}\sqrt{n}\simeq 0.84\sqrt{n},$$

(78)

while the expected radial distance, i.e., the size of the real BH excited at level n, is equal to $1.5$ times its classical (fixed) gravitational radius, that is

$$\overline{R_n}\left(BH\right)={\displaystyle \frac{3}{2}}{b}_n\left(BH\right)=3*2^{-{\displaystyle \frac{5}{4}}}\sqrt{n}\simeq 1.26\sqrt{n}.$$

(79)

Therefore, even in the case of a BH excited at the n level, the most probable distance at which the thin spherical shell can be found corresponds exactly to half the value of the BH classical (fixed) gravitational radius in Eq. (77). Thus, in general, the size of a real Schwarzschild BH—even one with a very large mass $M_n$ corresponding to a generic high excited state with large n —turns out to be exactly three-quarters the size of its classical counterpart. Again, one emphasizes that this is the price to be paid for Vaz’s mechanism [2], which eliminates both the event horizon and the singularity. The size of the corresponding quantum BH is smaller because its strong self-interacting gravitational potential energy causes it to oscillate within a range smaller than what would be its classical gravitational radius.

However, the analysis conducted thus far is valid only for non-relativistic oscillation speeds of the thin quantum spherical shell. Since BHs are often ultra-relativistic systems, relativistic corrections to the theory are necessary. This will be the subject of the next section of this paper.

4. Dirac Theory of Quantum Black Holes

A special relativistic extension to the BH Schrödinger theory has been proposed in [9] in terms of a Klein-Gordon theory. On the other hand, the Klein-Gordon theory is only valid for scalar particles with zero spin. If one wishes to continue following the analogy between the fictitious system and the hydrogen atom that emerged from the previous analysis, then one must interpret the “electron” in terms of a $spin-1/2$ particle (a fermion). Thus, in order to also include relativistic rotations and relativistic energies in the analysis, one must move from the Schrödinger equation to the Dirac equation. It follows that, while in Schrödinger’s theory the BH is interpreted as a quantum orbital, it must now be interpreted as a four-component spinor. Following [26], if one uses the scalar potential (it should be noted here that, within the framework of the Dirac equation and, more generally, relativistic quantum mechanics, the terms scalar and vector do not refer to the simple three-dimensional geometric nature of the potential, but rather to their Lorentz behavior—that is, how they transform when moving from one reference frame to another—and, consequently, how they couple mathematically to the Dirac spinor; see [26] for details) in Eq. (39), the BH Dirac equation can be written as

$$\left[\alpha ·\mathbf{p}+\beta \left(M-{\displaystyle \frac{M^2}{R}}\right)\right]\psi =E\psi ,$$

(80)

where are $\alpha$ and $\beta$ the famous $4\times 4$ Dirac matrices, $\mathbf{p}\equiv i\nabla$ is the momentum operator, and $\psi$ is a 4-component spinor. In spherical coordinates $\left(R,\theta ,\phi \right)$, the stationary wave spinor associated with a bound energy level of the system is written by separating the radial component from the angular component:

$${\psi }_{n\kappa m_j}\left(\mathbf{R}\right)={\displaystyle \frac{1}{r}}\left(\begin{array}{ccc}{P}_{n\kappa }\left(R\right)& & {\chi }_{\kappa m_j}(\theta ,\phi )\\ \\ i{Q}_{n\kappa }\left(R\right)& & {\chi }_{-\kappa m_j}(\theta ,\phi )\end{array}\right).$$

(81)

The bispinor is conceptually divided into two parts:

1.

Large Component (Upper Spinor): The first two rows describe the dominant part of the “electron”’s wave function in the non-relativistic regime.

2.

Small Component (Lower Spinor): The last two rows are coupled to the kinetic energy and the potential, and they tend to zero in the non-relativistic limit. Physically, they describe relativistic coupling effects.

In the Dirac formalism, orbital angular momentum L and spin S are not separately constants of motion due to spin-orbit coupling. Instead, the system is described by the following quantum numbers:

• $n:$ the principal quantum number;
• $m_j$: the projection of the total angular momentum along the $z-$axis;
• $\kappa$ (kappa): the Dirac operator quantum number, which encapsulates information about the total angular momentum j and the parity of the state. It is defined as:

  $$\kappa \equiv \begin{array}{ccc}-\left(j+{\displaystyle \frac{1}{2}}\right)=-\left(l+1\right)& & ifj=l+{\displaystyle \frac{1}{2}}\\ \\ \left(j+{\displaystyle \frac{1}{2}}\right)=l& & ifj=l-{\displaystyle \frac{1}{2}}.\end{array}$$

  (82)

The angular functions ${\chi }_{\kappa m_j}(\theta ,\varphi )$ are called spherical spinor harmonics; they are linear combinations of standard spherical harmonics and Pauli spinors. The exact radial solutions of the system —namely P and Q in Eq. (81)— are expressed in terms of associated Laguerre polynomials scaled by exponential factors and fractional powers of the radius. Using the standard method in spherical coordinates, see example 9.8 in [26], the exact solution for the energy eigenvalues is found (similar to the Darwin-Gordon formula) as

$$E_{n,j}=\pm M{\left[1-{\displaystyle \frac{M^4}{{\left(n-j-\frac{1}{2}+\sqrt{{\left(j+\frac{1}{2}\right)}^2+M^4}\right)}^2}}\right]}^{{\displaystyle \frac{1}{2}}}.$$

(83)

The quantum parameters inside the formula are:

• $n=1,2,3,...$ is the principal quantum number;
• $j={\displaystyle \frac{1}{2}},{\displaystyle \frac{3}{2}},{\displaystyle \frac{5}{2}},...$ is the quantum number of the total angular momentum $\mathbf{J}=\mathbf{L}+\mathbf{S},$ which can assume the values $j=l\pm {\displaystyle \frac{1}{2}}$;
• M is the mass of the “particle,” which is equal to the BH mass.

In this type of massive scalar potential, there are two branches of solutions for bound states: the solutions for positive and negative energies exhibit an identical character, since the interaction does not distinguish between positive and negative gravitational charges (mass is always positive) [26]. States with negative energy correspond to antiparticles. Hence, one needs to determine the correct sign of the solutions. To do this, one might think that the relationship between mass and energy in Eq. (61) should be replaced by its relativistic counterpart. Actually this is not necessary, since in the Schwarzschild coordinates that one is using here, in the OS gravitational collapse one has that $\dot{a}\to 0$ when collapsing matter approaches the gravitational radius [10]. This means that, if one wishes to use the conservation of energy at the moment the particle reaches the gravitational radius, starting from the non-relativistic Lagrangian in Eq. (55) is still perfectly fine. However, a correction must be made in Eq. (60). In fact, the rest energy of the “particle” M must be added. Ultimately, one obtains

$$E={\displaystyle \frac{3}{4}}M.$$

(84)

This means that one must choose the minus plus in Eq. (83), obtaining

$$E_{n,j}=M{\left[1-{\displaystyle \frac{M^4}{{\left(n-j-\frac{1}{2}+\sqrt{{\left(j+\frac{1}{2}\right)}^2+M^4}\right)}^2}}\right]}^{{\displaystyle \frac{1}{2}}}.$$

(85)

Imposing the condition in Eq. (84) into Eq. (85) now yields

$$\begin{array}{c}{\displaystyle \frac{3}{4}}M=\\ \\ =M{\left[1-{\displaystyle \frac{M^4}{{\left(n-j-\frac{1}{2}+\sqrt{{\left(j+\frac{1}{2}\right)}^2+M^4}\right)}^2}}\right]}^{{\displaystyle \frac{1}{2}}},\end{array}$$

(86)

which, looking for solutions for positive M, $j,$ and $n>j$, upon rearrangement, leads to the following equation for the fourth power of M:

$${\displaystyle \frac{9}{7}}{M}^4-{\displaystyle \frac{8\sqrt{7}}{7}}\left(n-j-{\displaystyle \frac{1}{2}}\right)M^2+{\left(n-j-{\displaystyle \frac{1}{2}}\right)}^2-{\left(j+{\displaystyle \frac{1}{2}}\right)}^2=0,$$

(87)

whose positive real solution gives one the spectrum of the BH mass as

$$M_{n,j}={\displaystyle \frac{\sqrt[4]{7}}{3}}{\left\{\left[\sqrt{9{\left(j+{\displaystyle \frac{1}{2}}\right)}^2+7{\left(n-j-{\displaystyle \frac{1}{2}}\right)}^2}+4\left(n-j-{\displaystyle \frac{1}{2}}\right)\right]\right\}}^{{\displaystyle \frac{1}{2}}}.$$

(88)

This allows one to also find the total relativistic energy as a function of n and j as

$$E_{n,j}={\displaystyle \frac{\sqrt[4]{7}}{4}}{\left\{\left[\sqrt{9{\left(j+{\displaystyle \frac{1}{2}}\right)}^2+7{\left(n-j-{\displaystyle \frac{1}{2}}\right)}^2}+4\left(n-j-{\displaystyle \frac{1}{2}}\right)\right]\right\}}^{{\displaystyle \frac{1}{2}}},$$

(89)

and the binding energy with the sign changed (negative binding energy)

$$▵E_{n,j}=-{\displaystyle \frac{\sqrt[4]{7}}{12}}{\left\{\left[\sqrt{9{\left(j+{\displaystyle \frac{1}{2}}\right)}^2+7{\left(n-j-{\displaystyle \frac{1}{2}}\right)}^2}+4\left(n-j-{\displaystyle \frac{1}{2}}\right)\right]\right\}}^{{\displaystyle \frac{1}{2}}}.$$

(90)

Let us now find the values of the mass and negative binding energy for the ground state ($n=1$, $j={\displaystyle \frac{1}{2}}$). One obtains (in Planck units)

$$M_{1,1/2}\simeq 0.94,$$

(91)

$$E_{1,1/2}\simeq -0.24.$$

(92)

By comparing Eqs. (91) and (92) with Eqs. (65) and (66), which correspond to the ground-state mass and energy in Schrödinger theory, one finds that the mass has increased slightly and the energy has decreased slightly (it is slightly more negative because its absolute value is slightly higher). Thus, high-speed effects and the spin of the electron modify the structure of the gravitational atom in the same way compared to what occurs with the hydrogen atom. Indeed, also in Dirac’s theory of the hydrogen atom, the system is more strongly bound than in Schrödinger’s theory, which means that its total energy levels (excluding rest energy) are lower (more negative). Clearly, if the approach proposed in this article—which aims to be pioneering—were to prove correct, further studies would be needed to better understand this analogy, as well as other analogies and/or differences, between the hydrogen atom and the BH gravitational atom. Now, let’s compare the theory with some astrophysical observations of rotating BHs. To do this, one recalls the well-known quantum relation between the total angular momentum and the quantum number j [26]:

$$\left|J\right|=\sqrt{j^2+j}.$$

(93)

Squaring both sides of this equation and solving for j one finds:

$$j={\displaystyle \frac{\sqrt{4{\left|J\right|}^2+1}-1}{2}},$$

(94)

which, for very large J is well approximated by $j\simeq \left|J\right|.$ Now one makes explicit n as a function of M and j from Eq. (86). Through a little algebra one gets

$$n-j-{\displaystyle \frac{1}{2}}+\sqrt{{\left(j+{\displaystyle \frac{1}{2}}\right)}^2+M^4}={\displaystyle \frac{4\sqrt{7}}{7}}{M}^2,$$

(95)

which leads to

$$n=j+{\displaystyle \frac{1}{2}}+{\displaystyle \frac{4\sqrt{7}}{7}}{M}^2-\sqrt{{\left(j+{\displaystyle \frac{1}{2}}\right)}^2+M^4}.$$

(96)

1.

One can start with the famous BH Cygnus X-1, which was the first astrophysical object globally recognized by the scientific community as a BH. It has a mass of $21.2\pm 2.2$ solar masses and a $J/M^2$ ratio of about $0.95$ [27]. From a quantum point of view, given the large value of J, it is therefore a highly excited, relativistic BH. For $J/M^2$ ratios close to unity, these are extremely rapid rotations close to the relativistic limit. One has: $j\simeq 0.95*{\left(21.2M_{\odot }\right)}^2\simeq 3.57*{10}^{78}.$ If one now inserts the values of M and j into Eq. (96), one finds $n\simeq 4.06*{10}^{78}$. The BH Cygnus X-1 thus turns out to be a highly excited quantum state, a gravitational spinor solution to the Dirac equation, having principal quantum number $n\simeq 4.06*{10}^{78}$ and total angular momentum quantum number $j\simeq 3.56*{10}^{78}.$ By inserting the determined quantum number values into Eq. (90), one finds a relativistic negative binding energy value $▵E\simeq -4.84*{10}^{38}{E}_p,$ where $E_p\equiv 1.96*{10}^9$ Joules is the Planck energy. Thus, the negative binding energy of Cygnus X-1 in standard units is about $▵E\simeq -9.49*{10}^{47}$ joules, and this has, in absolute value, roughly the same order of magnitude as its rest energy, $M{c}^2\simeq 3.80*{10}^{48}$ joules, and seems, therefore, consistent with astrophysical observations. This will also work for all the other 19 BHs that will be examined.

2.

Another stellar BH is M33 X-7 (located in the Triangulum Galaxy): it has a moderately high spin, estimated at about $J/M^2\simeq 0.84\pm 0.05$ and a mass of $11.4$ solar masses, with an uncertainty range of $-1.7$ to $+3.3$ solar masses [28]. Reasoning as in the previous case one has $j\simeq 0.84*{\left(11.4M_{\odot }\right)}^2\simeq 9.12*{10}^{77}$, $n\simeq 1.13*{10}^{78},$$▵E\simeq -2.60*{10}^{38}$ Planck energies, that is $▵E\simeq -5.10*{10}^{47}$ joules in standard units, with $M{c}^2\simeq 2.04*{10}^{48}$ joules. The data seem, therefore, consistent with astrophysical observations.

3.

A third stellar BH is GRO J1655—40, which has a mass of about $5.31\pm 0.07$ solar masses [29] and a lower spin than the previous ones, estimated at about $J/M^2\simeq 0.29\pm 0.003$. Now, one has $j\simeq 0.29*{\left(5.31M_{\odot }\right)}^2\simeq 6.83*{10}^{76}$. Reasoning as before, one finds $n\simeq 1.79*{10}^{77},$$▵E\simeq -1.21*{10}^{38}$ Planck energies, that is $▵E\simeq -2.38*{10}^{47}$ joules in standard units, with $M{c}^2\simeq 9.51*{10}^{47}$ joules. The results seem, therefore, consistent with astrophysical observations.

4.

Now, one can try to apply the Dirac theory for BHs that has been developed in this paper to the large BH located in the heart of our galaxy. The data for Sagittarius A* are as follows [30]: $4.0*{10}^6$ solar masses with an uncertainty range of $-0.6*{10}^6$ to $+1.1*{10}^6$ solar masses, $J/M^2\simeq 0.90\pm 0.06.$ Thus, it is: $j\simeq 0.90*{(4.0*{10}^6{M}_{\odot })}^2\simeq 1.20*{10}^{89}$, $n\simeq 1.42*{10}^{89},$$▵E\simeq -9.14*{10}^{43}$ Planck energies, which implies $▵E\simeq -1.79*{10}^{53}$ joules in standard units, with $M{c}^2\simeq 7.16*{10}^{53}$ joules. The results seem, therefore, consistent with astrophysical observations.

5.

The galactic BH M87* is the first BH ever imaged by the Event Horizon Telescope. Its data are: $M\simeq 6.5*{10}^9$ solar masses, $J/M^2\simeq 0.90\pm 0.05$ [31]. Thus, one gets: $j\simeq 0.90*{(6.5*{10}^9{M}_{\odot })}^2\simeq 3.18*{10}^{95}$, $n\simeq 3.76*{10}^{96},$$▵E\simeq -1.48*{10}^{47}$ Planck energies, which implies $▵E\simeq -2.91*{10}^{56}$ joules in standard units, with $M{c}^2\simeq 1.16*{10}^{57}$ joules. Thus, the results seem consistent with astrophysical observations.

6.

Another stellar BH is GRS 1915+105, which is one of the fastest-spinning BHs known. It indeed has a very high spin, estimated at about $J/M^2\gtrsim 0.98$ and a mass of $14.0\pm 4.4$ solar masses [32]. In this case one has $j\simeq 0.98*{\left(14.0M_{\odot }\right)}^2\simeq 1.60*{10}^{78}$, $n\simeq 1.79*{10}^{78}$, $▵E\simeq -3.20*{10}^{38}$ Planck energies, which implies $▵E\simeq -6.27*{10}^{47}$ joules in standard units, with $M{c}^2\simeq 2.51*{10}^{48}$ joules. Hence, the results seem consistent with astrophysical observations.

7.

LMC X-1 is a stellar-mass X-ray binary located in the Large Magellanic Cloud (a satellite galaxy of the Milky Way). It contains a rotating stellar-mass BH orbiting an O-type blue giant star. The BH data are: $M\simeq 10.91\pm 1.41$ solar masses [33], $J/M^2\simeq 0.92$ with an uncertainty range of $-0.07$ to $0.05$ [34]. Thus, one finds: $j\simeq 0.92*{\left(10.91M_{\odot }\right)}^2\simeq 9.14*{10}^{77}$, $n\simeq 1.07*{10}^{78},$$▵E\simeq -2.49*{10}^{38}$ Planck energies, which implies $▵E\simeq -4.88*{10}^{47}$ joules in standard units, with $M{c}^2\simeq 1.95*{10}^{48}$ joules. The results seem, therefore, consistent with astrophysical observations.

8.

XTE J1550-564 is another stellar-mass X-ray binary. The BH data are: $M\simeq 10.56$ solar masses with a $1\sigma$ interval between $9.68$ and $11.58$ solar masses, $J/M^2\simeq 0.34\pm 0.1$ [35]. Then, one has: $j\simeq 0.34*{\left(10.56M_{\odot }\right)}^2\simeq 3.17*{10}^{77}$, $n\simeq 7.4*{10}^{77},$$▵E\simeq -2.41*{10}^{38}$ Planck energies, which implies $▵E\simeq -4.73*{10}^{47}$ joules in standard units, with $M{c}^2\simeq 1.89*{10}^{48}$ joules. Thus, the results seem consistent with astrophysical observations.

9.

Another stellar-mass X-ray binary is 4U 1543-47. The BH data are: $M\simeq 9.4\pm 1$ solar masses, $J/M^2\simeq 0.46\pm 0.12$ [36]. Therefore, it is: $j\simeq 0.46*{\left(9.4M_{\odot }\right)}^2\simeq 3.39*{10}^{77}$, $n\simeq 6.43*{10}^{77},$$▵E\simeq -2.15*{10}^{38}$ Planck energies and $▵E\simeq -4.21*{10}^{47}$ joules in standard units, with $M{c}^2\simeq 1.68*{10}^{48}$ joules. Thus, the results seem consistent with astrophysical observations.

10.

MAXI J1820+070 is another stellar-mass X-ray binary. The BH data are: $M\simeq 8.48$ solar masses with a $1\sigma$ uncertainty between $-0.72$ and $0.79$ solar masses and a low spin-to-mass ratio $J/M^2\simeq 0.14\pm 0.09$ [37]. Hence, one finds: $j\simeq 0.14*{\left(8.48M_{\odot }\right)}^2\simeq 8.41*{10}^{76}$, $n\simeq 3.85*{10}^{77},$$▵E\simeq -1.94*{10}^{38}$ Planck energies and $▵E\simeq -3.80*{10}^{47}$ joules in standard units, with $M{c}^2\simeq 1.52*{10}^{48}$ joules. Then, the results seem consistent with astrophysical observations.

11.

Let us now consider the supermassive BH in Fairall 9. Its data are: $M\simeq \left(2.55\pm 0.56\right)*{10}^8$ solar masses and a spin-to-mass ratio $J/M^2\simeq 0.{52}_{-0.15}^{+0.19}$ (in this case, the measurement excludes extreme values in favor of an intermediate rotation) [38]. Thus, one obtains: $j\simeq 0.52*{(2.55*{10}^8{M}_{\odot })}^2\simeq 2.82*{10}^{92}$, $n\simeq 4.91*{10}^{92},$$▵E\simeq -5.83*{10}^{45}$ Planck energies and $▵E\simeq -1.14*{10}^{55}$ joules in standard units, with $M{c}^2\simeq 4.57*{10}^{55}$ joules. Hence, the results seem consistent with astrophysical observations.

12.

LMC X-3 is a stellar-mass BH located in the Large Magellanic Cloud. The BH data are: $M\simeq 6.98\pm 0.56$ solar masses and a low spin-to-mass ratio $J/M^2\simeq 0.{21}_{-0.22}^{+018}$ calculated at a $90\%$ confidence level, depending on the systematic error models applied [39]. Then, it is: $j\simeq 0.21*{\left(6.98M_{\odot }\right)}^2\simeq 8.54*{10}^{76}$, $n\simeq 2.85*{10}^{77},$$▵E\simeq -1.16*{10}^{38}$ Planck energies, which implies $▵E\simeq -3.13*{10}^{47}$ joules in standard units, with $M{c}^2\simeq 1.25*{10}^{48}$ joules. The results seem, therefore, consistent with astrophysical observations.

13.

XTE J1859+226 is a stellar-mass BH whose astrophysical parameters have been determined with extreme precision through the application of the Relativistic Precession Model based on data from the Rossi X-ray Timing Explorer satellite [40]. The BH data are: $M\simeq 7.85\pm 0.46$ solar masses and a low spin-to-mass ratio $J/M^2\simeq 0.149\pm 0.05$ [40]. Thus, one has: $j\simeq 0.149*{\left(7.85M_{\odot }\right)}^2\simeq 7.67*{10}^{76}$, $n\simeq 3.34*{10}^{77},$$▵E\simeq -1.79*{10}^{38}$ Planck energies, which implies $▵E\simeq -3.51*{10}^{47}$ joules in standard units, with $M{c}^2\simeq 1.41*{10}^{48}$ joules. Then, the results seem consistent with astrophysical observations.

14.

The physical data regarding the very first BH detected directly via gravitational waves (the historic event GW150914, resulting from the merger of two BHs) indicate the following values for the final BH (remnant) [41]: $M\simeq 62.2_{-3.4}^{+3.7}$ solar masses and a spin-to-mass ratio $J/M^2\simeq 0.{68}_{-0.06}^{+0.05}$. Therefore, one gets: $j\simeq 0.68*{\left(62.2M_{\odot }\right)}^2\simeq 2.20*{10}^{79}$, $n\simeq 3.17*{10}^{79},$$▵E\simeq -1.42*{10}^{39}$ Planck energies, which implies $▵E\simeq -2.79*{10}^{48}$ joules in standard units, with $M{c}^2\simeq 1.11*{10}^{49}$ joules. The results seem, therefore, consistent with gravitational wave observations.

15.

The microquasar H1743-322 is a BH with the following data [42]: $M=8.0\pm 1.2$ solar masses, $J/M^2\simeq 0.2_{-0.3}^{+0.7}$ at 90% confidence. Thus, it is: $j\simeq 0.2*{\left(8.0M_{\odot }\right)}^2\simeq 1.07*{10}^{77}$, $n\simeq 3.70*{10}^{77},$$▵E\simeq -1.83*{10}^{38}$ Planck energies, which implies $▵E\simeq -3.58*{10}^{47}$ joules in standard units, with $M{c}^2\simeq 1.43*{10}^{48}$ joules. Thus, the results seem consistent with astrophysical observations.

16.

The parameters of the BH A0620-00 (also known as V616 Monocerotis) are as follows [43]: $M=6.6\pm 0.25$ solar masses, $J/M^2\simeq 0.12\pm 0.19$. Thus, it is: $j\simeq 0.12*{\left(6.6M_{\odot }\right)}^2\simeq 4.36*{10}^{76}$, $n\simeq 2.27*{10}^{77},$$▵E\simeq -1.51*{10}^{38}$ Planck energies and $▵E\simeq -2.95*{10}^{47}$ joules in standard units, with $M{c}^2\simeq 1.18*{10}^{48}$ joules. Thus, the results seem consistent with astrophysical observations.

17.

The final BH produced by the event GW190521 has the following data [44]: $M={142}_{-16}^{+28}$ solar masses, $J/M^2\simeq 0.{72}_{-0.12}^{+0.09}$. Thus, one finds: $j\simeq 0.72*{\left(142M_{\odot }\right)}^2\simeq 1.21*{10}^{80}$, $n\simeq 1.68*{10}^{80},$$▵E\simeq -3.24*{10}^{39}$ Planck energies and $▵E\simeq -6.36*{10}^{48}$ joules in standard units, with $M{c}^2\simeq 2.54*{10}^{49}$ joules. Hence, the results seem consistent with gravitational wave observations.

18.

The stellar-mass BH MAXI J1535–571, located in our galaxy, has the following data [45]: $M=10.{39}_{-0.62}^{+0.61}$ solar masses, $J/M^2\simeq 0.{67}_{-0.04}^{+0.16}$. Thus, it is: $j\simeq 0.67*{\left(10.39M_{\odot }\right)}^2\simeq 6.04*{10}^{77}$, $n\simeq 8.82*{10}^{77},$$▵E\simeq -2.37*{10}^{38}$ Planck energies and $▵E\simeq -4.65*{10}^{47}$ joules in standard units, with $M{c}^2\simeq 1.86*{10}^{48}$ joules. Then, the results seem consistent with astrophysical observations.

19.

The physical data regarding the BH located at the center of the Seyfert galaxy NGC 3783 are [46] $M=3*{10}^7$ solar masses, with an uncertainty typical of optical reverberation methods, estimated within a factor of 2 ($\sim 0.3$ dex), $J/M^2\ge 0.88$ at the $99\%$ confidence level. Thus, it is: $j\simeq 0.88*{(3*{10}^7{M}_{\odot })}^2\simeq 6.61*{10}^{90}$, $n\simeq 7.96*{10}^{90},$$▵E\simeq -6.85*{10}^{44}$ Planck energies and $▵E\simeq -1.34*{10}^{54}$ joules in standard units, with $M{c}^2\simeq 5.37*{10}^{54}$ joules. Thus, the results seem consistent with astrophysical observations.

20.

The BH Swift J1753.5—0127 is part of a stellar-mass X-ray binary system located in our galaxy. Its data are [47]: $M=7.4\pm 1.2$ solar masses, $J/M^2\simeq 0.{76}_{-0.15}^{+0.11}$. Thus, it is: $j\simeq 0.76*{\left(7.4M_{\odot }\right)}^2\simeq 3.48*{10}^{77}$, $n\simeq 4.64*{10}^{77},$$▵E\simeq -1.69*{10}^{38}$ Planck energies and $▵E\simeq -3.31*{10}^{47}$ joules in standard units, with $M{c}^2\simeq 1.33*{10}^{48}$ joules. Hence, the results seem consistent with astrophysical observations.

It would therefore appear that Dirac’s quantum theory, applied to the new BH framework proposed by Vaz [2] and consistent with Einstein’s initial 1939 calculation [4], is in good agreement with astrophysical and/or gravitational wave observations of rotating BHs. Data concerning 20 rotating BHs—whether of stellar origin, located in galactic nuclei, or identified via gravitational wave astronomy—were analyzed. It emerged that all 20 correspond to highly excited quantum states, specifically, gravitational spinor solutions to the Dirac equation for specific values of the principal quantum number and the total angular momentum quantum number.

5. Concluding Remarks

In summary, this paper has shown that the new BH framework, proposed by Vaz in 2014 [2] and inspired by an idea of Hawking’s [1], is compatible with the quantum theories of Schrödinger and Dirac. According to this framework, the final outcome of gravitational collapse is not an event horizon, but rather an apparent horizon on which collapsing matter condenses, consistent with what Einstein had already predicted in 1939 regarding the localization of collapsing particles within a thin spherical shell [4]. This result generalizes previous analyses in which the BH mass and energy spectra were obtained using a Schrödinger-type approach in the non-relativistic regime [8] and a Klein-Gordon approach in the relativistic regime [9]. The analysis in this paper further supports previous conclusions that, rather than a spacetime singularity covered by an event horizon, the final result of gravitational collapse is an essentially quantum object, an extremely compact dark star, a thin spherical shell subject to quantum oscillations. This thin spherical shell should therefore not be interpreted as a simple oscillating shell in classical mechanics. Rather, via the introduction of a fictitious system composed of a “particle” and a “central field,” it is interpreted as a quantum orbital in Schrödinger’s theory and as a four-component spinor in Dirac’s theory. This gravitational atom is not supported by any degeneracy pressure, but rather, by quantum gravity, just as ordinary atoms are supported by quantum mechanics. Previous analyses focused on the Schwarzschild BH, i.e., the static BH. This paper takes the next step, introducing spin number rotations and relativistic energies from the Schrödinger equation to the Dirac equation. The results obtained seem consistent with astrophysical observations of rotating BHs.

Finally, an important implication for the BH information paradox must be emphasized. The analysis carried out in this paper has shown that the real BH quantum state is in terms of an object free of horizons and singularities; that is, a self-interacting, highly excited massive quantum shell, generated by matter condensing on the apparent horizon. Thus, the internal states of the oscillating quantum shell, obeying Schrödinger’s non-relativistic theory and Dirac relativistic theory must be in causal connection with distant observers, in agreement with the strong equivalence principle, which states that special relativity must hold locally for all of the laws of physics in all of spacetime as seen by timelike observers. This guarantees preservation of the physical information. Indeed, although, as it has been shown in [11,12] and summarized in section 1 of this paper, the Hawking mechanism occurs in an almost identical manner just outside the surface of the spherical shell, the paradox disappears for the following structural reasons, in a way similar to what happens in the fuzzball paradigm [56]:

1.

Absence of an internal vacuum region. In fact, in Hawking’s model, virtual particle pairs separate at the horizon: one falls into the vacuum beyond the horizon (becoming inaccessible) while the other escapes as radiation. In the approach presented in this paper, there is neither real horizon nor empty interior; instead, there are quantum and probabilistic oscillations of the spherical shell. Infalling matter does not cross a horizon to end up in a future singularity; rather, it strikes the shell’s surface, shifting it to a different energy level.

2.

The radiation is not pure quantum vacuum. Indeed, since Hawking-like radiation is generated just outside the physical surface of the spherical shell (rather than at a mathematical, geometric horizon in a vacuum) [11,12], it is directly influenced by the body’s internal structure. The members of the formed particle pair do not fall into the cosmic vacuum but instead interact with the spherical shell, modifying its oscillation modes.