Very High-Energy Cosmic Ray Particles from the Kerr Black Hole at the Galaxy Center

Abstract

After a just tribute to Guido Barbiellini, we show how the notion of a maximum force ($F_{\max }=c^4/\left(4G\right)\simeq 3\times {10}^{43}$ Newtons) present on the event horizon of a black hole (BH) can be used in conjunction with the Wilson area rule to obtain the surface confinement of the mass of a BH analogous to the surface confinement of quarks. This is then translated into the central result of the paper that PeV scale protons exist on the surface of the Kerr BH residing at our galactic center, a result in complete agreement with the HAWC Collaboration result of a Pevatron at the galactic center. We conjecture that the supermassive BHs present at the center of most galaxies are not born out of a galactic collapse but that they must have been present since the formation of their hosting galaxy.

1. A Tribute to Guido Barbiellini

One of the authors (YS) had the great fortune to have met Guido early in his career at the National INFN labs in Frascati, Italy. We soon became very close friends, and this friendship was renewed when two decades later Guido came to Northeastern for an astrophysics seminar (attended—amongst others—by his son Bernardo and by YS).

In an article written in memory of Giordano Diambrini Palazzi [1], Guido provides a very touching description of the birth and development of the Italian national lab at Frascati, and he reminisces about a most friendly ambiance prevailing between the physicists in the late fifties, during the sixties, and beyond. We learn how coherent polarized photon sources were invented there that would later be copied at other labs. This story is not too dissimilar to Touscheck’s invention of the first electron–positron colliding beam machine ADA at Frascati (along with its larger avatar ADONE at which Guido worked) that would be followed by bigger and more powerful machines elsewhere, culminating in the LHC (Large Hadron Collider) at CERN [2].

Guido also describes his participation in the first ever hadron collider at the CERN ISR experiment as well [3]. He notes the following: The experience was again very interesting also from a human perspective. Carlo Rubbia was leading the experiment with his style, while Giordano, very thoughtful according to his character, was perhaps the most followed.

We shall not recall here the plethora of research avenues followed by Guido but narrow it to areas of interest of the present paper. In 2015, we had made an ab initio calculation of the cosmic ray energy spectrum index for electrons and positrons and arrived at the value 3.151 [4]. In 2017, the AMS experiment had measured its value to be (3.170 ± 0.008 ± 0.008); with similar results (3.08 ± 0.05) obtained by HESS(+LAT) Collaborations [5]. At the time, YS was unaware of Guido’s participation in LAT; otherwise, he would have gone to him to discuss the theoretical result. Alas, YS presented the theoretical result to a prominent member of the AMS Collaboration, whose sole response to the extraordinary agreement was Yogi, it must be a cosmic coincidence!

Amen.

The HESS measurements gave the first indication of a cutoff at 2 TeV. These results can be interpreted as local cosmic ray electron sources with a spectral cutoff at this energy. The excess events found by HESS(+LAT) collaborations [6,7] would eventually lead HESS and HAWC Collaborations to find the extraordinary results about very high-energy cosmic ray events coming from our galactic center supermassive black hole (BH): a leitmotif of the present paper. A theoretical explanation of this important result was presented in our earlier paper [8]; it is summarized here first, and further issues will be discussed in a later section.

2. Introduction

It is by now common knowledge that in a seminal paper [9], Hawking showed that contrary to the classical notion in which a BH can only absorb particles, a quantum BH can also create and emit particles as if its surface were at a temperature $T_H$ satisfying $k_B{T}_H=\hslash \kappa /\left(2\pi \right)$, where $c\kappa$ is the surface gravity of the BH. For a Schwarzschild BH of mass M and radius $R_s=2GM/c^2$, $\kappa =c/\left(2R_s\right)$. Such quantum evaporation and radiative processes have been studied and confirmed by Bekenstein [10] and Unruh [11] among others. On the other hand, in a series of papers [12,13,14,15,16] by ’t Hooft and co-workers, it has been argued that for a Schwarzschild BH, the radiation temperature should be doubled, from $T_H$ to $2T_H$, since the BH entropy is halved due to a decrease in the number of linearly independent quantum states by precisely the same amount.

An approach using two apparently different inputs from the above was employed in [17] to obtain the Hawking result for the evaporation process of a BH. The first input is that of a maximum gravitational tension $F_{\max }=\tau =c^4/\left(4G\right)\simeq 3\times {10}^{43}$ Newtons, such a force is only realized at the horizon of a BH [18,19,20]. The second input employs a gravitational Wilson closed-loop action to obtain the central result that all matter is confined on the horizon surface by the action area law via the gravitational tension in the closely analogous sense that the Wilson action area law also describes a surface confinement of quarks in QCD [21].

Thus, through completely different reasonings, both ’t Hooft and reference [17] converged on the basic result that the dynamics of a BH is essentially on its surface—a result substantially different from the standard 3-dimensional description of it.

However, the detailed formalism and the tools employed by ’t Hooft and that in reference [17] being different, they both need to be discussed and understood. For example, in the Abstract of [16], it is stated that …the quantum black hole has no interior, or equivalently, the black hole interior is a quantum clone of the exterior region. On the other hand, perhaps more prosaically, we consider a BH as a perfect mass conductor with all its mass on its (event horizon) surface. The rather striking results derived in Section 4 from this notion of surface confinement are the central results of the present paper. For example, we are able to show that by virtue of such a surface confinement, the rotating Kerr BH at the galactic center of the Milky Way is driven to become a powerful source of extremely high-energy cosmic ray protons. Our calculations in Section 5 quantitatively bear out the recent experimental result from the HAWC Collaboration that calls for the existence of a PeVatron at GC (the galactic center) [22].

3. Black Hole Maximum Tension, Wilson Area Law and Surface Confinement

Conventional general relativity supplies the notion of a vacuum tension $\tau =c^4/\left(4G\right)\simeq 3\times {10}^{43}$ Newtons. This vacuum tension determines the maximum force (see Refs. [11,18,19,20]) $F\le \tau$, that can be exerted on any material body, with equality realized on bodies confined to a horizon surface. This allowed us to prove (rigorously, if anything is rigorous in relativistic quantum field theory!) the gravitational Wilson action area result [21] for matter confined on horizons [17]. In Euclidean field theory it led us to the well-known entropy area theorem on black hole horizons. Recalling our horizon thermodynamics via Equations (19)–(27) from Ref. [17]:

• the entropy area relation

  $$\begin{array}{c}\hfill {\displaystyle \frac{d\mathcal{S}}{k_B}}={\displaystyle \frac{\tau }{\hslash c}}d\mathcal{A},\end{array}$$

  where $\mathcal{A}$ is the area;
• the spherical BH radius

  $$\begin{array}{c}\hfill R_s={\displaystyle \frac{2GM}{c^2}}={\displaystyle \frac{\mathcal{E}}{2\tau }},\end{array}$$

  where $\mathcal{E}=M{c}^2$ is the BH energy;
• the area of the BH horizon

  $$\begin{array}{c}\hfill \mathcal{A}=4\pi R_s^2=\pi {\left({\displaystyle \frac{\mathcal{E}}{\tau }}\right)}^2;\end{array}$$
• the horizon entropy

  $$\begin{array}{c}\hfill \mathcal{S}=\pi k_B{\displaystyle \frac{{\mathcal{E}}^2}{\hslash c\tau }};\end{array}$$
• the temperature

  $$\begin{array}{c}\hfill T={\left({\displaystyle \frac{d\mathcal{S}}{d\mathcal{E}}}\right)}^{-1}\end{array}$$

  so that

  $$\begin{array}{c}\hfill k_{B}T={\displaystyle \frac{\tau }{2\pi }}{\displaystyle \frac{\hslash c}{\mathcal{E}}}={\displaystyle \frac{\hslash c}{4\pi R_s}};\end{array}$$
• finally, the free energy

  $$\begin{array}{c}\hfill \mathcal{F}=\mathcal{E}-T\mathcal{S}=T\mathcal{S}={\displaystyle \frac{\mathcal{E}}{2}}.\end{array}$$

There is surface tension $\sigma$ of the BH given by

$$\begin{array}{c}\hfill \sigma ={\displaystyle \frac{\mathcal{F}}{\mathcal{A}}}={\displaystyle \frac{k_{B}T}{\hslash c}}\tau ={\displaystyle \frac{\tau }{4\pi R_s}}.\end{array}$$

(1)

If one draws an equator around the sphere, then the two halves of the sphere attract each other via the vacuum gravitational tension. Equation (1) is central to our discussion. The surface tension of the horizon is from the confinement of both energy and entropy on the surface of the black hole. There is no need to discuss what is “internal” to the black hole. All of the physical quantities are confined to the horizon surface by the gravitational tension. A BH with all its mass–energy on the surface is the best analogue to an ideal conductor in electro-dynamics with all its charge on the surface. [Of course, with the most important difference that due to charges of both signs in electro-dynamics, charge neutral objects such as atoms and molecules are not confined to be on the surface of an ideal conductor; clearly, for gravity with positive masses only, all black hole mass is confined to the horizon surface of the BH.]

4. Practical Consequences of All BH Mass Lying on Its Surface 

Going beyond Ref. [17], we find that the inter-particle dynamics is dramatically changed as the BH mass is changed from being distributed (spherically uniformly) in 3-dimensions to being uniformly distributed on the 2-dimensional surface of the BH sphere. We shall repeat the argument from [8] that we expect to hold only for supermassive SMBH (of mass $\ge {10}^5$$M_{\odot }$).

Consider the inter-particle distance $L_2$ in 2 and $L_3$ in 3 dimensions for the same radius $R_s$ of the BH and the same total number $N_T$ of the particles (here assumed to be nucleons), which, in terms the BH mass M is

$$\begin{array}{c}\hfill N_T=N_{\odot }{\displaystyle \frac{M}{M_{\odot }}}\simeq 1.2\times {10}^{57}{\displaystyle \frac{M}{M_{\odot }}}.\end{array}$$

The distances are

$$\begin{array}{ccc}\hfill L_2& =& {\left({\displaystyle \frac{\left[\mathrm{area}\right]}{N_T}}\right)}^{1/2}={\left({\displaystyle \frac{4\pi R_s^2}{N_T}}\right)}^{1/2}\simeq \left(3\times {10}^{-10}\mathrm{fm}\right){\left({\displaystyle \frac{M}{M_{\odot }}}\right)}^{1/2},\hfill \\ \hfill L_3& =& {\left({\displaystyle \frac{\left[\mathrm{volume}\right]}{N_T}}\right)}^{1/3}={\left({\displaystyle \frac{4\pi R_s^3/3}{N_T}}\right)}^{1/3}\simeq \left(0.45\mathrm{fm}\right){\left({\displaystyle \frac{M}{M_{\odot }}}\right)}^{2/3},\hfill \end{array}$$

and their ratio

$$\begin{array}{c}\hfill {\displaystyle \frac{L_3}{L_2}}={\left({\displaystyle \frac{N_T}{36\pi }}\right)}^{1/6}\simeq 1.5\times {10}^9{\left({\displaystyle \frac{M}{M_{\odot }}}\right)}^{1/6}\end{array}$$

As expected, the inter-particle spacing $L_2$ (when the mass is spread out over the 2-dimensional surface of the BH) compared to $L_3$ (when the BH mass is spread uniformly over a 3-dimensional volume of the BH sphere) is smaller by factors of over a billion; the precise value depends upon how massive the BH is with respect to the solar mass $M_{\odot }$. We can obtain an estimate of the scale of mean particle energies $\langle E\rangle$ for a given length scale L through the uncertainty principle:

$$\begin{array}{c}\hfill \langle E\rangle \ge {\displaystyle \frac{\hslash c}{L}}.\end{array}$$

Some limiting cases such as the SMBH mass varies between say ${10}^5{M}_{\odot }$ to a super massive BH of mass $6.5\times {10}^9{M}_{\odot }$ (that has been found by the Event Horizon Telescope to reside at the center of the giant galaxy M87 [23]), illustrating the range of variation in the energy scale (and thus the corresponding standard model induced particle dynamics) when all the mass is concentrated at the surface of the BH versus a 3-dimensional spherically symmetric distribution.

There is a factor of 2 difference in the horizon radius between a Schwarzschild and a Kerr BH [24]. Explicitly, $R_s=2GM/c^2=2R_k$, where $R_s$ and $R_k$ are the Schwarzschild and Kerr radii, respectively. Estimates of the mean energies in 3 mass hypotheses for rotating Kerr BHs, in the 2 and 3 dimensional cases are given in

Table 1. Mean energies for three mass hypotheses for 2 and 3 dimensional Kerr BH.

	$\mathit{M}/{\mathit{M}}_{\odot }=10$	$\mathit{M}/{\mathit{M}}_{\odot }=4.3·{10}^6$	$\mathit{M}/{\mathit{M}}_{\odot }=6.5·{10}^9$
${\displaystyle {\langle E\rangle }_2=\frac{2\hslash c}{(3·{10}^{-10}\mathrm{fm})}{\left(\frac{M_{\odot }}{M}\right)}^{1/2}}$	420 PeV	$0.630$ PeV	$0.016$ PeV
${\displaystyle {\langle E\rangle }_3=\frac{2\hslash c}{\left(0.45\mathrm{fm}\right)}{\left(\frac{M_{\odot }}{M}\right)}^{2/3}}$	190 keV	33 keV	0.25 keV

.

The values of Table 1 illustrate how different the dynamics near the surface of a BH becomes both because the inter-particle dynamics is highly peaked (for our own galactic center Kerr BH in the PeV scale for protons) if there is only a surface mass and also because the heavier the BH the less active it would be energetically. The super massive BH at the center of the M87 galaxy would be almost atomic were the mass spherically distributed. Clearly, if the surface dynamics effects discussed here are indeed realized, they would provide a clean experimental signal about how the mass is distributed over a BH by virtue of the intensity of particle interactions near the surface of a BH. In particular, we have found—as shown above in Equation (2)—that only if the mass is distributed over the surface of our galactic black hole, we shall have a powerful source of cosmic PeVatron protons.

In the following Section 5, we shall discuss experimental data from the HAWC Collaboration that distinctly shows that our Kerr BH at the galactic center is a cosmic source of PeV protons thereby confirming our hypothesis that this BH is indeed a perfect mass conductor. Please note that were it much more massive, say the BH had a mass of $6.5\times {10}^9{M}_{\odot }$ such as the one at the center of galaxy M87, the energy scale for cosmic protons would be meager tens of TeV, and thus it would not match the experimental UHE gamma ray energy spectrum found between ($6÷118$ TeV) [22].

An attentive reader may have wondered about the fate of much less massive BHs, say with a mass of $10M_{\odot }$, that should be radiating 100 PeV protons. It has been shown that such BHs would have a rather short half-life and would not last long. Thus, the mass of BHs that are long lasting but at the same time are also producers of very high-energy protons is in a very limited region [8]. Further work is required to investigate this interesting subject along with a discussion of the critical indices (parameter $\gamma$ in Equation (2)) in the energy spectrum of high-energy cosmic ray particles [4,25,26,27,28].

5. Experimental Verification of Our Predictions by the HAWC Collaboration

We summarize here the salient aspects of 7-year data about ultra high-energy (UHE) gamma rays from the galactic center (GC) obtained by the HAWC (High Altitude Water Cherenkov) Observatory [22].

• Very high-energy ($6÷118$ TeV) gamma ray data are best described as originating from a point like source [HAWC J1746-2856] with a power law spectrum for N (i.e., the number current flux density per unit area per unit time) that has been parametrized as

  $$\begin{array}{c}\hfill {\displaystyle \frac{dN}{dE}}=\varphi {\left({\displaystyle \frac{E}{26TeV}}\right)}^{\gamma },\end{array}$$

  where

  $$\begin{array}{c}\hfill \gamma =-2.88\pm 0.15,\varphi =\left(1.5\pm 0.3\right)\times {10}^{-15}{\left({\mathrm{TeV}\mathrm{cm}}^2\mathrm{s}\right)}^{-1}.\end{array}$$

  (2)
• There is no evidence of a spectral cut-off [29,30] up to 100 TeV in the HAWC data.
• The HAWC Collaboration concludes that the UHE gamma rays detected by them originate via hadronic interaction of PeV cosmic ray protons with the dense ambient gas and confirms the presence of a proton PeVatron at the GC, but they do not provide a mechanism for it.

These experimental results require a dynamical mechanism for the existence of a proton Pevatron, and it is pleasing to note that such a Pevatron emerges naturally under our central notion in the present paper that the Kerr BH at the center of our galaxy is indeed a perfect mass conductor, in that all its mass is concentrated at its event horizon (see Table 1).

6. On the Origins of Supermassive BH

Most BHs are supposed to be formed through the gravitational collapse of a massive star [24,31]. It is difficult to imagine that supermassive BHs [with mass ranging between (${10}^5÷{10}^9$)$M_{\odot }$)] present at the center of most galaxies originate similarly. That is, SMBHs were formed through the gravitational collapse—unless of course we extend the definition of collapse to include within it the very birth of the mother galaxy.

Consider our own Milky Way galaxy that is supposed to contain about ${10}^8$ BHs with masses ranging between $(10÷100)M_{\odot }$ and exactly one supermassive BH of mass $4.3\times {10}^6{M}_{\odot }$ at the galactic center. One fundamental difference between our low mass LMBHs and the SMBH is that while the surface of the latter is acting as a PeVatron, there is apparently no similar radiational activity from an LMBH. Radiational energy considerations and the center position they occupy force us to conclude that SMBH must have been formed around the time of the formation of its hosting galaxy, whereas there are no such restrictions on LMBH. In this respect, both the conjectured mini black holes (lying in the mass range say of a proton or a similar nuclear particle) and SMBHs (in the mass range $({10}^5÷{10}^9)M_{\odot }$) may have been created around the birth of the parent galaxy. Admittedly a difficult question to answer but that is not beyond all conjecture is that there exists a critical mass $M^{*}$ separating the LMBH branch from the SMBH branch. The point being made here is that a configuration of protons (and electrons) all lying on the horizon surface of a BH have quite high individual kinetic energies compared to the kinetic energies of particles when they are uniformly distributed in three dimensions. Were there a phase transition (brought about through an interplay of gravitational and nuclear interactions), we could associate this difference in the kinetic energy to the latent heat of the phase transition.

Only further research will tell.

7. Conclusions

For (super)massive BHs that are supposed to be at the center of most galaxies, we have shown that the inter-particle distance can indeed become very small when all mass is distributed over the horizon surface. This implies that the mean energy of the particles can be driven to extremely large values. And we have explored here the notion that cosmic rays are emitted from the surfaces of BHs just as they are from neutron stars. The cosmic rays themselves are in the stellar atmospheric winds blowing away from the source. The cosmic rays are equivalently nuclei which are evaporating from the surface. The method of computing the energy distribution of the evaporated cosmic rays is closely analogous to those employed by Landau and Fermi for the evaporation of nucleons in the Bohr–Mottelson liquid drop model [4,25,26,27,28].

Our prediction about PeV-scale protons at the surface of the Kerr black hole situated at our galactic center agrees with that of the HAWC Collaboration claiming the existence of a PeVatron proton source at the galactic core (through their observation of ultra high-energy photons with energies between ($6÷118$) TeV). In view of the discovery of a PeVatron at the galactic core through the observation of UHE photons, it tempts us to call super massive BHs bright holes rather than black holes.

It seems inescapable to generalize Wheeler’s no hair theorem for a BH to read as follows: A BH is characterized by its mass; its angular momentum and its total charge: all lying on its (event horizon) surface.