Kerr Black Hole as a Pevatron at the Galaxy Center

Abstract

Conventional general relativity supplies the notion of a vacuum tension and thus a maximum force $F_{\max }=c^4/\left(4G\right)\simeq 3·{10}^{43}$ N that is realized for a black hole. In conjunction with the Wilson area rule, we are thus led to the surface confinement of the mass of a black hole analogous to the surface confinement of quarks. The central result of our paper is that PeV scale protons can be generated on the surface of a Kerr black hole. This result is in concert with the presence at the galactic center of that Pevatron accelerating mechanism first suggested by the H.E.S.S. Collaboration and further confirmed by the HAWC Observatory.

1. Introduction

In a seminal paper [1], Hawking proposed that, contrary to the classical notion in which a black hole (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 $R_s=(2GM/c^2)$, the surface gravity is $c\kappa =c^2/\left(2R_s\right)$. Such quantum evaporation and radiative processes have been studied and confirmed by Bekenstein [2] and Unruh [3] among others. On the other hand, in a series of papers [4,5,6,7,8] by ’t Hooft and co-workers, it has been argued that for a Schwarzschild BH, the radiation temperature should be doubled, i.e., $2T_H$, 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 Ref. [9] to obtain the Hawking’s result for the evaporation process of a BH. The first input is that of a maximum gravitational tension $F_{\max }=\tau =c^4/\left(4G\right)\simeq 3·{10}^{43}$ N; such a force is only realized at the horizon of a BH [10,11,12]. 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 [13].

Thus, through completely different reasonings, both ’t Hooft and reference [9] converged on the basic result that the dynamics of a BH are essentially on its surface—a result substantially different from the standard three-dimensional description of it.

However, with the detailed formalism and the tools employed by ’t Hooft and those in Ref. [9] being different, they both need to be discussed and understood. For example, in the abstract of Ref. [8], 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 3 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 (GC) of the Milky Way is driven to become a powerful source of extremely high energy cosmic ray protons. Our calculations in Section 5 qualitatively bear out the recent observational result from the H.E.S.S. Collaboration and HAWC Observatory that calls for the existence of a PeVatron at the GC [14,15].

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

Conventional general relativity supplies the notion of a vacuum tension $\tau =c^4/\left(4G\right)\simeq 3·{10}^{43}$ N. This vacuum tension determines the maximum force (see Refs. [3,10,11,12]), $F_{\max }=\tau$, that can be exerted on any material body; such a force is maximum on bodies confined to a horizon surface. This allows us to prove (rigorously, if anything is rigorous in relativistic quantum field theory) the gravitational Wilson action-area result [13] for matter confined on horizons [9]. In Euclidean field theory, it leads us to the well-known entropy area theorem on black hole horizons. Recalling our horizon thermodynamics via Equations (19)–(27) of Ref. [9], the relation between the entropy $\mathcal{S}$ and the area $\mathcal{A}$ is

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

and the spherical BR radius is

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

where $\mathcal{E}=M{c}^2$ is the BH energy, the area of the BH horizon and its entropy is

$$\begin{array}{c}\hfill \mathcal{A}=4\pi R_s^2,\mathcal{S}=\pi k_B{\displaystyle \frac{{\mathcal{E}}^2}{\hslash c\tau }},\end{array}$$

(1)

and the temperature is

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

so that, from Equation (1),

$$\begin{array}{c}\hfill k_{b}T={\displaystyle \frac{\tau }{2\pi }}{\displaystyle \frac{\hslash c}{\mathcal{E}}}={\displaystyle \frac{\hslash c}{4\pi R_s}},\end{array}$$

and finally, the free energy is

$$\begin{array}{c}\hfill \mathcal{F}=\mathcal{E}-T\mathcal{S}=T\mathcal{S}={\displaystyle \frac{1}{2}}\mathcal{E}.\end{array}$$

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

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

(2)

If one draws an equator around the sphere, then the two halves of the sphere attract each other via the vacuum gravitational tension. Equation (2) is central to our discussion. The surface tension of the horizon is from the confinement of both energy and entropy on the BH surface. There is no need to discuss what is “internal” to the BH. 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 electrodynamics with all its charge on the surface. Of course, with the most important difference that due to charges of both signs in electrodynamics, 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 BH mass is confined to the horizon surface of the BH itself.

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

Going beyond Ref. [9], we find that the inter-particle dynamics are dramatically changed as the BH mass is changed from being distributed (spherically uniformly) in three dimensions to being uniformly distributed on the two-dimensional surface of the BH sphere.

The inter-particle distances $L_2$ and $L_3$ of a BH of radius $R_s$ in two and three dimensions, respectively, are

$$\begin{array}{c}\hfill L_2={\left({\displaystyle \frac{\left[\mathrm{area}\right]}{N_T}}\right)}^{1/2}={\left({\displaystyle \frac{4\pi R_s^2}{N_T}}\right)}^{1/2},L_3={\left({\displaystyle \frac{\left[\mathrm{volume}\right]}{N_T}}\right)}^{1/3}={\left({\displaystyle \frac{4\pi R_s^3/3}{N_T}}\right)}^{1/3},\end{array}$$

where $N_T$ is the number of particles, here assumed to be nucleons. Using the value

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

and the radius $R_s=2GM/c^2$, the inter-distances in terms of powers of the relative mass $M/M_{\odot }$ are

$$\begin{array}{ccc}\hfill L_2& =& {\left({\displaystyle \frac{16\pi G^2{M}_{\odot }^2}{c^4{N}_{\odot }}}\right)}^{1/2}{\left({\displaystyle \frac{M}{M_{\odot }}}\right)}^{1/2}\simeq \left(3·{10}^{-25}\mathrm{m}\right){\left({\displaystyle \frac{M}{M_{\odot }}}\right)}^{1/2}=\left(3·{10}^{-10}\mathrm{fm}\right){\left({\displaystyle \frac{M}{M_{\odot }}}\right)}^{1/2},\hfill \\ \hfill L_3& =& {\left({\displaystyle \frac{32\pi G^3{M}_{\odot }^3}{3c^6{N}_{\odot }}}\right)}^{1/3}{\left({\displaystyle \frac{M}{M_{\odot }}}\right)}^{2/3}\simeq \left(4.5·{10}^{-16}\mathrm{m}\right){\left({\displaystyle \frac{M}{M_{\odot }}}\right)}^{2/3}=\left(0.45\mathrm{fm}\right){\left({\displaystyle \frac{M}{M_{\odot }}}\right)}^{2/3},\hfill \end{array}$$

and their ratio is

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

As expected, the inter-particle spacing in the two-dimensional BH is smaller compared to that in the three-dimensional BH by factors of over a billion. The precise value depends upon how massive the BH is with respect to the solar mass $M_{\odot }$. An estimate of the mean particle energies $\langle E\rangle$ for a given length scale L can be obtained through the uncertainty principle, i.e.,

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

(3)

Some limiting cases where the BH mass varies between say $10M_{\odot }$ to a super massive BH of mass $6.5·{10}^9{M}_{\odot }$ have been found by the Event Horizon Telescope to reside at the center of the giant galaxy M87 [16], and they illustrate 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 three-dimensional spherically symmetric distribution.

There is a factor of two difference between Schwarzschild, $R_s$, and Kerr, $R_k$, BH–horizon radii [17], i.e., $R_s=2GM/c^2=2R_k$. Estimates of the mean energies in three mass hypotheses for rotating Kerr BH, in the two- and three-dimensional cases, are given in

Table 1. Mean energies for three mass hypotheses for two- and three-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 factor of two in the formulas of the mean energies reported in the first column of Table 1 is due to the Kerr nature of the considered BHs.

These numbers illustrate how different the dynamics near the surface of a BH become both because the inter-particle dynamics are highly peaked (for our own GC 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 observational 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, see Table 1, that only if the mass is distributed over the surface of our galactic BH shall we have a powerful source of cosmic PeV protons.

Our model is unable to predict or even give some hint on the microscopic mechanism underlying the interaction of PeV particles with the BH surrounding matter. Nevertheless, our formalism does bring out an important result, i.e., the evasion from Hawking–Penrose (HP) no-go singularity theorems for BH. The explanation of this phenomenon lies on the simple fact that the HP singularities arise due to divergent mass density at the origin. But, as a perfect mass conductor has no mass inside it, it has no HP singularities either.

Unfortunately, we have no insight at present about possible spikes in the DM mass distribution caused by the horizon. Our instinct—without proof—is that strict General Relativity does not distinguish between the nature of a mass object but only the mass, any special effects are highly unlikely. Of course, going beyond General Relativity is going beyond the scope of our paper as well. In the following Section 5, we shall discuss observational data from the HAWC Observatory. This observation does not exclude the possibility that the Kerr BH lying at the GC could be a source of PeV protons, supporting our hypothesis that it would behave as a perfect mass conductor. Please note that were it much more massive, say the BH had a mass of $6.5·{10}^9{M}_{\odot }$ such as the one at the center of galaxy M87, the energy scale for cosmic protons would be a meager tens of TeV and thus could not match the observational UHE gamma ray energy spectrum found between 6 and 118 TeV [15].

An attentive reader may have wondered about the fate of a much less massive BH, say with a mass of $10M_{\odot }$, that according to Table 1, should be radiating several hundreds PeV protons. It can be easily inferred 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. In the next section, we discuss this matter briefly and present lifetime estimates for S-stars orbiting our own Milky Way BH. We shall return to this interesting subject elsewhere along with a discussion of the critical indices (parameter $\gamma$ in Equation (7)) in the energy spectrum of high energy cosmic ray particles [18,19,20,21,22].

4. Gravitational Radiation Induced Lifetime for a Star Orbiting a Black Hole

Classically, the rate of gravitational radiation $P_G$ emitted due to a star of mass m in a stable circular orbit of radius R around an object of mass M is given by the expression (see Equation (4.25) in Ref. [23]):

$$\begin{array}{c}\hfill P_G={\displaystyle \frac{d{E}_G}{dt}}={\displaystyle \frac{32}{5}}{\displaystyle \frac{G}{c^5}}{\left(m{R}^2\right)}^2{\left({\displaystyle \frac{GM}{R^3}}\right)}^3,\end{array}$$

(4)

where $E_G$ is the radiated energy. Using the expression of the gravitational constant in terms of the gravitational radius of M, $R_M=GM/c^2$, the rate $P_G$ can be written as

$$\begin{array}{c}\hfill P_G={\displaystyle \frac{32}{5}}m{c}^2{\displaystyle \frac{c}{R}}{\displaystyle \frac{m}{M}}{\left({\displaystyle \frac{R_M}{R}}\right)}^4.\end{array}$$

(5)

As discussed rather resignedly by Weinberg (who stated: I have no idea what it is good for [23]), the above rate is quite low for celestial objects in our solar system (typically ∼${10}^5$ Watts). However, the radiated gravitational power appears to be quite substantial for a star orbiting a BH at a relatively close distance. We illustrate it through a realistic example provided by S-type stars revolving at near distances around the massive BH Sagittarius A* at the center of the Milky Way [24,25,26,27].

For purposes of illustration, consider the BH to have a mass $M=4\times {10}^6{M}_{\odot }$ with a star of mass $m=1M_{\odot }$ revolving around at a (minimum) distance of $R=1200$ AU $\simeq 1.8\times {10}^{11}$ km. Such a system would (classically) radiate the gravitational radiation power of about

$$\begin{array}{c}\hfill P_G(R=1.8\times {10}^{11}\mathrm{km})\simeq 5.75\times {10}^{17}\mathrm{Watts}.\end{array}$$

(6)

On the other hand, if a similar solar-sized star were in the lowest allowed circular stable orbit around an extreme Kerr BH ($J=G{M}^2/c$), the distance would be reduced enormously $R\to R_M$ [17,28,29]. Keeping the same mass $M=4\times {10}^6{M}_{\odot }$, the gravitational power would be enhanced to

$$\begin{array}{c}\hfill P_G(R=R_M)=1.45\times {10}^{40}\mathrm{Watts}\simeq {\displaystyle \frac{M{c}^2}{(1.57\times {10}^6)\mathrm{years}}},\end{array}$$

which is roughly 12 orders of magnitude higher than in the previous case in Equation (6). All the mass-energy of the BH shall have evaporated away in about 1.6 million years. If a star were to be found that is essentially hugging the BH horizon for the same mass M, it would imply that the BH must be quite young. Thus, it is comforting to find that the S-stars found orbiting Sagittarius A* are all quite far, i.e., at a distance R form the GC, such that $R>>R_M$.

A preliminary calculation shows that the lowest circular orbit of a solar mass star around an extreme Kerr BH studied by Ruffini and Wheeler [28,29] can remain stable for over a billion years only for a super massive BH of mass $M=3.42\times {10}^7{M}_{\odot }$. We shall provide proofs of these statements and enlarge upon them in a future work.

The expression Equation (4) may similarly be used to study not only the members of the S2 list but also those of the extensive catalogue of $1.2\times {10}^8$ single BH with average mass of about $14M_{\odot }$ and $9.3\times {10}^6$ BHs in binary systems with an average mass of $19M_{\odot }$ that exist in the Milky Way [30].

5. Observational Verification of Our Predictions by the HAWC Observatory

Salient aspects of seven-year data about ultra high energy (UHE) gamma rays from the GC obtained by the HAWC (High Altitude Water Cherenkov) Observatory [15] may be summarized as follows:

1.

Very high energy (6–118 TeV) gamma ray data are best described as originating from a point-like source (HAWCJ1746-2856 [15]) with a power law spectrum, i.e., the current flux density per unit area per unit time, that has been parametrized as

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

(7)

where

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

2.

We mention in passing that this index $\gamma$ is—as expected for a bremsstrahlung photon spectrum—close to the high-energy cosmic ray index for protons ${\gamma }_p\simeq -2.71$ found in our references [18,19,20,21,22].

3.

The HAWC Observatory concludes that the UHE gamma rays detected by them originate via hadronic interaction of PeV cosmic ray protons with dense ambient gas and confirms the presence of a PeV proton at the GC, but they do not provide a mechanism for it.

These observational results require the existence of a dynamical mechanism for accelerating PeV protons, and it is pleasing to note that such a mechanism 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. Conclusions

For (super)massive BH 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 that employed by Landau and Fermi for the evaporation of nucleons in the Bohr–Mottelson liquid drop model [18,19,20,21,22].

Classically, a BH only absorbs particles/radiation. But, as Hawking showed, quantum mechanically, a BH can emit and absorb particles/radiation. Thus, he found the gravitational temperature T on the surface of a BH of mass M to be (see Equation (2))

$$\begin{array}{c}\hfill k_{B}T={\displaystyle \frac{\hslash c^3}{8\pi GM}}\simeq \left(5.3\mathrm{pico} \mathrm{electronvolt}\right){\displaystyle \frac{M_{\odot }}{M}},\end{array}$$

at which the surface of a BH radiates gravitationally (the presence of ℏ in Hawking’s result above confirms its inherently quantum nature). Similarly, our calculation shows that for the extremely small distance packing of nucleons on the surface of a supermassive BH mass M, a surface temperature $T_{\mathrm{nucl}}$ for nucleons is given by

$$\begin{array}{c}\hfill k_B{T}_{\mathrm{nucl}}={\displaystyle \frac{\hslash c}{L_2}}\simeq (6.6\times {10}^3\mathrm{Peta} \mathrm{electronvolt}){\left({\displaystyle \frac{M_{\odot }}{M}}\right)}^{1/2}.\end{array}$$

Such a nuclear surface temperature, which is approximately equivalent to the energy of $0.63$ PeV per nucleon for BH at our GC, does compare favorably with the existence of a proton PeVatron, as could be deduced by the photon revealed by the HAWC Observatory.

Our prediction of a mechanism able to accelerate PeV scale protons at the surface of the Kerr BH lying at the GC seems to agree with a possible interpretation of the observational data by the HAWC Observatory. In particular, our model could help in interpreting the presence of photons with ultra-high energies between 6 and 118 TeV. The discovery of a PeVatron at the galactic core through the observation of UHE photons tempts us to call super massive BHs bright holes rather than black holes.

It seems inescapable to generalize Wheeler’s statement: 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.