A better understanding of the nature of black holes (BHs) has been sought ever since Schwarzschild’s discovery Schwarzschild, 1916) of a solution of Einstein’s equations (Einstein, 1915) with an essential singularity (black hole), and Kerr’s generalization of the solution to a rotating BH (Kerr, 1963; Rees et al., 1974; Rueda et al., 2022). The most significant flaw is the failure to merge gravitational physics with quantum physics, with some convincing first steps (Penrose and Floyd, 1971; Bekenstein, 1973; Bardeen et al., 1973; Hawking, 1974; Hawking, 1975; Rueda and Ruffini, 2020; Rueda and Ruffini 2021); some early and recent books are Misner et al. (1973), Misner et al. (2017), Rees et al. (1974), Joshi (1993), Joshi (2007), Joshi (2011), Joshi (2014), Joshi (2015). The best hope to explore BH physics is to consider more detailed observations e.g. Mirabel et al. (2011). The goal of this paper is to further the understanding of BHs by exploring the observations of Super-Nova Remnants (SNRs) which are produced in those SN explosions which lead to BHs. These sources are referred to as Radio Super-Novae (RSNe). Numerous observational data have been obtained for these stellar explosions, which make BHs, at various wavelengths.

There are a number of samples of Radio Super Novae (RSNe): First is the large set of RSNe in the discovery paper (Kronberg et al., 1985), 28 sources certain, and 43 possible. Then there are the independent observations by the team of Muxlow, (Muxlow et al., 1994; McDonald et al., 2002; Muxlow et al., 2005; Muxlow et al., 2010), of the same population of RSNe in M82 (30 classified as SNR). There is the newly observed list of the M82 RSNe collected and analyzed in Allen and Kronberg (1998). Then there are the lists assembled in Biermann et al. (2018) of Red Super Giant (RSG) and Blue Super Giant (BSG) RSNe, all from the literature. For the RSNe collected in Biermann et al. (2018) we know the moment of explosion, with all accompanying information; the radio interferometric observations (VLBI) have followed the expansion to a radial scale of order 1 0 16   c m . For the RSNe in M82 we have only estimates when the explosion occurred, but from ram pressure arguments (Biermann et al., 2019) one can show that these RSNe must have originated in most cases from the explosion of a Blue Super Giant (BSG) star; in these cases the expansion can be followed to a radial scale of order 1 0 18.5   c m . All these RSNe are consistent with just different stages of the same kind of explosions, from RSG as well as BSG stars, for the large radial scales mostly BSG star explosions. Considering the independent data shown in the papers by Muxlow and his group (e.g., Muxlow et al., 1994; McDonald et al., 2002; Muxlow et al., 2005; Muxlow et al., 2010), they give the same value of ( B   ×   r ) , just with a larger error bar, as the data obtained and analyzed by Allen and Kronberg (1998). The collection in Biermann et al. (2019) is based on the Allen and Kronberg (1998) analysis and data. Moreover, there is an independent discussion using another data set of very energetic explosions by Soderberg et al. (2010), leading to about the same value for ( B   ×   r ) , as shown in Biermann et al. (2018).

It has been argued that very massive star SN lead to a BH by direct collapse, without leaving a visible trace (e.g., Smartt, 2009; Smartt, 2015; Van, 2017; Humphreys et al., 2020). These arguments are based on visual and infrared data, and are influenced by obscuration and selection effects. However, gamma-ray line data and radio data (e.g.; Diehl et al., 2006; Diehl et al. 2010; Diehl et al. 2011; Prantzos et al., 2011; Diehl, 2013; Siegert et al., 2016b; Biermann et al., 2018) clearly give much more accurate SN statistics data, unaffected by obscuration. These data show for instance [summarized in Biermann et al. (2018)], that Blue Super Giant star explosions happen in our Galaxy about once every 600 years, and in other galaxies at corresponding frequencies, scaled with the star formation rate, derivable from both far-infrared and radio observations, as they scale with each other (e.g., Tabatabaei et al., 2017).

These RSN range from RSG star explosions to BSG star explosions, which cover vastly different environments in density. Among the BSG star explosions they probably cover the entire range of masses [summarized in Biermann et al. (2018) based on the work of Chieffi and Limongi (2013), Limongi and Chieffi (2018), Limongi and Chieffi (2020)], which can be derived from the now many lists in LIGO/VIRGO-Coll (2019), LIGO/VIRGO-Coll (2021a), LIGO/VIRGO-Coll (2021b), LIGO/VIRGO/KAGRA-Coll (2021c). Of course, the lists of observed mergers of stellar mass BHs encompasses second generation mergers, and that is why the BH mass can reach relatively high values, up to four times the highest single BH mass.

The only common feature of all these explosions is that they form a BH, and the explosions happen into a wind. SN-explosions that make a neutron star explode into the ISM. Here we consider explosions into a wind: and yet, the quantity ( B   ×   r ) is consistent with having the same value for all explosions. We note that the EHT data for M87 are consistent with the same number; the radio galaxy M87 harbors a central black hole with a mass approaching 1 0 10   M ⊙ , suspected to be near maximal rotation (Daly, 2019; EHT-Coll, 2019b). The minimum jet powers in Punsly and Zhang (2011) are also consistent with the same values. So, we explore the possibility that this quantity is actually related to the BH in the sense that this quantity refers to a near maximal rotation of the black hole, independent of the mass, but with energetically negligible accretion.

In the following we will assume that the physics around black holes scales such that fundamental principles carry over across all masses observed (Merloni et al., 2003; Merloni et al., 2006; Falcke and Markoff, 2004; Markoff et al., 2015; Gültekin et al., 2019); this is commonly referred to as the “Fundamental plane of black hole accretion”. Much of the accretion physics is mass-invariant. As a consequence we will assume the same physical concepts across all masses of black holes discussed in the following.

The magnetic field observed via non-thermal radio emission in the winds of massive stars (Abbott et al., 1984; Drake et al., 1987; Churchwell et al., 1992) can be attributed to the dynamo process working in the central convection zone of massive stars (Biermann and Cassinelli, 1993). The rotation and convection allows the magnetic field to be amplified right up to the stress limit. Then the magnetic field can meander in flux tubes through the radiative zone, and penetrate into the wind. The estimate gives the right order of magnitude, but does not allow to comprehend, that the resulting magnetic field observed in the post-shock region of the SN-explosion racing through the wind is the same number for very different stars, RSG and BSG stars, with extremely different wind properties.

At this point there are some important questions:

1 : What is the reason for the observed specific number ( B   ×   r )   =   1 0 16.0 ± 0.12 Gauss  ×  cm ? It can be written as an energy flow with ( B   ×   r ) 2   c   =   { ℏ   c } / e 2    { m X   c 2 } / τ P l with m X close to the proton or neutron mass, and τ P l the Planck time (see, e.g., Rueda and Ruffini, 2021). Below, in the paragraph headed by “Frequency of the Penrose process” we will derive such a relationship based on angular momentum flow; this relationship supported by observations requires the Planck time, and so connects gravitation and quantum mechanics.

  2 : Is there a possible physical connection to a relationship between magnetic field and rotational frequency (here equivalent to radius at maximum spin) also well known for super-conducting spheres (Hirsch, 2014; Hirsch, 2019)?

  3 : Does this also explain that knee and ankle energy are independent of the mass of the star which explodes and makes a BH? This has in fact been proposed (e.g., Biermann, 1993; Biermann and Astroph, 1993; Biermann and Cassinelli, 1993; Biermann and Strom, 1993; Stanev et al., 1993; Biermann et al., 2018; Biermann et al., 2019). Finding the relationship between magnetic field and angular momentum transport, as explained here in this paper, provides this argument.

  4 : Do all BHs near maximal rotation have the same magnetic field in terms of ( B   ×   r ) independent of mass? That does seem to be the case, comparing magnetic field strength numbers in RSNe and in M87 (EHT-Coll, 2019b) and the inferred energy flow in radio quasars (Punsly and Zhang, 2011). The relationship derived below supports this conclusion.

  5 : What is the magnetic field at lower spin? Here the Galactic Center SMBH will be a useful test. This will be derived in a subsequent paper. Some dependencies on spin are derived below.

  6 : What is the effect of electric drift currents (Northrop, 1963; Equation 1.79) allowed by an energetic population of E − 2 particles? Such electric drift currents can occur in electrically neutral plasmas and can be extremely fast. This was worked out in Gopal-Krishna and Biermann (2024), where it was shown that electric gradient drift currents, electric fields, and violent discharges are quite common in variable jets and winds.

All this provides motivation for deeper study.

At first we consider a Parker limit approximation to understand what is required at the inner boundary even in the simple Newtonian limit approximation. In this case we can include the ϕ -dependence, which we cannot do in the GR approximation. We posit r 2   B r   =   B 0   r H 2   H r − r H   cos ⁡ θ   cos ⁡ ϕ (1) r   B θ   = − B 0   r H 2   δ r − r H   sin ⁡ θ 2   cos ⁡ ϕ + B 1   r H   H r − r H   sin ⁡ θ 2   sin ⁡ ϕ (2) r   B ϕ   =   B 1   r H   H r − r H   sin ⁡ θ   cos ⁡ θ   cos ⁡ ϕ (3) r H is the radius of the horizon, assumed at first to be independent of θ . H ( r − r H ) is the Heaviside function, and its derivative is the δ -function δ ( r − r H ) . This allows the angular momentum transport B ϕ   B r   r 3 to be of the same sign everywhere. This construction immediately allows the divergence equation to be satisfied, and avoids any requirement for a monopole. In this solution the magnetic field stops at r H , and does not penetrate inside. It is obvious that the magnetic field could be expanded into a long series, just as in Parker (1958), but these are simple first terms. This results in B 1 r H H ( r − r H ) 2 cos 2 ⁡ θ − sin 2 ⁡ θ { cos ⁡ ϕ } − B 1 r H H ( r − r H ) { cos ⁡ ϕ } 2 − B 0 r H 2 δ ( r − r H ) { sin ⁡ ϕ } 2 = 4 π r 2 c j r

This allows the surface integral of the radial current j r to be zero, separately in θ and ϕ . It also shows that the electric current runs in the same direction, both at θ   =   0 and at θ   =   π , both either outwards or inwards. The current scales with +   B 1 near the two poles, and is negative with −   B 1 at the equator, with negative values in a broad equatorial band. − B 1 r H δ ⁡ r − r H sin ⁡ θ cos ⁡ θ cos ⁡ ϕ − B 0 r H 2 H r − r H r 2 cos ⁡ θ sin ⁡ θ sin ⁡ ϕ = 4 π r c j θ (4)

and B 0 r H 2 δ r − r H r − r H 1 2 + B 0 r H 2 H r − r H r 2 sin ⁡ θ cos ⁡ ϕ + B 1 r H δ ⁡ r − r H sin ⁡ θ sin ⁡ ϕ 2 = 4 π r c j ϕ (5)

Considering the δ -function as a narrow Gaussian this suggests a double-layer in the ϕ -current, plus an asymmetric term.

This clearly shows that already in this simple approximation we get a δ -function term, and even the derivative of a δ -function term for the electric current at the inner boundary. It also demonstrates that the density of the current carrying charged particles diverges at the boundary, which implies that collisions also diverge in this approximation. One part of the end-product of these collisions is accreted to the BH, and the other part is ejected in the wind with a known magnetic power flow independent of BH mass.

Here we derive the angular momentum transport in the terms of General Relativity, so allowing to treat the behavior of the magnetic field close to the black hole, for any rotation. In this section we set the speed of light c to unity for simplicity.

The metric tensor elements for the Kerr metric are given in Boyer - Lindquist coordinates by d s 2 = d ϕ 2 ⁡ sin 2 ( θ ) a 2 + r 2 2 − a 2 ⁡ sin 2 ( θ ) Δ ( r ) ρ ( r , θ ) 2 − ( d t d ϕ + d t d ϕ ) 2 a G N M B H r ⁡ sin 2 ( θ ) ρ ( r , θ ) 2 + d θ 2 ⁡ ρ ( r , θ ) 2 + d r 2 ⁡ ρ ( r , θ ) 2 Δ ( r ) + d t 2 − 1 − 2   G N   M B H r ρ ( r , θ ) 2 . (6) where G N is the universal gravitational constant, M B H is the mass of the black hole and ρ ( r , θ ) 2   =   r 2 + a 2   cos 2 ( θ ) , Δ ( r )   =   r 2 − 2   G N   M B H   r + a 2 . (7)

The electromagnetic tensor is F μ ν   =   0 0 E ̃ θ ( r , θ ) 0 0 0 B ̃ ϕ ( r , θ ) − B ̃ θ ( r , θ ) − E ̃ ( r , θ ) − B ̃ ϕ ( r , θ ) 0 B ̃ r ( r , θ ) 0 B ̃ θ ( r , θ ) − B ̃ r ( r , θ ) 0 , (8) and the components of F μ ν are determined from the vector potential components A μ F μ ν   =   ∂ μ   g ν ν   A ν ( r , θ ) − ∂ ν   g μ μ   A μ ( r , θ ) . (9) The measured components of the electric and magnetic fields are related to the tilde components in F μ ν by the relations E ̃ θ ( r , θ ) = E θ ( r , θ ) B ̃ r ( r , θ ) = g θ θ   g ϕ ϕ   B r ( r , θ ) B ̃ θ ( r , θ ) = − g r r   g ϕ ϕ   B θ ( r , θ ) B ̃ ϕ ( r , θ ) = g r r   g θ θ   B ϕ ( r , θ ) (10) These expressions are based on the definitions of the electric and magnetic fields given in Komissarov (2004). They have the asymptotic forms given in Weber and Davis (1967). We are assuming that the r - and ϕ -components of the electric field are zero. The E θ ( r , θ ) component of the electric field can be determined for the case of a static magnetic field, ∂   B ⃗ / ∂   t   =   0 , from the relation ∇ ×   E ⃗ ϕ   =   0 . (11) This relation requires that E θ ( r , θ )   =   E 0 ρ ( r , θ ) , (12) where E 0 is a constant. The B r ( r , θ ) component of the magnetic field is obtained from the divergence relation ∇   ⋅   B ⃗ ( r , θ )   =   0 . (13) For B θ ( r , θ )   =   0 (Weber and Davis, 1967) this relation requires that B r ( r , θ )   =   B 0 g r r   g θ θ   g ϕ ϕ , (14) where B 0 is a constant. The remaining components of the magnetic field are undetermined. Based on observational radio data extensively discussed in Biermann et al. (2018), Biermann et al. (2019), we assume that g r r   g θ θ   B ϕ ( r , θ )   =  constant  =   B p 0 . (15)

Here both B r and B ϕ   ∼   Δ 1 / 2 . The ratio between B ϕ and B r is given by the Parker model, and this indicates that B p 0 / B 0   ∼   χ / M B H , where χ is the dimensionless spin (i.e., maximum unity), so χ   =   a / M B H . Furthermore we assume that the total radial magnetic field energy is proportional to the available rotational energy, which results in B 0   ∼   χ   M B H , in the χ   ≪   1 approximation. From this it follows that B p 0   ∼   χ 2 . Using observations of radio loud quasars (Punsly and Zhang, 2011) we can check on the implications, since GR solutions and far-distant solutions have to be consistent in their dependence on χ and M B H , namely, L jet   ∼   χ 4 , independent of BH mass M B H . Furthermore E 0   ∼   χ 2   M B H from the consistency requirement of the energy flow and angular momentum flow, worked out below.

The energy flux is obtained from the contraction of the covariant form of the Killing vector k t μ with the electromagnetic energy-momentum tensor E μ   =   T μ ν   ( k t ) ν , (16) and the angular momentum flux is obtained from the contraction of the covariant form of the Killing vector k ϕ μ with the electromagnetic energy-momentum tensor L μ   =   T μ ν   ( k ϕ ) ν . (17) The r - and θ -spatial components of the energy flux and the angular momentum flux are given by E r = B p 0   E 0   Δ ( r ) ρ ( r , θ ) 5 E θ = 0 L r = B 0   B p 0   Δ ( r ) 3 / 2 ρ ( r , θ ) 5 L θ = 0 . (18) The energy flux and the angular momentum flux are related via the expression E r   =   ω ( r , θ )   L r , (19) where ω = E θ ̃ B r ̃   = E 0 B 0 Δ . (20) This is the same relation as the one in Equation 4.4 of Blandford and Znajek (1977). Here ω   ∼   χ / M B H .

The location of the horizon is determined by the condition Δ ( r )   =   0 , so the flux components E r and L r vanish on the horizon. On the equator of the black hole ( θ   =   π / 2 ) the radial component of the angular momentum flux reaches a maximum at a radius of slightly less than three horizon radii, Figure 2. These expressions are similar to the ones obtained by Blandford and Znajek (1977), but there are significant differences due to the differences between our model and theirs. In the BZ model both of the poloidal components of the energy flux are non-zero, while in our model both of the fluxes in the θ -direction (polar direction) are zero. The vanishing of the θ -component of the energy flux in our model is due to setting the r - and ϕ -components of the electric field equal to zero, and the vanishing of the θ -component of the angular momentum flux is due to setting the θ -component of the magnetic field equal to zero, following Weber and Davis (1967).

As seen by an observer at infinity the rate of energy extraction is given by E ̇ rad   =   ∫ E r ⁡ ρ ( r , θ ) 2   d Ω , (21) and the rate of angular momentum extraction is given by L ̇ rad   =   ∫ L r ρ ( r , θ ) 2   d Ω , (22) where d Ω is the infinitesimal solid angle. The evaluation of these integrals gives (note that the radius r refers to the BH mass, so that spin a , radius r , and G N M B H have the same unit in this section) E ̇ rad = 4 π B p 0   E 0   a 2 + r ( r − 2 G N M B H ) r 2 a 2 + r 2 L ̇ rad = 4 π B 0   B p 0   a 2 + r ( r − 2 G N M B H ) 3 / 2 r 2 a 2 + r 2 . (23)

Here E ̇ rad   ∼   χ 4 , and L ̇ rad   ∼   χ 3   M B H , consistent with a derivation following (Weber and Davis, 1967; Falcke and Biermann, 1995).

Here the power is proportional to χ 4 and the angular momentum transport to χ 3   M B H . Since the power put out via magnetic fields is also proportional to B 0 2 Falcke and Biermann, 1995 this is consistent. The angular momentum transport by magnetic fields (Weber and Davis, 1967; Equation 9, integrated over 4   π   r 2 ) runs as B 0 × B p 0 ∼ χ 3   M B H , so this is also consistent. In these graphs (Figures 2–4) the lower limit of r is given by the condition Δ ( r )   =   0 , so for maximal spin, that radius is r   =   { G N   M B H } / c 2 , the Kerr radius.

The current can be calculated from the covariant divergence of the electromagnetic field tensor ∇ μ F μ ν   =   J ν (24)

For the radial and theta components of the current this calculation gives J r = − 4   a 2   B p 0 ⁡ sin ( θ ) cos ( θ ) a 2 + r   ( r − 2   G N M B H ) a 2 ⁡ cos 2 ( θ ) + r 2 3 J θ = − 2   B p 0 a 2 ⁡ cos 2 ( θ ) ( G N M B H − r ) + r 2   a 2 + r   ( r − 3   G N M B H ) a 2 ⁡ cos 2 ( θ ) + r 2 3 (25) The J t and J ϕ components are non-zero, but their expressions are much longer. The latter two components decrease much more rapidly with r than either J r or J θ .

The expression for the charge density as obtained from the covariant divergence relation is given by J 0 = 2 ( 2 ) a 2 sin ( 2 θ ) ⁢ 6 a 4 E 0 cos 2 ( θ ) − 20 a B 0 G N M B H r a 2 + r ( r − 2 G N M B H ) a 2 + r ( r − 2 G N M B H ) a 2 ⁡ cos ( 2 θ ) + a 2 + 2 r 2 7 / 2 + 2 E 0 r 3 4 G N M B H + 3 r a 2 + r ( r − 2 G N M B H ) a 2 ⁡ cos ( 2 θ ) + a 2 + 2 r 2 7 / 2 + 2 ( 2 ) a 2 sin ( 2 θ ) ⁢ ( a 2 E 0 r ) ⁢ ( 14 G N M B H + 9 r + 3 ( − 2 G N M B H + r ) cos ( 2 θ ) ) a 2 + r ( r − 2 G N M B H ) a 2 cos ( 2 θ ) + a 2 + 2 r 2 7 / 2 (26)

This shows that in terms of the local charge density we also get a divergence at the inner boundary, at the horizon. This is proportional to χ 4   M B H − 2 . In more detail the leading terms with B 0 as well as E 0 run as χ 4   M B H − 2 , while the terms with E 0 have two further terms running as χ 8   M B H − 2 and χ 6   M B H − 2 . It follows that the density may get high enough for lots of energetic collisions.

Furthermore the term running with B 0 has the factor Δ − 1 / 2 , while the terms running with E 0 all have the factor Δ − 1 . When Δ approaches a value small compared to radius r , and writing the spin parameter as χ   =   1 − δ χ with δ χ   ≪   1 , then Δ   =   ( r − r g ( 1 + 2   δ χ ) ) × ( r − r g ( 1 − 2   δ χ ) ) . Writing the first term in brackets as δ r , then Δ becomes δ r × ( δ r + 2 2   δ χ ) . If we could constrain the collision rate then it follows that we could also constrain δ χ to be a possibly small number.

To work out the numbers we note that B p 0 is observed to be 1 0 16.0 ± 0.12 G a u s s × c m (Biermann et al., 2018; Biermann et al., 2019); writing all other terms with their proper dimensions using the equatorial outer radius of the ergo-region of a 10   M ⊙ BH, so 1 0 6.4   c m , gives a charged particle density of about 1 0 14.0   c m − 3 , ignoring here the factors with some power of Δ , and adopting the limit χ   ≃   1 .

This suggests that collisions could an important process, and this is what we explore further.

There is an inconsistency between what the mass transport is in the wind (assuming equipartition with the observed magnetic fields) and what accretion to the BH is needed to sustain the luminosity of ∼   1 0 43 .   e r g / s , if one were to power this emission simply by accretion, as equality would require 100% efficiency. This inconsistency can be resolved by considering the pure spin-down mode (Blandford and Znajek, 1977), which implies very little accretion. Here we note that in the pair production variant to the Penrose process, this could imply that the BH accretes predominantly particle/anti-particle pairs, most of which never get out. The creation of such pairs costs at least two proton masses in energy, but energetically pion production dominates by far (below we use a factor of about 30 based on the ratio of cross sections to make pions and to make proton-anti-proton pairs from p-p collisions). They are available from interaction with magnetic irregularities and non-linear waves, such as shock waves. In fact, from the mismatch in mass turnover, one might speculate that the energetic protons initiate a cascade process similar to the interaction of ultra high energy CR particles entering the atmosphere of the Earth. In such a cascade a very large number of secondary particles is produced. By analogy with the Penrose argument one may expect that half the cascade particles are directly on orbits falling into the BH; the other half are initially on orbits to escape. These particles interact with the magnetic field. At the outer boundary of the ergo-region, the particles may transfer a significant fraction of their energy and angular momentum to the magnetic fields and fall back down in accretion to the BH (see Penrose and Floyd, 1971). In processes such as p ̄ ’s colliding with p ’s, pions and multiple neutrinos are produced. These neutrinos have a good chance to escape altogether. All this should be re-evaluated using proper frames (e.g., Bardeen et al., 1972; Shaymatov et al., 2015; Bambhaniya et al., 2021), although a collision-dominated gas with a magnetic field, in which some energetic particles have Larmor radii which are close to the scale of the system, is a challenge. What we present here is a detailed balancing of different particle species in the local frame.

Many different losses go into production of pions, which quickly decay into energetic electrons, positrons, photons and neutrinos. In the model proposed the photons are optically thick in their propagation. This is akin to the model published for blazars, and their neutrino emission in Kun et al. (2021). The electrons/positrons and neutrinos have a chance of escaping. Based on the ratio of cross-sections for p-p-collisions to make pions versus p-p-collisions to make proton-anti-proton pairs, about 30 times as much energy goes into an electron/positrons pair plasma from pion decay (ratio of cross sections and energy turnover), and neutrinos, as goes into proton-anti-proton pairs, in terms of what gets out. The neutrinos - in the model proposed - range from MeV to very much higher energy, and for those the IceCube data provide a serious upper limit, if the model is used at TeV energies and beyond. Other than an electron/positron plasma neutrinos could be a second main escape path. That is a main point of the model.

A check with data can be done: the proposal is consistent with IceCube-Coll et al. (2016), IceCube-Coll et al. (2021) and INTEGRAL data (Diehl et al., 2006; Diehl et al., 2010; Siegert et al., 2016a; Siegert et al., 2016b):

In the model proposed the cosmic ray flux of the component going to EeV energies is about 1 0 − 2.8 of the normal CR flux at GeV energies (numbers taken from Gaisser et al. (2013), Table 3, CR components 3 or 3 * ; pop three contains all elements (Table 2) and pop 3 * contains only protons); correcting for a slightly flatter spectrum assumed here, anchored at EeV, gives about 1 0 − 3.4 . This implies 1 0 37.6   e r g / s , again using Gaisser’s et al. numbers for the entire Galaxy of 1 0 41   e r g / s . Falcke and Markoff (2013) give an estimate of the accretion rate measured close to the central BH in our Galaxy, and it corresponds to a power of about 1 0 37.8   e r g / s , consistent with the number above. INTEGRAL (Siegert et al., 2016b) gives a positronium production of 1 0 43.5   s − 1 in a very large region, with a scale height of several kpc and along the plane from a larger region than any other recognizable source class, corresponding to about 1 0 37.5   e r g / s , again consistent with the Gaisser et al. (2013) number. The papers by Diehl et al. support the point of view that there could be plenty more electrons and positrons that escape from the Galactic disk unseen. The production of a large number of electrons/positrons is demonstrated by observations of the BH V404 Cyg (Siegert et al., 2016a). The electron/positron pair plasma production in our Galaxy appears to be due to many sources, possibly the Galactic Center black hole (GC BH) and most probably many stellar/SN/BH sources, including microquasars and SN Ia supernovae (Martin et al., 2010; Prantzos et al., 2011; Prantzos, 2017; Mera Evans et al., 2022). Diehl et al. propose that all black holes produce an electron/positron pair plasma, often in outbursts. Based on gamma-ray line spectroscopy (Diehl et al., 2006; Diehl et al., 2010, Diehl, 2017) give a SN rate of those SNe making black holes in the Galaxy of about 1 SN per 400 years (again, with an uncertainty of 1 0 ± 0.11 ); this has been worked through in Biermann et al. (2018); this includes both Red Super Giant and Blue Super Giant star progenitors, both of which produce black holes, or short BH-SNe. The time scale of the activity is at least 30 years (1 parsec at 0.1   c ), as observed numbers from Radio Super-Novae given in Biermann et al. (2019), based on the M82 data of Radio Super-Novae (RSNe) (Allen and Kronberg, 1998; Kronberg et al., 1985; Kronberg et al., 2000). It ensues that each BH-SN contributes - again using the numbers in Gaisser et al. (2013) - about 1 0 50.8   e r g in CRs, as shown above. For this specific low level HE CR component this translates to 1 0 47.4   e r g , as well as 1 0 48.9   e r g in e + e − plasma and MeV neutrinos, by virtue of the 30 times larger cross section (p-p collisions making pions versus p-p collisions making p - p ̄ pairs). This translates into a maximal flux, using the shortest reasonable time scale - of 1 0 39.9   e r g / s initially. The observed power of about 1 0 37.5   e r g / s in the Galactic Center region (by INTEGRAL) in electron/positron plasma means, if produced by a SN, that the activity could be down now by e − 400 / 30   ≃   1 0 − 5.8 , for a possible initial power of 1 0 43.3   e r g / s for all SN contributors summed together. This in fact approximately matches the spin-down power seen in both M87 (EHT-Coll, 2019a; EHT-Coll, 2019b), many other radio galaxies in their minimum jet power (e.g., Punsly and Zhang, 2011), and in Radio Super-Nova Remnants interpreting them as driven by a relativistic wind from a spinning compact object, presumably a BH.

Using the starburst galaxy M82 (Kronberg et al., 1985; Allen and Kronberg, 1998; Kronberg et al., 2000) itself as our IceCube limit for point sources (IceCube-Coll et al., 2016; IceCube-Coll et al., 2020; IceCube-Coll et al., 2021) gives about 1 0 − 12.0 TeV events c m − 2   s − 1 , assuming a E − 2 spectrum, corresponding to a limit of about 1 0 − 11.0   e r g   c m − 2   s − 1 at GeV for a E − 7 / 3 spectrum assumed here for the relevant CR spectrum, where the Gaisser et al. (2013) numbers are anchored. This corresponds to a limiting luminosity at TeV of 1 0 39.1   e r g / s at the distance of M82, and 1 0 33.9   e r g / s at the distance of the Galactic Center. Since there is evidence from the Telescope Array et al. (2020) as well as Auger-Coll (2018), that both starburst galaxies M82 in the North and NGC253 in the South may have been detected in UHECRs, we assume that the detailed analysis of recent Radio Super-Novae (RSNe) in M82 applies also to NGC253 (Kronberg et al., 1985; Allen and Kronberg, 1998; Kronberg et al., 2000), where the specific IceCube limit mentioned above applies and so a limit for all sources is <   1 0 39.1   e r g / s . In M82 there are about 40 such sources (Kronberg et al., 1985), so the limit per source is <   1 0 37.5   e r g / s , if all sources contribute equally. However, again, for a possible decay time of 30 years, only one source may contribute, and this possibility would imply a luminosity of <   1 0 39.1   e r g / s for that one source. To within the large errors of such an estimate this is still consistent with the data, which give an expectation for a single contributing source at 1 0 39.9   e r g / s . Allowing for a slightly steeper spectrum would loosen these constraints, as would an even faster change with time of any single source. The age of the youngest source 41.9 + 58 is sufficiently large so that it may have decayed already significantly. Of course, if the HE neutrinos were pointed in their emission, then their luminosity could be quite a bit higher without showing up in our observations.

To do a further test: Applying the same neutrino flux limit to possible sources in the Galactic Center (GC) region gives a limit of about 1 0 5.1 times stronger, so <   1 0 34.4   e r g / s . As shown above this is fully consistent with the rate of BH-SNe occurring; the expected flux reduction is 1 0 − 5.8 for an initial luminosity limit of <   1 0 40.3   e r g / s , again consistent. One problem in such an argument is that the sources are known to be highly fluctuating (e.g., Siegert et al., 2016a). It is possible to repeat this exercise for the Cyg region, which is much closer than the Galactic Center. This gives a limiting luminosity of 1 0 33.2   e r g / s , and it is again consistent, since the BH-SN rate is very low near to us, 1 BH-SN per about 1 0 5 years, so predicting a huge reduction from the expected initial luminosity of 1 0 39.9   e r g / s worked out above; Cygnus might be close enough to provide an actual source of the Galactic EeV CRs identified by Gaisser et al. (2013).

Analyses of particle collisions near to BHs and singularities have been carried out, (Patil et al., 2010; Patil and Joshi, 2011a; Patil and Joshi, 2011b; Patil and Joshi, 2012; Patil et al., 2012; Patil and Joshi, 2014; Patil et al., 2015; Patil et al., 2016; Liu et al., 2011; Banados et al., 2009; Banados et al., 2011). These papers did not have the benefit of insight provided by the RSN observations, the most detailed of which by Allen and Kronberg (1998), Allen (1999), Kronberg et al. (2000). The latter provide a newer solid foundation to develop the approach.

As an example, we calculate the particle density and flux for the ergo-region around a stellar mass BH of 10   M ⊙ : The magnetic field, extrapolated to near the BH, at radius R   =   1 0 6.4   c m , is about 1 0 9.6   G a u s s . In equipartition, ( B 2 ) / ( 8   π )   ≃   n   k B   T , leading to a particle density of n   ≃   1 0 21.6   c m − 3 at a weakly relativistic temperature of ∼   1 0 12   K . This, in turn, allows a flow of particles of 4   π   R 2    n   c   ≃   1 0 46   s − 1 . Interactions give a similar number, using a cross section of 1 0 − 27   c m 2 (valid for making proton-anti-proton pairs (Winkler, 2017; Reinert and Winkler, 2018); the inelastic cross-section is about 30 times higher well above threshold), as obtained from 4   π   R 3   n 2   σ   c   ≃   1 0 47   s − 1 , which is more than what is needed to explain the observations; as even a smaller cross-section could be accommodated. This latter quantity cannot be readily extrapolated to a higher BH mass, as we discuss below.

Using the general approach of EHT-Coll (2019b) we can show that this optical depth may reach order 10, independent of radius. This means that the interaction time to produce proton-anti-proton pairs is less than the residence time, possibly considerably less.

The observations show that B   =   1 0 16.0 ± 0.12 / r   G a u s s , with r in c m . This relationship has been observed over the range of radius from about 1 0 18.5   c m down to order 1 0 16   c m , with the highest resolution observations done by radio interferometry (VLBI). Using the analogy with the Solar wind (Parker, 1958; Weber and Davis, 1967) we extrapolate it down for the case of fast rotation. The EHT observations of M87 suggest that such an extrapolation is reasonable (EHT-Coll, 2019b): There the product ( B   ×   r ) has about the same value as in RSNe at about five gravitational radii; the M87 black hole has been suspected to be in substantial rotation, perhaps near maximal (Daly, 2019; EHT-Coll, 2019b and later). The jet power of M87 is consistent with what is derived for RSNe using the available energy content of a maximally rotating black hole, and the time-scale derived from angular momentum transport (Weber and Davis, 1967). This suggests that the jet power far outside the ergo-region is already visible at five gravitational radii.

Putting in numbers as observed (EHT-Coll, 2019b) extrapolated to a stellar mass BH suggests that the production time scale for making proton-anti-proton pairs is safely of order <   1 of the resident time scale in the inner region around the ergo-region.

This argument works for stellar mass black holes, and we can speculate here that the model proposed would allow this to work also for more massive black holes.

The concept is that the energetic particles are confined by the magnetic field and so stay in the ergo-region; the magnetic field is due to electric currents in the (weakly relativistic) thermal matter, which is held in the gravitational field. In momentum phase space there is a cone, inside of which all particles are on orbit to accrete to the BH. This is akin to arguments in Hills (1975), Bahcall and Wolf (1976), Frank and Rees (1976). In that approach, stars interact with molecular clouds to fill a cone in momentum phase space which allows accretion to a central BH. This is referred to as the loss cone mechanism. Here, charged particles interact with the magnetic fields (Strong et al., 2007; Moskalenko and Seo, 2019), and also with each other, to also finally accrete to the BH.

Given all the above arguments, what are the predictions in these scenarios? In these conditions, one can ask what the fraction of anti-protons n p ̄ / n p might be. The observed fraction of anti-protons is about 1 0 − 3.7 (AMS Coll, 2016), with a spectral shape dependence of about E − 2.7 for both protons and anti-protons. We assume that this spectrum changes for both towards a flatter spectrum at higher energy since at lower energies, both components have other contributions (see, e.g., Biermann et al., 2018). Could this match the observed flux of anti-protons? Fitting above 200 GeV, the CR flux is about 1 0 − 3 relative to other CR-populations from the similar SN-explosions. Using a spectrum such as E − 7 / 3 , this modifies the factor of 1 0 − 3 to 1 0 − 3.3 to 1 0 − 3.4 . However, at EeV, the sum of protons and anti-protons is observed, while at lower energy, anti-protons are observed separately. Thus, correcting the prediction by another factor of order two gives 1 0 − 3.6 to 1 0 − 3.7 , which allows the observed 1 0 − 3.7 . Consequently, we propose a model to explain the flux, energy content, spectrum, maximal particle energy, and particle/anti-particle ratio of highly energetic protons. It follows then, that the spectrum of anti-protons continues all the way to ankle energies, with a spectral shape near E − 7 / 3 . The energetic protons would approach the spectrum of the anti-protons at some energy slightly above PeV. AMS may well detect some of these anti-protons among its highest energy particles, around TeV.

One may well ask whether anti-protons survive their path to us: Their cross-section to interaction is the same as for protons, and since we see protons at EeV (Auger-Coll, 2020a) without being able to distinguish protons and anti-protons, the particles detected may well contain anti-protons, in this proposal here possibly half.

If there are in fact large numbers of cascades, then many of the secondaries, including electrons and positrons might also escape, creating a funnel in the Galactic disk which allows them to flow out (see Diehl et al., 2006; Diehl et al., 2011; Diehl, 2013; Siegert et al., 2016b). The total positron production in a large region around the Galactic Center corresponds to a power on the order of 1 0 37.1   e r g / s . This is 1 0 − 4.6 of the maximal energetic particle flow, of order 1 0 51   e r g in about 1 0 9.3   s (see above) even for a single massive star SN event, suggesting that much of the energy is vented out to the Galactic halo. Even allowing for a reduction by about a factor of 100, to account for the difference in CR electron fluxes from CR proton and Helium, would still leave a factor of 1 0 − 2.6 . The contribution from the Galactic Center BH seems to be less than that which any possible surrounding sources could contribute.

In the balance between production of anti-protons from p - p collisions, as well as p ̄ - p ̄ collisions, the annihilation process p - p ̄ dominates. Those interactions will limit not only the p ̄ net production, but will also produce large numbers of neutrinos. On the other hand, the p vs p ̄ interaction decreases with energy, while the p vs p interaction cross-section to produce p - p ̄ pairs, rises with energy. These neutrinos will be crudely commensurate with the Poynting flux energy flow. They, however, could exceed the Poynting flux, if the production and immediate destruction of p ̄ greatly exceed the rate of accretion of p ̄ , as this runs with the ratio of the cross sections. Consequently, this process could emit a significant fraction of the rotational energy of the BH via neutrinos.

We consider the following reactions: first for creating and annihilating anti-protons; here we include the primary protons. Note that these densities represent integrals over the momentum distribution, and the cross-sections include weighting due to the momentum phase-space distribution:

1) p + p   →   p + p + p   + p ̄ (27) with cross section σ p r , p p ̄ ; (28) protons have density n p and anti-protons density n p ̄ ;

2) p + p ̄   →   m u l t i p l e   π (29) with cross-section σ d e , p p ̄ . (30) – The pions decay into neutrinos and other leptons.

3) The reaction p ̄ + p ̄   →   p ̄ + p ̄ + p   + p ̄ (31) has the same cross section as above for protons, σ p r , p p ̄ . (32)

4) The production of anti-neutrons p ̄ + p ̄   →   p ̄ + n ̄   + π ̄ (33) has the cross section σ n ̄ , p ̄ p ̄ . (34)

There are corresponding analogous processes for producing or destroying protons.

The detailed balance equations are (adopting c as an approximate typical velocity for the particles): d   n p ̄ d   t   =   σ p r , p p ̄   c   n p 2   − σ d e , p p ̄   c   n p ̄   n p + σ p r , p p ̄   c   n p ̄ 2   −   σ n ̄ , p ̄ p ̄   c   n p ̄ 2 − n p ̄ τ B H , (35) and d   n p d   t   =   σ p r , p p ̄   c   n p 2   − σ d e , p p ̄   c   n p ̄   n p + σ p r , p p ̄   c   n p ̄ 2   −   σ n , p p   c   n p 2 − n p τ B H + n p τ gal . (36)

Here the last term in the previous equation, and the last two terms in this equation, represent accretion to the BH, and accretion from the outside, from an accretion disk for instance. Accretion from outside constitutes positive baryon number accretion. If many secondaries are created and accreted, their net baryon number is zero. Baryon number accretion derives from both populations.

Initially, we assume that the accretion terms are negligible. By virtue of particles and anti-particles behaving the same in corresponding cross-sections, we can now consider two situations:

First we consider the case, where n p ̄   ≪   n p . In this case, the production of anti-protons via pair creation dominates, and for protons the reaction leading to neutron production dominates. So, in this case, the anti-protons grow in number, and the protons decrease in number. The situation is not stationary.

Next, the condition of exact stationarity can be required, and the two equations above can be subtracted from each other: By virtue of the symmetry of cross-sections between particles and anti-particles, the first three terms in the first equation are equal to the first three terms in the subsequent equation, leaving the fourth term. This gives n p ̄ 2   σ n ̄ , p ̄ p ̄ − σ n , p p   n p 2   =   0 . (37)

By virtue of the equivalence between particles and anti-particles, the two cross-sections are identical and can be cancelled out. The result of the above operation is n p ̄ 2 − n p 2   =   0 , (38) thus the density of protons and anti-protons is the same in stationarity, neglecting accretion both from outside and to the BH. This does not violate baryon number conservation since in this model, both protons and anti-protons are secondary; the baryon number is exactly zero.

It follows that the ratio of neutrino production via pion decay to p p ̄ pair-production runs with the ratio of the two cross-sections, which is large; however, the cross-sections have to be weighted with the momentum phase space distribution as noted above. It follows that the time scale for refilling the momentum phase space necessary to yield large interaction rates is key to the effective neutrino luminosity. Correspondingly, the ratio of neutron production to p p ̄ pair-production runs with the ratio of the two cross-sections, which is also large. The cross-section to make pions and ensuing neutrinos starts at small energy and is large, and so dominates over the neutron production. In this simplified picture creation and destruction balance, and so the momentum distribution adjusts itself to make the effective cross-sections match, moderated by the time scales of redistributing particles in momentum phase space.

Second, we allow for the accretion terms to be relevant. Then the difference of the two terms leads to n p − n p ̄   σ n , p p   n p + n p ̄ + 1 τ B H   =   n p τ disk . (39)

This means if the sum of the neutron production and the BH net accretion is much larger than the outside accretion (from, e.g., an accretion disk), then the relative difference n p − n p ̄ / n p (40) is small. The anti-proton density approaches the proton density. Next consider the sum of the two equations: A solution is possible, in which the creation of secondaries is mostly balanced by destruction, with some accreting to the BH, and an even smaller number providing net loss of particles to the outside.

The pion decay leading to neutrino production can be approximated well by the approach of Penrose and Floyd (1971), leading to an accretion of neutrinos to the BH. It also leads to a corresponding luminosity of outgoing neutrinos.

In summary, the test is clearly to determine the anti-proton fraction at the EeV energy scale. If that fraction is half of the sum of protons and anti-protons, then the neutrino luminosity is predicted to be large, with most neutrinos near GeV energies. We observe TeV energies in neutrinos, and above.

All these arguments depend on the Penrose process (Penrose and Floyd, 1971, Bardeen et al., 1972). However, the main difference to the collisional Penrose process (e.g., Bejger et al., 2012; Hod, 2016; Leiderschneider and Piran, 2016; Schnittman, 2018) is that in our approach, based on the magnetic field observations, particles are scattered by magnetic field irregularities frequently and throughout the ergo-region. We can write the spectrum of magnetic field irregularities I ( k ) k as energy density with wavenumber k , so that the mean free path can be written as r g   B 2 / 8   π I k   k (41) where r g is the Larmor radius of the motion of a charged particle. This mean free path is far smaller then the scale of the ergo-region except for the very highest particle energies, spanning more than nine orders of magnitude (from the values of B   ×   r ) observed, as discussed above and in Biermann et al. (2018), Biermann et al. (2019).

Here we focus on the angular momentum transport and work out, how frequently the data show that the Penrose process happens; however, first we have to comment on orbits of particles versus the local 3D momentum phase space distribution.