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It has recently been postulated that protons are accelerated up to energies of from nearby Seyfert galaxies which harbor Kerr holes in the center of their hosts. Here, we point out that the relativistic force equation of protons is governed by the Lorentz force when using the post-Newtonian approximation. The synchroton emission from nonthermal relativistic protons has to be ruled out. We carefully discuss the origin of the magnetic field close to the event horizon of a black hole.

The origin of the ultra-high energy cosmic rays (UHECRs) with energies greater than 10 19 eV is an important topic in astrophysics. The sources of the UHECRs have to be very powerful, because they have to be able to accelerate cosmic rays to very high energies. The sources would have to be located in the range of distances of d < 100 Mpc : cosmic rays from more distant sources would have to lose energy when interacting with cosmic microwave background photons. This is the GZK cutoff (see e.g. [

In Wirsich [

The full relativistic equation of a motion of a particle of mass m p , charge e 0 , velocity v and momentum p including the Lorentz force is [

d p d t = e 0 ( E + v × B ) + m p γ p g + H _ _ p + f rad (1)

where p = m p γ p v is the paricle momentum, γ p the Lorentz factor, g the gravitational acceleration and H _ _ is the tensor of the gravimagnetic force [

f rad = 2 e 0 4 γ p 2 3 m p 2 [ ( E + v × B ) 2 − [ v ⋅ ( E + v × B ) ] 2 ] v | v | . (2)

E is the electric field and B the magnetic field. f rad represents the energy loss by synchrotorn radiation and the curvature energy (“tidal radiation”) [

is the gravitational radius and M the mass of the black hole. G means the gravitational constant. An approximation like PNA signifies that scale length 10 r g is large compared to r + , but in the range of the capture radius of a solar-type star r C = 2 r g − 11 r g for a Kerr hole with a mass of 10 8 M ⊙ that is rotating with a spin-parameter near a ≈ 0.9 [

r + = G M c 2 + [ ( G M c 2 ) 2 − a 2 ] 1 / 2 (3)

of a Kerr hole. r = r + means that a particle entering it is unable to escape. Therefore, r + is defined as the horizon of a Kerr hole, which forms a boundary around the black hole singularity. Penrose [

The PNA means moreover that 10 r g is well below the influence radius r inf = G M σ 2 , where σ is t the velocity dispersion of stars in any direction [

The calculation of the PNA-metric is given in [

g ≈ − G M r 2 e r (4)

and for the field H

H ≈ − 4 J r 3 (5)

where M is the mass of the black hole. Note that the angular momentum J is weak far from the black hole.

In conclusion, the first term in Equation (1) remains the leading form, since the second and the third term are small at 10 r g compared to the Lorentz force, when using 10 8 M ⊙ for the mass of the black hole. On the other hand, neither a synchrotron radiation nor a curvature radiation of relativistic protons is ever observed, although several authors have postulated such an effect [

Furthermore, models for proton synchrotron radiation from hot disks around neutron stars are not able to produce the observed X-ray radiation of L X ≈ 10 36 erg × s − 1 . On the other hand, nonthermal relativistic protons should be able to produce a luminosity of L s y n . ≈ 10 33 erg × s − 1 in optical and UV bands [

In all, the last term in Equation (1) is f rad = 0 .

Studying

It appears that jets are only driven from central engines and massive black holes in host galaxies of elliptical Hubble type. But there are some signatures that nuclear jets are common features in Seyfert galaxies [

The no-hair theorem tells us that a black hole is uniquely fixed by its mass, spin and charge even the black hole spins fast and is strongly deformed by its spin e.g. to Kerr radius r + . On the other hand, black holes are viewed as dynamic objects which are able to storing 29% of the rotational energy, magnetic extraction of the rotational energy (horizon as a conducting membrane), emit gravitatonal waves and allow an escape of thermal quanta (“evaporation of black holes”). All of these processes take place in the horizon and explain the name event horizon, which covers the singularity. The event horizon guarantees the virginity of the singularity.

The strongly increasing loss of radiation energy at high redshifts signifies that protons with their ultrahigh energies have their origin at a depth of a few Mpc. Therefore, cosmic ray protons of E p ≥ 1.5 × 10 19 eV have their source in nearby Seyfert galaxies that harbor Kerr holes with a spin-parameter of a ≥ 0.9 [

The relativistic force equation of the protons is governed by the Lorentz force (see Equation (10) in [

We have discussed the origin of the magnetic field in connection with the membrane-paradigm [

The author is very grateful to Dr. Wolfgang Hasse (Technischen Universität Berlin and Wilhelm-Foerster-Sternwarte Berlin) and to Kevin Klinik (Technische Universität Berlin and Wilhelm-Foerster-Sternwarte Berlin) for their help in converting my manuscript into LaTeX version.

Wirsich, J. (2017) Sources of the Ultra-High Energy Cosmic Rays (UHECRs) (Part II). Open Access Library Journal, 4: e3107. http://dx.doi.org/10.4236/oalib.1103107

The Kerr hole and the membrane paradigm

A New Zealand mathematician, named Roy Kerr, discovered a solution to Einstein’s field equation [

The gravitational field is described by metric tensor g μ υ . It is used to give the invariant interval d s 2 of a spacetime instead of the metric tensor.

The Kerr solution is a metric that describes the time independent gravitational field of collapsed objects. The Kerr metric takes the form given in equation (12.7.1) of [

The nature of the Kerr geodesics plays an important role in the process e.g. of extraction of energy from black holes. An example for this is given by the Blandford-Znajek process (BL-process) (see ref. [

As explained by the BZ-process, energy entering near the horizon, a strong flux of angular momentum leaves the horizon. That ensures that energy is extracted from the black hole. The membrane viewpoint e.g. motivate Hawking [

In 1986, the model of the BZ-process was recalculated by Straumann (see ref. [

Let us calculate the magnetic field at a distance of 10 r g from the center of the black hole. For this, we use the scattering process of free electrons in an ionized plasma (“Thomson scattering”).

The luminosity in which the radiation pressure on free electrons balances gravity is the well known Eddington limit [

L E = 4 π G M m p c σ T , (A)

where M is the mass of the black hole, m p the mass of the proton and σ T = 6.65 × 10 − 25 cm 2 means the Thomson cross section. For particle density n E (in cm − 3 ) we have at 10 r g

n E = 1 σ T 10 r g = 10 10 . (B)

This is the density at 10 r g , where r g = G M c 2 and M = 10 8 M ⊙ . Assumig a balance between the energy densities

B 2 8 π = n E m p c 2 , (C)

we find at 10 r g the result

B ≈ 1.95 × 10 4 G . (D)

However, it cannot be excluded that scattering process is dominated by cyclotron scattering instead of Thomson scattering. Unfortunately, the opacity of cyclotron scattering near a magnetosphere of a Kerr hole is unknown. Instead, we use data from magnetospheres of neutron stars. We find at 10 r g the result

B ≈ 6.19 × 10 5 G , (E)

where σ c y l = 0.00 σ T was used [

In summary, our approximations lead to the results in the range of those of Straumann (see Equation (11) in [