# Nature of the ultrarelativistic prompt emission phase of GRB 190114C

R. Moradi  ICRA and Dipartimento di Fisica, Università di Roma “La Sapienza”,
Piazzale Aldo Moro 5, I-00185 Roma, Italy  International Center for
Relativistic Astrophysics Network, Piazza della Repubblica 10, I-65122
Pescara, Italy  INAF – Osservatorio Astronomico d’Abruzzo,Via M. Maggini snc,
I-64100, Teramo, Italy<EMAIL_ADDRESS>J. A. Rueda  ICRA and
Dipartimento di Fisica, Università di Roma “La Sapienza”, Piazzale Aldo Moro
5, I-00185 Roma, Italy  International Center for Relativistic Astrophysics
Network, Piazza della Repubblica 10, I-65122 Pescara, Italy  ICRANet-Ferrara,
Dipartimento di Fisica e Scienze della Terra, Università degli Studi di
Ferrara, Via Saragat 1, I–44122 Ferrara, Italy  Dipartimento di Fisica e
Scienze della Terra, Università degli Studi di Ferrara, Via Saragat 1, I–44122
Ferrara, Italy  INAF, Istituto di Astrofisica e Planetologia Spaziali, Via
Fosso del Cavaliere 100, 00133 Rome, Italy<EMAIL_ADDRESS>R. Ruffini
ICRA and Dipartimento di Fisica, Università di Roma “La Sapienza”, Piazzale
Aldo Moro 5, I-00185 Roma, Italy  International Center for Relativistic
Astrophysics Network, Piazza della Repubblica 10, I-65122 Pescara, Italy
INAF, Viale del Parco Mellini 84, 00136 Rome, Italy<EMAIL_ADDRESS>Liang Li
ICRA and Dipartimento di Fisica, Università di Roma “La Sapienza”, Piazzale
Aldo Moro 5, I-00185 Roma, Italy  International Center for Relativistic
Astrophysics Network, Piazza della Repubblica 10, I-65122 Pescara, Italy
INAF, Viale del Parco Mellini 84, 00136 Rome, Italy<EMAIL_ADDRESS>C.
L. Bianco  ICRA and Dipartimento di Fisica, Università di Roma “La Sapienza”,
Piazzale Aldo Moro 5, I-00185 Roma, Italy  International Center for
Relativistic Astrophysics Network, Piazza della Repubblica 10, I-65122
Pescara, Italy  INAF, Istituto di Astrofisica e Planetologia Spaziali, Via
Fosso del Cavaliere 100, 00133 Rome, Italy  S. Campion  ICRA and Dipartimento
di Fisica, Università di Roma “La Sapienza”, Piazzale Aldo Moro 5, I-00185
Roma, Italy  International Center for Relativistic Astrophysics Network,
Piazza della Repubblica 10, I-65122 Pescara, Italy  C. Cherubini
International Center for Relativistic Astrophysics Network, Piazza della
Repubblica 10, I-65122 Pescara, Italy  ICRA, University Campus Bio-Medico of
Rome, Via Alvaro del Portillo 21, I-00128 Rome, Italy  Department of Science
and Technology for Humans and the Environment and Nonlinear Physics and
Mathematical Modeling Lab,
University Campus Bio-Medico of Rome, Via Alvaro del Portillo 21, 00128 Rome,
Italy  S. Filippi  International Center for Relativistic Astrophysics Network,
Piazza della Repubblica 10, I-65122 Pescara, Italy  ICRA, University Campus
Bio-Medico of Rome, Via Alvaro del Portillo 21, I-00128 Rome, Italy
Department of Engineering, University Campus Bio-Medico of Rome, Nonlinear
Physics and Mathematical Modeling Lab,
Via Alvaro del Portillo 21, 00128 Rome, Italy  Y. Wang  ICRA and Dipartimento
di Fisica, Università di Roma “La Sapienza”, Piazzale Aldo Moro 5, I-00185
Roma, Italy  International Center for Relativistic Astrophysics Network,
Piazza della Repubblica 10, I-65122 Pescara, Italy  INAF – Osservatorio
Astronomico d’Abruzzo,Via M. Maggini snc, I-64100, Teramo, Italy
<EMAIL_ADDRESS><EMAIL_ADDRESS>S. S. Xue  ICRA and Dipartimento di
Fisica, Università di Roma “La Sapienza”, Piazzale Aldo Moro 5, I-00185 Roma,
Italy  International Center for Relativistic Astrophysics Network, Piazza
della Repubblica 10, I-65122 Pescara, Italy

(/ Received date / Accepted date )

###### Abstract

We address the physical origin of the ultrarelativistic prompt emission (UPE)
phase of GRB 190114C observed in the interval $t_{\rm rf}=1.9$–$3.99$ s, by
the Fermi-GBM in 10 keV–10 MeV energy band. Thanks to the high signal–to–noise
ratio of Fermi-GBM data, a time-resolved spectral analysis has evidenced a
sequence of similar blackbody plus cutoff power-law spectra (BB+CPL), on ever
decreasing time intervals during the entire UPE phase. We assume that during
the UPE phase, the “ _inner engine_ ” of the GRB, composed of a Kerr black
hole (BH) and a uniform test magnetic field $B_{0}$, aligned with the BH
rotation axis, operates in an overcritical field $|{\bf E}|\geq E_{c}$, where
$E_{c}=m_{e}^{2}c^{3}/(e\hbar)$, being $m_{e}$ and $-e$ the mass and charge of
the electron. We infer an $e^{+}~{}e^{-}$ pair electromagnetic plasma in
presence of a baryon load, _a PEMB pulse_ , originating from a vacuum
polarization quantum process in the inner engine. This initially optically
thick plasma self-accelerates, giving rise at the transparency radius to the
MeV radiation observed by Fermi-GBM. At times $t_{\rm rf}>3.99$ s, the
electric field becomes undercritical, $|{\bf E}|<E_{c}$, and the inner engine,
as previously demonstrated, operates in the classical electrodynamics regime
and generate the GeV emission.  During both the “quantum” and the “classical”
electrodynamics processes, we determine the time varying mass and spin of the
Kerr BH in the inner engine, fulfilling the Christodoulou-Hawking-Ruffini
mass-energy formula of a Kerr BH. For the first time, we quantitatively show
how the inner engine, by extracting the rotational energy of the Kerr BH,
produces a series of PEMB pulses. We follow the quantum vacuum polarization
process in sequences with decreasing time bins. We compute the Lorentz
factors, the baryon loads and the radii at transparency, as well as the value
of the magnetic field, $B_{0}$, assumed to be constant in each sequence. The
fundamental hierarchical structure, linking the quantum electrodynamics regime
to the classical electrodynamics regime, is characterized by the emission of
“blackholic quanta” with a timescale $\tau\sim 10^{-9}$ s, and energy
$\mathcal{E}\sim 10^{45}$ erg.

##  I Introduction

It is by now clear that gamma-ray bursts (GRBs), far from being a short single
elementary process lasting less than 102 s, are possibly the most complex
astrophysical systems in the Universe, an authentic fundamental physics
laboratory. A series of Episodes, corresponding to new different physical
laws, take place on vastly different characteristic time scales ranging from
quantum electrodynamics (QED) time scales of $10^{-21}$ s, to classical
electrodynamics processes of $10^{-14}$ s, as well as to gravitational
processes of $10^{-6}$ s, and to hydrodynamics timescales of $1$ s and of
$10^{7}$ s, and the GRB source lifetime can indeed be as long as $10^{17}$ s,
i.e. the Universe lifetime (see Ruffini  _et al._ (2021), and references
therein).

One of the most intriguing phenomena occurring in the most energetic long GRBs
is the ultrarelativistic prompt emission (UPE) phase. In the case of GRB
190114C: 1) it takes place on a $2$ s rest-frame time ($t_{\rm rf}$) interval,
2) it encompasses $40\%$ of the GRB isotropic energy, and 3) it occurs in an
originally optically thick domain reaching transparency in the keV-MeV energy
range.

We address in this article the challenge of inferring, from spectral
properties, on an ever-decreasing time scales, the nature of this new and yet
unexplained process. We use: A) The concepts previously developed for a self-
accelerating optically thick $e^{+}~{}e^{-}$ pair-electromagnetic-baryon
plasma (PEMB pulse) originated from vacuum polarization produced by a
overcritically charged Kerr-Newman black hole (BH) Damour and Ruffini (1975);
Ruffini (1998); Damour  _et al._ (1978); Ruffini  _et al._ (1999, 2000, 2010).
B) A specific property of the PEMB pulse: the reaching of the transparency
radius with $\Gamma\sim$ 100 (Ruffini, 1998), which is essential to overcome
the compactness problem of the UPE phase; see e.g. (Ruderman, 1975; Piran,
2004) C) The Papapetrou-Wald solution (Papapetrou, 1966; Wald, 1974) as an
alternative to the charged Kerr-Newman BH as the source of vacuum polarization
process. D) The concept of an “effective charge”, $Q_{\rm eff}$, which
overcomes the difficulty of adopting the unexplained origin of a charged BH.
This concept allows to explain in an “effective way” the electric field which
arises from the gravitomagnetic interaction of a Kerr BH with a surrounding
magnetic field, $B_{0}$. The effective charge has been used in the study of
the “ _inner engine_ ” of GRB 130427A Ruffini  _et al._ (2019a) and GRB
190114C Moradi  _et al._ (2021).

We address the UPE phase of GRB 190114C that, owing to its morphology, can be
identified as a canonical binary driven hypernova (BdHN) of type I (see
details below), observed with a viewing angle orthogonal to the orbital plane
of the GRB binary progenitor. Indeed, a variety of Episodes of GRB 190114C
have been already identified and duly explained (Ruffini  _et al._ , 2021),
including the X-ray afterglow (Rueda  _et al._ , 2020) and the GeV emission
(Moradi  _et al._ , 2021).

It has been possible since the beginning of 2018 (Ruffini  _et al._ , 2018a;
Wang  _et al._ , 2019a; Rueda  _et al._ , 2020; Rueda and Ruffini, 2020;
Ruffini  _et al._ , 2021) to obtain specific new results thanks to a variety
of factors, including the identification of new GRB paradigms, a novel time-
resolved spectral analysis fulfilling stringent criteria of statistical
significance (Li  _et al._ , 2019a; Li, 2019a, b, 2020; Li  _et al._ , 2020;
Li and Zhang, 2021), and three-dimensional, numerical smoothed-particle-
hydrodynamic (SPH) simulations of BdHNe presented in Becerra  _et al._ (2019).
From these results, it has been concluded:

A) There is clear evidence that the progenitors of long GRBs are binary
systems composed of a carbon-oxygen (CO) star and a neutron star (NS)
companion: the BdHN. The gravitational collapse of the iron core of the CO
star leads to the SN and forms the newborn NS ($\nu$NS). When the binary
period is short i.e. $\sim$ 5 min, the SN ejecta hypercritically accretes onto
the companion NS, leading to the formation of a BH (Ruffini  _et al._ , 2021).
These systems are known as BdHN of type I (BdHN I). This approach was
successfully adopted in explaining the physical origin of the X-ray flares
(Ruffini  _et al._ , 2018b), further confirmed in Ruffini  _et al._ (2018c).

B) The accretion of the SN ejecta onto the $\nu$NS in BdHNe has given the
opportunity to explain the underlying physical nature of the X-ray afterglow
in GRB 130427A, GRB 160509A, GRB 160625B, GRB 180720A and GRB 190114C; see
Ruffini  _et al._ (2018a) and Rueda  _et al._ (2020).

C) The observations of a mildly relativistic phase in the GRB plateau and in
the afterglow (Ruffini  _et al._ , 2018b, c), have motivated the use of BdHNe
model in order to explain the energetic of the GeV emission as originating
from the extraction of rotational energy of a Kerr BH very close to the BH
horizon, described by the inner engine, addressed in Ruffini  _et al._
(2019a); Rueda and Ruffini (2020).

The inner engine is composed of: 1) a Kerr BH with mass of $M$ and angular
momentum of $J$, 2) an asymptotically uniform magnetic field, $B_{0}$ aligned
with the BH rotation axis, the Papapetrou-Wald solution (Wald, 1974), and 3) a
very low density plasma composed of ions and electrons with density of
$10^{-14}$ g cm-3; see Ruffini  _et al._ (2019a). The effective charge,
$Q_{\rm eff}$, of this system:

$\displaystyle{\color[rgb]{0,0,0}Q_{\rm eff}=2B_{0}JG/c^{3},}$ (1)

originates from the gravitomagnetic interaction of the Kerr BH with the
surrounding magnetic field, $B_{0}$, being $c$ and $G$ the speed of light in
vacuum and the gravitational constant, respectively; see Ruffini  _et al._
(2019a), Rueda and Ruffini (2020) and Moradi  _et al._ (2021).

In order to explain the GeV emission the _inner engine_ operates in an
undercritical electric field regime, i.e. $|{\bf E}|<E_{c}$, where
$E_{c}=m_{e}^{2}c^{3}/(e\hbar)$, being $m_{e}$ and $-e$ the electron mass and
charge, respectively, in presence of a magnetic field of $B_{0}\sim 10^{10}$
G, assumed to be constant in the entire process of emission. During this
process:

  1. 1.

electrons are injected close to the horizon with an initial Lorentz factor of
$\gamma=1$;

  2. 2.

electrons are accelerated by the electromagnetic fields of the _inner engine_
and radiate synchrotron photons of GeV energies;

  3. 3.

the radiation does not occur continuously but is emitted in elementary events
(quanta) of $\sim 10^{37}$ erg on a time scale of $\sim 10^{-14}$ s. This
energy is paid by the rotational energy of the Kerr BH implying a
corresponding decrease of the angular momentum $J$ of the Kerr BH (Ruffini
_et al._ , 2019a; Rueda and Ruffini, 2020; Moradi  _et al._ , 2021).

The emission of the quanta is repetitive. After the emission of each quanta, a
new process occurs starting from a new value $J^{*}=J-\Delta J$ of the angular
momentum, with $\Delta J/J\sim 10^{-16}$, being $\Delta J$ the angular
momentum extracted to the Kerr BH by the event in each repetitive step
(Ruffini  _et al._ , 2019a; Moradi  _et al._ , 2021).

In this article, we address the study of the UPE phase utilizing our previous
background and being guided by: 1) the hierarchical structure of the UPE in
GRB 190114C with characteristic spectral signature of a cutoff power-law and a
blackbody component (CPL$+$BB); see Ruffini  _et al._ (2019b); 2) the _inner
engine_ model which has been already well tested for the GeV radiation in GRB
130427A (Ruffini  _et al._ , 2019a) and GRB 190114C (Moradi  _et al._ , 2021).

We recall that the electric field in the Papapetrou-Wald solution in the slow-
rotation approximation is given by (Ruffini  _et al._ , 2019a):

${\color[rgb]{0,0,0}E_{r}\approx-\frac{1}{2}\alpha
B_{0}\,\frac{r_{+}^{2}}{r^{2}},}$ (2)

where $r_{+}$ is the outer event horizon and $\alpha\equiv cJ/(GM^{2})$ is the
dimensionless BH spin parameter.

The profound novelty characterizing the UPE phase is the assumption of an
overcritical field, i.e., $|{\bf E}|\geq E_{c}$ around the Kerr BH in the
_inner engine_. This overcritical field generates, via vacuum polarization, an
optically thick PEMB pulse which owing to its high density (here $\sim 10^{8}$
g cm-3) and high interior pressure, self-accelerates to an ultrarelativistic
regime and finally reaching the transparency point (Ruffini  _et al._ , 2010).

The hydrodynamic equations of the relativistic expanding PEMB pulses are
integrated until the point of transparency when the MeV radiation becomes
observable. The radius of transparency and Lorentz factor are explicitly
evaluated. This solution was first addressed in Ruffini (1998); Ruffini  _et
al._ (1999, 2000, 2010). This is the fundamental physical process which is
assumed to be at the very ground of the description of the UPE phase and its
spectral properties. Again, the energy in the overcritical field originates
from the rotational energy of the Kerr BH in the Papapetrou-Wald solution.

An additional necessary step is how to carry out the matching of the
overcritical regime, characterizing the UPE phase, its MeV radiation, and its
intrinsic quantum nature, with the already analysed undercritical regime
following the UPE phase. This undercritical regime describes the GeV radiation
and is dominated in the _inner engine_ by a classical electrodynamics nature
with very low density surrounding plasma.

For the determination of the parameters of the inner engine, we are guided by
the time-resolved spectra analysis and existence of the hierarchical structure
found in the UPE phase (Ruffini  _et al._ , 2019b). Each successive iteration
(rebinning) fulfills the total energy requirement and spectral structures in
different timescales. We select as the fundamental iterative process the
“only” one which allows the electric field to fulfill at the end of the UPE
phase the constraint $|{\bf E}|=E_{c}$. This boundary condition determines the
value of $B_{0}$ and is necessary to join the UPE phase to the classical
electrodynamics regime, originating the GeV radiation.

Similar to the case of the generation of GeV radiation from the _inner engine_
, also the emission of the MeV radiation during the UPE phase is not
continuous:

  1. 1.

$e^{+}~{}e^{-}\gamma$ plasma, in presence of the baryon load, is generated by
the vacuum polarization close to horizon with initial bulk Lorentz factor
$\Gamma=1$ on a characteristic timescale of $\sim\hbar/(m_{e}c^{2})\approx
10^{-21}~{}\rm s$,

  2. 2.

these PEMB pulses self-accelerate all the way to the point of transparency at
which emit MeV radiation in an ultrarelativistic regime,

  3. 3.

the process is again repetitive; at the end of each step the process restarts
with a value of electric field given by Eq. (2), keeping the magnetic field
constant, but with a new value of the BH dimensionless spin parameter
$\alpha^{*}=\alpha-\Delta\alpha$, with $\Delta\alpha/\alpha\sim 10^{-9}$,
being $\Delta\alpha$ the amount of dimensionless BH spin extracted to the Kerr
BH by the event in each repetitive step.

The UPE phase stops in the sequence which allows the condition $|{\bf
E}|=E_{c}$ to be reached at the right time.

In Sec. II, we recall three different Episodes identified in time-resolved
spectral analysis of GRB 190114C. We focus on the spectral analysis of the
Fermi-GBM (keV-MeV) and the Fermi-LAT (GeV) data during and after the UPE
phase.

In Sec. III, we present the time-resolved analysis of the UPE phase as well as
the appearance of the hierarchical structure of its spectra. These results
were announced in Ruffini  _et al._ (2019b) and here presented in an improved
numerical analysis, with their theoretical modeling.

In Sec. IV, we outline the properties of the _inner engine_. This is composed
of a uniform magnetic field aligned with the rotation axis of a Kerr BH,
following the exact, mathematical solution of the Einstein-Maxwell equations
given by Wald (1974). We here apply this solution to the astrophysical
conditions occurring in a BdHN I.

In Sec. V, for $t_{\rm rf}>3.99$ s, namely after the UPE phase, following
Ruffini  _et al._ (2019a, 2021), Rueda and Ruffini (2020) and Moradi  _et al._
(2021), we proceed to the self-consistent determination of a) the mass and
spin of the BH, b) the magnetic field $B_{0}$. These parameters are determined
to fulfill the energetics of GeV radiation and its transparency with respect
to the process of pair production by photon-magnetic field interaction. The
mass and spin of BH at $t_{\rm rf}$=$3.99$ s are, respectively,
$M=4.45M_{\odot}$ and $\alpha=0.41$, and magnetic field is $B_{0}\sim 10^{10}$
G.

In Sec. VI, we determine the mass and spin of BH at $t_{\rm rf}$=$1.9$ s,
$M=4.53M_{\odot}$ and $\alpha=0.54$. This result is consistent with the
luminosity obtained from the time-resolved spectral analysis, during the UPE
phase, and the above values of the mass and spin for $t_{\rm rf}>$ 3.99 s
given in Sec. V; see Figs. 3 and 4.

In Sec. VII, we address the overcritical regime, $|{\bf E}|\geq E_{c}$, in
order to have the vacuum polarization via Schwinger $e^{+}~{}e^{-}$ pair
production (see discussion in Cherubini  _et al._ , 2009), in the UPE phase;
see also Ruffini (1998); Ruffini  _et al._ (1999, 2000, 2010).

In Sec. VIII, we assess the general formulation of the transparency of the MeV
photons during the UPE phase.

In Sec. IX, we determine the magnetic field, $B_{0}\sim 10^{17}~{}$G, inferred
from the time-resolved spectral analysis with $\Delta t=0.125$ s resolution,
represented in Sec. III, corresponding to the emission at the transparency
point of the $16$ PEMB pulses with the repetition time of $\tau=0.125~{}$s.
This sequence does not fulfill the boundary condition of the UPE phase i.e.,
$|{\bf E}|=E_{c}$ at $t_{\rm rf}=3.99~{}$s; see Fig. 5

In Sec. X, we obtain the lower limit of magnetic filed around the BH,
$B_{0}=2.3\times 10^{14}~{}$G, during the UPE phase by imposing $|{\bf
E}|=E_{c}$ at $t_{\rm rf}=3.99~{}$s, marking the end of UPE phase. We infer
that the UPE phase results from emission at the transparency point of the
$\sim 10^{9}$ PEMB pulses, with radiation timescale of $\tau_{q}\sim
10^{-9}~{}$s; see Fig. 6.

In Sec. XI, we make a comparison with other approaches

In section XII, we outline the conclusions of this article.

##  II Fermi data of GRB 190114C

Figure 1:  Luminosity of the Fermi-GBM in the $10$ keV–$10$ MeV energy band
together with the luminosity of Fermi-LAT during and after UPE phase expressed
in the rest-frame of the source. The light grey part shows the $\nu$NS–rise
from $t_{\rm rf}=$0.79 s to $t_{\rm rf}=$1.18 s. The light blue part shows the
UPE phase which is in the time interval $t_{\rm rf}=1.9$–$3.99$ s, whose lower
and upper edges correspond, respectively, to the moment of BH formation and to
the moment which blackbody component disappears from the GBM data. The
corresponding analysis for GRB 130427A, GRB 160509A and GRB 160625B is
presented in (Li  _et al._ , 2019b). The red part shows the Fermi-GBM the
cavity introduced in Ruffini  _et al._ (2019c). The rest-frame $0.1$–$100$ GeV
luminosity light-curve of GRB 190114C after UPE phase is best fitted by a
power-law with slope of $1.2\pm 0.04$ and amplitude of $7.75\times 10^{52}$
erg s-1

At 20:57:02.63 UT on 14 January 2019, Fermi-GBM was triggered by GRB 190114C
(Hamburg  _et al._ , 2019). The Fermi-LAT had a boresight angle of $68$
degrees at the trigger time, the GRB remained in the field of view of Fermi-
LAT for $150$ s. With the redshift of $z=0.424$ (J. Selsing  _et al._ , 2019)
the isotropic energy of this burst is $E_{\rm iso}=(2.48\pm 0.22)\times
10^{53}$erg. Since BdHNe I are characterized by $E_{\rm iso}\gtrsim 10^{52}$
erg, we have identified GRB 190114C as a BdHN I and predict the occurrence of
an associated SN (Ruffini  _et al._ , 2019d). This prediction was followed by
the successful observation of the SN associated with this burst (Melandri  _et
al._ , 2019). The first GeV photon with probability more than 90$\%$ belonging
to this GRB is a $\sim~{}0.9$ GeV photon observed at $t_{\rm rf}=1.9$ s after
the GBM trigger. The highest-energy photon is a $22.9$ GeV event which is
observed $15$ s after the GBM trigger (D. Kocevski  _et al._ , 2019). GRB
190114C has become since a prototype for identifying the BdHN I Episodes.

Three different Episodes have been identified in the Fermi-GBM data; see Fig.
1:

Episode 1 with an isotropic energy of $E_{\rm iso}=(1.0\pm 0.12)\times
10^{53}$ erg, occurs in the rest-frame time interval $t_{\rm rf}=[0,1.9]$ s,
being $t_{\rm rf}=0$ s the Fermi-GBM rest-frame trigger time. It reveals a
thermal component from $t_{\rm rf}=0.79$ s to $t_{\rm rf}=1.18$ s, in its
spectral analysis, marking the rise of newly born NS ($\nu$NS–rise) with a
corresponding isotropic energy of
$E^{\rm{\color[rgb]{0,0,0}iso}}_{\rm{\color[rgb]{0,0,0}\nu NS}}=(2.82\pm
0.13)\times 10^{52}$ erg.

Episode 2, with an isotropic energy of $E_{\rm iso}=(1.47\pm 0.2)\times
10^{53}$ erg, equivalent to 40% of isotropic energy of the GRB, lasts only $2$
s. It occurs in the rest-frame time interval $t_{\rm rf}=[1.9,3.99]$ s. It
encompasses three major events: a) The formation of the BH , observation of
the first GeV photon at $t_{\rm rf}=1.9$ s , see details in Ruffini  _et al._
(2019a). b) An increase of the 0.1-100 GeV luminosity following a power-law of
$L_{\rm GeV}=8.7\times 10^{50}~{}t^{+(1.77\pm 0.28)}$ erg s-1. c) The
energetically dominant UPE phase observed by Fermi-GBM in the $10$ keV–$10$
MeV band, occurring in the entire interval t${}_{\rm rf}=$1.9s-3.99 s
fulfilling a hierarchical structure signed by a spectrum composed of a thermal
emission and a cutoff power-law component (CPL$+$BB); see section III.

Episode 3, the “cavity”, starts at $~{}t_{\rm rf}=11$ s and ends at $t_{\rm
rf}=20$ s. The presence of a “cavity” in GRB 190114C, carved out in the SN
ejecta by the BH formation, has been confirmed in Ruffini  _et al._ (2019c).

The GeV luminosity following the UPE phase is best fitted by the _decreasing_
power-law of $L_{\rm GeV}=\left(7.75\pm 0.44\right)\times
10^{52}~{}t^{-(1.2\pm 0.04)}$ erg s-1, with an isotropic energy of $E_{\rm
GeV}=(1.8\pm 1.3)\times 10^{53}$ erg. The spectrum of Fermi-LAT in the
$0.1$–$100$ GeV energy band, after the UPE phase, is best fitted by a power-
law (Wang  _et al._ , 2019b); see Fig. 1 and Moradi  _et al._ (2021) for more
details.

All these results have been presented in Ruffini  _et al._ (2019b), Ruffini
_et al._ (2019c), Ruffini  _et al._ (2019a) and Moradi  _et al._ (2021).

##  III The time-resolved spectral analysis, the hierarchical structure and
the MeV luminosity of the UPE phase

Following the spectral analysis performed over the UPE phase from $t_{\rm
rf}=1.9$ s to $t_{\rm rf}=3.9$ s (first iteration), we perform the spectral
analysis over the 1 second intervals ($\Delta t_{\rm rf}=1$ s), namely
[$1.9$s–$2.9$s] and [$2.9$s–$3.9$s](second iteration).

Each half intervals are further divided in half (third iteration), i.e.,
$\Delta t_{\rm rf}=0.5$ s: [$1.9$s–$2.40$s], [$2.40$s–$2.9$s], [$2.9$s–$3.4$s]
and [$3.4$s–$3.9$s] and the corresponding spectral analysis is performed over
each interval.

We further divide the UPE into $8$ intervals of $\Delta t_{\rm rf}=0.25$ s (
fourth iteration): [$1.9$s–$2.15$s],[$2.15$s–$2.40$s], [$2.40$s–$2.65$s],
[$2.65$s–$2.9$s], [$2.9$s–$3.15$s], [$3.15$s–$3.4$s], [$3.4$s–$3.65$s] and
[$3.65$s–$3.9$s], and perform the same spectral analysis over each interval.

We continue until the final iteration (fifth iteration), where the adequate
signal-to-noise ratio S/N is fulfilled. The UPE is divided into $16$ time
intervals of $\Delta t_{\rm rf}=0.125$ s:
[$1.896$s–$2.019$s],[$2.019$s–$2.142$s],[$2.142$s–$2.265$s],
[$2.265$s–$2.388$s], [$2.388$s–$2.511$s], [$2.511$s–$2.633$s],
[$2.633$s–$2.756$s], [$2.756$s–$2.87$s], [$2.879$s–$3.002$s],
[$3.002$s–$3.125$s], [$3.125$s–$3.248$s], [$3.248$s–$3.371$s],
[$3.371$s–$3.494$s], [$3.494$s–$3.617$s], [$3.617$s–$3.739$s] and
[$3.739$s–$3.862$s] and perform the spectral analysis. After dividing into
subintervals of 0.125 s one extra time interval of [$3.862$s–$3.985$s] has
been added.

The spectral fitting of a cutoff power law plus black body (CPL+BB) is
confirmed in each time interval and for each iterative process; see Table 1,
Fig. 2, Ruffini  _et al._ (2019b, for more details).

Figure 2: Time-resolved spectral analysis of UPE phase GRB 190114C: from $t=2.7$ s ($t_{\rm rf}=1.9$ s) to $t=5.5$ s ($t_{\rm rf}=3.9$ s). For the second iteration; (a) the time interval is divided into two parts, four parts for the  third iteration; (b), eight parts for the  fourth iteration; (c), and sixteen parts for the fifth iteration; (d), respectively. The spectral fitting parameters for each iteration are reported in Table 1. Plots are taken from Ruffini  _et al._ (2019b) with permission of authors. Table 1: The parameters of the time-resolved spectral fits of the UPE phase of GRB 190114C, performed from $\Delta t_{\rm rf}=2$ s down to subintervals of $\Delta t_{\rm rf}=0.125$ s. The UPE phase extends from $t=2.7$ s ($t_{\rm rf}=1.9$ s) to $t=5.5$ s ($t_{\rm rf}=3.99$ s). Column 1; represents the time intervals in the observer’s frame (Obs), column 2; the time intervals in the rest-frame (rf), column 3; the statistical significance ($S$) for each time interval,  column 4; the power-law index of the cutoff power-law (CPL) component, column 5; the rest-frame cutoff energy, column 6; the rest-frame black body (BB) temperature, column 7; the Akaike Information Criterion/Bayesian Information Criterion (AIC/BIC), column 8; the BB flux ($F_{\rm BB}$), column 9; the CPL+BB or total flux ($F_{\rm tot}$), column 10; the ratio of the BB flux to the total flux, $F_{\rm BB}/F_{\rm tot}$ and, finally column 11; the isotropic energy in each time interval. As it can be seen from column 10, the $F_{\rm BB}/F_{\rm tot}$ remains almost constant in each iteration. The AIC (Akaike 1974) and the BIC (Schwarz  _et al._ 1978) methods were used to select non-nested and nested models, respectively (see Li  _et al._ , 2019a; Li, 2019a, b, 2020; Li  _et al._ , 2020; Li and Zhang, 2021, for more information about these methods). Table is taken from Ruffini  _et al._ (2019b) with permission of authors. $t_{1}$$\sim$$t_{2}$ |  $t_{rf,1}$$\sim$$t_{rf,2}$ | $S$ | $\alpha$ | $E_{\rm c}$ | $kT$ |  $\Delta$DIC | $F_{\rm BB}$ | $F_{\rm tot}$ | $F_{\rm ratio}$ | $E_{\rm tot}$
---|---|---|---|---|---|---|---|---|---|---
(s) | (s) |  |  | (keV) | (keV) |  |  (10-6) |  (10-6) |  | (erg)
Obs | Rest-frame |  |  |  |  |  |  (erg cm-2 s-1) |  (erg cm-2 s-1) |  |
2.700$\sim$5.500 |  1.896$\sim$3.862 | 418.62 |  -0.71${}^{+0.02}_{-0.02}$ |  717.6${}^{+25.4}_{-25.4}$ |  159.0${}^{+3.6}_{-3.6}$ | -3344/6697/6719 |  22.49${}^{+3.21}_{-2.65}$ |  111.10${}^{+11.60}_{-10.40}$ | 0.20 | 1.50e+53
2.700$\sim$4.100 |  1.896$\sim$2.879 | 296.60 |  -0.51${}^{+0.02}_{-0.02}$ |  696.6${}^{+31.9}_{-32.4}$ |  209.7${}^{+9.3}_{-9.1}$ | -2675/5360/5381 |  24.67${}^{+6.93}_{-5.35}$ |  142.50${}^{+23.90}_{-21.00}$ | 0.17 | 9.64e+52
4.100$\sim$5.500 |  2.879$\sim$3.862 | 318.07 |  -0.90${}^{+0.02}_{-0.02}$ |  639.3${}^{+31.9}_{-31.6}$ |  130.6${}^{+2.5}_{-2.5}$ | -2529/5069/5090 |  25.55${}^{+2.97}_{-2.75}$ |  80.98${}^{+9.68}_{-8.07}$ | 0.32 | 5.48e+52
2.700$\sim$3.400 |  1.896$\sim$2.388 | 204.30 |  -0.59${}^{+0.03}_{-0.03}$ |  724.7${}^{+44.5}_{-45.5}$ |  220.0${}^{+17.1}_{-17.2}$ | -1882/3774/3796 |  18.55${}^{+9.42}_{-7.40}$ |  123.90${}^{+29.20}_{-22.30}$ | 0.15 | 4.19e+52
3.400$\sim$4.100 |  2.388$\sim$2.879 | 225.88 |  -0.46${}^{+0.04}_{-0.04}$ |  699.8${}^{+47.8}_{-48.3}$ |  196.7${}^{+8.9}_{-8.7}$ | -2032/4074/4095 |  31.78${}^{+9.60}_{-7.31}$ |  161.40${}^{+47.10}_{-32.40}$ | 0.20 | 5.46e+52
4.100$\sim$4.800 |  2.879$\sim$3.371 | 233.97 |  -0.84${}^{+0.03}_{-0.03}$ |  608.1${}^{+42.1}_{-42.2}$ |  130.4${}^{+3.7}_{-3.9}$ | -1880/3770/3792 |  23.94${}^{+4.20}_{-4.22}$ |  85.37${}^{+14.83}_{-12.27}$ | 0.28 | 2.89e+52
4.800$\sim$5.500 |  3.371$\sim$3.862 | 227.90 |  -0.96${}^{+0.03}_{-0.03}$ |  679.5${}^{+49.1}_{-48.7}$ |  130.6${}^{+3.1}_{-3.2}$ | -1809/3628/3649 |  27.18${}^{+4.01}_{-3.73}$ |  78.20${}^{+11.40}_{-9.66}$ | 0.35 | 2.65e+52
2.700$\sim$3.050 |  1.896$\sim$2.142 | 148.59 |  -0.59${}^{+0.03}_{-0.03}$ |  547.7${}^{+44.2}_{-44.9}$ |  240.8${}^{+29.2}_{-29.1}$ | -1187/2384/2406 |  19.67${}^{+17.96}_{-8.88}$ |  103.20${}^{+30.60}_{-20.28}$ | 0.19 | 1.75e+52
3.050$\sim$3.400 |  2.142$\sim$2.388 | 145.04 |  -0.60${}^{+0.02}_{-0.02}$ |  965.2${}^{+28.5}_{-30.1}$ |  203.5${}^{+14.8}_{-14.8}$ | -1320/2650/2671 |  22.87${}^{+8.88}_{-7.23}$ |  152.00${}^{+24.00}_{-21.00}$ | 0.15 | 2.57e+52
3.400$\sim$3.750 |  2.388$\sim$2.633 | 134.60 |  -0.63${}^{+0.04}_{-0.04}$ |  885.7${}^{+70.9}_{-70.1}$ |  240.6${}^{+10.5}_{-10.6}$ | -1224/2458/2480 |  41.02${}^{+11.09}_{-7.91}$ |  129.10${}^{+32.40}_{-23.40}$ | 0.32 | 2.18e+52
3.750$\sim$4.100 |  2.633$\sim$2.879 | 187.77 |  -0.35${}^{+0.06}_{-0.05}$ |  607.8${}^{+57.1}_{-60.1}$ |  151.5${}^{+12.4}_{-14.2}$ | -1428/2866/2887 |  23.92${}^{+12.46}_{-10.40}$ |  192.00${}^{+101.70}_{-60.30}$ | 0.12 | 3.25e+52
4.100$\sim$4.450 |  2.879$\sim$3.125 | 171.81 |  -0.69${}^{+0.04}_{-0.04}$ |  515.9${}^{+43.6}_{-43.6}$ |  117.3${}^{+5.0}_{-5.0}$ | -1271/2552/2573 |  19.19${}^{+4.89}_{-4.40}$ |  92.71${}^{+27.69}_{-22.43}$ | 0.21 | 1.57e+52
4.450$\sim$4.800 |  3.125$\sim$3.371 | 230.14 |  -0.98${}^{+0.04}_{-0.04}$ |  702.0${}^{+78.1}_{-78.2}$ |  141.3${}^{+5.8}_{-5.8}$ | -1254/2518/2539 |  26.76${}^{+6.41}_{-5.47}$ |  80.73${}^{+17.95}_{-14.95}$ | 0.33 | 1.37e+52
4.800$\sim$5.150 |  3.371$\sim$3.617 | 166.30 |  -0.97${}^{+0.04}_{-0.04}$ |  685.1${}^{+69.4}_{-68.6}$ |  140.8${}^{+4.6}_{-4.6}$ | -1218/2447/2468 |  31.83${}^{+6.85}_{-4.98}$ |  82.51${}^{+15.62}_{-12.33}$ | 0.39 | 1.40e+52
5.150$\sim$5.500 |  3.617$\sim$3.862 | 161.51 |  -0.95${}^{+0.04}_{-0.04}$ |  692.2${}^{+79.1}_{-77.7}$ |  120.0${}^{+4.0}_{-4.0}$ | -1203/2416/2438 |  23.19${}^{+5.38}_{-3.81}$ |  73.57${}^{+18.69}_{-12.93}$ | 0.32 | 1.24e+52
2.700$\sim$2.875 |  1.896$\sim$2.019 | 117.09 |  -0.58${}^{+0.05}_{-0.05}$ |  470.5${}^{+74.4}_{-83.7}$ |  261.5${}^{+29.0}_{-27.9}$ | -640/1291/1311 |  33.68${}^{+20.39}_{-14.33}$ |  112.30${}^{+28.37}_{-25.73}$ | 0.30 | 9.50e+51
2.875$\sim$3.050 |  2.019$\sim$2.142 | 94.40 |  -0.68${}^{+0.04}_{-0.05}$ |  627.6${}^{+87.0}_{-91.5}$ |  258.0${}^{+30.1}_{-28.7}$ | -664/1337/1359 |  28.45${}^{+20.42}_{-12.51}$ |  98.14${}^{+33.56}_{-26.44}$ | 0.29 | 8.30e+51
3.050$\sim$3.225 |  2.142$\sim$2.265 | 106.62 |  -0.59${}^{+0.03}_{-0.03}$ |  957.1${}^{+34.1}_{-34.9}$ |  245.3${}^{+21.5}_{-21.0}$ | -768/1547/1568 |  25.71${}^{+13.87}_{-9.03}$ |  169.30${}^{+38.20}_{-31.60}$ | 0.15 | 1.43e+52
3.225$\sim$3.400 |  2.265$\sim$2.388 | 100.40 |  -0.73${}^{+0.06}_{-0.06}$ |  1275.9${}^{+208.9}_{-215.4}$ |  208.6${}^{+9.1}_{-9.2}$ | -669/1349/1369 |  36.78${}^{+9.54}_{-8.93}$ |  144.90${}^{+33.02}_{-27.63}$ | 0.25 | 1.23e+52
3.400$\sim$3.575 |  2.388$\sim$2.511 | 98.23 |  -0.59${}^{+0.05}_{-0.05}$ |  804.0${}^{+86.7}_{-82.3}$ |  255.9${}^{+17.4}_{-17.4}$ | -702/1414/1436 |  42.19${}^{+19.41}_{-13.59}$ |  139.30${}^{+48.30}_{-35.60}$ | 0.30 | 1.18e+52
3.575$\sim$3.750 |  2.511$\sim$2.633 | 93.84 |  -0.65${}^{+0.04}_{-0.04}$ |  916.3${}^{+64.6}_{-67.7}$ |  229.3${}^{+13.6}_{-13.5}$ | -730/1471/1492 |  39.25${}^{+11.97}_{-10.71}$ |  119.50${}^{+32.90}_{-25.45}$ | 0.33 | 1.01e+52
3.750$\sim$3.925 |  2.633$\sim$2.756 | 126.63 |  -0.51${}^{+0.02}_{-0.02}$ |  960.9${}^{+30.9}_{-31.4}$ |  204.6${}^{+9.9}_{-10.0}$ | -808/1627/1648 |  57.70${}^{+15.81}_{-12.25}$ |  221.10${}^{+35.60}_{-31.50}$ | 0.26 | 1.87e+52
3.925$\sim$4.100 |  2.756$\sim$2.879 | 141.61 |  -0.27${}^{+0.06}_{-0.06}$ |  412.7${}^{+12.2}_{-11.9}$ |  196.8${}^{+14.0}_{-16.1}$ | -729/1468/1488 |  32.20${}^{+19.05}_{-18.86}$ |  176.50${}^{+12.91}_{-11.21}$ | 0.18 | 1.49e+52
4.100$\sim$4.275 |  2.879$\sim$3.002 | 122.91 |  -0.54${}^{+0.06}_{-0.06}$ |  474.1${}^{+45.5}_{-46.2}$ |  162.6${}^{+14.9}_{-14.8}$ | -758/1526/1547 |  24.26${}^{+17.09}_{-10.09}$ |  116.10${}^{+52.40}_{-35.12}$ | 0.21 | 9.82e+51
4.275$\sim$4.450 |  3.002$\sim$3.125 | 122.62 |  -0.64${}^{+0.08}_{-0.08}$ |  365.0${}^{+44.9}_{-48.5}$ |  107.5${}^{+15.7}_{-12.6}$ | -675/1360/1380 |  9.04${}^{+9.47}_{-5.69}$ |  72.20${}^{+19.06}_{-14.95}$ | 0.13 | 6.11e+51
4.450$\sim$4.625 |  3.125$\sim$3.248 | 111.94 |  -1.04${}^{+0.05}_{-0.05}$ |  640.0${}^{+108.7}_{-106.1}$ |  161.0${}^{+11.1}_{-10.8}$ | -640/1290/1310 |  22.34${}^{+9.36}_{-6.65}$ |  68.54${}^{+11.70}_{-11.21}$ | 0.33 | 5.80e+51
4.625$\sim$4.800 |  3.248$\sim$3.371 | 123.33 |  -0.95${}^{+0.05}_{-0.05}$ |  694.2${}^{+96.8}_{-94.2}$ |  146.3${}^{+6.7}_{-6.6}$ | -734/1477/1499 |  35.59${}^{+9.47}_{-8.00}$ |  89.91${}^{+27.59}_{-18.82}$ | 0.40 | 7.60e+51
4.800$\sim$4.975 |  3.371$\sim$3.494 | 129.65 |  -0.85${}^{+0.05}_{-0.05}$ |  564.5${}^{+68.9}_{-71.9}$ |  135.3${}^{+7.5}_{-7.6}$ | -744/1498/1519 |  30.78${}^{+11.12}_{-8.55}$ |  96.58${}^{+31.02}_{-23.68}$ | 0.32 | 8.17e+51
4.975$\sim$5.150 |  3.494$\sim$3.617 | 107.36 |  -1.10${}^{+0.04}_{-0.04}$ |  820.5${}^{+115.0}_{-111.2}$ |  149.7${}^{+5.9}_{-5.8}$ | -683/1376/1398 |  32.76${}^{+6.98}_{-5.92}$ |  71.57${}^{+16.74}_{-11.99}$ | 0.46 | 6.05e+51
5.150$\sim$5.325 |  3.617$\sim$3.739 | 108.96 |  -1.04${}^{+0.05}_{-0.05}$ |  765.2${}^{+119.0}_{-115.8}$ |  130.9${}^{+5.8}_{-5.8}$ | -697/1404/1426 |  26.14${}^{+7.02}_{-5.96}$ |  66.70${}^{+20.48}_{-14.17}$ | 0.39 | 5.64e+51
5.325$\sim$5.500 |  3.739$\sim$3.862 | 121.57 |  -0.88${}^{+0.06}_{-0.06}$ |  635.3${}^{+88.7}_{-92.0}$ |  108.9${}^{+5.3}_{-5.4}$ | -736/1483/1504 |  20.90${}^{+6.51}_{-5.15}$ |  79.48${}^{+28.02}_{-21.03}$ | 0.26 | 6.72e+51

[a] [b]

Figure 3: Luminosity [a] and rest-frame temperature [b] during the UPE
observed by Fermi-GBM, obtained from analyses with $\Delta t=0.125$ s (blue
circles), $\Delta t=0.25$ s (black circles) and $\Delta t=0.5$ s (red circles)
time resolutions reported in Table 1. The luminosity is best fitted by a
power-law of amplitude $(3.5\pm 1.1)\times 10^{53}$ erg s-1 and power-law
index $-1.50\pm 0.30$. The best fit of luminosity, obtained from $\Delta
t_{\rm rf}=0.125$ s time-resolved analysis, is in principle independent of the
resolution of data analysis and is fulfilled in all iterative sequences.

From a time-resolved analysis of the UPE phase, performed down to the fifth
iteration, a hierarchical structure is obtained. It reveals a common spectral
feature for each subinterval characterized by the CPL$+$BB best-fit model with
a rest-frame temperature of $kT=100\sim 300$ keV and the ratio of blackbody
flux ($F_{\rm BB}$) to the total flux ($F_{\rm tot}$) of:

$0.1\lesssim\frac{F_{\rm BB}}{F_{\rm tot}}\lesssim 0.5.$ (3)

see Table 1,  Fig. 2 and Ruffini  _et al._ (2019b, for more information)

During the UPE phase the MeV luminosity is best fitted by

$L_{\rm MeV}=A_{\rm MeV}~{}t^{-\alpha_{\rm MeV}},$ (4)

with slope $\alpha_{\rm MeV}=1.5\pm 0.3$, and amplitude $A_{\rm MeV}=(3.5\pm
1.1)\times 10^{53}$ erg s-1. This best fit is obtained from $\Delta t_{\rm
rf}=0.125$ s time-resolved analysis and is fulfilled in all iterative
sequences; see Table 1 and Figs. 1 and 3.

The existence of the thermal and the cutoff power-law components in the
spectra of the UPE phase have been identified as the characteristic signature
of $e^{+}~{}e^{-}$ pair creation in presence of baryons (PEMB pulse)
originating from the vacuum polarization process Ruffini  _et al._ (1999,
1999, 2000, 2010); see Sec. VII.

##  IV The properties of _inner engine_

The discovery of the _inner engine_ obtained by incorporating the Papapetrou-
Wald solution (Papapetrou, 1966; Wald, 1974; Ruffini  _et al._ , 2019a; Rueda
and Ruffini, 2020; Moradi  _et al._ , 2021), around the newborn Kerr BH in the
BdHNe I, in presence of the low density plasma of the cavity, see (Ruffini
_et al._ , 2019c), was operative in the classical electrodynamics with $|{\bf
E}|<E_{c}$ process and leading to the generation by synchrotron radiation of
the GeV emission in GRB 130427A and GRB190114C. We also apply here this inner
engine in the $|{\bf E}|>E_{c}$ regime to describe the quantum electrodynamics
process.

Wald’s work is based on the Papapetrou discovery Papapetrou (1966) that
Killing vectors are vector potential solutions of sourceless Maxwell equations
in vacuum spacetimes in the test field approximation (i.e. no metric
backreaction). A linear combination of these two Killing vector solutions led
Wald to the solution for a rotating BH immersed in a uniform magnetic field
$B_{0}$, aligned with the rotation axis of the Kerr BH at infinity.

The electromagnetic field of the _inner engine_ in the Carter’s orthonormal
tetrad is:

$\displaystyle E_{\hat{r}}$ $\displaystyle=$
$\displaystyle\frac{\hat{a}B_{0}}{\Sigma}\left[r\sin^{2}\theta-\frac{\hat{M}\left(\cos^{2}\theta+1\right)\left(r^{2}-\hat{a}^{2}\cos^{2}\theta\right)}{\Sigma}\right],$
(5) $\displaystyle E_{\hat{\theta}}$ $\displaystyle=$
$\displaystyle\frac{\hat{a}B_{0}}{\Sigma}\sin\theta\cos\theta\sqrt{\Delta},$
(6) $\displaystyle B_{\hat{r}}$ $\displaystyle=$
$\displaystyle-\frac{B_{0}\cos\theta}{\Sigma}\left[-\frac{2\hat{a}^{2}\hat{M}r\left(\cos^{2}\theta+1\right)}{\Sigma}+\hat{a}^{2}+r^{2}\right],$
(7) $\displaystyle B_{\hat{\theta}}$ $\displaystyle=$
$\displaystyle\frac{B_{0}r}{\Sigma}\sin\theta\sqrt{\Delta},$ (8)

where $\Sigma=r^{2}+\hat{a}^{2}\cos^{2}\theta$,
$\Delta=r^{2}-2\hat{M}r+\hat{a}^{2}$, $\hat{M}=GM/c^{2}$,
$\hat{a}=a/c=J/(M\,c)$, being $M$ and $J$ the mass and angular momentum of the
Kerr BH. The (outer) event horizon is located at
$r_{+}=(\hat{M}+\sqrt{\hat{M}^{2}-\hat{a}^{2}})$.

The electromagnetic field in the polar direction $\theta=0$ and at small
angles from it is well approximated by (Ruffini  _et al._ , 2019a; Rueda and
Ruffini, 2020):

$\displaystyle E_{\hat{r}}$ $\displaystyle=$
$\displaystyle-\frac{2B_{0}J\,G}{c^{3}}\frac{\left(r^{2}-\hat{a}^{2}\right)}{\left(r^{2}+\hat{a}^{2}\right)^{2}}$
(9) $\displaystyle E_{\hat{\theta}}$ $\displaystyle=$ $\displaystyle 0$ (10)
$\displaystyle B_{\hat{r}}$ $\displaystyle=$
$\displaystyle\frac{B_{0}\left(-\frac{4\,G\,J^{2}r}{M\left(r^{2}+\hat{a}^{2}\right)}+a^{2}+r^{2}\right)}{\left(r^{2}+\hat{a}^{2}\right)}$
(11) $\displaystyle B_{\hat{\theta}}$ $\displaystyle=$ $\displaystyle 0.$ (12)

Equation (9) is the same as the radial electric field of the Kerr-Newman
metric in the same tetrad just substituting to the charge $Q$ of the Kerr-
Newman solution the effective charge $Q_{\rm eff}$, given by Eq. (1), see e.g.
Giacconi and Ruffini (1978). Therefore, up to linear order in $\theta$ and in
the dimensionless BH spin parameter $\alpha\equiv\hat{a}/(GM/c^{2})$, the
electric field can be written as

$E_{\hat{r}}=-\frac{2B_{0}J\,G}{c^{3}}\frac{\left(r^{2}-\hat{a}^{2}\right)}{\left(r^{2}+\hat{a}^{2}\right)^{2}}\approx-\frac{1}{2}\alpha
B_{0}\frac{r_{+}^{2}}{r^{2}},$ (13)

which for spin values $\alpha\lesssim 0.7$, the available electrostatic energy
is well approximated by

$\mathcal{E}\approx\frac{\left(2B_{0}JG/c^{3}\right)^{2}}{2\,r_{+}}={\color[rgb]{0,0,0}\frac{Q_{\rm
eff}^{2}}{2\,r_{+}}}=1.25\times
10^{43}\frac{\beta^{2}\alpha^{2}\mu^{3}}{1+\sqrt{1-\alpha^{2}}}\textrm{ erg},$
(14)

where we have normalized the mass and magnetic field strength by,
respectively, $\mu=M/M_{\odot}$ and $\beta=B_{0}/B_{c}$, being

$B_{c}=E_{c}=\frac{m_{e}^{2}c^{3}}{e\hbar}\approx 4.41\times 10^{13}\,\rm G,$
(15)

the critical field for vacuum polarization; see Ruffini  _et al._ (2019a) for
more details.

The values $\mu$ and $\beta$ in this general equations will be determined as a
function of astrophysical process operative in the inner engine in GRB
190114C. The corresponding description of the overcritical and the
undercritical fields are represented in the next sections.

We adopt that the magnetic field and BH spin are parallel, therefore along the
symmetry axis direction electrons in the surrounding ionized medium are
repelled, while protons are pulled into the BH (see Ruffini  _et al._ , 2019a,
for additional details).

##  V The mass and spin of the BH of GRB 190114C

We here recall the self-consistent solution following the UPE phase, well
tested in the case of GRB 130427A (Ruffini  _et al._ , 2019a) and GRB 190114C
(Moradi  _et al._ , 2021) which fulfills three conditions: 1) The GeV
energetics observed by the Fermi-LAT is paid by the extractable energy of the
BH, i.e.: $E_{\rm GeV}=E_{\rm extr}$. 2) The magnetic $e^{+}~{}e^{-}$ pair
production (MPP) process does not occur around the BH, therefore the GeV
photons fulfill the transparency condition. 3) The timescale of the
synchrotron radiation determines the time scale of observed GeV radiation.

Having these conditions, and assuming the minimum energy budget requirement;,
the _inner engine_ parameters at $t>t_{\rm rf}=3.99~{}$s, i.e., after the UPE
phase, are: magnetic field strength $B_{0}\approx 3.9\times 10^{10}$ G, spin
and BH mass, respectively, $\alpha=0.41$ and $M=4.45~{}M_{\odot}$. The
corresponding BH irreducible mass is $M_{\rm irr}=4.35~{}M_{\odot}$ see Moradi
_et al._ (2021) for more details.

##  VI Determination of the mass and spin of the BH during the UPE phase

We have obtained in the previous section at $t_{\rm rf}=3.99$ s the values of
mass and spin parameters of the BH and the magnetic field: $M=4.45M_{\odot}$,
$\alpha=0.41$, and $B_{0}=3.9\times 10^{10}$ G, respectively. We now turn to
the determination of the mass and spin of the BH during the UPE phase. The
mass-energy formula of the Kerr BH (Christodoulou, 1970; Christodoulou and
Ruffini, 1971; Hawking, 1971; see also ch.33 in (Misner  _et al._ , 1973)) is
given by:

$M^{2}=\frac{c^{2}J^{2}}{4G^{2}M^{2}_{\rm irr}}+M_{\rm irr}^{2}.$ (16)

We require that the energetics of the MeV radiation be explained by the
extractable rotational energy of the Kerr BH, i.e.,

$E_{\rm MeV}=E_{\rm extr}=(M-M_{\rm irr})c^{2}.$ (17)

Therefore, the extractable energy is given by:

$E_{\rm extr}=(M-M_{\rm
irr})c^{2}=\left(1-\sqrt{\frac{1+\sqrt{1-\alpha^{2}}}{2}}\right)Mc^{2}.$ (18)

The time derivative of Eq. (18) gives the luminosity

${L_{\rm MeV}=-\frac{dE_{extr}}{dt}=-\frac{dM}{dt},}$ (19)

in which we assume that $M_{irr}$ is constant for BH during the energy
emission process.

From the luminosity of MeV radiation expressed in the rest-frame of the GRB,
given by Eq. (4) in the time interval of the UPE phase (see also Fig. 3), and
from the values of the spin and of the mass of the BH at $t_{\rm rf}=3.99$ s,
we can now work backward by integrating Eq. (19) and determine the BH mass and
spin at the beginning of the UPE, when the BH is formed, namely at $t_{\rm
rf}=1.9$ s. We obtain $M=4.53M_{\odot}$ and $\alpha=0.54$, respectively.

This assumption demands that all the luminosity of UPE phase originates from
the rotational energy of the BH. This point is going to be justified in the
next sections.

##  VII Vacuum polarization, dyadoregion and the UPE phase

The UPE phase is characterized by an electric field $|{\bf E}|>E_{c}$ (Ruffini
_et al._ , 2010, 2018c). The problem of vacuum polarization due to the
overcritical field has a vast literature which dates back to the concept of
the the dyadosphere (Ruffini, 1998) and dyadotorous in the Kerr-Newman
geometry, developed in Cherubini  _et al._ (2009). Dyado is from the Greek
word “duados” for pair, indicating here the $e^{+}~{}e^{-}$ pairs. The
dyadotorus is the region where the vacuum polarization processes occur around
a rotating charged BH, leading to the production of $e^{+}~{}e^{-}$ pairs; see
also (Ruffini  _et al._ , 2010) for details.

In order to evaluate this process in the present case, we adopt a description
using the Kerr-Newman geometry for which an analytic formula for the energy
contained in the dyadoregion has been derived in Cherubini  _et al._ (2009).
We have checked numerically that the energy of the dyadoregion in the Kerr-
Newman geometry (see Eq. 20 below), setting the BH charge as the effective
charge of the Papapetrou-Wald solution, $2B_{0}JG/c^{3}$, is a good
approximation of the one estimated numerically with the Papapetrou-Wald
solution. We have verified that the quantitative difference is at most $30\%$,
which implies that this approximation does not affect our conclusions.

We can now evaluate the energy of $e^{+}~{}e^{-}$ pairs generated in the
Papapetrou-Wald solution using the Kerr-Newman analogy. We use the Carter
orthonormal frame, in which the flat spacetime Schwinger framework can be
locally applied and determine the dyadoregion energy (see discussion in
Cherubini  _et al._ , 2009):

$\displaystyle E_{(r_{+},r_{\rm d})}$ $\displaystyle=$
$\displaystyle\frac{(2B_{0}JG/c^{3})^{2}}{4r_{+}}\left(1-\frac{r_{+}}{r_{\rm
d}}\right)+\frac{(2B_{0}JG/c^{3})^{2}}{4\hat{a}}$ (20) $\displaystyle\times$
$\displaystyle\left[\left(1+\frac{\hat{a}^{2}}{r_{+}^{2}}\right)\arctan\left(\frac{\hat{a}}{r_{+}}\right)\right.$
$\displaystyle-$
$\displaystyle\left.\left(1+\frac{\hat{a}^{2}}{r_{d}^{2}}\right)\arctan\left(\frac{\hat{a}}{r_{d}}\right)\right],$

where $r_{d}$ is the radius of the dyadoregion

$\left(\frac{r_{d}}{\hat{M}}\right)^{2}=\frac{1}{2}\frac{\lambda}{\mu\epsilon}-\alpha^{2}+\left(\frac{1}{4}\frac{\lambda^{2}}{\mu^{2}\epsilon^{2}}-2\frac{\lambda}{\mu\epsilon}\alpha^{2}\right)^{1/2}\,$
(21)

with $\epsilon=E_{c}M_{\odot}G^{3/2}/c^{4}\approx 1.873\times 10^{-6}$, and

$\lambda=\frac{2B_{0}JG/c^{3}}{\sqrt{G}M}={\color[rgb]{0,0,0}\frac{Q_{\rm
eff}}{\sqrt{G}M}},$ (22)

is the effective charge-to-mass ratio.

The characteristic width of the dyadoregion, i.e. the region around the BH
where the electric field overcritical is

$\Delta_{\rm d}(t)=r_{d}(t)-r_{+}(t).$ (23)

[a] [b]

Figure 4:  The decrease of the BH spin and mass, as a function of rest-frame
time for GRB 190114C during the UPE phase, namely in the rest-frame time
interval $t_{\rm rf}=1.9$–$3.99$ s. The values of spin and mass at the moment
when BH formed are, respectively, $M=4.53M_{\odot}$ and $\alpha=0.54$. At the
moment when the UPE is over, i.e. at $t_{\rm rf}=3.99$ s, are: $\alpha=0.41$
and $M=4.45M_{\odot}$.

##  VIII Transparency condition in the UPE phase

The existence of overcritical fields in the UPE phase and the consequent
production of an $e^{+}~{}e^{-}~{}\gamma$ plasma, have been addressed in Sec.
VII.

In presence of an overcritical electric field around the BH, a sequence of
events occur:

1) An optically thick $e^{+}~{}e^{-}~{}\gamma$ plasma of total energy
$E_{e^{+}~{}e^{-}}^{\mathrm{tot}}=E_{\gamma,\rm iso}$ endowed with baryon load
with a mass of $M_{B}$. The self-acceleration and expansion of such PEMB
pulses has been described in Ruffini  _et al._ (1999). The dynamics of the
PEMB pulses due to the effect of baryonic matter (the remnant of the collapsed
object) has been considered in Ruffini  _et al._ (2000). The thermalization of
the pair plasma is achieved almost instantaneously ($\sim 10^{-13}$ s) and
expands due to its self-acceleration up to ultrarelativistic velocities
($\Gamma\sim 100$ in the case of long GRBs; Aksenov  _et al._ , 2007, 2009).

2) The transparency of the $e^{+}~{}e^{-}~{}\gamma$ plasma. When the PEMB
pulses expand with ultrarelativistic velocities, the $e^{+}~{}e^{-}~{}\gamma$
plasma becomes optically thin, a thermal radiation that has been called the
Proper-GRB (P-GRB) is emitted (Ruffini  _et al._ , 1999, 2000). The P-GRB is
characterized by the observed thermal component; see Sec. II and Sec. III. The
dynamics of the expanding plasma from the vicinity of the BH up to the
transparency point is described by the plasma energy,
$E_{e^{+}e^{-}}^{\mathrm{tot}}$ and the baryon load parameter, ${\cal
B}=M_{B}c^{2}/E_{e^{+}e^{-}}^{\mathrm{tot}}$(Ruffini  _et al._ , 1999, 2000).

The total P-GRB energy in the comoving frame of each impulsive process is

$\displaystyle E^{com}_{\rm P-GRB}$ $\displaystyle=$ $\displaystyle\int
aT^{4}_{com}dV_{com},$ (24) $\displaystyle=$ $\displaystyle
aT^{4}_{com}V_{com}$

where $a$ is radiation constant, $T_{com}$ is the P-GRB temperature in the
comoving frame and $V_{com}$ is the volume of the PEMB pulses in the comoving
frame.

Dividing Eq. (24) by the Doppler factor $\Gamma(1-v/c)$ at transparency, i.e.
when the P-GRB is emitted, being $\Gamma$ and $v$ the Lorentz factor and speed
of the PEMB pulses, and assuming head-on emission; namely $\cos\vartheta=1$
one can obtain:

$\frac{E^{com}_{\rm
P-GRB}}{\Gamma(1-v/c)}=\frac{aT^{4}_{com}}{\Gamma(1-v/c)}V_{com}\,,$ (25)

where we have assumed head-on emission and therefore fixed $\cos\vartheta=1$
in the Doppler factor.

Since:

$\displaystyle T_{obs}$ $\displaystyle=$
$\displaystyle\frac{T_{com}}{\Gamma(1-v/c)}\,,$ $\displaystyle E^{obs}$
$\displaystyle=$ $\displaystyle\frac{E^{com}}{\Gamma(1-v/c)}\,,$
$\displaystyle V_{lab}$ $\displaystyle=$
$\displaystyle\frac{V_{com}}{\Gamma}\,,$ (26)

we have that:

$\displaystyle E^{obs}_{\rm P-GRB}$ $\displaystyle=$ $\displaystyle
aT^{4}_{obs}\Gamma^{3}(1-v/c)^{3}\Gamma V_{lab}$ (27) $\displaystyle=$
$\displaystyle aT^{4}_{obs}\Gamma^{4}(1-v/c)^{3}4\pi R^{2}\Delta_{lab},$

where we have used the fact that $V_{lab}=4\pi R^{2}\Delta_{lab}$, where
$\Delta_{lab}$ is the thickness of the PEMB pulses, and $a=4\sigma/c$, being
$\sigma$ the Stefan-Boltzmann constant.

Moreover, we know from the condition of transparency

$\displaystyle\tau$ $\displaystyle=$
$\displaystyle\sigma_{T}(n_{e^{+}e^{-}}+\bar{Z}n_{B})\Delta_{lab}\approx\sigma_{T}(\bar{Z}n_{B})\Delta_{lab},$
(28) $\displaystyle=$ $\displaystyle\sigma_{T}\frac{\bar{Z}M_{B}}{m_{N}4\pi
R^{2}\Delta_{lab}}\Delta_{lab}=1,$

where $\sigma_{T}$ is the Thomson cross section, $\bar{Z}$ is the average
atomic number of baryons ($\bar{Z}=1$ for Hydrogen atom and $\bar{Z}=1/2$ for
general baryonic matter), $m_{N}$ is nucleon mass and $M_{B}$ is the baryon
mass. Since the value of number density of $e^{+}~{}e^{-}$ can only be
obtained numerically, for simplicity we assume here $n_{e^{+}e^{-}}\ll n_{B}$
and we have numerically checked that this assumption is indeed valid for the
values of ${\cal B}$ considered here; namely ${\cal B}=10^{-3}$–$10^{-2}$. In
addition, we assume the constant slab approximation with a constant width
$\Delta_{lab}$ in the laboratory frame following Ruffini  _et al._ (1999,
2000).

Therefore, the lower bound of transparency radius is

$R^{\rm tr}=\left(\frac{\sigma_{T}}{8\pi}\frac{M_{B}}{m_{N}}\right)^{1/2}.$
(29)

By substituting Eq. (29) in Eq. (27), and dividing it by $E_{\rm iso}$, one
obtains:

$\frac{E^{obs}_{\rm P-GRB}}{E_{\rm
iso}}=\frac{1}{2}aT^{4}_{obs}\Gamma^{4}(1-v/c)^{3}\sigma_{T}\frac{{\cal
B}}{m_{N}c^{2}}\Delta_{lab}\,,$ (30)

where we used the fact that, by definition, ${\cal B}\equiv M_{B}c^{2}/E_{\rm
iso}$.

Using the fact that:

$1-v/c=\frac{1}{(1+v/c)\Gamma^{2}}\simeq\frac{1}{2\Gamma^{2}}\,,$ (31)

where we assumed $v/c\sim 1$, that is certainly accurate at the transparency
of the PEMB pulses, we have that:

$\frac{E^{obs}_{\rm P-GRB}}{E_{\rm
iso}}=\frac{aT^{4}_{obs}}{16\Gamma^{2}}\sigma_{T}\frac{{\cal
B}}{m_{N}c^{2}}\Delta_{lab}\,.$ (32)

From the total energy conservation we have that:

$E_{\rm iso}=E^{obs}_{\rm P-GRB}+E_{\rm Kinetic}\,,$ (33)

therefore

$1=\frac{E^{obs}_{\rm P-GRB}}{E_{\rm iso}}+\frac{E_{\rm Kinetic}}{E_{\rm
iso}}\,$ (34)

where $E_{\rm Kinetic}$ is the kinetic energy of the baryonic PEMB pulses:

$E_{\rm Kinetic}=(\Gamma-1)M_{B}c^{2}\,.$ (35)

By substituting Eq. (35) in Eq. (34) we have

${\cal B}=\frac{1}{\Gamma-1}\left(1-\frac{E^{obs}_{\rm P-GRB}}{E_{\rm
iso}}\right),$ (36)

or, equivalently:

$\Gamma=1+{\cal B}^{-1}\left(1-\frac{E^{obs}_{\rm P-GRB}}{E_{\rm
iso}}\right).$ (37)

The radius of transparency, $R^{\rm tr}$, is given by Eq. (29) in this
theoretical approach:

$R^{\rm tr}=\left(\frac{\sigma_{T}}{8\pi}\frac{{\cal B}E_{\rm
iso}}{m_{N}c^{2}}\right)^{1/2}.$ (38)

In general, from Eqs. (32) and (36), the values of ${\cal B}$ and $\Gamma$ can
be estimated by the values of $E^{obs}_{\rm P-GRB}/E_{\rm iso}$, $T_{obs}$ and
$\Delta_{\rm lab}$. Also, having $E_{\rm iso}$ and ${\cal B}$, we can obtain
the transparency radius from Eq. (38).

[a] [b] [c] [d] [e] [f]

Figure 5:  The parameters of _inner engine_ and transparency condition as a
function of rest-frame time for GRB 190114C during the UPE phase, namely in
the rest-frame time interval $t_{\rm rf}=1.9$–$3.99$ s, obtained from a time-
resolved analysis down to a $\Delta t=0.125$ s time resolution reported in
Table 1. (a): The electric field during UPE phase which is clearly
overcritical. (b): The energy of dyadoregion during the UPE phase obtained
from Eq. 20. (c): The width of dyadoregion obtained from Eq. 23. [d]:
Timescale of radiation during the UPE phase. (e): The decrease of the Lorentz
gamma factor, $\Gamma$, as a function of rest-frame time. (f): The evolution
of transparency radius in the UPE phase of GRB 190114C. All values are
obtained for magnetic field of, $B_{0}=1.8\times 10^{17}$ G, calculated in
Sec. IX.

##  IX The magnetic field inferred from the $\Delta{t}=0.125$ s time resolved
spectral analysis

As a specific example, we calculate the magnetic field and transparency
parameters for the $\Delta t_{\rm rf}=0.125$ s time resolved interval, namely
the fifth iteration analysis reported in Sec. III. Therefore, the UPE phase is
assumed to be composed of $16$ expanding PEMB pulses emitting an average
isotropic energy of $E_{\rm iso}\sim 10^{52}$ erg, with radiation time scale
of $\tau_{q}=0.125~{}$s, as reported in Table 1.

Therefore, from 1) the ratio $E_{\rm\rm P-GRB}/E_{\rm iso}=0.3$, and 2)
$E_{\rm\rm P-GRB}=E_{(r_{+},r_{\rm d})}$, the electromagnetic energy stored in
each expanding PEMB pulse should be $E_{(r_{+},r_{\rm d})}=0.3\times E_{\rm
iso}$. Consequently, from Eq. (20) and the value of mass and spin parameter of
the BH, the magnetic field needed to fulfill this energetic is
$B_{0}=1.85\times 10^{17}$ G. The Lorentz factor, $\Gamma\sim 1000$, the
baryon load, ${\cal B}\sim 2\times 10^{-3}$, and the radius of transparency,
$R^{\rm tr}\sim 10^{13}~{}\rm cm$, are obtained using:

  * •

the isotropic energy of each time interval, $E_{\rm iso}\sim 10^{52}$ erg;

  * •

the ratio of black body energy to isotropic energy $E^{\rm obs}_{\rm
P-GRB}/E_{\rm iso}\sim 0.3$;

  * •

the value of black body temperature in keV reported in Table 1;

  * •

the width of the dyadoregion at decoupling, $\Delta_{\rm lab}=\Delta_{\rm d}$
obtained from Eq. (23) for magnetic field of $B_{0}=1.85\times 10^{17}~{}$G.

The results are shown in Fig. 5. It is appropriate to notice that the magnetic
field of $B_{0}=1.85\times 10^{17}$ G, obtained from the $\Delta t_{\rm
rf}=0.125$ s time-resolved analysis, does not fulfill the boundary condition
of the UPE phase, $|{\bf E}|=E_{c}$ at $t_{\rm rf}=3.99$ s. In the next
section, we calculate the lowest limit of magnetic field and the minimum
repetition time which fulfill the required boundary condition $|{\bf
E}|=E_{c}$ at $t_{\rm rf}=3.99$ s.

##  X The lower limit of magnetic field during the UPE phase

Having determined the boundary value of the magnetic field at $t_{\rm
rf}=3.99$ s to be $B_{0}=3.9\times 10^{10}$ G, we must now require that at
$t_{\rm rf}>3.99$ s the electric to be undercritical, and overcritical inside
the UPE phase. In section VI, we have determined the overall behavior of the
mass and spin of BH during the UPE since the moment of the formation of BH;
see Fig. 4.

We set the value of $B_{0}$ in the UPE phase, i.e. at times $t_{\rm rf}<3.99$
s, such that the electric field therein is overcritical. The lower limit of
the magnitude of magnetic field is determined in a way that in Eq. (13),
$|E_{r_{+}}|=E_{c}$ at the end of the UPE phase; at $t_{\rm rf}=3.99$ s. For
BH mass and spin parameter at the end of UPE, it implies a magnetic field of
$\beta=B_{0}/B_{c}=5.1$ or $B_{0}=2.3\times 10^{14}$ G; see Fig. 6.

[a] [b] [c] [d] [e] [f]

Figure 6:  The parameters of _inner engine_ and transparency point, obtained
for $B_{0}=2.3\times 10^{14}$ G, as a function of rest-frame time for GRB
190114C during the UPE phase, namely in the rest-frame time interval $t_{\rm
rf}=1.9$–$3.99$ s. (a): The electric field during UPE phase which is clearly
overcritical and reaches its critical value at the end of UPE phase (t${}_{\rm
rf}=$3.99 s). (b): The energy of dyadoregion during the UPE phase obtained
from Eq. (20). (c): The width of dyadoregion obtained from Eq. (23) which
tends to zero at the end of UPE phase, indicating that the number of
$e^{+}e^{-}$ pairs are suppressed and the UPE phase is over. (d): Repetition
time scale of the _inner engine_ during the UPE phase obtained from Eq. (44).
(e): The decrease of the Lorentz factor, $\Gamma$, as a function of rest-frame
time. This indicates the fact that $\Gamma$ tends to unity for the last layers
which confirms the the end of UPE is reached. (f): The evolution of
transparency radius; see Sec. X.

For $B_{0}=2.3\times 10^{14}$ G and at the moment which BH is formed, namely
$t_{\rm rf}=1.9$ s, $\lambda\approx 4.7\times 10^{-5}$ (from Eq. 22), which
leads to $r_{d}=1.15~{}r_{+}$. Having these values, the energy of dyadoregion
at $t_{\rm rf}=1.9$ s is $E_{d}\approx 6.27\times 10^{44}$ erg. The evolution
of energy of dyadoregion is shown in Fig. 6 [b].

The evolution of characteristic width of the dyadoregion is shown in Fig. 6
[c]. At $t_{\rm rf}=3.99$ s, the extent of the dyadoregion tends to zero
confirming that no enough $e^{+}~{}e^{-}$ pairs are created and the UPE phase
is finished.

###  X.1 The transparency condition obtained from the lower limit of magnetic
field, $B_{0}=2.3\times 10^{14}$ G

For $B_{0}=2.3\times 10^{14}$ G, each expanding PEMB pulse; which are produced
via vacuum polarization and self-expanded with different Lorentz factors, has
an isotropic energy $\sim 10^{45}$ erg obtained from Eq. (20); see Fig. 6 The
total isotropic energy of the UPE phase is $E^{\rm UPE}_{\rm iso}=1.47\times
10^{53}$ erg, therefore, this phase consists of $\sim 10^{8}$ impulses in the
time interval $1.9$–$3.99$ s. Radiation from each one of these PEMB pulse can
be interpreted as a blackholic quantum introduced in Rueda and Ruffini (2020).

As expressed in Sec. VIII, the key parameters for calculating the transparency
radius of each impulse are: (1) its isotropic energy, $E_{\rm iso}$, (2) the
blackbody to isotropic energy ratio, $E^{\rm obs}_{\rm P-GRB}/E_{\rm iso}$,
(3) the blackbody temperature, ($T_{\rm obs}$), and finally (4) the width
$\Delta_{\rm lab}=\Delta_{\rm d}$.

From the _inner engine_ theory, as presented in the previous subsection, for
each impulse we have $E_{\rm iso}\sim 10^{45}$ erg and the width of the
dyadoregion at decoupling is $\Delta_{\rm d}=1.9\times 10^{5}$ cm. From the
hierarchical structure of UPE phase in this GRB presented by Eq. 3, we have
$E^{\rm obs}_{\rm P-GRB}/E_{\rm iso}\sim 0.3$ and the temperature $kT_{\rm
obs}\sim 150$ keV.

With these values of $E^{\rm obs}_{\rm P-GRB}/E_{\rm iso}$, $\Delta_{\rm d}$,
$T_{\rm obs}$, and $E_{\rm iso}$, we obtain via Eqs. (32), (36) and (38), the
transparency radius of

$R^{\rm tr}=9.4\times 10^{9}~{}\rm cm,$ (39)

the baryon load parameter

${\cal B}=5.1\times 10^{-3},$ (40)

and finally the Lorentz factor of

$\Gamma\approx 139.$ (41)

We have checked that these estimated values are in good agreement with the
corresponding ones obtained from the numerical simulation of the PEMB pulses
evolution. The corresponding values from the numerical simulation are: $R^{\rm
tr}=9.3\times 10^{9}$ cm, temperature $kT=150$ keV, and Lorentz factor
$\Gamma\sim 140$.

The evolution of the Lorentz gamma factor, $\Gamma$ is shown in Fig. 6(e)
indicating the fact that $\Gamma$ tends to $\sim$ 4 for the last shell which
confirms the end of UPE is reached.

The evolution of transparency radius in the UPE of GRB 190114C, using the
exact numerical values of energy and width of dyadoregion are also shown in
Fig. 6 (f).

###  X.2 The repetition time of sequence of the blackholic quanta

We now study the timescale of each blackholic quanta in which the system
starts over. The new value of the electric field is set by the new values of
the BH angular momentum and mass, $J=J_{0}-\Delta J$ and $M=M_{0}-\Delta M$,
keeping the magnetic field value constant $B_{0}$ in which at $t_{\rm rf}=1.9$
s, i.e.

$\frac{\Delta J}{J}\approx\frac{\Delta M}{M}=3.1\times 10^{-9}.$ (42)

Regarding the presence of the baryon load obtained from Eq. (36) in the
acceleration process, we infer from the MeV luminosity, the evolution of the
timescale $\tau_{q}(t)$ of the blackholic quantum by requiring it to explain
the MeV emission, i.e.:

$L_{\rm MeV}=\frac{[1-{\cal
B}(\Gamma-1)]~{}E_{(r_{+}(t),r_{d}(t))}}{\tau_{q}(t)}.$ (43)

In fact, the effect of baryon load is $(1-{\cal B}~{}\Gamma)\approx 0.3$.
Therefore, we obtain for the timescale

$\tau_{q}(t)=\frac{0.3~{}E_{(r_{+}(t),r_{d}(t))}}{L_{\rm MeV}}.$ (44)

where the $E_{(r_{+}(t),r_{d}(t))}$ is the energy of dyadoregion obtained from
Eq. (20), determined from the new values of $J$ and $M$ for each blackholic
quanta and $L_{\rm MeV}$ is the MeV luminosity obtained from best fit
represented by Eq. (4); the evolution of the blackholic timescale is shown in
Fig. 6 (d).

###  X.3 The approach of the $|{\bf E}|=E_{c}$ at the UPE phase boundary

From the above theoretical derivation, we can explicitly see that, for an
iteration, such that the duration of each elementary process of the nth
iteration is $10^{-9}$ s, namely after $10^{9}$ iterations, the physical model
can be consistently implemented, deriving the necessary parameters
characterizing the process, namely the energy of each PEMB pulse, the baryon
load, the Lorentz factor, and the radius at transparency. From Fig. 6, it
becomes clear that after $t_{\rm rf}\sim$ 3.7 s, the emission of quanta by the
QED process becomes not effective and the classical regime is soon approached.

The lowest limit of the magnetic field to reach $|{\bf E}|=E_{c}$ occurs in an
_inner engine_ composed of a Kerr BH of initial mass of $M=4.53M_{\odot}$ and
$\alpha=0.51$, immersed with a uniform magnetic of $B_{0}=2.3\times 10^{14}$ G
with a radiation timescale of $\sim 10^{-9}$ s.

Indeed, the decrease of the magnetic field from $\beta=5.1$ to
$\beta=8.9\times 10^{-4}$ at $t_{\rm rf}=3.99$ s, can be explained as the
result of the induced current created by pairs in the inward electric field,
which screens the original magnetic field. This is a very interesting process
that has consequences in different astrophysical scenarios. Therefore, we here
limit ourselves to the above explanation and refer the reader for further
details in the dedicated, separated publication (Campion  _et al._ , 2021).

All the above results: (1) are in perfect agreement with observational data;
see Fig. 3 and, (2) overcome the compactness problem of the UPE phase. It is
appropriate to mention all these results have been obtained guided by the
hierarchical structure of the UPE phase.

##  XI Comparison with other approaches

The magnetohydrodynamics of plasma accretion onto the Kerr BH was first
addressed in Ruffini and Wilson (1975) assuming he infinite conductivity
condition, $F_{\rm\mu\nu}U^{\rm\nu}=0$ implying $\mathbf{E}\cdot\mathbf{B}=0$.
In view of zero net charge on the surface of the BH, no process of energy
extraction, neither by vacuum polarization nor by electromagnetic process was
there possible (Cherubini  _et al._ , 2018).

Blandford and Znajek (1977) returned on the same process and in order to
overcome the difficulty of extracting energy and they introduced, in analogy
with pulsar, the presence “gaps” (Sturrock, 1971; Ruderman and Sutherland,
1975).

Thorne and MacDonald (1982), following Hanni and Ruffini (1973), calculated
the surface charge induced on the horizon of the Kerr BH immersed in the
magnetic field in the Papapetrou-Wald solution Wald (1974). Miniutti and
Ruffini (2000) explicitly manifested that the Papapetrou-Wald solution Wald
(1974) implies $\mathbf{E}\cdot\mathbf{B}\neq 0$, and identified that the
induced surface charge implies a quadrupolar distribution of electric field
around the BH. These results were confirmed in (Punsly, 2008). Applying these
works to the case of GRBs, it has been shown that the mathematical Papapetrou-
Wald solution can be used in order to describe the inner engine of a GRB
130427A (Ruffini  _et al._ , 2019a), which presents mechanism to extract the
rotational energy of the Kerr BH. The process which occurs in the
undercritical field regime leads to the emission of synchrotron radiation in
the GeV domain as well as ultrahigh energy cosmic rays (UHECRs) (Moradi  _et
al._ , 2021). The synchrotron emission of the inner engine occurs near the BH
horizon and is emitting in blackholic quanta (Rueda and Ruffini, 2020).

The extrapolation to overcritical field regime, presented in this paper, leads
to the explanation of the MeV radiation during the UPE phase.

In recent years, in parallel to the theoretical progresses in the field,
computer simulations were also developed. These simulations point the fact
that present plasma in any energy extracting scheme would screen the
background electric field of the vacuum solution of Papapetrou-Wald from the
magnetosphere; see e.g. Komissarov (2005); Parfrey  _et al._ (2019). These
simulation mainly address the physics of active galactic nuclei (AGNs) and
particularly attentive has been the review of their theoretical models
indicated in Komissarov (2005). Their choice of parameters and physical
processes are quite different from the ones we have used for the GRB analysis.
In our GRB approach we have been guided by the theoretical explanation of a
vast number of observations obtained from: 1) the unprecedented time-resolved
spectral analysis of the UPE phase; 2) the power-law MeV luminosity observed
by Fermi-GBM; and 3) the power-law GeV luminosity observed by Fermi-LAT. This
allows us to identify the physical processes and parameters which had to be
fulfilled in order to obtain the detailed acquired data. Their choice of
parameters enforce $\mathbf{E}\cdot\mathbf{B}=0$ condition so different from
$\mathbf{E}\cdot\mathbf{B}\neq 0$ which has allowed us to obtain our results.

In our model, the magnetic field is left over by the collapse of the accreting
NS to the BH, rooted in the surrounding material, and the electric field is
created by the gravitomagnetic interaction of the spacetime rotation with the
present magnetic field; see, e.g., Rueda  _et al._ (2020). Following this
procedure, and since the electric field is assumed to be overcritical, in a
very short timescale $\sim\hbar/(m_{e}c^{2})\approx 10^{-21}$ s, much shorter
than any electromagnetic process, a dyadoregion originate dominated by the
high density and high pressure of the neutral $e^{+}e^{-}\gamma$ plasma
Ruffini  _et al._ (2010).

The optically thick pair electromagnetic-baryon (PEMB) pulse self-accelerates
to the ultrarelativistic regime and finally reaches the transparency point at
the radius of $\sim 10^{10}$ cm. These classical results were obtained thanks
to a collaboration with Wilson at Lawrence Livermore National Laboratory
(Ruffini  _et al._ , 1999, 2000)

As soon as the BH is formed, the first and the most efficient process in
action to produce the $e^{+}e^{-}$ plasma and, consequently decreasing the
rotational energy of BH, occurs through the Schwinger critical field pair
production. Since an overwhelming amount of pair plasma is created in quantum
timescales, the plasma expansion by its internal pressure starts well before
any electric field screening.

This process takes a fraction of angular momentum of the Kerr BH. The BH then
is left with a slightly smaller angular momentum $J^{*}=J-\Delta J$, with
$\Delta J/J\sim 10^{-9}$, being $\Delta J$ the angular momentum, and the same
magnetic field which leads to a new electric field created by the space-time
rotation. As a result, the system starts a new process in presence of the same
magnetic field $B_{0}$, kept rigorously constant and a new effective charge of
$Q^{*}_{\rm eff}=Q_{\rm eff}-\Delta Q_{\rm eff}$ which $\Delta Q_{\rm
eff}=2B_{0}\Delta J$.

This process continues till the moment that electric field is not overcritical
anymore, and after that the sole electromagnetic process is at work. The
expanding $e^{+}e^{-}\gamma$ plasma sweeps away the matter in the cavity whose
density after this process becomes $\sim 10^{-14}$ g cm-3, and an
undercritical electromagnetic field is left; see Ruffini  _et al._ (2019c).
This low-density ionized plasma is needed to fulfill an acceleration of
charged particles leading to the electrodynamical process around a newborn BH.
In fact, this density is much below the Goldreich-Julian density $\rho_{\rm
GJ}=8\times 10^{-12}$ g cm-3 obtained for the $B_{0}=3.9\times 10^{10}~{}$ G
and $M=4.45M_{\odot}$ and $a=0.41M$. Moreover, the matter energy density
inside the cavity is negligible comparing to the electromagnetic energy
density, namely $\rho_{M}/(|B|^{2}-|E|^{2})\sim 10^{-14}$, while in Komissarov
(2005) this ratio is $0.05$ or higher.

It is interesting that the inner engine operates as well in the supermassive
BHs in active galactic nuclei in the $|{\bf E}|<E_{c}$ regime. In the case of
M87, with a mass of a few $10^{9}~{}M_{\odot}$, the repetition timescale is
$0.68$ d in the polar direction, with a quanta of $\mathcal{E}\sim 10^{45}$
erg (Moradi  _et al._ , 2021)

##  XII Conclusions

GRB 190114C has offered already the possibility of testing different Episodes
of the BdHN I sequence by a time-resolved spectral analysis (Ruffini  _et al._
, 2021); the $\nu$NS-rise [Becerra et al. in preparation], the formation of
the BH triggering the UPE phase and the associated GeV emission (see Fig. 1),
the formation of the cavity (Ruffini  _et al._ , 2019c), the long-lasting
emission in the x-ray afterglow from the spinning $\nu$NS (Rueda  _et al._ ,
2020),  and in the GeV emission from the newly-formed BH (Moradi  _et al._ ,
2021).

The long lasting GeV radiation, with a luminosity following a power-law of
$L_{\rm GeV}=\left(7.75\pm 0.44\right)\times 10^{52}~{}t^{-(1.2\pm 0.04)}$ erg
s-1, has been shown to originate from the extraction of the rotational energy
of a Kerr BH in a sequence of discrete “blackholic quanta” emission (Rueda and
Ruffini, 2020). This process occurs in an inner engine, which is composed of
an uniform magnetic field aligned with the rotation axis of the Kerr BH
described by the Papapetrou-Wald solution (Papapetrou, 1966; Wald, 1974) and
immersed in a very low density fully ionized plasma with density as low as
$10^{-14}$ g cm-3 (Ruffini  _et al._ , 2019c; Rueda and Ruffini, 2020; Moradi
_et al._ , 2021; Ruffini  _et al._ , 2021).

One of the main results has been the concept of effective charge $Q_{\rm
eff}$, given by Eq. 1 driving the acceleration process in the _inner engine_.

The most unexpected result has been the discovery of hierarchical structure in
the time-resolved spectral analysis on ever-decreasing time scales of the UPE
phase of GRB 190114C by Ruffini  _et al._ (2019b) and here updated in section
III. There, we have determined the spectral properties and luminosities during
and after the UPE phase, of the MeV emission observed by Fermi-GBM, and of the
GeV emission observed by Fermi-LAT.

A new arena is open in this article linking the macroscopic hierarchical
structure of the UPE phase to a microphysical sequence of discrete elementary
events in a QED regime.

For the first time, we have here approached the energy extraction process from
a Kerr BH by the general relativistic QED process occurring in the _inner
engine_.

We have assumed that the electric field of the _inner engine_ operates in an
overcritical $|{\bf E}|>E_{c}$ during the UPE phase, and in an undercritical
$|{\bf E}|<E_{c}$ just after the end of the UPE phase. A sharp separatix both
in the theoretical treatment and in the observational properties of these two
domains are evidenced.

The main result of this article is to have compared and contrasted the two
different processes for explaining the MeV and GeV emissions of GRBs.

The first process, originating the MeV radiation, is dominated by the vacuum
polarization originating from the overcritical field in the UPE phase. The
overcritical field generates an initially optically thick
$e^{+}~{}e^{-}~{}\gamma$-baryon plasma, which self-accelerates until reaching
the point of transparency, a PEMB pulse. Typical values of $\Gamma\sim 100$
guarantee the avoidance of the compactness problem (Ruderman, 1975; Piran,
2004) in the UPE phase. We have shown that the magnetic field $B_{0}$ keeps a
constant value during the UPE phase of order of $\sim 10^{14}$ G and reduces
to $3.9\times 10^{10}$ G after the UPE phase.

The second process, originating the GeV emission, is based on the classical
ultrarelativistic electrodynamics generated from the electrons injected in the
magnetic field emitting synchrotron radiation close to the BH horizon, in
selected energies with specific pitch angle dependence, see Fig. 9 in Ruffini
_et al._ (2019a).

Both these processes originate from the rotational energy of the Kerr BH
acting on a uniform magnetic field, aligned with the BH rotation axis, within
Papapetrou-Wald solution.

The results presented here were expected since fifty years when the
Christodoulou-Hawking-Ruffini mass-energy formula of the BH Christodoulou,
1970; Christodoulou and Ruffini, 1971; Hawking, 1971, as well as some of the
pioneering works, using the vacuum polarization process of a BH, were
established (Damour and Ruffini, 1976; Ruffini, 1998). They were followed by
fundamental contributions on the self-acceleration process of the
$e^{+}~{}e^{-}~{}\gamma$ optically thick plasma, PEMB pulses (Ruffini, 1998;
Ruffini  _et al._ , 1999, 2000), and by the concepts of dyadosphere and
dyadotorus (see Ruffini  _et al._ , 2010, and references therein), which are
the fundamental conceptual framework of this article. The revival of these
concepts, as we explained in this article, has been made possible by the
fundamental introduction of “ _the effective charge_ ” overcoming the concept
of a net charged BH and fulfilling, nevertheless, all the necessary
electrodynamical process of an electrically charged BH.

The fact that all the properties of GRB 190114C have been confirmed to occur
in GRB 130427A, GRB 160509A and GRB 160626B; see (Li  _et al._ , 2019b), allow
us to extend and apply the analysis here performed for the _inner engine_ ,
generally to all BdHNe I.

This has introduced a radical change by modifying the traditional energetic
arguments based on the gravitational binding energy of massive particles
geodesics, following a classical electrodynamics process in the Kerr metric
occurring at very high density. Indeed, in Ruffini  _et al._ (2019c), it has
been shown how this _inner engine_ operates most efficiently in a cavity in
presence a very tenuous ionized plasma with density of $10^{-14}$ g cm-3
following a classical electrodynamics process. The _inner engine_ equally
works at high densities of the PEMB pulses in the quantum electrodynamics
process. In both processes, the fundamental energetic role is being played by
the rotational energy of the Kerr BH, which is converted by associated
classical and quantum ultrarelativistic acceleration processes into the
observed multi-wavelength energy emissions and UHECRs. The application of the
classical work of the innermost stable circular orbit (ISCO) of massive
particles around Kerr BH, introduced in Ruffini and Wheeler (1971), has been
superseded in this new approach. The concepts of the dyadosphere and (Ruffini,
1998; Ruffini  _et al._ , 1999, 2000) and dyadotorous (Cherubini  _et al._ ,
2009)s are the fundamental ones in this new electrodynamical scenario.

The most important result in this paper has been the understanding the role of
hierarchical structure discovered in the time-resolved spectral analysis of
the UPE phase, finally explained by their underlying quantum nature.

This long march was started by the intuitions announced in Ruffini  _et al._
(2019b). They have been here expanded and approached in their theoretical
implication in this article. Although the motivations were clear, their
detailed comprehension has needed further work which is here presented. We are
ready to look at the implications of these results.

Thanks to the observation of GRB 190114C, which is by far the most complex
fundamental physical system ever approached in Science, a new scenario is now
open. The most unique complexity of BdHNe, their enormous energy emitted in an
observer homogeneous Universe, see e.g., Ruffini  _et al._ (1988),  and the
special quantum ans classical electrodynamics nature of their radiation make
us wonder about the role GRB may play in the appearance of life in the
Universe (Chen and Ruffini, 2015). This new overarching conceptual description
appears to be in sight thanks to the observation of GRB 190114C.

###### Acknowledgements.

We thank the referee for important remarks which have improved the
presentation of our results. We are grateful to Prof. Roy Kerr for discussion
on the energy extraction process from Kerr BH, to Prof. Sang Pyo Kim for
fruitful discussion about overcritical field and Schwinger pair production in
such a field and to Prof. Narek Sahakyan for studying the extension of these
results from GRBs to AGNs.

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