#  Black holes solutions in power-law Maxwell-$f(T)$ gravity in diverse
dimensions

G.G.L. Nashed<EMAIL_ADDRESS>Centre for Theoretical Physics, The British
University in Egypt, P.O. Box 43, El Sherouk City, Cairo 11837, Egypt
Int. Lab. Theor. Cosmology, Tomsk State University of Control Systems and
Radioelectronics (TUSUR), 634050 Tomsk, Russia  Kazuharu Bamba
<EMAIL_ADDRESS>Division of Human Support System, Faculty of
Symbiotic Systems Science, Fukushima University, Fukushima 960-1296, Japan

###### Abstract

We investigate the solutions of black holes in $f(T)$ gravity with nonlinear
power-law Maxwell field, where $T$ is the torsion scalar in teleparalelism. In
particular, we introduce the Langranian with diverse dimensions in which the
quadratic polynomial form of $f(T)$ couples with the nonlinear power-law
Maxwell field. We explore the leverage of the nonlinear electrodynamics on the
space-time behavior. It is found that these new black hole solutions tend
towards those in general relativity without any limit. Furthermore, it is
demonstrated that the singularity of the curvature invariant and the torsion
scalar is softer than the quadratic form of the charged field equations in
$f(T)$ gravity and much milder than that in the classical general relativity
because of the nonlinearity of the Maxwell field. In addition, from the
analyses of physical and thermodynamic quantities of the mass, charge and the
Hawking temperature of black holes, it is shown that the power-law parameter
affects the asymptotic behavior of the radial coordinate of the charged terms,
and that a higher-order nonlinear power-law Maxwell field imparts the black
holes with the local stability.

##  I Introduction

Over the past years, black holes (BHs) possessing linear and nonlinear Maxwell
fields have attracted tremendous attention. The study of charged BH solutions
is relevant because BH created in a collider may generally possess an electric
field. Moreover, the difference between the electroweak and Planck scales is
still an unsolved challenge that is known as the hierarchy problem. This
problem has been tackled in the frame of theories with extra spatial
dimensions. Further, this work is an attempt at elucidating higher-dimension
charged BH solutions. The first higher-dimension spherically symmetric BH
solution was derived in Tangherlini (1963), after which it was generalized in
Myers and Perry (1986). Moreover, a higher-dimension charged BH solution,
which is a generalization of the Reissner-Nordström BH, was derived in Fadeev
_et al._ (1991). There are different sets of the charged BH solutions, which
were derived by the Brans-Dicke and Lovelock theories; their physical
properties have been studied Dehghani and Hendi (2007); Dehghani  _et al._
(2007, 2008); Hendi (2010). The research on the higher dimensions of the Kerr-
Newman solution is in progress, although the slowly rotating BH solution has
been derived in Mignemi and Stewart (1993); El Hanafy and Nashed (2016);
Volkov and Straumann (1997); Ghosh and SenGupta (2007); Kim and Cai (2008);
Sheykhi and Allahverdizadeh (2008).

It is known that the Maxwell theory is invariant under conformal
transformation in four dimensions, although it is not invariant in a higher
dimension. The lack of this transformation has been explicitly studied in
higher dimensions employing the nonlinear power-law Maxwell field Hassaine and
Martinez (2007, 2008). The conformal invariance in a higher dimension was
considered to derive a similar four-dimensional Reissner-Nordström BH with
extra dimensions. Thus, the present study is aimed at deriving diverse-
dimensional charged BH solutions within the frame of the modified teleparallel
equivalent of general relativity (TEGR), namely, f(T)-gravitational theory.

There are various reasons, ranging from our accelerated expansion of the
universe and its dark energy in the astrophysical tests, for researchers to
consider modifying general relativity (GR). Among the modifications, the
Brans-Dicke Brans and Dicke (1961); Dicke (1962), Lovelock Lovelock (1971),
$f(R)$ Buchdahl (1970); Sotiriou and Faraoni (2010); Nojiri  _et al._ (2017),
and $f(T)$ (refer to Cai  _et al._ (2016) for more details) gravitational
theories have gained much attraction for different reasons. In the present
study, we concentrate on the $f(T)$ gravity for many reasons, such as the fact
that the Lagrangian of this theory which depends on the torsion scalar only,
which makes it easy to handle compared with other modified gravitational
theories ($f(R)$ Cembranos  _et al._ (2012); Nojiri and Odintsov (2011)).
Another major reason for focusing on $f(T)$ is the point that the
gravitational field equations are of the second order, unlike those in the
other modified theories Ferraro and Fiorini (2007); Awad and Nashed (2017); Li
_et al._ (2011).

TEGR is a theory, which was developed by Einstein to unify the gravitational
and electromagnetic fields Nashed (2006, 2008, 2010, 2007a); Unzicker and Case
(2005); Nashed (2007b). The TEGR theory could be applied to calculate the
conserved quantities, mass and angular momentum employing the energy-momentum
tensor Maluf (1995); Maluf  _et al._ (2002). The main motivation of the
modification of TEGR theory was the issues recently appeared in observations
that TEGR cannot explain Ferraro and Fiorini (2007). $f(T)$, which exhibits
many viable applications, is the modification of the TEGR theory Capozziello
_et al._ (2015); Nashed (2002); Iorio and Saridakis (2012); Gamal (2012);
Bamba  _et al._ (2014). By employing this theory, not only inflation Ferraro
and Fiorini (2007) in the early universe but also the late-time cosmic
acceleration could be explained Bengochea and Ferraro (2009); Linder (2010);
Wu and Yu (2011); Dent  _et al._ (2011); Awad and Nashed (2017); Bamba  _et
al._ (2011, 2010, 2012); Aviles  _et al._ (2013); Jamil  _et al._ (2012);
Ferraro and Fiorini (2011a, b); Sebastiani and Zerbini (2011); Salako  _et
al._ (2013); Haghani  _et al._ (2012, 2013). There exist a number of
interesting applications of $f(T)$ gravity to the realm of cosmology Bamba
_et al._ (2013a, b); Bamba (2011); Bamba  _et al._ (2013c); Bengochea and
Ferraro (2009); Linder (2010); Geng  _et al._ (2011); Otalora (2013);
Chattopadhyay and Pasqua (2013); Dent  _et al._ (2011); Yang (2011); Bamba
_et al._ (2011); Capozziello  _et al._ (2011); Awad  _et al._ (2018);
Capozziello  _et al._ (2011); Geng  _et al._ (2012); Farajollahi  _et al._
(2012); Cardone  _et al._ (2012); Bahamonde  _et al._ (2015), as well as in
the domain of astrophysics Capozziello  _et al._ (2013); Paliathanasis  _et
al._ (2014); Gonzalez  _et al._ (2012); Boehmer  _et al._ (2012); Nashed
(2013); Ruggiero and Radicella (2015). In the astrophysical domain, $f(T)$
possesses a new exact charged BH solution, which involves, in addition to the
monopole term, a quadruple term whose contribution accrues from the quadratic
$f(T)$ form, i.e., $T^{2}$. Based on the achievements recorded in $f(T)$
gravity, we explore the implication of the nonlinear power-law Maxwell field.
Particularly, we derive the field equations of $f(T)$ coupled with the
nonlinear power-law Maxwell field and apply their quadratic form, that is,
$f(T)=T_{0}+\alpha T-\beta T^{2}$, to diverse-dimensional flat transverse
sections.

The arrangements of the study are the followings. In Sec. II, we explain
$f(T)$ gravity and show the field equations with the Maxwell field. In Sec.
III, we investigate the solutions of the charged static BH with the Anti-de-
Sitter/de-Sitter (AdS/dS) behavior. We analyze the singularity and horizon
structure of these BH solutions and calculate their energies in Sec. III. In
Sec. IV, we explore the charged AdS solutions of the rotating BH in power-law
Maxwell-$f(T)$ gravity. In Sec. V, we calculate different thermodynamical
quantities and establish the local stability of our BH solutions. Finally, the
conclusions and discussions are given in Sec. VI.

##  II Basic formulations of $f(T)$ gravity

The f(T)-gravity is a kind of extension of TEGR. In this theory, it is
suitable to apply the vielbein (tetrads) fields, $e_{i}^{\mu}$ , as dynamic
variables (the Greek indices run for the coordinate space and the Latin one
spans for the tangent one) that constitute orthonormal basis of the tangent
space at the spacetime each point. The relations between the covariant and
contravariant tetrads and between the tetrads and metrics are represented by
the following equation:

$e^{i}_{\mu}e_{i}^{\nu}:={\delta_{\mu}}^{\nu},\qquad\qquad
e^{i}_{\mu}e_{j}^{\mu}:={\delta_{j}}^{i},\qquad\qquad{\it
g_{\mu\nu}:=\eta_{ij}{e^{i}}_{\mu}{e^{j}}_{\nu},}\qquad\qquad\eta_{ij}:={e_{i}}^{\mu}{e_{j}}^{\nu}g_{\mu\nu},$
(1)

where $\eta_{ij}=(+,-,-,-\cdots)$ is the $d$-dimensional Minkowskian metric of
the tangent space. Unlike the symmetric Levi-Civita connection in GR, the
(non-symmetric) Weitzenböck connection in TEGR is defined as Weitzenböck
(1923)

$\overset{{\bf{w}}}{\Gamma}^{\lambda}_{\mu\nu}:={e_{i}}^{\lambda}~{}\partial_{\nu}e^{i}_{\mu}.$
(2)

It follows from Eq. (2) that the torsion tensor is given by

${T^{\alpha}}_{\mu\nu}:=\overset{{\bf{w}}}{\Gamma}^{\alpha}_{\nu\mu}-\overset{{\bf{w}}}{\Gamma}^{\alpha}_{\mu\nu}={e_{i}}^{\alpha}\left(\partial_{\mu}{e^{i}}_{\nu}-\partial_{\nu}{e^{i}}_{\mu}\right).$
(3)

This tensor encodes all the information of the gravitational field. The
difference between the Levi-Civita and the Weitzenböck connections defines the
contorsion tensor, which is expressed by Eq. (4):

${K^{\lambda}}_{\mu\nu}:=\overset{{\bf{w}}}{\Gamma}^{\lambda}_{\mu\nu}-\overset{{\bf{\circ}}}{\Gamma}^{\lambda}_{\mu\nu},$
(4)

where $\overset{{\bf{\circ}}}{\Gamma}^{\lambda}_{\mu\nu}$ is the symmetric
Levi-Civita connection. The super-potential can be defined by the foregoing
equations, as follows:

${S_{\lambda}}^{\mu\nu}:={K^{\mu\nu}}_{\lambda}+\delta^{\mu}_{\lambda}{T^{\alpha\nu}}_{\alpha}-\delta^{\nu}_{\lambda}{T^{\alpha\mu}}_{\alpha},$
(5)

that has a skew symmetry in the last two indices. The torsion scalar with the
following form can be defined by Eqs. (3) and (5):

$T=\frac{1}{2}{S_{\lambda}}^{\mu\nu}{T^{\lambda}}_{\mu\nu}\equiv\frac{1}{4}T^{\rho\mu\nu}T_{\rho\mu\nu}+\frac{1}{2}T^{\rho\mu\nu}T_{\nu\mu\rho}-T_{\rho\mu}^{\
\ \rho}T_{\ \ \ \nu}^{\nu\mu}.$ (6)

The Lagrangian of TEGR mainly depends on the torsion scalar $T$ and the
variation of the the Lagrangian with respect to the vierbeins can lead to the
same field equations in GR.

With the same spirit of $f(R)$ gravity, the Lagrangian of TEGR described by
$T$ can be expanded to an arbitrary function $f$ of $T$ as $f(T)$ as follows
Cai  _et al._ (2016):

${\cal L}=\frac{1}{2\kappa}\int|e|f(T)~{}d^{d}x.$ (7)

Here, $|e|=\sqrt{-g}=\det\left({e^{a}}_{\mu}\right)$ is the determinant of the
tetrad. Moreover, $\kappa$ is a constant in the $d$-dimensions with the form
$\kappa=2(d-3)\Omega_{d-1}G_{d}$, where $G_{d}$ the Newtonian constant in the
$d$-dimensions and $\Omega_{d-1}$ is the volume of the unit sphere in the
$(d-1)$-dimensions, given by

$\Omega_{d-1}=\frac{2\pi^{(d-1)/2}}{\Gamma[(d-1)/2]},$ (8)

with $\Gamma$ a $\Gamma$-function.

The action (7) coupled with the power law-Maxwell Lagrangian is represented as

${\cal L}=\frac{1}{2\kappa}\int|e|f(T)~{}d^{d}x+\int|e|{\cal
L}_{em}~{}d^{d}x,$ (9)

where ${\cal L}_{em}={\cal F}^{s}$, with ${\cal F}=dQ$ and $Q=Q_{\mu}dx^{\mu}$
is the electromagnetic potential 1-form Capozziello  _et al._ (2013) and $s$
is a power-law parameter in terms of the Maxwell field. When the power for the
Maxwell field is equal to unity ($s=1$), the Lagrangian describes the ordinary
Maxwell theory Nashed (2013).

By varing the action in Eq. (9) in terms of the tetrads, we acquire Cai  _et
al._ (2016):

$\displaystyle
I^{\nu}{}_{\mu}={S_{\mu}}^{\rho\nu}\partial_{\rho}Tf_{TT}+\left[e^{-1}{e^{i}}_{\mu}\partial_{\rho}\left(e{e_{i}}^{\alpha}{S_{\alpha}}^{\rho\nu}\right)-{T^{\alpha}}_{\lambda\mu}{S_{\alpha}}^{\nu\lambda}\right]f_{T}-\frac{1}{4}f\delta^{\nu}_{\mu}+\frac{1}{2}\kappa{{{\mathrm{T}}^{{}^{{}^{{\\!\\!\\!\\!\scriptstyle{em}}}}}}}^{\nu}_{\mu}\equiv
0.$ (10)

Here, we show $f(T)$ as $f$, and we describe $f_{T}:=\frac{\partial
f(T)}{\partial T}$ and $f_{TT}:=\frac{\partial^{2}f(T)}{\partial T^{2}}$.
Moreover,
${{{\mathrm{T}}^{{}^{{}^{{\\!\\!\\!\\!\scriptstyle{em}}}}}}}^{\nu}_{\mu}$
means the energy-momentum tensor of the power law-Maxwell field defined by

${{{\mathrm{T}}^{{}^{{}^{{\\!\\!\\!\\!\scriptstyle{em}}}}}}}^{\nu}_{\mu}=s{\cal
F}_{\mu\alpha}{\cal F}^{\nu\alpha}{\cal
F}^{s-1}-\frac{1}{4}\delta_{\mu}{}^{\nu}{\cal F}^{s}\,,$ (11)

where ${\cal F}={\cal F}_{\mu\nu}{\cal F}^{\mu\nu}$.

Furthermore, the variation of Eq. (9) with respect to $\bm{Q_{\mu}}$, which is
the 1-form of the gauge potential, yields

$\displaystyle\partial_{\alpha}\left(\sqrt{-g}{\cal F}^{\mu\nu}{\cal
F}^{s-1}\right)=0\;.$ (12)

It follows from Eq. (12) clearly that when the power law $s=1$ the energy-
momentum tensor of Eq. (12) coincides with the linear form of Maxwell field
Nashed (2013). Eq. (12) determines the power law of Maxwell field in arbitrary
dimensions.

In addition, Eq. (10) can take the following form:

$\partial_{\nu}\Biggl{[}e{S}^{a\rho\nu}f_{T}\Biggr{]}=\kappa
e{e^{a}}_{\mu}\Biggl{[}t^{\rho\mu}+{{{\mathrm{T}}^{{}^{{}^{{\\!\\!\\!\\!\scriptstyle{em}}}}}}}^{\rho\mu}\Biggr{]},$
(13)

where $t^{\nu\mu}$ is the energy-momentum tensor of the gravitational
configuration, given by

$t^{\nu\mu}=\frac{1}{\kappa}\Biggl{[}4f_{T}{S^{\alpha}}^{\nu\lambda}{T_{\alpha\lambda}}^{\mu}-g^{\nu\mu}f\Biggr{]}.$
(14)

The anti-symmetry of the tensor ${S}^{a\nu\lambda}$ leads

$\partial_{\mu}\partial_{\nu}\left[e{S}^{a\mu\nu}f_{T}\right]=0,$ (15)

from which we have

$\partial_{\mu}\left[e\left(t^{a\mu}+{{{\mathrm{T}}^{{}^{{}^{{\\!\\!\\!\\!\scriptstyle{em}}}}}}}^{a\mu}\right)\right]=0.$
(16)

Hence, from Eq. (16) we find

$\frac{d}{dt}\int_{V}d^{(d-1)}x\ e\
{e^{a}}_{\mu}\left(t^{0\mu}+{{{\mathrm{T}}^{{}^{{}^{{\\!\\!\\!\\!\scriptstyle{em}}}}}}}^{0\mu}\right)+\oint_{\Sigma}\left[e\
{e^{a}}_{\mu}\
\left(t^{j\mu}+{{{\mathrm{T}}^{{}^{{}^{{\\!\\!\\!\\!\scriptstyle{em}}}}}}}^{j\mu}\right)\right]=0.$
(17)

Equation (17) denotes the conservation law of
${{{\mathrm{T}}^{{}^{{}^{{\\!\\!\\!\\!\scriptstyle{em}}}}}}}^{\lambda\mu}$ as
well as the pseudo tensor $t^{\lambda\mu}$, which describes the energy-
momentum tensor of gravitation in $f(T)$ gravity Ulhoa and Spaniol (2013).
Thus, the energy-momentum tensor of $f(T)$ gravity in the ($d-1$)–dimensions
contained in the volume $V$ reads

$P^{a}=\int_{V}d^{(d-1)}x\ e\
{e^{a}}_{\mu}\left(t^{0\mu}+{{{\mathrm{T}}^{{}^{{}^{{\\!\\!\\!\\!\scriptstyle{em}}}}}}}^{0\mu}\right)=\frac{1}{\kappa}\int_{V}d^{(d-1)}x\partial_{\nu}\left[e{S}^{a0\nu}f_{T}\right],$
(18)

which corresponds to TEGR when $f(T)=T$ Maluf  _et al._ (2002). The above
equation (18) is the conserved four-momentum equation for any configuration
that behaves as a flat spacetime. In this research, we derived a class of BH
solutions, which behaves asymptotically as an AdS spacetime. Therefore, it was
necessary to conduct the calculations of the conserved quantities concerning a
pure AdS/dS space to avoid the conserved quantities having infinite value
because the asymptotic behavior of the solutions of BH is similar to that of
AdS. With the difference of the energy of the pure AdS BH solution from that
of the AdS space, the total energy of the AdS BH could be found. Hence, for
the calculation of the conserved quantities, we subtract the effects of the
AdS space, which we describe by using the subscription “$r$” to regularized
value. From Eq. (17), we get

$\frac{d}{dt}\int_{V}d^{(d-1)}x\ e\
{e^{a}}_{\mu}\left(t_{r}^{0\mu}+{{{\mathrm{T}}^{{}^{{}^{{\\!\\!\\!\\!\scriptstyle{em}}}}}}}^{0\mu}\right)+\oint_{\Sigma}\left[e\
{e^{a}}_{\mu}\
\left(t_{r}^{j\mu}+{{{{\mathrm{T}}_{r}^{{}^{{}^{{\\!\\!\\!\\!\scriptstyle{em}}}}}}}}{}^{j\mu}\right)\right]=0.$
(19)

Thus Eq. (19) is the regularized conservation law of any spacetime that
behaves as (A)dS.

##  III New Anti-de-Sitter solutions of black holes in power law
Maxwell-$f(T)$ gravity

In this section, we will derive the AdS charged BH solutions in $d$–dimensions
for the power-law Maxwell-$f(T)$-gravity. For this aim, we will use the flat
transverse sections in the $d$–dimensions ($t$, $r$, $\theta_{1}$,
$\theta_{2}$, $\cdots$, $\theta_{i}$, $z_{1}$, $z_{2}$ $\cdots$ $z_{k}$),
where $k=1,2\cdots$ $d-i-2$, $0\leq r<\infty$, $-\infty<t<\infty$,
$0\leq\theta_{i}<2\pi$, $-\infty<z_{k}<\infty$, with the following vielbein
form Capozziello  _et al._ (2013); Awad  _et al._ (2017):

$\left({e^{i}}_{\mu}\right)=\left(\sqrt{N(r)},\;\frac{1}{\sqrt{N(r)g(r)}},\;r,\;r,\;r\;\cdots\right)$.
---
(20)

Here, $t$ and $r$ are time and the radial coordinate, respectively. The metric
associated of the vielbein in Eq. (20) assumes the following form:

$ds^{2}=N(r)dt^{2}-\frac{1}{N(r)g(r)}dr^{2}-r^{2}\left(\sum_{i=1}^{n}d\theta^{2}_{i}+\sum_{k=1}^{d-n-2}dz_{k}^{2}\right),$
(21)

with $N(r)$ and $g(r)$ two unknown functions $r$111In 4-dimension the metric
(21) yields
$ds^{2}=N(r)dt^{2}-\frac{1}{N(r)g(r)}dr^{2}-r^{2}\gamma_{ij}dx^{i}dx^{j}\,.$
where $\gamma_{ij}dx^{i}dx^{j}$ represents the line element of a two-
dimensional surface with constant curvature $k=-1,0,1$, and the indices
$(i,j)=1,2$. The well-known solutions of GR, such as the Schwarzschild and the
Reissner-Nordstr¨om geometries, correspond to spherical horizon structure
where $k=1$. In this work, however, we shall consider solutions with a flat
horizon structure where $k=0.$. By combining the tetrad in Eq. (20) into the
scalar torsion $T$ in Eq. (6), we obtain

$T=\frac{(d-2)g}{r}\Big{[}N^{\prime}+\frac{(d-3)N}{r}\Big{]},$ (22)

where $N^{\prime}=\frac{dN(r)}{dr}$ and $g^{\prime}=\frac{dg(r)}{dr}$. In the
following, we omit the arguments of $N(r)$, $g(r)$, $N^{\prime}(r)$ and
$g^{\prime}(r)$. Owing to the success of the power-law form of $(T)$ to
describe cosmology Nesseris  _et al._ (2013); Nunes  _et al._ (2016);
Basilakos  _et al._ (2018), we focus on the quadratic form

$f(T)=T_{0}+\alpha T-\beta T^{2},$ (23)

where $T_{0}$, $\alpha$ and $\beta$ are constants.

###  III.1 Asymptotically static AdS BHs with the power-law Maxwell field

Regarding the vanishing of the electromagnetic sector, i.e.,
${{{\mathrm{T}}^{{}^{{}^{{\\!\\!\\!\\!\scriptstyle{em}}}}}}}^{\nu}_{\mu}=0$,
the results are identical to those obtained in Ref. Capozziello  _et al._
(2013); Awad  _et al._ (2017); Nashed and Saridakis (2018). However, novel
results can be found for the non-vanishing of the electromagnetic field, and
this could be explained by the tetrads in Eq. (20) in the field equations (10)
and (12) by utilizing the vector potential 1-form222Through out the rest of
this study we will put $\kappa=1$..

$Q(r)=\phi(r)dt.$ (24)

The non-zero components of the field equations read

$\displaystyle I^{r}{}_{r}=2Tf_{T}-f-(2s-1)\Big{\\{}-2\phi^{\prime
2}g(r)\Big{\\}}^{s}=0,$ $\displaystyle
I^{z_{1}}{}_{z_{1}}=I^{z_{2}}{}_{z_{2}}=\cdots\cdots=I^{z_{d-n-2}}{}_{z_{d-n-2}}=\frac{f_{TT}[r^{2}T+(d-2)(d-3)N]gT^{\prime}}{r(d-2)}+\frac{f_{T}}{2r^{2}}\Biggl{\\{}2r^{2}gN^{\prime\prime}+2(3d-8)rgN^{\prime}$
$\displaystyle+4(d-3)^{2}Ng+2(d-3)rNg^{\prime}+r^{2}N^{\prime}g^{\prime}\Biggr{\\}}-f+\Big{\\{}-2\phi^{\prime
2}g(r)\Big{\\}}^{s}=0,$ $\displaystyle
I^{t}{}_{t}=\frac{2(d-2)Ngf_{TT}T^{\prime}}{r}+\frac{(d-2)f_{T}[2\\{(d-3)Ng+rgN^{\prime}\\}+rNg^{\prime}]}{r^{2}}-f-(2s-1)\Big{\\{}-2\phi^{\prime
2}g(r)\Big{\\}}^{s}=0,$

where $\phi^{\prime}=\frac{d\phi}{dr}$. Various observations including the
followings are extracted from Eq. (III.1):
(i)-Equation (III.1) is reduced to those derived in Awad  _et al._ (2017) for
$s=1$.
(ii)-When $s=1/2$, the above system does not yield any solution because the
charged terms of Eq. (III.1) are imaginary. Hence, the case (ii) is excluded
in this study.

To derive an exact and well-behaved physical solution for the above system, we
considered $T_{0}=-\frac{1}{12\beta}$ and $\alpha=1$ and assumed that $s$
possessed odd values. Other factors prevented the good behavior of the
solution and imparted it with an imaginary value. By employing the pervious
constraints, we acquire the following general solution of the above
differential equations in $d$-dimensions

$\displaystyle
N(r)=\frac{r^{2}}{6(d-1)(d-2)\beta}+\frac{c_{1}}{r^{d-3}}+\frac{3\;2{{}^{{}^{s}}}(2s-1)^{2}c_{2}{}^{2s}}{2(d-2)(2s+1-d)r^{\frac{2[1+(d-4)s]}{2s-1}}}+\frac{2^{{}^{\frac{3s-1}{2}}}(2s-1)^{\frac{5}{2}}c_{2}{}^{3s}\sqrt{6\beta}}{(d-2)[(d-1)+(d-4)s]r^{\frac{[2+(3d-10)s]}{2s-1}}},$
$\displaystyle
g(r)=\frac{1}{\Bigg{[}c_{2}{}^{s}\sqrt{\frac{2{{}^{{}^{(s-1)}}}\;(-1)^{{}^{1+s}}\;6\;\beta(2s-1)}{r^{{}^{\frac{2s(d-2)}{2s-1}}}}}+1\Bigg{]}^{2}},\qquad\qquad{\cal
F}_{tr}=\frac{c_{2}}{r^{{}^{\frac{(d-2)}{2s-1}}}}+\frac{c_{2}{}^{s+1}\sqrt{2{{}^{{}^{(s-1)}}}\;(-1)^{{}^{1+s}}\;6\;\beta(2s-1)}}{{r^{{}^{\frac{(s+1)(d-2)}{2s-1}}}}},$
$\displaystyle\phi(r)=\frac{c_{2}(2s-1)}{(d-1-2s){r^{{}^{\frac{(d-1-2s)}{2s-1}}}}}+\frac{c_{2}{}^{s+1}\sqrt{2{{}^{{}^{(s-1)}}}\;(-1)^{{}^{1+s}}\;6\;\beta(2s-1)^{5/2}}}{{[d-1+s(d-4)]r^{{}^{\frac{[d-1+s(d-4)]}{2s-1}}}}},$
(26)

where $s$ takes a odd value, ${\cal F}_{tr}$ is the electric field, and
$\phi(r)$ is the gauge potential 1-form. Equation (III.1) clearly expresses
that $s$ must not be equal to half ($s\neq 1/2$). Additionally, Eq. (III.1) is
a generalization of the one presented in Awad  _et al._ (2017) and is reduced
to them at $s=1$. For an asymptotic AdS/dS spacetime, we have

$\Lambda_{\mathrm{eff}}=\frac{1}{6(d-1)(d-2)\beta}.$ (27)

Equation (27) ensures that black hole solution (III.1) has no corresponding in
TEGR upon taking the limit $\beta\rightarrow 0$, which means this charged
black hole solution has no analogue in GR or TEGR.

By combining Eqs. (III.1) and (27), we get

$\displaystyle
N(r)=\Lambda_{\mathrm{eff}}r^{2}-\frac{m}{r^{d-3}}+\frac{3\;2{{}^{{}^{s}}}(2s-1)^{2}q^{2s}}{2(d-2)(2s+1-d)r^{\frac{2[1+(d-4)s]}{2s-1}}}+\frac{2^{{}^{\frac{3s-1}{2}}}(2s-1)^{\frac{5}{2}}q^{3s}\sqrt{6\beta}}{(d-2)[(d-1)+(d-4)s]r^{\frac{[2+(3d-10)s]}{2s-1}}},$
(28) $\displaystyle
g(r)=\frac{1}{\Bigg{[}q{}^{s}\sqrt{\frac{2{{}^{{}^{(s-1)}}}\;(-1)^{{}^{1+s}}\;6\;\beta(2s-1)}{r^{{}^{\frac{2s(d-2)}{2s-1}}}}}+1\Bigg{]}^{2}},\qquad\qquad{\cal
F}_{tr}=\frac{q}{r^{{}^{\frac{(d-2)}{2s-1}}}}+\frac{q{}^{s+1}\sqrt{2{{}^{{}^{(s-1)}}}\;(-1)^{{}^{1+s}}\;6\;\beta(2s-1)}}{{r^{{}^{\frac{(s+1)(d-2)}{2s-1}}}}},$
$\displaystyle\phi(r)=\frac{q(2s-1)}{(d-1-2s){r^{{}^{\frac{(d-1-2s)}{2s-1}}}}}+\frac{q{}^{s+1}\sqrt{2{{}^{{}^{(s-1)}}}\;(-1)^{{}^{1+s}}\;6\;\beta(2s-1)^{5/2}}}{{[d-1+s(d-4)]r^{{}^{\frac{[d-1+s(d-4)]}{2s-1}}}}},$

where $c_{1}=-m$, and $c_{2}=q$. It follows from Eq. (28) that a kind of
cosmological constant can appear in $f(T)$ gravity Iorio and Saridakis (2012);
Kofinas  _et al._ (2015). Moreover, it is seen from Eq. (28) that
$s\leq\frac{d-1}{2}$ so that the monopole term, first term of $\phi(r)$ in Eq.
(28), could exhibit finite behavior. However, the second term of $\phi(r)$ in
Eq. (28) is always finite at $r\rightarrow 0$.

To discuss some of the physical properties of the aforementioned BH solution,
we substituted Eq. (28) into Eq. (21) and obtained the metric spacetime in the
following form:

$\displaystyle\\!\\!\\!\\!\\!\\!\\!ds{}^{2}=\Biggl{[}\Lambda_{eff}r^{2}-\frac{m}{r^{d-3}}+\frac{3\;2{{}^{{}^{s}}}(2s-1)^{2}q^{2s}}{2(d-2)(2s+1-d)r^{\beta_{1}}}+\frac{2^{{}^{\frac{3s-1}{2}}}(2s-1)^{\frac{5}{2}}q^{3s}\sqrt{6\beta}}{(d-2)[(d-1)+(d-4)s]r^{\beta_{2}}}\Biggr{]}dt^{2}-r^{2}\left(\sum_{i=1}^{n}d\theta^{2}_{i}+\sum_{k=1}^{d-n-2}dz_{k}^{2}\right)$
$\displaystyle\ \ \
-\frac{\\!dr^{2}}{\Biggl{[}\Lambda_{eff}r^{2}-\frac{m}{r^{d-3}}+\frac{3\;2{{}^{{}^{s}}}(2s-1)^{2}q^{2s}}{2(d-2)(2s+1-d)r^{\beta_{1}}}+\frac{2^{{}^{\frac{3s-1}{2}}}(2s-1)^{\frac{5}{2}}q^{3s}\sqrt{6\beta}}{(d-2)[(d-1)+(d-4)s]r^{\beta_{2}}}\Biggr{]}\Bigg{[}q{}^{s}\sqrt{\frac{2{{}^{{}^{(s-1)}}}\;(-1)^{{}^{1+s}}\;6\;\beta(2s-1)}{r^{\beta_{3}}}}+1\Bigg{]}^{2}},\\!$
(29)

where $\beta_{1}=\frac{2[1+(d-4)s]}{2s-1}$,
$\beta_{2}=\frac{[2+(3d-10)s]}{2s-1}$ and $\beta_{3}={\frac{2s(d-2)}{2s-1}}$.
As expected, we have obtained a solution, which behaves asymptotically as
(A)dS dS according to the sign of the dimensional parameter $\beta$ because
the constants $\beta_{1}$, $\beta_{2}$ $\beta_{3}$ are always positive.
Additionally, Eq. (28) is a generalization of the ones derived in Ref.
Capozziello  _et al._ (2013); Awad  _et al._ (2017), owing to the application
of a more general power-law Maxwell-$f(T)$-gravity.

Furthermore, we investigate the singularity structure of the BH solution in
Eq. (28),by calculating the curvature and torsion invariants. The curvature
scalars were calculated from the metric in Eq. (III.1, whereas the torsion
scalar was calculated employing the vierbeins in Eq. (20). By calculating the
Ricci scalar, the Ricci tensor square, and the Kretschmann scalar, we find

$\displaystyle
R\approx\frac{C_{1}(r)}{\sqrt{\beta}r^{{}^{\frac{{{s(d-2)}}}{2s-1}}}},\qquad
R^{\mu\nu}R_{\mu\nu}\approx\frac{C_{2}(r)}{\sqrt{\beta}r^{{}^{\frac{{{2s(d-2)}}}{2s-1}}}},\qquad
K\approx
R^{\mu\nu\lambda\rho}R_{\mu\nu\lambda\rho}\approx\frac{C_{3}(r)}{\sqrt{\beta}r^{{}^{\frac{{{2s(d-2)}}}{2s-1}}}},$
(30)

and the torsion scalar exhibited the following form:

$\displaystyle
T(r)\approx\frac{C_{4}(r)}{\sqrt{\beta}r^{{}^{\frac{{{s(d-2)}}}{2s-1}}}},$
(31)

where $C_{i}(r)$ is the lengthy polynomial function of $r$. The foregoing
invariants clearly indicate that there is singularity at $r=0$. At
$limit_{r=0}$, , the above invariants exhibited the following form
$(K,R_{\mu\nu}R^{\mu\nu})\sim r^{{}^{\frac{{{-2s(d-2)}}}{2s-1}}}$ and
$(R,T)\sim r^{{}^{\frac{{{-s(d-2)}}}{2s-1}}}$ (dissimilar to the BH solutions
that were derived from the linear Maxwell-$f(T)$ in which
$(K,R_{\mu\nu}R^{\mu\nu})\sim r^{{}^{{{{-2(d-2)}}}}}$ and $(R,T)\sim
r^{{}^{{{{-(d-2)}}}}}$. the asymptotic behaviors of the curvature and torsion
invariants of the solution (Eq. 28) ) were different from those of the BH
solution that were derived from the Einstein-Max- ell theory in the GR and
TEGR formulations, which behaved as $(K,R_{\mu\nu}R^{\mu\nu})\sim r^{-2d}$ and
$(R,T)\sim r^{-d}$. This clearly indicated that the singularity of the charged
BH solution (28) was softer than that, which was obtained in the linear
Maxwell-$f(T)$ and much softer than those of GR and TEGR for the charged case.
Finally, it is noteworthy that although the solution (28) possessed different
components, $g_{tt}$ and $g_{rr}$, its Killing vector and event horizons were
equal.

To investigate the horizons of the solution (28), it was necessary to
calculate the roots of the function $N(r)=0$. We plotted the function $N(r)$
versus the radial coordinate, $r$, in four and five dimensions for the various
values of the model parameters, as depicted in figure 1.

(a) The $N(r)$ in 4-dim.

(b) The $N(r)$ in 5-dim.

Figure 1: The function $N(r)$ of solution (28) of power law Maxwell-$f(T)$
gravity in four and five dimensions, for various values of the power law
parameter $s$. The term $r_{h}$ denotes the black hole of inner Cauchy
horizon.

Furthermore, the aforementioned properties of the BH solution were explained
differently by expressing the horizon mass–radius, $m_{h}$, of $m$ which
corresponds to $r_{h}$, that representing the global properties of a horizon
as obtained by setting $N(r_{h})=0$, namely

${m_{h}}=\frac{r_{h}^{(d-3)}\left(36\beta(2s-1)^{2}q^{2s}2^{s}\mathrm{A}+(d-1-2s)\Big{[}24\;2^{s}\beta
q^{3s}(2s-1)^{5/2}\sqrt{6\beta}\;r_{h}{}^{\frac{-(d-2)s}{2s-1}}-r_{h}{}^{\frac{2[2+(d-4)s]}{2s-1}}(d-2)\mathrm{A}\Big{]}\right)}{24r_{h}{}^{\frac{2[1+(d-4)s]}{2s-1}}\beta(d-2)(d-1-2s)\mathrm{A}},$
(32)

where $\mathrm{A}=(1+s)d-1-4s$. Noteworthily, at $s=1$ and $d=4$ we obtained
$m_{h}$ of the BH solutions that were derived in Awad  _et al._ (2017).

(a) The value $m_{h}$ of the parameter $m$ in 4-dim.

(b) The value $m_{h}$ of the parameter $m$ in 5-dim.

Figure 2: The value $m_{h}$ of the parameter $m$ that corresponds to the
horizon $r_{h}$, of solution (28) of power law Maxwell-$f(T)$ gravity in four
and five dimensions, for various values of the power law parameter $s$.

Eq. (32) is plotted in Fig. 2 when $d=4$ and $d=5$.

###  III.2 Energy of the black hole (28)

In this subsection, the energy of the BH solution (28) was calculated from Eq.
(18), and the necessary component of the super-potential $S^{\mu\nu\rho}$ of
the BH solution (28) is given as follows:

$S^{001}=\frac{g(r)}{r},$ (33)

where $g(r)$ is expressed by Eq. (28). The regularized expression of Eq. (19)
takes the following form:

$P^{a}:=\int_{V}d^{d-2}x\left[e{S}^{a00}f_{T}\right]-\int_{V}d^{d-2}x\left[e{S}^{a00}f_{T}\right]_{AdS/dS},$
(34)

where AdS/dS means calculations for pure AdS/dS space. Using Eq. (34) in
solution (28), we get

$\displaystyle
E=\frac{(d-2)m}{2}+\frac{2(d-3)^{2}(2s-1)2^{s-2}3^{s-1}q^{2s}}{(2s-3)r^{\frac{[d-1-2s]}{2s-1}}}+\frac{2^{s+1}(3s-1)(2s-1)(d-3)^{4}\sqrt{3(2s-1)\beta}q^{3s}}{3(2s-3)r^{\frac{(2d-5)}{2s-1}}}+\cdots\,.$
(35)

Eq. (35) indicates that the modification of $f(T)$ did not affect the mass
term of the energy of standard TEGR Maluf  _et al._ (2002); Maluf (1995),
although it affected the charge terms. The charge terms would aid the
calculation of energy starting from
${\mathcal{O}}\Bigg{(}\frac{1}{r}\Bigg{)}$, contrary to Reissner-Nordström
spacetime. This difference is due to the contribution of the function $g(r)$
given in Eq. (28). Moreover, as already stated, $s$ must satisfy
$s\leq\frac{d-1}{2}$ so that the second term in Eq. (35) could be finite at
$r\rightarrow 0$.

##  IV Rotating BHs in the power law Maxwell-$f(T)$ gravity

Further, the rotating BH solutions were derived to satisfy the field equations
of the power-law $f(T)$ gravity. Thus, we utilized the static BH solutions
that were derived in the previous section and applied the following
transformations with $n$-rotation parameters, as follows:

$\bar{\theta}_{i}=-\aleph~{}{\theta_{i}}+\frac{\omega_{i}}{\lambda^{2}}~{}t,\qquad\qquad\qquad\bar{t}=\Omega~{}t-\sum\limits_{i=1}^{n}\omega_{i}~{}\theta_{i},$
(36)

where $\omega_{i}$ are the rotation parameters (their number is
$n=\lfloor(d-1)/2\rfloor$ where $\lfloor...\rfloor$ represents the integer
part), and $\lambda$ is related to the parameter $\Lambda_{eff}$ of the static
solution (28) through Eq. (37):

$\displaystyle\lambda=-\frac{(d-2)(d-1)}{2\Lambda_{eff}}.$ (37)

Additionally, the parameter $\Omega$ is defined as follows:

$\Omega:=\sqrt{1-\sum\limits_{j=1}^{{n}}\frac{\omega_{j}{}^{2}}{\lambda^{2}}}.$

Applying the transformation (36) to the vielbeins (20) we obtained the
following equation:

$\displaystyle\left({e^{i}}_{\mu}\right)=\left(\begin{array}[]{cccccccccccccc}\Omega\sqrt{N(r)}&0&-\omega_{1}\sqrt{N(r)}&-\omega_{2}\sqrt{N(r)}\cdots&-\omega_{n}\sqrt{N(r)}&0&0&\cdots&0\\\\[5.0pt]
0&\frac{1}{\sqrt{N(r)g(r)}}&0&0\cdots&0&0&0&\cdots&0\\\\[5.0pt]
\frac{\omega_{1}r}{\lambda^{2}}&0&-\Omega r&0\cdots&0&0&0&\cdots&0\\\\[5.0pt]
\frac{a_{2}r}{\lambda^{2}}&0&0&-\Omega r\cdots&0&0&0&\cdots&0\\\\[5.0pt]
\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\cdots&\vdots\\\\[5.0pt]
\frac{\omega_{n}\;r}{\lambda^{2}}&0&0&0\cdots&-\Omega
r&0&0&\cdots&0\\\\[5.0pt] 0&0&0&0\cdots&0&r&0&\cdots&0\\\\[5.0pt]
0&0&0&0\cdots&0&0&r&\cdots&0\\\\[5.0pt] 0&0&0&0\cdots&0&0&0&\cdots&r\\\
\end{array}\right),$ (47)

where $N(r)$ and $g(r)$ are given by Eq. (28). Hence, for the electromagnetic
potential (24), we obtained the following form:

$\bar{\phi}(r)=-\phi(r)\left[\sum\limits_{j=1}^{n}\omega_{j}d\theta^{\prime}_{j}-\Omega
dt^{\prime}\right].$ (49)

Notably, although the transformation (36) did not alter the local properties
of the spacetime, it changed them globally, as reported in Lemos (1995), since
it mixed the compact and noncompact coordinates. Thus, the vielbeins, (20) and
(47) could only be locally mapped into each other Lemos (1995); Awad (2003).

The metric, which corresponds to the vielbein, (47) is written as follows:

$\displaystyle ds^{2}=N(r)\left[\Omega
d{t^{\prime}}-\sum\limits_{i=1}^{n}\omega_{i}d{z}^{\prime}\right]^{2}-\frac{dr^{2}}{N(r)g(r)}-\frac{r^{2}}{\lambda^{2}}\sum\limits_{i=1}^{n}\left[\omega_{i}d{t}^{\prime}+\Omega\lambda^{2}d{\theta}^{\prime}_{i}\right]^{2}-\sum\limits_{k=1}^{d-n-2}r^{2}dz_{k}^{2}-\frac{r^{2}}{\lambda^{2}}\sum\limits_{i<j}^{n}\left(\omega_{i}d{\theta}^{\prime}_{j}-\omega_{j}d{\theta}^{\prime}_{i}\right)^{2},$

where $0\leq r<\infty$, $-\infty<t<\infty$, $0\leq\theta_{i}<2\pi$,
$i=1,2\cdots n$ and $-\infty<z_{k}<\infty$, where $dz_{k}^{2}$ is the
Euclidean metric on $(d-2)$ dimensions with $k=1,,2\cdots d-3$. As mentioned
earlier, the static configuration (21) could be recovered as a special case of
the aforementioned general metric if we choose to vanish the rotation
parameters $\omega_{j}$.

Finally, following the procedure in subsection III.2 , the energy of the
rotating charged AdS BH (LABEL:m1), as calculated as follows:

$E=\frac{(d-2)[1+2(d-1)(d-2)\Lambda_{eff}\beta][\Lambda_{eff}{}^{2}\omega_{j}\sum\limits_{j=1}^{{n}}\omega_{j}+3\Omega^{2}]m}{12(d-3)G_{d}}+\cdots.$
(51)

Equation (51) is the energy of spacetime (47) which contains the rotation
parameters $\omega_{j}$ and when these parameters are vanishing we get the
value of energy given by Eq. (35) provided the use of Eq. (27).

##  V Thermodynamics of the derived black holes

To study different thermodynamical properties Hunter (1999); Hawking  _et al._
(1999); Bekenstein (1972, 1973); Gibbons and Hawking (1977) of BH solution
(28) we start by defining the Bekenstein–Hawking entropy of $f(T)$ as reported
in Karami and Abdolmaleki (2012); Bamba  _et al._ (2012); Nashed (2015a, b)

$S(r_{h})=\frac{1}{4}Af_{T}=\pi r_{h}{}^{2}(1-2\beta T),$ (52)

where $A$ is the area of the event horizon and $T$ is the scalar torsion,
which is given by Eq. (31). We plotted the entropy relation in Fig. 3 for
$d=4$ and $d=5$. The figure 3 2(a) revealed that we obtained a negative
entropy in the region when $r<r_{dg}$. At $r>r_{dg}$, we obtained a positive
value. Eq. (52) revealed that the entropy was not proportional to the area
because of the appearance of $\beta$. However, when $d=5$ we always obtain a
positive entropy as 3 2(b) shows. The heat capacity, $C_{h}$ which is valuable
for examining the stability of BH, was defined thermodynamically according to
its sign. $C_{h}$ is defined as reported in Nouicer (2007); Dymnikova and
Korpusik (2011); Chamblin  _et al._ (1999); Nashed  _et al._ (2018)

$C_{h}=\frac{\partial m}{\partial T}\equiv\frac{\partial m}{\partial
r_{h}}\left(\frac{\partial r_{h}}{\partial T}\right).$ (53)

Therefore, if the heat capacity is is positive, $C_{h}>0$, BH would
thermodynamically stable. Conversely, at ($C_{h}<0$), BH would be
thermodynamically unstable.

To calculate Eq. (53) we calculate the mass of BH mass in $r_{h}$ as obtained
in Eq. (32). The Hawking temperature of BHs can be defined as reported in
Hawking (1975)

$T=\frac{\kappa}{2\pi},\qquad\textmd{where}\quad\kappa\quad\textmd{ is the
surface gravity which is defined as,
}\qquad\qquad\kappa=\frac{N^{\prime}(r_{h})}{2}.$ (54)

The Hawking temperatures which is associated with the BH solution (28), is
expressed as follows:

$T_{h}=\frac{(d-1)(d-2)r_{h}{}^{2}-24\;2^{s}(2s-1)q^{3s}\beta\sqrt{2^{s}\;(-1)^{{}^{s+1}}\;3\;\beta(2s-1)}\;r^{-\beta_{2}}-36\beta
q^{2s}(2s-1)2^{s}\;r^{-\beta_{1}}}{96r_{h}\beta(d-2)},$ (55)

where ${T_{h}}$ is the Hawking temperature at the inner horizon. We plotted
$T_{h}$ in Fig. 4 for $d=4$ and $d=5$.

(a) The value of the entropy, $S_{h}$ in 4-dim.

(b) The value of the entropy, $S_{h}$ in 5-dim.

Figure 3: The value of the entropy, $S_{h}$, that corresponds to the horizon
$r_{h}$, of solution (28) of power law Maxwell-$f(T)$ gravity in four and five
dimensions, for various values of the power law parameter $s$.

(a) The value of the Hawking temperature, $T_{h}$ in 4-dim.

(b) The value of the Hawking temperature, $T_{h}$ in 5-dim.

Figure 4: The value of the Hawking temperature, $T_{h}$, that corresponds to
the horizon $r_{h}$, of solution (28) of power law Maxwell-$f(T)$ gravity in
four and five dimensions, for various values of the power law parameter $s$.

Further, we calculated $C_{h}$ after substituting Eq. (32) and (55) into (53)
as follows:

$\displaystyle
C_{h}=\Big{(}\frac{4\pi\sqrt{2s-1}r_{h}{}^{d-2}}{A(24\;2^{s}\;q^{3s}\;(2s-1)\sqrt{2^{s}\;(-1)^{{}^{s+1}}\;3\;\beta}(1+3sd-8s)\beta
r_{h}{}^{-\beta_{2}}+\sqrt{2s-1}\\{r_{h}{}^{2}(d-1)(d-2)-36\;q^{2s}\;2^{s}\;\beta\;r_{h}{}^{-\beta_{1}}\\})}\Big{)}$
$\displaystyle\times\Big{(}24\;q^{3s}\;2^{s}\;\sqrt{2^{s}\;(-1)^{{}^{s+1}}\;3\;\beta(2s-1)}(2s-1)[(2+3sd-10s)-(2s-1)(d-3)]\beta\;r_{h}{}^{-\beta_{2}}$
$\displaystyle-A[r_{h}{}^{2}(d-1)(d-2)-36\;q^{2s}\;2^{s}\;(2s-1)\;\beta\;r_{h}{}^{-\beta_{1}}]\Big{)},$
(56)

It was challenging to extract any information from Eq. (V), therefore, we
plotted it in 4-dimension as illustrated in Fig. 5, for particular values of
the BH parameters. As figure 5 4(b) shows that we have always positive heat
which means a stable BH in 5-dimension.

(a) The value of the heat capacity, $C_{h}$ in 4-dim.

(b) The value of the heat capacity, $C_{h}$ in 5-dim.

Figure 5: The value of the heat capacity, $C_{h}$, that corresponds to the
horizon $r_{h}$, of solution (28) of power law Maxwell-$f(T)$ gravity in four
and five dimensions, for various values of the power law parameter $s$.

Although Fig. 5 indicates that $C_{h}$ of the linear case was always positive,
it was negative in the nonlinear case and diverged at some critical values,
$r_{h}<r_{min}$ a, thus exhibiting a positive value. These imply that BH of
the linear Maxwell–$f(T)$ exhibited local stability, whereas the nonlinear
case exhibited local stability only when $r_{h}>r_{min}$, otherwise, it did
not.

The free energy in the grand canonical ensemble, which is called the Gibbs
free energy, is defined as reported in Nashed  _et al._ (2018):

$G(r_{h})=M(r_{h})-T(r_{h})S(r_{h}),$ (57)

where $M(r_{h})$, $T(r_{h})$ and $S(r_{h})$ are the mass of the BH, the
temperature and entropy at the event horizon, respectively. Inserting Eqs.
(32), (52) and (55) into (57) ) affords a lengthy expression. Here we just
demonstrated the behavior of the free energy in 4-dimension, as illustrated in
Fig. 6.

(a) The value of the free energy, $G_{h}$ in 4-dim.

(b) The value of the free energy, $G_{h}$ in 5-dim.

Figure 6: The value of the free energy, $G_{h}$, that corresponds to the
horizon $r_{h}$, of solution (28) of power law Maxwell-$f(T)$ gravity in four
and five dimensions, for various values of the power law parameter $s$.

Notably, when $d=4$, the linear Maxwell field had a positive value, followed
by a negative one in the range of $0.2<r<0.8$ and a positive one again
Altamirano  _et al._ (2014). However, the nonlinear Maxwell field was always a
positive quantity, and this indicates that the nonlinear Maxwell field was
always locally stable. Same discussion can be applied for $d=5$ but in that
case the linear one has a positive value then a negative value in the range
$0.2<r<0.4$ and the always positive value.

##  VI Conclusions and Discussions

We investigated the effect of the nonlinear power law of electrodynamics in
the context of the modified TEGR theory, $f(T)$-gravity. To do this, we
applied the flat horizon spacetime in a diverse dimension and applied it to a
specific form of $f(T)=T_{0}+\alpha T-\beta T^{2}$, where $T_{0}$ and $\alpha$
are constants and $\beta$ is a dimensional parameter. The obtained nonlinear
second-order differential equations were solved in an exact approach. The BH
solution of these differential equations was characterized by two integration
constants, in addition to the nonlinear parameter that describes the
nonlinearity of the Maxwell field equation. The two constants were explained
to represent the mass and charge of BH. This BH solution is a generalization
to that, which was presented in Awad  _et al._ (2017) owing to the existence
of the nonlinear parameter. When it was equal to 1, we returned to BH that was
discussed in Awad  _et al._ (2017).

To investigate the physical properties of this generalized BH, we calculated
the scalar invariants (the ones that are related to the curvature and those
that are related to the torsion) and demonstrated that the singularities of
this BH were much softer than those of Awad  _et al._ (2017) owing to the
contribution of the nonlinear electromagnetic parameter. Moreover, we
demonstrated that the singularities of our BH were much milder than those of
GR BH. This result is considered to be the main merit of this study, in
addition to the fact that the calculations of the energy confirmed that $s$
affects the asymptotic behavior of the charged terms as shown in Eq. (35).

To explore our BH in detail, we applied a coordinate to create an exact
rotating BH with nonlinear electrodynamics in the $f(T)$ frame. The features
of this rotating BH are that it possessed $d$-dimensional rotating parameters,
and could easily return to the nonrotating BH if all the rotation parameters
were set to zero. Notably, all the features of the singularities of the
nonrotating BH were present in the rotating one.

Finally, we calculated some thermodynamic quantities of the nonrotating case
and revealed that the entropy was not proportional to the area. The entropy
was non-proportional to the area because our BH possessed a non-vanishing
value of $T$. Our calculations revealed that the entropy might have a negative
value when $r_{h}<r_{dg}$ otherwise, it would have a positive value. A
negative entropy has been obtained and explained Cvetic  _et al._ (2002);
Nojiri  _et al._ (2001); Nojiri and Odintsov (2002, 2017); Clunan  _et al._
(2004); Nojiri  _et al._ (2002). The entropy might be negative because the
dimensional parameter, $\alpha$, had entered an unpermitted phase.

Further, we computed the temperature of BH and demonstrated that its value was
negative. This value mainly accounts for the structure of ultracold BH Davies
(1977). The computations of the thermodynamics in the context of the
$f(T)$-gravity for a non-trivial BH, which possessed a non-vanishing value of
$T$, achieved a limit for validation. This statement required further study,
which would be conducted in another work.

### Acknowledgments

The authors acknowledge the anonymous referee for improving the presentation
of the manuscript. G.N. would like to thank TUSUR for visiting fellowship. The
work of KB has partially been supported by the JSPS KAKENHI Grant Number
JP21K03547.

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