# Revisiting diagonal tetrads:
New Black Hole solutions in f(T) gravity

Adel Awad1,2, Alexey Golovnev2, María-José Guzmán3, and Waleed El Hanafy2

1Department of Physics, Faculty of Science, Ain Shams University,
Cairo 11566, Egypt
2Centre for Theoretical Physics, The British University in Egypt,
P.O. Box 43, El Sherouk City, Cairo 11837, Egypt
3Laboratory of Theoretical Physics, Institute of Physics, University of Tartu,
W. Ostwaldi 1, Tartu 50411, Estonia

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###### Abstract

We study various forms of diagonal tetrads that accommodate Black Hole
solutions in $f(T)$ gravity with certain symmetries. As is well-known, vacuum
spherically symmetric diagonal tetrads lead to rather boring cases of constant
torsion scalars. We extend this statement to other possible horizon
topologies, namely, spherical, hyperbolic and planar horizons. All such cases
are forced to have constant torsion scalars to satisfy the anti-symmetric part
of the field equations. We give a full classification of possible vacuum
static solutions of this sort. Furthermore, we discuss addition of time-
dependence in all the above cases. We also show that if all the components of
a diagonal tetrad depend only on one coordinate, then the anti-symmetric part
of the field equations is automatically satisfied. This result applies to the
flat horizon case with Cartesian coordinates. For solutions with a planar
symmetry (or a flat horizon), one can naturally use Cartesian coordinates on
the horizon. In this case, we show that the presence of matter is required for
existence of non-trivial solutions. This is a novel and very interesting
feature of these constructions. We present two new exact solutions, the first
is a magnetic Black Hole which is the magnetic dual of a known electrically
charged Black Hole in literature. The second is a dyonic Black Hole with
electric and magnetic charges. We present some features of these Black holes,
namely, extremality conditions, mass, behavior of torsion and curvature
scalars near the singularity.

##  1 Introduction

The motivations for studying modifications of general relativity (GR) are
multiple, including the well-known phenomenological problems of cosmology as
well as theoretical issues of singularities and quantisation of gravity. One
particular branch of this research teeming with alternative candidates
corresponds to the geometric foundations of gravity, that is dropping the
assumption of the Levi-Civita connection. An interesting subset of such
modifications consists of theories based on the teleparallel framework, where
the spacetime connection is flat, but allowed to have torsion and/or non-
metricity. In the particular case when only torsion is nonvanishing, we
retrieve the well-known teleparallel equivalent of general relativity (TEGR),
an equivalent version of GR with the same equations of motion [1, 2], and its
modifications such as $f(T)$ which are being intensively studied now [3, 4, 5,
6, 7].

The usual approach to teleparallel theories requires working with a tetrad
instead of just the metric. One can also write the standard GR in this way,
and then it has an additional symmetry group of local Lorentz transformations
of the tetrad field, the new dynamical variable. At the same time, the action
of TEGR preserves this invariance only up to a surface term which is omitted,
and at this cost it gets rid of the second derivatives in the Lagrangian. One
of its simplest generalisations uses a non-linear function of the torsion
scalar $T$ which was in the action of TEGR, and is therefore dubbed $f(T)$
gravity, in analogy with the famous $f(R)$ gravities. The mentioned above
difference from GR leads to second order equations of motion, unlike in metric
$f(R)$, though with a much more complicated dynamics which has been
extensively studied in the recent literature, even though with not too much
clarity yet.

Despite all the known foundational problems of these theories [8], their
cosmological applications receive lots of attention nowadays. At the same
time, another interesting and important area of applications corresponds to
the search for astrophysical solutions, and unfortunately it is less studied,
and also with much less success. Moreover, to the best of our knowledge, these
studies were almost fully restricted to static spherically symmetric solutions
[9, 10, 11, 12, 13, 14, 15]. We would like to mention that, in fact, other
topologies of horizons of Black Hole (BH) type solutions might also be
possible, which will be one of the main topics of the present paper.

Of course, the search for spherically symmetric solutions is anyway a
reasonable first step towards realistic astrophysical compact objects, and is
therefore of utmost importance for testing alternative gravity models. In
$f(T)$ gravity this search was rather complicated throughout the time, due to
the fact that the dynamical variable is the tetrad instead of the metric, and
the theory possesses extra degrees of freedom that produce nontrivial dynamics
on top of the usual metric geometry [16, 17, 18] . It means that only very
special cases of tetrads among those which reproduce a given metric are
solutions of the corresponding equations of motion, unfortunately with the
simplest choices often not being any good.

After adding the spin connection into the game, like in the covariant version
of the theory [19, 20], any possible tetrad can be used, but the problem
translates then to determination of an adequate spin connection for the tetrad
preferred. In any case, the enhanced Lorentz symmetry of TEGR is lost, and it
equally alters the dynamics in both versions of the theory, as has been proven
before [21]. Therefore, in this paper we use the simpler, pure tetrad version
of $f(T)$ gravity, or equivalently, the zero spin connection gauge for those
who prefer the covariant approach.

Solutions of $f(T)$ gravity equations of motion do qualitatively depend on
behaviour of the torsion scalar. Several vacuum solutions with constant
torsion scalar have been discussed in the past [22, 23, 24, 25, 26, 27]. Such
solutions coincide with the usual GR ones (unless with the vanishing
derivative of the function, $f_{T}=0$, which gives a pathology of switched-off
gravity), possibly with a cosmological constant, and therefore are not very
interesting, unless for studying perturbations around them. With non-constant
torsion scalar, the equations become very complicated, even in the case of
static ansätze with spherical symmetry, and in the spherical coordinates they
require non-diagonal tetrads.

Usually, the spherically symmetric solutions are found in terms of series
expansions. At first glance, they are not very plausible to be found since
there are three equations for two arbitrary functions. However, as it has
already been shown, there is a differential relation between them, the
generalised Bianchi identity [28]. Nevertheless, exact non-GR solutions are
still elusive, with probably the only explicit spherically symmetric example
employing an everywhere complex tetrad [29] which is of course highly
problematic, even though the metric and the torsion scalar are real.

Our aim in this paper is to improve the knowledge about spherically symmetric
tetrads, and to generalise it to other possible horizon topologies. We will
show that in the case of flat, or axially symmetric, horizons there is a
natural choice of coordinates in which the antisymmetric equations of motion
are automatically satisfied by the diagonal tetrad. In case of magnetic and
dyonic BHs, it allowed us to find exact non-trivial solutions, with a non-
constant torsion scalar. Electrically charged solutions of this sort were
previously found in Ref. [30].

The paper is organised as follows. In Section 2 we briefly introduce the
theoretical framework of teleparallel gravity and $f(T)$ gravity. In Section 3
we summarise what is know about the tetrad choice for spherically symmetric
solutions, including the subsection 3.1 about new static ansätze with
spherical, axial and hyperbolic symmetry, while in Section 4 we extend this
analysis to time-dependence in the tetrad. In Section 5 we present the new
choice of the tetrad for the case of axial symmetry which allowed us to find
new exact non-constant-$T$ solutions, magnetic in the Section 6 and dyonic in
the Section 7. We present our conclusions in Section 8.

##  2 Teleparallel geometry and $f(T)$ gravity

In this section we briefly describe the teleparallel space which we refer to
as a pair ($M$, $e_{a}$) where $e_{a}\,(a=1,\cdots,n)$ are $n$ independent
vector fields defined on an $n$ dimensional smooth manifold $M$. Let
$e_{a}{{}^{\mu}}\,(\mu=1,\cdots,n)$ be their components with respect to a
coordinate basis $x^{\mu}$. This defines $n$ independent 1-forms
$e^{a}=e^{a}{{}_{\mu}}dx^{\mu}$ such that111Both the Greek (coordinate) and
the Latin (Lorentz) indices are subject to the Einstein summation convention.

$e_{a}{{}^{\mu}}e^{a}{{}_{\nu}}=\delta^{\mu}_{\nu},\qquad
e_{a}{{}^{\mu}}e^{b}{{}_{\mu}}=\delta^{b}_{a}$ (1)

where $\delta$ is the Kronecker delta tensor. On the teleparallel space ($M$,
$e_{a}$) one can define Weitzenböck linear connection,

$\Gamma{{}^{\alpha}}_{\mu\nu}:=e{{}_{a}}{{}^{\alpha}}\,\partial_{\mu}e{{}^{a}}{{}_{\nu}},$
(2)

with respect to which the vector fields fulfill the relation

$\nabla_{\mu}e^{a}{{}_{\nu}}:=\partial_{\mu}e^{a}{{}_{\nu}}-\Gamma{{}^{\alpha}}_{\mu\nu}e^{a}{{}_{\alpha}}=0$
(3)

where $\nabla$ is the covariant derivative associated with the Weitzenböck
connection, and the Latin index in $e^{a}_{\mu}$ taken as simply numbering the
parallelly transported 1-forms, therefore with no connection coefficient
associated to it.

In this sense, the above constraint is called the teleparallelism condition.
We write the non-commutation of covariant derivatives acting on arbitrary
vector fields $V_{a}$ as

$\nabla_{\nu}\nabla_{\mu}V_{a}{{}^{\alpha}}-\nabla_{\mu}\nabla_{\nu}V_{a}{{}^{\alpha}}=R^{\alpha}{{}_{\epsilon\mu\nu}}V_{a}{{}^{\epsilon}}+T^{\epsilon}{{}_{\mu\nu}}\nabla_{\epsilon}V_{a}{{}^{\alpha}}.$
(4)

It can be easily shown that the curvature tensor of Weitzenböck linear
connection vanishes identically $R^{\alpha}{{}_{\epsilon\mu\nu}}\equiv 0$ and
the associated non vanishing torsion tensor is given by

$T{{}^{\alpha}}_{\mu\nu}=\Gamma{{}^{\alpha}}_{\mu\nu}-\Gamma{{}^{\alpha}}_{\nu\mu}=e{{}_{a}}{{}^{\alpha}}\left(\partial_{\mu}e{{}^{a}}{{}_{\nu}}-\partial_{\nu}e{{}^{a}}{{}_{\mu}}\right)$
(5)

which is antisymmetric in the last two indices by definition.

In addition, one can define a pseudo-Riemannian metric tensor on $M$ in terms
of the $\\{e_{a}\\}$ as

$g_{\mu\nu}:=\eta_{ab}e^{a}{{}_{\mu}}e^{b}{{}_{\nu}}$ (6)

with inverse metric

$g^{\mu\nu}=\eta^{ab}e_{a}{{}^{\mu}}e_{b}{{}^{\nu}}$ (7)

where $\eta_{ab}$ is the Minkowski metric of the tangent space with signature
$(-,+,+,+)$. Then one can obtain Levi-Civita connection associated with the
metric as

$\accentset{\circ}{\Gamma}{{}^{\alpha}}{{}_{\mu\nu}}:=\frac{1}{2}g^{\alpha\sigma}\left(\partial_{\nu}g_{\mu\sigma}+\partial_{\mu}g_{\nu\sigma}-\partial_{\sigma}g_{\mu\nu}\right).$
(8)

It is straightforward to show that both Levi-Civita and Weitzenböck linear
connections are metric ones, i.e.

$\accentset{\circ}{\nabla}_{\alpha}g_{\mu\nu}=0={\nabla}_{\alpha}g_{\mu\nu},$
(9)

where $\accentset{\circ}{\nabla}$ is the covariant derivative associated with
the Levi-Civita connection.

In the context of the teleparallel geometry one can define an invariant,
namely the torsion scalar,

$T=T{{}^{\alpha}}_{\mu\nu}\,S{{}_{\alpha}}^{\mu\nu},$ (10)

which differs from Ricci scalar $\accentset{\circ}{R}$, the double contraction
of the curvature tensor of Levi-Civita connection, by a total derivative term,
where

$S{{}_{\alpha}}^{\mu\nu}=\frac{1}{4}\left(T{{}_{\alpha}}^{\mu\nu}+T^{\nu\mu}{{}_{\alpha}}-T^{\mu\nu}{{}_{\alpha}}\right)+\frac{1}{2}\left(\delta^{\mu}_{\alpha}\,T^{\sigma\nu}{{}_{\sigma}}-\delta^{\nu}_{\alpha}\,T^{\sigma\mu}{{}_{\sigma}}\right).$
(11)

This tensor is commonly called the superpotential. It is anti-symmetric in the
last pair of indices. Of course, the teleparallel torsion scalar can be used
instead of the usual scalar curvature in the Hilbert-Einstein action giving
the field equations equivalent to those of the General Relativity (GR) theory,
so that we get the Teleparallel Equivalent of General Relativity (TEGR).

One of the simplest generalisations of TEGR is known as $f(T)$ gravity. Its
action is given by

$\mathcal{S}=-\frac{1}{2\kappa}\int d^{4}x\left[ef(T)+\mathcal{L}_{m}\right].$
(12)

where the constant factor is related to the Newton’s gravitational constant
$G$ by $\kappa=8\pi G$, and $e$ stands for the tetrad $e^{a}_{\mu}$
determinant. The variation of this action (12) with respect to the tetrad
yields the following field equation:

$f_{T}\mathfrak{G}_{\mu\nu}-\frac{1}{2}g_{\mu\nu}\left(T\,f_{T}-f\right)+2S_{\mu\nu}{{}^{\sigma}}\,\partial_{\sigma}f_{T}=\kappa\mathcal{T}_{\mu\nu}$
(13)

where $\mathfrak{G}_{\mu\nu}$ is Einstein tensor, $\mathcal{T}_{\mu\nu}$ is
the matter energy-stress tensor, and $f_{T}\equiv\frac{df(T)}{dT}$. It is to
be noted that the TEGR theory is invariant under local Lorentz transformation
at the level of the field equations, while this feature is not inherited by
the $f(T)$ extension [31].

This equation can be rewritten in other ways. For example, one can divide it
by $f_{T}$ to see the change in the effective gravitational constant. On the
other hand, for example in cosmology one can go for effective energy density
and pressure in modified Friedman equations by rewriting it as

$\mathfrak{G}_{\mu\nu}=\kappa\left(\mathcal{T}_{\mu\nu}+\mathfrak{T}_{\mu\nu}\right)$
(14)

with

$\mathfrak{T}_{\mu\nu}=\frac{1}{\kappa}\left[\frac{1}{2}g_{\mu\nu}\left(T\,f_{T}-f\right)-2S_{\mu\nu}{{}^{\sigma}}\,\partial_{\sigma}f_{T}+(1-f_{T})\mathfrak{G}_{\mu\nu}\right].$
(15)

The $\mathfrak{T}_{\mu\nu}$ tensor then represents the modifications of
Einstein’s field equations due to the $f(T)$ type of action.

As, unlike all the other terms, the term with the superpotential
$2S_{\mu\nu}{{}^{\sigma}}\partial_{\sigma}f_{T}$ is clearly non-symmetric, the
field equations (13)

${\mathfrak{L}}_{\mu\nu}=\kappa\mathcal{T}_{\mu\nu}$ (16)

can be split into symmetric and antisymmetric parts with the latter being

${\mathfrak{A}}_{\mu\nu}\equiv\frac{1}{2}({\mathfrak{L}}_{\mu\nu}-{\mathfrak{L}}_{\nu\mu})=(S_{\mu\nu}{{}^{\sigma}}-S_{\nu\mu}{{}^{\sigma}})\,\partial_{\sigma}f_{T}=(S_{\mu\nu}{{}^{\sigma}}-S_{\nu\mu}{{}^{\sigma}})\,f_{TT}\cdot\partial_{\sigma}T=0.$
(17)

It is straightforward to verify that, once the antisymmetric equations are
satisfied, then the symmetric part of equations satisfies the generalised
Bianchi identity, i.e. its Levi-Civita covariant divergence vanishes [28].

For an $f(T)$ gravity solution with $T=T_{c}$ constant, the equations of
motion (13) take the simple form of

$f_{T}(T_{c})\mathfrak{G}_{\mu\nu}-\frac{1}{2}g_{\mu\nu}\left(T_{c}\,f_{T}(T_{c})-f(T_{c})\right)=\kappa\mathcal{T}_{\mu\nu}.$
(18)

These equations are Einstein equations with a rescaled Newton constant
$\tilde{G}=\frac{G}{f_{T}(T_{c})}$ and a cosmological constant

$\Lambda=\dfrac{1}{2}\left(T_{c}\ -\frac{f(T_{c})}{f_{T}(T_{c})}\right).$ (19)

Therefore, if $f(T_{c})=T_{c}f_{T}(T_{c})$, then the GR/TEGR solutions having
$T=T_{c}$ are also solutions of $f(T)$ gravity. And in the case of non-vacuum
solutions, the Newton constant should be adjusted. More generally, any TEGR
solution having constant $T=T_{c}$ is a solution of $f(T)$ equations of motion
with proper values of effective cosmological and Newton constants.

Note that above we assumed that $f_{T}\neq 0$. Otherwise it would be a
pathological case of switched-off gravity. In particular, if there exists such
$T_{c}$ that $f(T_{c})=f_{T}(T_{c})=0$, then any geometrical construction with
this constant value of $T$ solves the equations of motion (13) in vacuum. The
fundamental reason is that every possible linear variation of the $f(T)$
action around it obviously vanishes.

##  3 The current situation with Black Hole solutions

The initial quest for BH solutions of $f(T)$ gravity went a wrong way since it
neglected the antisymmetric part of the equations of motion. Therefore, it
used diagonal tetrads in spherical coordinates which were not solutions of the
equations (13) at all, like for example in the paper [32]. However, it is
possible to construct solutions using quite a sophisticated tetrad of the form

$e^{a}_{\mu}=\left(\begin{array}[]{cccc}{A}(r)&0&0&0\\\
0&{B}(r)\sin\vartheta\cos\varphi&r\cos\vartheta\cos\varphi&-r\sin\vartheta\sin\varphi\\\
0&{B}(r)\sin\vartheta\sin\varphi&r\cos\vartheta\sin\varphi&r\sin\vartheta\cos\varphi\\\
0&{B}(r)\cos\vartheta&-r\sin\vartheta&0\\\ \end{array}\right).$ (20)

A review of what is known about that can be found, for example, in the Ref.
[29]. The equations take a complicated form, so that exact solutions are not
known beyond TEGR or constant $T$ constructions. The only exception is an
exact solution [29] with a non-real tetrad, i.e. a tetrad with some components
being imaginary. Both the metric and the torsion scalar are real; but since
the tetrad is a dynamical variable in the modified teleparallel framework,
this situation can be taken as problematic, and in any case it requires more
work of defining the model, especially if one wants to study the gravitational
perturbations around such solutions.

We now come back to a diagonal tetrad choice and aim at figuring out what can
be done for novel BH solutions with that. Of course, in the pure tetrad
formulation of a theory like $f(T)$, it is some physical restriction on the
ansatz, not simply a gauge choice. Obviously, our motivation is related with
the equations which are then easier to be solved. In the covariant language,
it is also a simplifying assumption of a diagonal tetrad and zero spin
connection, or to put it in another way, a tetrad which is diagonal in the
zero spin connection gauge.

In Sec. 7, we will show that one can find an exact solution with a diagonal
tetrad in the case of flat horizons. Unlike the exact solution from the Ref.
[29], the tetrad will be real everywhere outside the outer horizon (and inside
the inner one), or fully real for a naked singularity solution. In between the
two horizons, we will have it complex. This is actually the minimal price to
pay. Since we are constructing the standard horizons across which the signs of
$g_{tt}$ and $g_{rr}$ change, there is no other way to have it with a diagonal
(in these coordinates) tetrad but by using an excursion to imaginary numbers.

Our first task is to see whether any non-trivial new solutions are possible
with our choice of diagonal tetrads. In this context, people often talk about
”good” and ”proper” tetrads. A tetrad is called ”good” if its ansatz
automatically satisfies the antisymmetric part of equations of motion [33]. We
must say that it would be hard to give a precise definition to it, as well as
that it actually seems quite strange to have a special name for satisfying a
particular part of equations. On the other hand, another approach assumes that
a procedure of ”switching off gravity” could be used to find the correct
tetrad [34] called ”proper”. It hinges upon an unjustified assumption that it
is possible to objectively separate gravity from inertia, and its whole
philosophy neglects the new degrees of freedom. At the same time, not
surprisingly, the results of using this recipe have already appeared to be not
unique [35] and even not always correct [36], in the sense of giving a
”proper” tetrad which is not ”good”, i.e. not a solution at all.

In other words, explicit solutions are very difficult to find, and it is well-
known by now that the spherically symmetric solutions of $f(T)\neq$ TEGR
theories can not be found in the simple form of

$e^{a}_{\mu}=\text{diag}\left(A(r),\ B(r),\ r,\
r\sin(\theta)\vphantom{\int}\right),$ (21)

due to the anti-symmetric part of equations not being satisfied. Note that as
long as we use usual matter sources, i.e. those with symmetric energy-momentum
tensor, it cannot be resolved by considering non-vacuum solutions.

A very important point is that all this was done in spherical coordinates. No
claim like that can be made irrespective of the chosen coordinates. Let us
state it here explicitly: an abstract claim of impossibility of diagonal
tetrads for this kind of solutions is an absolutely wrong statement in itself.
The theory is diffeomorphism invariant, and the property of a tetrad being
diagonal depends on the choice of coordinates. What is shown in the literature
is impossibility of diagonal spherically symmetric tetrads in the standard
spherical coordinates. At the same time, one can show that diagonal tetrads in
Cartesian coordinates are possible for this metric [37]. They are related to
the usual non-diagonal tetrads used in most of the current literature on this
topic via the corresponding coordinate transformations.

In this paper, we will rewrite the diagonal tetrads in spherical symmetry with
a slightly different choice of coordinates, with no change to the results.
However, also hyperbolic and axially symmetric (flat horizon) solutions are to
be tried. In the case above, the hyperbolic symmetry corresponds to the simple
change of $\sin\longrightarrow\sinh$, and therefore, as is almost immediately
obvious, it is of no help. But then we will see that the flat case does give
interesting solutions of this sort, in coordinates which correspond to the
ones above by substituting unity instead of $\sin$ or $\sinh$ of $\theta$.

###  3.1 Generalisation to other horizon topologies

Let us rewrite the usual spherically symmetric ansatz for the metric as

$ds^{2}=-A(r)dt^{2}+B(r)dr^{2}+r^{2}\left[\dfrac{d\rho^{2}}{1-k\rho^{2}}+\rho^{2}d\phi^{2}\right]$
(22)

with $k=1$ and $\rho\in[-1,+1]$. In principle, a change of coordinates can
change the conclusions about possibility of using a diagonal tetrad. However,
it is not the case now. We have made a very simple transformation of
coordinates, $\theta\longrightarrow\rho=\sin(\theta)$ with no change in $t$,
$r$ or $\phi$, nor with any dependence on them in the $\rho$. A tetrad is
diagonal in the new coordinates if and only if it was diagonal in the old
ones.

Therefore, the diagonal tetrad

$e^{a}_{\mu}=\text{diag}\left(\sqrt{A(r)},\ \sqrt{B(r)},\
\frac{r}{\sqrt{1-k\,\rho^{2}}},\ r\rho\right)$ (23)

still cannot give us any non-constant $T$ solutions. However, it allows us to
treat the cases of hyperbolic ($k=-1$) and flat ($k=0$) horizons uniformly
with the usual spherical ones. Moreover, let us also do another innocent
coordinate change, $r\longrightarrow{\tilde{r}}(r)$, so that the angular part
of the metric has some arbitrary function of the radial coordinate in front of
it, instead of the standard $r^{2}$.

All in all, we consider the metric

$ds^{2}=-A(r)dt^{2}+B(r)dr^{2}+h^{2}(r)\left[\dfrac{d\rho^{2}}{1-k\rho^{2}}+\rho^{2}d\phi^{2}\right]$
(24)

with the following general tetrad which can describe a static gravitational
solution in $f(T)$ gravity with spherical, planar or hyperbolic symmetry

$e^{a}_{\mu}=\text{diag}\left(\sqrt{A(r)},\ \sqrt{B(r)},\
\frac{h(r)}{\sqrt{1-k\,\rho^{2}}},\ h(r)\rho\right)$ (25)

with an arbitrary function $h(r)$. If such solution possesses a horizon, this
will be the symmetry of this null surface. The torsion scalar, in this case,
is given by

$T=-2\frac{h^{\prime}\,(A\,h)^{\prime}}{ABh^{2}}$ (26)

with $h^{\prime}=\frac{dh(r)}{dr}$.

If the function $h(r)$ is supposed to describe our freedom of choosing the
radial variable, then we must assume $h^{\prime}\neq 0$. And in this case, we
can actually safely put $h=r$ because $T$ is a scalar, and its behaviour
cannot be changed by a coordinate transformation. This is indeed true. If we
put $r=r(\tilde{r})$ in the metric (24), the functions $A$ and $h$ remain the
same, just calculated at another value of their argument, i.e.
$A(r)={\tilde{A}}(\tilde{r}(r))$, while the function $B$ also gets multiplied
by ${{\tilde{r}}^{\prime}}^{2}$. The function $B$ is present in the
denominator of the formula (26) for $T$. At the same time, each one of the two
derivatives in its numerator gets multiplied by ${\tilde{r}}^{\prime}$.
Therefore, the derivatives of the new radial coordinate with respect to the
old one cancel each other, and the quantity $T$ is indeed just a scalar.

Having calculated the anti-symmetric part of the field equations, one gets the
following single non-vanishing component

${\mathfrak{A}}_{r\rho}=\frac{1}{2\rho}\cdot f_{TT}T^{\prime}=0$ (27)

which, if $f_{TT}\neq 0$, can only be resolved by finding a constant torsion
scalar

$-2{h^{\prime}\,(A\,h)^{\prime}\over ABh^{2}}=T_{c}.$ (28)

From the equations of motion Eq. (18) we see that such solutions of $f(T)$
gravity are inherited from GR/TEGR, possibly with a cosmological constant,
unless the rather pathological case of $f_{T}(T_{c})=0$ is encountered. In
other words, these tetrads do not allow us to get non-trivial new solutions.

We would like to mention here that the coordinates we use look similar to the
standard treatment of spatially curved cosmologies. Therefore, it is also
related to the known fact that these cosmologies do require non-diagonal
tetrads [38], while the case of $k=0$ corresponds to the need for non-diagonal
tetrads in the spatially flat cosmology, or even Minkowski space, when written
in spherical coordinates.

###  3.2 A collection of constant $T$ solutions.

As was already mentioned above, it is often much easier to find solutions with
constant $T$ because in this case the equations reduce to the simple GR ones,
or even fully trivialise in the pathological case of $f_{T}(T_{c})=0$. Though
those solutions are therefore not very interesting in themselves, let us first
discuss possible choices of that sort. We rewrite the eq. (27) as the
condition of constant torsion scalar (26):

$-2(\ln{h})^{\prime}\,(\ln(A\,h))^{\prime}=T_{c}\,B.$ (29)

Obviously, the case of $T_{c}=0$ is very elementary. It is either
$h^{\prime}=0$ which goes against our initial assumption of having $h(r)=r$
with some choice of the radial coordinate, or it can be $(Ah)^{\prime}=0$. Let
us call these cases ${\bf(i)}$ and ${\bf(ii)}$ respectively. Finally, the case
${\bf(iii)}$ will be about $T_{c}\neq 0$ which immediately implies non-
constant $h$. In this case, suppose we have chosen some function $B(r)$, then
define another function $\eta(r)$ such that

$\eta^{\prime}=\frac{B}{(\ln{h})^{\prime}}.$ (30)

With some arbitrary integration constant in the definition of $\eta$, it
transforms the equation (29) into the one

$\left(T_{c}\,\eta+2\ln(A\,h)\vphantom{\int}\right)^{\prime}=0$ (31)

which can immediately be solved as $A(r)=C\cdot\frac{\exp{\left(-{T_{c}\over
2}\eta(r)\right)}}{h(r)}$. Note that an additive integration constant in the
definition of $\eta$ can then be absorbed into the constant $C$.

We have just described the geometrical constructions of constant $T$. The
equations of motion then take the form of Eq. (18), and in the cases (i) and
(ii) of $T=0$ it even simplifies to
$f_{T}(0){\mathfrak{G}}_{\mu\nu}+\frac{1}{2}f(0)g_{\mu\nu}=\kappa\mathcal{T}_{\mu\nu}$.
The only non-trivial part which remains is the usual computation of the
Einstein tensor. Let us have a closer look at all these cases in vacuum.

####  3.2.1 Case (i)

This case of $h^{\prime}=0$ is not about any BHs. The spacetime is a direct
product of two 2-dimensional spaces. Here we have $T=0$ and constant
$h=h_{c}$, and the field equations read (all the functions $f$ and $f_{T}$
stand for their values at the $T=0$ point)

$\displaystyle
2{\mathfrak{L}}_{t}^{t}=2{\mathfrak{L}}_{r}^{r}=f-2f_{T}\cdot{k\over
h_{c}^{2}}=0,$ $\displaystyle
2{\mathfrak{L}}_{\rho}^{\rho}=2{\mathfrak{L}}_{\phi}^{\phi}=f+f_{T}\cdot{2AA^{\prime\prime}B-A^{\prime
2}B-A^{\prime}AB^{\prime}\over 2A^{2}\,B^{2}}=0.$ (32)

Let us first assume that $k\neq 0$. Note that for a non-pathological solution
we must have $f_{T}(0)\neq 0$, and in the case (i) it immediately requires
also $f(0)\neq 0$ if $k\neq 0$. The value of $h_{c}$ trivially follows from
the temporal or radial equation, and existence of a real solution obviously
depends on the signs of $f(0)$, $f_{T}(0)$, and $k$. Since $f\neq 0$, the
angular equation then requires $A^{\prime}\neq 0$. Therefore, it can be easily
solved by multiplying it by this non-zero factor and presenting in the shape
of $2fA^{\prime}+f_{T}\left(\frac{{A^{\prime}}^{2}}{AB}\right)^{\prime}=0$.
The answer is

$h_{c}=\pm\sqrt{2k\cdot\frac{f_{T}(0)}{f(0)}}\qquad\mathrm{and}\qquad
B(r)={-{A^{\prime}}^{2}(r)\cdot f_{T}(0)\ \over 2A^{2}(r)\cdot f(0)-CA(r)}$
(33)

with an arbitrary constant $C$.

For $k=0$ and $h(r)=h_{c}$, we must have $f(0)=0$ and also satisfy
$2AA^{\prime\prime}B-A^{\prime 2}B-A^{\prime}AB^{\prime}=0$. In case of a non-
degenerate metric with $A^{\prime}\neq 0$, it can be rewritten as $\left(\ln
B\right)^{\prime}=\left(\ln\frac{{A^{\prime}}^{2}}{A}\right)^{\prime}$ giving
the obvious solution of $B=C\,{{A^{\prime}}^{2}\over A}$ with an arbitrary
constant $C$ and an arbitrary function $A(r)$, as can also be found as the
$f(0)=0$ case of above. Another available option is to take $A^{\prime}=0$,
i.e. a constant $A$ and an arbitrary function $B(r)$.

In particular, one message we have from the last option above is that one can
describe Minkowski space by a diagonal tetrad

$e^{a}_{\mu}=\text{diag}\left(1,\ 1,\ 1,\ \rho\right),$ (34)

with one of its planes given in polar coordinates:

$ds^{2}=-dt^{2}+dr^{2}+d\rho^{2}+\rho^{2}d\phi^{2}.$ (35)

The same as for the trivial Minkowski tetrad in Cartesian coordinates, this
tetrad solves the vacuum equations of motion222Actually, its components depend
on only one of the coordinates, and therefore, as will be shown below, it
would solve the antisymmetric equations automatically, even if it was not a
case of constant $T$. if $f(0)=0$. In line of the discussion in the beginning
of this Section, this Minkowski tetrad is an example of a tetrad which is
”good” but not ”proper”. On the other hand, here we have different possible
tetrads solving the equations of motion and having the same (Minkowski) metric
and the same $T=0$, a fact that represents the so-called remnant symmetries
[39]. However such solutions are usually not physically equivalent, for they
acquire different properties of perturbations [40]. It should be taken as
manifestation of extra degrees of freedom.

####  3.2.2 Case (ii)

In this case, $A(r)=\frac{A_{0}}{h(r)}$ with an arbitrary constant $A_{0}$,
which has $T=0$ as well. This leads to the following field equations

$\displaystyle
2{\mathfrak{L}}_{t}^{t}=f+2f_{T}\cdot\frac{B\,{h^{\prime}}^{2}-hh^{\prime}B^{\prime}+2Bhh^{\prime\prime}-\,kB^{2}}{h^{2}B^{2}}=0$
$\displaystyle 2{\mathfrak{L}}_{r}^{r}=f-2f_{T}\cdot\frac{k}{h^{2}}=0,$ (36)
$\displaystyle
2{\mathfrak{L}}_{\rho}^{\rho}=2{\mathfrak{L}}_{\phi}^{\phi}=f+f_{T}\cdot\frac{B\,{h^{\prime}}^{2}-hh^{\prime}B^{\prime}+2Bhh^{\prime\prime}}{2h^{2}B^{2}}=0.$

Note that the equations with mixed (lower and upper) indices do not have
$A_{0}$ constant in them (though ${\mathfrak{L}}_{tt}$ would get an overall
factor of $A_{0}$). The reason is that this constant can be absorbed by a
constant re-scaling of the time variable, while the solution is static.

By subtracting ${\mathfrak{L}}^{r}_{r}$ from ${\mathfrak{L}}^{t}_{t}$, we see
that the second term in
${\mathfrak{L}}^{\rho}_{\rho}={\mathfrak{L}}^{\phi}_{\phi}$ must be zero. It
requires then $f(0)=0$. And applying the ${\mathfrak{L}}^{r}_{r}$ equation
again, we see that also $f_{T}(0)=0$ in the cases of $k\neq 0$. In other
words, for a non-flat horizon, only the option of a rather pathological
solution with switched-off gravity is left, with no further restriction for
functions $B(r)$ and $h(r)$.

However, for $k=0$, one gets only $f(0)=0$ with no restriction on $f_{T}(0)$
if $B{h^{\prime}}^{2}-hh^{\prime}B^{\prime}+2Bhh^{\prime\prime}=0$. The latter
equality can be written as $(\ln B)^{\prime}=(h(h^{\prime})^{2})^{\prime}$ and
solved as $B=C\,h\,(h^{\prime})^{2}$, with an arbitrary constant $C$, and
$h(r)$ being an arbitrary function. By a radial coordinate choice such that
$h=r$, $A=\frac{A_{0}}{r}$ and $B=Cr$, it leads to a known flat-horizon
solution in GR/TEGR which obviously possesses a naked singularity. By constant
rescaling of the time variable, we can set $A_{0}=C$, and therefore
$A(r)=\frac{1}{B(r)}$. This solution can be obtained from Schwarzschild-AdS
with flat horizon in GR/TEGR after sending the cosmological constant to zero,
for example, see the solution in [41].

To summarise, in the case (ii) we always face either the switched-off gravity
and a very arbitrary solution then, or a naked singularity which is often
considered as a serious pathology.

####  3.2.3 Case (iii)

In this case, $T=T_{c}\neq 0$ and therefore $h^{\prime}\neq 0$ at any point.
We can take an initially arbitrary function $\eta(r)$ where

$A(r)=C\,{\exp\left(-{T_{c}\over 2}\,\eta(r)\right)\over h(r)},$ (37)

for some arbitrary constant $T_{c}$. This leads to a torsion scalar

$T={T_{c}h^{\prime}\eta^{\prime}\over hB},$ (38)

therefore, if $T=T_{c}\neq 0$, we have

$B(r)=\eta^{\prime}(r)\cdot(\ln{h(r)})^{\prime}=\frac{h^{\prime}\eta^{\prime}}{h}.$
(39)

Upon substituting the expressions for $A$ and $B$ into the equations of
motion, and massaging the results a bit, the equations can be written as

$\displaystyle
2{\mathfrak{L}}_{t}^{t}=f-f_{T}\cdot\left(\frac{2k}{h^{2}}+2T_{c}-\frac{T_{c}hh^{\prime}{\eta^{\prime}}^{2}-2hh^{\prime}\eta^{\prime\prime}+2hh^{\prime\prime}\eta^{\prime}+4{h^{\prime}}^{2}\eta^{\prime}}{hh^{\prime}{\eta^{\prime}}^{2}}\right)=0,$
$\displaystyle
2{\mathfrak{L}}_{r}^{r}=f-f_{T}\cdot\left(\frac{2k}{h^{2}}+2T_{c}\right)=0,$
(40) $\displaystyle
2{\mathfrak{L}}_{\rho}^{\rho}=2{\mathfrak{L}}_{\phi}^{\phi}=f-f_{T}\cdot\left(2T_{c}-\frac{T_{c}h\eta^{\prime}+2h^{\prime}}{8h{h^{\prime}}^{2}{\eta^{\prime}}^{2}}\cdot\left(T_{c}hh^{\prime}{\eta^{\prime}}^{2}-2hh^{\prime}\eta^{\prime\prime}+2hh^{\prime\prime}\eta^{\prime}+4{h^{\prime}}^{2}\eta^{\prime}\right)\right)=0.$

Combination of ${\mathfrak{L}}_{t}^{t}$ with ${\mathfrak{L}}_{r}^{r}$ shows
that either $f_{T}(T_{c})=0=f(T_{c})$, i.e. the pathology of switched-off
gravity again, or we need to require

$T_{c}hh^{\prime}{\eta^{\prime}}^{2}-2hh^{\prime}\eta^{\prime\prime}+2hh^{\prime\prime}\eta^{\prime}+4{h^{\prime}}^{2}\eta^{\prime}=0.$

It can be easily taken care of by writing it as
$\left(\ln\eta^{\prime}\right)^{\prime}=\left(\frac{T_{c}}{2}\eta+\ln\left(h^{2}h^{\prime}\right)\right)^{\prime}$
which means that
$\left(e^{-\frac{T_{c}}{2}\eta}\right)^{\prime}\propto\left(h^{3}\right)^{\prime}$
where we used the necessary conditions: $h\neq 0$, $h^{\prime}\neq 0$,
$\eta^{\prime}\neq 0$.

Even $k\neq 0$ would anyway not help, since the angular equation contradicts
then the radial one. However, in case of $k=0$, and if the theory satisfies
the condition $f(T_{c})=2T_{c}\cdot f_{T}(T_{c})$ coming from the radial
equation, we can solve all the equations by

$\eta(r)=-\frac{2}{T_{c}}\ln\left(C_{1}h^{3}(r)+C_{2}\right)$ (41)

with two arbitrary integration constants $C_{1}$ and $C_{2}$, of which
$C_{1}\neq 0$ must be required, in order to not make $B$ vanish. Of course,
one could set one of the two constants to unity, and add an additive arbitrary
constant to $\eta$ instead.

Since, unlike for $B(r)$, the overall factor in $A(r)$ is not determined by
the function $\eta(r)$, this result can equivalently be written down in terms
of three arbitrary constants

$A(r)=C_{1}h^{2}(r)+\frac{C_{2}}{h(r)}\qquad\mathrm{and}\qquad
B(r)=C_{3}\frac{{h^{\prime}}^{2}(r)}{A(r)}$ (42)

of which the $C_{3}$ is not of a big interest since it can be set to $1$ by a
simple re-scaling of the time variable by a constant, for it obviously changes
the value of $A$ without affecting $B$. By choosing $h=r$, one can check that
this is nothing but the known GR/TEGR BH solution in asymptotically de Sitter
or anti de Sitter space with flat horizon [41, 42, 43], after a simple re-
scaling of coordinates. And this is the best of what can be done with this
kind of tetrads.

##  4 Switching the time dependence on

In this section we take the diagonal tetrad case even further by showing that
for a general time-dependent solution of this type the vanishing of the
antisymmetric part of the field equations still leads to a constant torsion
scalar. In other words, this case cannot be improved by allowing time
dependence333In the next section, we will show a successful tetrad
configuration for planar horizons, considering it in both static and time-
dependent cases..

###  4.1 The classical tetrad

For clarity, let us first illustrate the failure in the classical case. If we
take the time-dependent generalisation of the standard (unsuccessful) tetrad
(21)

$e^{a}_{\mu}=\text{diag}\left(A(t,r),\ B(t,r),\ r,\
r\sin(\theta)\vphantom{\int}\right)$ (43)

which gives the metric

$ds^{2}=-A^{2}(t,r)dt^{2}+B^{2}(t,r)dr^{2}+r^{2}d\Omega^{2},$ (44)

then the nonvanishing components of the torsion tensor are easily found to be

$T_{ttr}=A\cdot A^{\prime},\quad T_{rtr}=B\cdot{\dot{B}},\quad T_{\theta
r\theta}=r,\quad T_{\phi r\phi}=r\sin^{2}(\theta),\quad
T_{\phi\theta\phi}=r^{2}\cos(\theta)\sin(\theta)$

where $A^{\prime}\equiv\dfrac{\partial A(t,r)}{\partial r}$ and
${\dot{B}}\equiv\dfrac{\partial B(t,r)}{\partial t}$. It yields the following
nonvanishing components of the superpotential:

$S_{ttr}=-\dfrac{A^{2}}{r},\quad
S_{tt\theta}=-\dfrac{A^{2}\cot(\theta)}{2},\quad
S_{rr\theta}=\dfrac{B^{2}\cot(\theta)}{2},\quad S_{\theta
t\theta}=-\dfrac{r^{2}\dot{B}}{2B},\quad S_{\theta
r\theta}=-\dfrac{r\left(A+rA^{\prime}\right)}{2A},$ $S_{\phi
t\phi}=\sin^{2}(\theta)S_{\theta t\theta},\quad S_{\phi
r\phi}=\sin^{2}(\theta)S_{\theta r\theta}.$

Note that both the torsion and the superpotential tensor are antisymmetric in
the last two indices, therefore we do not explicitly give here those non-zero
components which can be obtained from the ones above by simple interchange of
these indices.

We readily see from it that such a tetrad is not possible beyond TEGR and/or
constant $T$. For the $f_{TT}S^{\mu\nu\alpha}\partial_{\alpha}T$ term in the
equations, i.e. the antisymmetric equations ${\mathfrak{A}}_{\mu\nu}=0$, the
problematic components are $S_{trt},\ S_{t\theta t},S_{r\theta r}$. They give
non-symmetric part of equations of motion in $tr,\ t\theta,\ r\theta$
components. The first two require $\partial_{t}T=0$, and the last one –
$\partial_{r}T=0$, and therefore the torsion scalar must be constant. Note
that if only one combination of these two derivatives was present in the anti-
symmetric equations, it would be possible to find a non-trivial solution,
possibly at the cost of some specific energy-momentum tensor of matter.

###  4.2 Our tetrad of different topologies

Our general case is not too different from above, even with all the freedom of
an arbitrary function $h(t,r)$. To see this, we take the following tetrad:

$e^{a}_{\mu}=\text{diag}\left(\sqrt{A(t,r)},\ \sqrt{B(t,r)},\
\frac{h(t,r)}{\sqrt{1-k\,\rho^{2}}},\ h(t,r)\rho\right).$ (45)

This tetrad leads to the metric

$dS^{2}=-A(t,r)\,dt^{2}+B(t,r)\,dr^{2}+h(t,r)^{2}\,\left[{d\rho^{2}\over
1-k\,\rho^{2}}+\rho^{2}d\phi^{2}\right]$ (46)

where the two-dimensional metric between the square brackets is a two-sphere
if $k=1$, a hyperboloid if $k=-1$ or a plane (or a cylinder, or a flat torus)
if $k=0$. Calculating the torsion scalar in this case one finds

$T(t,r)=-{2\over
A\,B\,h^{2}}\,\left({h^{\prime}\,(A\,h)^{\prime}}-{\dot{h}\,(B\,h\dot{)}}\right).$
(47)

The anti-symmetric part ${\mathfrak{A}}_{\mu\nu}=0$ of the field equations
leads to

${\mathfrak{A}}_{tr}=f_{TT}\,\left((\ln h\dot{)}\cdot T^{\prime}-(\ln
h)^{\prime}\cdot{\dot{T}}\right)=0,$ (48) ${\mathfrak{A}}_{\rho\,t}={1\over
2\rho}\cdot f_{TT}{\dot{T}}=0,$ (49) ${\mathfrak{A}}_{\rho\,r}={1\over
2\rho}\cdot f_{TT}T^{\prime}=0$ (50)

where $\,A^{\prime}\equiv\partial_{r}A$ and $\dot{A}\equiv\partial_{t}A$.
Since, for the tetrad considered, the most general torsion scalar can only
depend on $t$ and $r$, $T=T(t,r)$, then (any two of) the above three field
equations lead to a constant torsion scalar, $T=T_{c}$.

We conclude that, equivalently to the static cases above, time-dependent
diagonal tetrads with spherical and hyperbolic symmetries lead to trivial
solutions in a generic $f(T)$ theory, i.e., solutions with constant torsion
scalar. The same is true of the axial symmetry in these (polar) coordinates.
However, this symmetry naturally allows another choice of coordinates, too.

##  5 Flat horizons with Cartesian coordinates

Now we will focus on the flat horizon ($k=0$) case which can be written in
radically different coordinates. The corresponding diagonal tetrad is
different from the one we considered in the previous sections. This is similar
to the change of coordinates between polar and Cartesian ones. It does not
preserve the property of a tetrad to be diagonal. In other words, we are going
to study a genuinely different tetrad,

$e^{a}_{\mu}=\text{diag}\left(\sqrt{A(r)},\ \sqrt{B(r)},\ h(r),\
h(r)\vphantom{\int}\right)$ (51)

which reproduces the metric

$ds^{2}=-A(r)dt^{2}+B(r)dr^{2}+h^{2}(r)(d\theta^{2}+d\phi^{2}).$

Let us mention that these coordinates are topologically more natural and can
cover the whole space, unlike the polar type of coordinates in case of torical
or cylindrical topology, as opposed to a whole infinite plane. In the context
of teleparallel studies, it also means that this tetrad is globally better
defined.

It yields the following torsion scalar

$T=-\dfrac{2h^{\prime}\left(hA\right)^{\prime}}{h^{2}AB},$ (52)

and the components of the three equations of motion are given by

$\displaystyle 2{\mathfrak{L}}_{t}^{t}$ $\displaystyle=$ $\displaystyle
f+\dfrac{f_{T}}{AB^{2}h^{2}}\cdot\left(2A^{\prime}h^{\prime}Bh-2B^{\prime}h^{\prime}hA+4h^{\prime
2}AB+4h^{\prime\prime}hAB\right)+\dfrac{4h^{\prime}f_{TT}}{Bh}\cdot
T^{\prime}=0,$ (53) $\displaystyle 2{\mathfrak{L}}_{r}^{r}$ $\displaystyle=$
$\displaystyle
f+\dfrac{4h^{\prime}f_{T}}{ABh^{2}}\cdot\left(hA^{\prime}+h^{\prime}A\right)=0,$
(54) $\displaystyle
2{\mathfrak{L}}_{\theta}^{\theta}=2{\mathfrak{L}}_{\phi}^{\phi}$
$\displaystyle=$ $\displaystyle
f+\dfrac{\left(hA^{\prime}+2Ah^{\prime}\right)f_{TT}}{ABh}\cdot
T^{\prime}+\dfrac{f_{T}}{2A^{2}B^{2}h^{2}}\cdot\left(\vphantom{\int}-AhB^{\prime}\left(hA^{\prime}+2Ah^{\prime}\right)\right.$
(56) $\displaystyle\left.+B\left(4A^{2}h^{\prime 2}-h^{2}(A^{\prime
2}-2AA^{\prime\prime})+2Ah(3A^{\prime}h^{\prime}+2Ah^{\prime\prime})\vphantom{\frac{a}{a}}\right)\vphantom{\int}\right).$

Using this tetrad and these equations, we were able to find exact, and with
non-constant torsion scalar, solutions when coupling the $f(T)$ gravity to an
electromagnetic field. But to start with, let us mention that of course it is
possible to look for a constant $T=T_{c}$ solution also in this case. After
all, the functional shape of the torsion scalar is the same as before.

For example, for the function $f(T)=T_{0}+T+\beta\,T^{2}$ and with the
standard choice of the radial coordinate, which we will also use in the next
sections, a vacuum solution with constant torsion scalar (of our case (iii)
above) can be given by

$\displaystyle h(r)=r,\qquad A(r)={C_{1}\over r}+C_{2}\,r^{2},\qquad
B(r)=\frac{C_{3}}{A(r)}$ (57)

with initially arbitrary constants $C_{i}$ restricted by the torsion scalar
(52) value

$T_{c}=\dfrac{-1\pm\sqrt{1+12\beta T_{0}}}{6\beta}.$ (58)

This value of $T$ is a consequence of the radial equation requiring
$f(T_{c})=2T_{c}\cdot f_{T}(T_{c})$ again. And once the torsion scalar is made
to be constant, the equations cease to depend on which one of the $k=0$
tetrads was used.

However, in this case the anti-symmetric part of the field equations is
vanishing, without any constraint on the three functions in the metric, nor on
the value of the torsion scalar or the function $f$ defining the model. There
is another reason for which we need a matter source. Indeed, carefully looking
at the radial equation, we see that it identically takes the form of
$f(T)-2T\cdot f_{T}(T)=0$ when in vacuum. Generically, it is an algebraic
equation demanding some particular constant value of $T$. Since it is the
symmetric part of equations, what makes them not solvable in vacuum, then any
usual matter source can have its influence on the situation. We will use it to
give less trivial solutions in the sections to follow. Our exact solutions
there will be in a non-vacuum case, though with a very natural type of matter,
that of electromagnetic field.

Note also that, with the requirement of $f(T)=2T\cdot f_{T}(T)$, the only
chance to obtain non-constant $T$ solutions in vacuum is for the theories with
$f(T)\propto\sqrt{\pm T}$. This is an unpleasant situation when a theory
allows for only one particular sign of $T$. Also, presence of matter would be
allowed only with vanishing radial component of the energy-momentum tensor.
Interestingly, it has been noticed before that functions of this type are
special in $f(T)$. For example, any standard de Sitter, i.e.
$e^{a}_{\mu}=\mathrm{diag}\left(1,e^{Ht},e^{Ht},e^{Ht}\right)$ with constant
$H$, is a vacuum solution in this case [44]. Indeed, this is a constant $T$
construction, and being a maximally symmetric spacetime with
${\mathfrak{G}}_{\mu\nu}=-\frac{T}{2}\cdot g_{\mu\nu}$, it brings the
equations (18) to the form of $\left(\frac{1}{2}f-f_{T}T\right)\cdot
g_{\mu\nu}=\kappa\mathcal{T}_{\mu\nu}$. Only the energy-momentum of a
cosmological constant type is allowed then in any $f(T)$ gravity, and then
this is an algebraic equation for determining the Hubble constant $H$. So far,
this is nothing unusual, of course. However, these models of $\sqrt{T}$-type
make the l.h.s. vanish identically, and therefore allow only for fully vacuum
solutions, though with absolutely arbitrary $H$.

Before we describe our new solutions, let us discuss a bit more about the
general properties of this kind of tetrads.

###  5.1 The general theorem

The case of the tetrad from this Section can be generalised to the following
statement which also includes spatially flat, or even Bianchi I, cosmology:

Theorem. A diagonal tetrad with all components depending on only one of the
coordinates (in which it is diagonal) automatically satisfies the
antisymmetric part of $f(T)$ equations of motion.

To prove the theorem, we take a diagonal tetrad with components that depend
only on one coordinate $\xi$:

$e^{a}_{\mu}=A_{(\mu)}(x_{\xi})\cdot\delta^{a}_{\mu}.$ (59)

In other words, it has four non-zero components equal to four arbitrary
functions $A_{(\mu)}$ of one particular coordinate $x_{\xi}$. Obviously, we
will never assume summation over $\xi$; and any summation over an index which
also has some appearance in brackets will always be indicated explicitly.

The torsion tensor can be written as

$T^{\alpha}_{\hphantom{\alpha}\mu\nu}=\frac{A^{\prime}_{(\alpha)}}{A_{(\alpha)}}(\delta^{\xi}_{\mu}\delta^{\alpha}_{\nu}-\delta^{\xi}_{\nu}\delta^{\alpha}_{\mu})$
(60)

where $A^{\prime}\equiv\frac{\partial A}{\partial x_{\xi}}$, or

$T_{\alpha\mu\nu}=A^{\prime}_{(\alpha)}A_{(\alpha)}(\delta^{\xi}_{\mu}\eta_{\alpha\nu}-\delta^{\xi}_{\nu}\eta_{\alpha\mu})$
(61)

upon lowering the index.

The torsion vector is

$T_{\mu}\equiv
T^{\alpha}_{\hphantom{\alpha}\mu\alpha}=\delta^{\xi}_{\mu}\cdot\sum_{\alpha\neq\xi}\frac{A^{\prime}_{(\alpha)}}{A_{(\alpha)}}$
(62)

and has only one non-zero component, $T_{\xi}$.

The contortion is also easily found:

$K_{\alpha\mu\nu}=\frac{1}{2}(T_{\alpha\mu\nu}+T_{\nu\alpha\mu}+T_{\mu\alpha\nu})=A^{\prime}_{(\mu)}A_{(\mu)}(\delta^{\xi}_{\alpha}\eta_{\mu\nu}-\delta^{\xi}_{\nu}\eta_{\mu\alpha})$
(63)

which finally gives the superpotential

$S_{\alpha\mu\nu}=\frac{1}{2}\left(K_{\mu\alpha\nu}+g_{\alpha\mu}T_{\nu}-g_{\alpha\nu}T_{\mu}\right)=\frac{1}{2}(\eta_{\alpha\mu}\delta^{\xi}_{\nu}-\eta_{\alpha\nu}\delta^{\xi}_{\mu})\cdot
A^{2}_{(\alpha)}\sum_{\beta\neq\xi,\alpha}\frac{A^{\prime}_{(\beta)}}{A_{(\beta)}}.$
(64)

The right factor in the last expression can be easily derived assuming that
$\alpha\neq\xi$. However, in case of $\alpha=\xi$ its left factor is anyway
zero. And indeed $S_{\xi\mu\nu}=0$, so that this formula can always be used.

In particular, since $T$ can only depend on the $\xi$ coordinate, we see that
$S_{\mu\nu\alpha}\partial^{\alpha}T=S_{\mu\nu\xi}\partial^{\xi}T$, and it has
only symmetric, even diagonal, components which means that this is always a
”good tetrad”, in the sense of automatically solving the antisymmetric part of
equations. The theorem is proven.

This theorem applies to many different cases, for example to the standard
spatially flat cosmology. In our case of $A_{(t)}=\sqrt{A(r)},\
A_{(r)}=\sqrt{B(r)},\ A_{(\theta)}=A_{(\phi)}=r$ we get the only non-zero
components of $S_{\mu\nu\xi}$ for the $f_{TT}$ term: $S_{ttr}=-\frac{A}{r}$
and $S_{\theta\theta r}=S_{\phi\phi
r}=\frac{r^{2}}{2}(\frac{A^{\prime}}{2A}+\frac{1}{r})$ with the radial
coordinate in the role of $\xi$. Finally, for completeness of these
calculations, we present the torsion scalar:

$T=S_{\alpha\mu\nu}T^{\alpha\mu\nu}=-\frac{\eta^{\xi\xi}}{A^{2}_{(\xi)}}\cdot\sum_{\alpha\neq\xi}\left(\frac{A^{\prime}_{(\alpha)}}{A_{(\alpha)}}\sum_{\beta\neq\alpha,\xi}\frac{A^{\prime}_{(\beta)}}{A_{(\beta)}}\right)$
(65)

which then gives $T=-\frac{2}{Br}(\frac{A^{\prime}}{A}+\frac{1}{r})$.

Interestingly enough, if we take $A_{(\theta)}=A_{(\phi)}=h(r)$, then this
result is
$T=-\frac{2h^{\prime}}{Bh}(\frac{A^{\prime}}{A}+\frac{h^{\prime}}{h})$ which
coincides with our result of the previous section. The case of $k=0$ in those
tetrads gives another tetrad for the same metric we have here. Reproducing the
same torsion scalar then can be taken as yet another case of the remnant
symmetry [39]. At the same time, it shows that the actual meaning of remnant
symmetry is rather limited, especially in a non-constant $T$ scenario. Those
two cases of $k=0$ tetrad have the same metric and the same torsion scalar,
but one of the cases does solve the antisymmetric equations while another one
doesn’t.

###  5.2 Comment on time dependence

Notice that, the same as in the static case, the tetrad with a planar symmetry
of the form

$e^{a}_{\mu}=\text{diag}\left(\sqrt{A(t,r)},\ \sqrt{B(t,r)},\ h(t,r),\
h(t,r)\vphantom{\int}\right),$ (66)

can not be obtained by a coordinate transformation from the $k=0$ case of the
tetrad from the previous sections, for under the transformation relating the
two metrics, that tetrad would get off-diagonal terms. In other words, this is
a different tetrad.

With the non-trivial time-dependence, the anti-symmetric part of the field
equations becomes non-trivial even for this new tetrad. But it leads to just a
single equation in this sector, namely

${\mathfrak{A}}_{tr}=f_{TT}\,\left((\ln h\dot{)}\cdot T^{\prime}-(\ln
h)^{\prime}\cdot{\dot{T}}\right)=0$ (67)

which generically does not lead to a constant torsion scalar. For example, the
equation is satisfied if $h(t,r)\sim(rt)^{a}$ and $T\sim(rt)^{b}$, where $a$
and $b$ are some constants. Also, setting $h(t,r)=r$, we need a constant in
time torsion scalar but possibly dependent on $r$. This is a certain
restriction for the functions in the tetrad, also with an off-diagonal $tr$
component in the symmetric equations, so that the equations might well get too
restrictive in vacuum. However, having solved the $\mathfrak{A}$ part, one can
try to look for a solution with matter.

##  6 Magnetic static solutions with axial symmetries

In this section, we present a new magnetically charged solution in
Maxwell-$f(T)$ gravity. The equations we use are the $f(T)$ equations (12) of
motion with electromagnetic field as a source and Maxwell equations with no
charges or currents:

$f_{T}\mathfrak{G}_{\mu\nu}-\frac{1}{2}g_{\mu\nu}\left(T\,f_{T}-f\right)+2S_{\mu\nu}{{}^{\sigma}}\,\partial_{\sigma}f_{T}=\kappa\mathcal{T}_{\mu\nu},\hskip
36.135pt\accentset{\circ}{\nabla}_{\nu}F^{\mu\nu}=0,$ (68)

where the energy-momentum tensor of the electromagnetic field

$\mathcal{T}_{\mu\nu}=\frac{1}{4\pi}\left(F_{\mu\alpha}\,F_{\nu}^{\hphantom{\mu}\alpha}-\frac{1}{4}F_{\alpha\beta}F^{\alpha\beta}g_{\mu\nu}\right),$
(69)

is given in terms of its strength tensor
$F_{\mu\nu}\equiv\partial_{\mu}{\mathcal{A}}_{\nu}-\partial_{\nu}{\mathcal{A}}_{\mu}$.
There is also another, a slightly more mathematical way to write the
electromagnetic part of the equations by using the Hodge dual of the field
strength $H_{\mu\nu}\equiv*F_{\mu\nu}={1\over
2}\,\epsilon_{\mu\nu\alpha\beta}\,F^{\alpha\beta}$:

$\mathcal{T}_{\mu\nu}=\frac{1}{8\pi}\left(F_{\mu\alpha}\,F_{\nu}^{\hphantom{\mu}\alpha}+H_{\mu\alpha}\,H_{\nu}^{\hphantom{\mu}\alpha}\right),$
(70)

Notice that the equations of motion are therefore invariant under electric-
magnetic duality, or changing the 2-forms as $F\rightarrow H$ and
$H\rightarrow-F$. We will comment on this symmetry after presenting new
magnetic and dyonic solutions afterwards.

Note that, of course, in the theoretical physics literature the factors of
$4\pi$ are often omitted. In order to have our BH formulae in a more familiar
shape, we keep it and also take $\kappa=8\pi G$, so that the r.h.s. of the
equations acquires the form of $\kappa{\cal
T}_{\mu\nu}=2G\left(F_{\mu\alpha}\,F_{\nu}^{\hphantom{\mu}\alpha}-\frac{1}{4}F_{\alpha\beta}F^{\alpha\beta}g_{\mu\nu}\right)$
with $G$ being the gravitational constant. And for brevity, we put $G=1$. If
needed, the explicit factor of $G$ can be reconstructed in all expressions
below by multiplying each $q$ and $p$ by $\sqrt{G}$.

We are interested in solutions with axial symmetry which can be represented by
the diagonal tetrad (51) in which we chose the standard radial coordinate such
that $h(r)=r$. Also, it proves convenient to write $B(r)=\frac{W(r)}{A(r)}$ to
identify the asymptotic behaviour of the solution. Therefore, we take the
tetrad to be

$e^{a}_{\mu}=\text{diag}\left(\sqrt{A(r)},\ \sqrt{\frac{W(r)}{A(r)}},\ r\
,r\right),$ (71)

in the coordinates $(t,r,\theta,\phi)$, with $\theta$ and $\phi$ not
necessarily being angle-like coordinates. The metric is

$dS^{2}=-A(r)dt^{2}+\dfrac{W(r)}{A(r)}dr^{2}+r^{2}\left(d\theta^{2}+d\phi^{2}\right).$
(72)

Obviously, the usual (anti)-de-Sitter asymptotic behaviour corresponds to
$W(r)$ tending to a constant positive value which can be conveniently set to
unity.

This is the same geometry that we had in the previous Section. On top of that,
we will assume the following form of the gauge potential

${\cal A}={\cal A}_{4}(\theta)\,d\phi.$ (73)

The Maxwell equations are easy to solve in this case. Indeed, the only non-
vanishing component of the field strength is
$F_{\theta\phi}=-F_{\phi\theta}=\partial_{\theta}{\cal A}_{4}(\theta)$, and
since the metric is diagonal and with components depending only on $r$, the
only non-trivial equation tells us that

$0=\frac{1}{\sqrt{-g}}\partial_{\mu}\left(\sqrt{-g}F^{\mu\phi}\right)=\partial_{\theta}F^{\theta\phi}\propto\partial_{\theta}F_{\theta\phi}=\partial_{\theta}\partial_{\theta}{\cal
A}_{4}(\theta)$

which, modulo a gauge choice of ${\cal A}_{4}(0)=0$, gives ${\cal
A}_{4}(\theta)=p\theta$ with the constant $p$ representing the magnetic charge
of the BH.

###  6.1 GR/TEGR

Before we present our new magnetic solution, it is instructive to briefly
review a known magnetic solution in Maxwell-TEGR. Let us consider the TEGR
case, i.e. $f(T)=T_{0}+T$. Obviously, we should get magnetically charged flat-
horizon BHs of the usual GR with a cosmological constant given by
$\frac{T_{0}}{2}$.

For the tetrad (71), we obtain the components of the superpotential tensor
(11)

$S_{trt}=-S_{ttr}=\frac{A(r)}{r},\qquad S_{\theta\theta r}=S_{\phi\phi
r}=-S_{\theta r\theta}=-S_{\phi r\phi}=\dfrac{r}{4A}(rA^{\prime}+2A)$ (74)

and the torsion scalar (10) as above (52)

$T=-\dfrac{2}{r^{2}W}(rA^{\prime}+A).$ (75)

And the only non-zero component of the field strength is
$F_{\theta\phi}=-F_{\phi\theta}=\partial_{\theta}{\cal A}_{4}(\theta)=p$.
Therefore, the value of the torsion scalar can be derived from the radial
equation $\frac{1}{2}f-f_{T}T=\kappa{\cal T}^{r}_{r}$. With $f(T)=T_{0}+T$ and
$\kappa{\cal
T}^{r}_{r}=-\frac{1}{2}F_{\mu\nu}F^{\mu\nu}=-\frac{p^{2}}{r^{4}}$, we
immediately get

$T(r)=T_{0}+\frac{2p^{2}}{r^{4}}.$ (76)

The presence of magnetic charge allows us to have a non-constant $T$ solution.

Moreover, with $G=1$, we can easily find the full energy-momentum tensor of
the electromagnetic field as

$\kappa{\cal
T}^{\nu}_{\mu}=\mathrm{diag}\left(-\frac{p^{2}}{r^{4}},\,-\frac{p^{2}}{r^{4}},\,\frac{p^{2}}{r^{4}},\,\frac{p^{2}}{r^{4}}\right).$

Then the field equations can be solved as

$A=C\left(-{T_{0}\over 6}\,r^{2}\,-{2m\over r}+{p^{2}\over r^{2}}\right)\qquad
W=C,\qquad{\cal A}_{4}=p\,\theta$ (77)

where the cosmological constant is given by $\frac{T_{0}}{2}$, and there are
three arbitrary constants, with $p$ being the previously defined magnetic
charge and $m$ related to the BH mass. The arbitrary constant $C>0$ is a
proportionality coefficient between $B$ and $\frac{1}{A}$, and as such it can
be set to $C=1$ by simply re-scaling the time coordinate, and this is a
reasonable choice to make in order to have a familiar form of the asymptotic
behaviour.

###  6.2 Quadratic polynomial $f(T)$ gravity

Before moving to a magnetic solution for the quadratic polynomial gravity,

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

we need to set a value for $T_{0}$ in order to simplify the resulting
solutions, for otherwise these solutions will not have a simple analytic form.
Recall that the uncharged solution, for the tetrad ansatz (71), can be found
as (57)

$A(r)=C_{1}\,r^{2}+C_{2}/r,\hskip 36.135ptW(r)=C_{3}$ (78)

with a convenient choice of $C_{3}=1$ easily achievable by a constant time re-
scaling. This is irrespective of the particular choice of the function $f$.

A natural fine-tuning is suggested by the value
$T_{c}=\dfrac{-1\pm\sqrt{1+12\beta T_{0}}}{6\beta}$ of the torsion scalar (58)
in the vacuum solutions. Notice that by choosing

$T_{0}=-{1\over 12\,\beta}$ (79)

one gets rid of the square root term in the above expression, and therefore it
also simplifies a lot the form of the magnetic solution and the dyonic
solution in the coming section. For the uncharged case in this quadratic
polynomial $f(T)$ gravity, one then has the constant torsion scalar
$T_{c}=2T_{0}=-\frac{1}{6\beta}$.

In other words, our model is given by

$f(T)=T_{0}+T-\frac{T^{2}}{12T_{0}}=-\frac{1}{12\beta}+T+\beta T^{2}.$ (80)

The required torsion scalar can be found in this case by observing that
$\frac{1}{2}f-f_{T}T=\frac{1}{2T_{0}}\left(T_{0}-\frac{T}{2}\right)^{2}$.
Since according to the radial equation it must be equal to
$-\frac{p^{2}}{r^{4}}<0$, we need to demand $T_{0}<0$, or equivalently
$\beta>0$. Clearly the obtained solution has no TEGR limit since $\beta=0$ is
not allowed. This case is similar to the charged AdS BH solution in higher-
dimensional $f(T)$ gravity [47]. After that, the torsion scalar can be found
as

$T=2T_{0}\pm 2\sqrt{-2T_{0}}{\ |p\ |\over r^{2}}=-{1\over
6\beta}\pm\sqrt{\frac{2}{3\beta}}{\ |p\ |\over r^{2}}.$ (81)

Note that the two possible solutions come from the case of the quadratic
equations for finding $T$. However, we have there $\ |p\ |=\sqrt{p^{2}}$ which
is not surprising since the sign of the charge cannot influence the geometry,
with ${\cal T}_{\mu\nu}$ being quadratic in $F_{\mu\nu}$. Below we will omit
the plus-minus and the modulus signs, but one should keep in mind that the
sign of $p$ can be chosen arbitrarily, irrespective of the real sign of
magnetic charge of the BH, and it represents two different possible
gravitational solutions.

The full field equations can be solved as

$A=C\cdot\left({r^{2}\over 36\beta}-{2m\over r}+{3p^{2}\over
2r^{2}}+{\sqrt{6\beta}p^{3}\over 3r^{4}}\right)=C\cdot\left(-{T_{0}r^{2}\over
3}-{2m\over r}+{3p^{2}\over 2r^{2}}+{p^{3}\over 3\sqrt{-2T_{0}}\cdot
r^{4}}\right),$ (82) $W=C\cdot\left(1+{\sqrt{6\beta}p\over
r^{2}}\right)^{2}=C\cdot\left(1+{p\over\sqrt{-2T_{0}}\cdot
r^{2}}\right)^{2},\hskip 21.68121pt{\cal A}_{4}=p\,\theta.$ (83)

The constant $C>0$ can again be naturally set to $C=1$ by a proper re-scaling
of the time variable.

This solution is the magnetic dual of the solution presented in [30] and its
rotating version444The procedure to add angular momentum to 4-dimensioal
spacetime has been developed in GR context in [45] and generalized to
$N$-dimension in [46]. in [48]. Notably, in the quadratic polynomial $f(T)$
gravity, as $r\to 0$, the teleparallel torsion scalar diverges as $T\to{1\over
r^{2}}$ , which shows that the quadratic $f(T)$ gravity smooths the
singularity in comparison to Maxwell-TEGR case, namely Eq. (76). It is
important to comment here on the possibility of generating another solution
which in this case an electric solution. I.e., with electric charge $q$
instead of magnetic charge $p$ since the above field equations are invariant
under $F\rightarrow H$ and $H\rightarrow-F$ which takes $q\rightarrow p$ and
$p\rightarrow-q$. This produces the previously known solution [30]. Therefore,
no new solution is produced from this symmetry transformations.

##  7 Dyonic static solutions with axial symmetries

In this section, we consider the more general solution when both electric and
magnetic charges present. Similar to the previous section we obtain Maxwell-
TEGR and Maxwell-$f(T)$ gravity with axial symmetry. For this reason, we
generalise the gauge potential (73) to also include electric charge:

${\cal A}={\cal A}_{0}(r)\,dt+{\cal A}_{4}(\theta)\,d\phi.$ (84)

It adds one more non-zero component to the field strength:
$F_{rt}=-F_{tr}={\cal A}^{\prime}_{0}$. The Maxwell equations for this,
$r$-dependent, type of components become a little bit more complicated in our
metric. More precisely, the magnetic equation remains the same, while the
electric one takes the form of

$0=\frac{1}{\sqrt{-g}}\partial_{\mu}\left(\sqrt{-g}F^{\mu
t}\right)\propto\partial_{r}\left(\sqrt{W}r^{2}\cdot
F^{rt}\right)=-\partial_{r}\left(\sqrt{W}r^{2}\cdot\frac{1}{W}\cdot
F_{rt}\right)\propto\left(\frac{r^{2}}{\sqrt{W}}{\cal
A}^{\prime}_{0}\right)^{\prime}$

which means $F_{rt}={\cal A}^{\prime}_{0}=-q\frac{\sqrt{W(r)}}{r^{2}}$, where
the arbitrary constant is chosen as $-q$ so as to correspond to the usual
definition of charge with ${\cal A}_{0}=\frac{q}{r}$ in case of $W=1$.

We see that in case of non-constant $W(r)$, the electric potential behaves not
in the standard way which is familiar from the monopole law of
$\propto\frac{1}{r}$. This change is dictated by the Maxwell equations in our
metric. This important feature was fully ignored in the paper [49], therefore
producing wrong solutions.

We see that, superficially, electric part behaves differently. However, it is
very easy to see that the energy-momentum tensor acquires basically the very
same form as before:

$\kappa{\cal
T}^{\nu}_{\mu}=\mathrm{diag}\left(-\frac{p^{2}+q^{2}}{r^{4}},\,-\frac{p^{2}+q^{2}}{r^{4}},\,\frac{p^{2}+q^{2}}{r^{4}},\,\frac{p^{2}+q^{2}}{r^{4}}\right),$

which shows the electromagnetic duality in its full power. We present the
dyonic solutions below. But it must be clear by now that everything can be
obtained from the magnetic case by substituting
$p\longrightarrow\sqrt{p^{2}+q^{2}}$, and the same way as the sign of $p$ had
no relation to the actual sign of the magnetic charge but rather represented a
bifurcation of possible solutions, below one also has to bare in mind that
this square root can be defined with either plus or minus sign.

###  7.1 GR/TEGR

For GR/TEGR, $f(T)=T_{0}+T$, using the ansatz (71) for the tetrad and the
gauge potential (84) one can show that $A$, $B$, ${\cal A}_{0}$ and ${\cal
A}_{4}$ satisfying the field equations are

$A=C\cdot\left(-{T_{0}\over 6}\,r^{2}-{2m\over r}+{p^{2}+q^{2}\over
r^{2}}\right)$ (85) $W=C,\hskip 21.68121pt{\cal A}_{0}=\sqrt{C}\cdot{q\over
r},\hskip 21.68121pt{\cal A}_{4}=p\,\theta,$ (86)

where $q$ denotes the BH electric charge. Similar to Subsection 6.1 we can set
$C=1$ without any change to the solution. The torsion scalar (75) of this
solution is

$T(r)=T_{0}+\frac{2(q^{2}+p^{2})}{r^{4}}.$ (87)

We can illustrate this result by a very simple calculation again. In the
radial equation we simply equate the quantity
$\frac{1}{2}f-f_{T}T=\frac{1}{2}(T_{0}-T)$ to the matter source $\kappa{\cal
T}^{r}_{r}=2\left(F_{rt}F^{rt}-\frac{1}{4}\left(2F_{rt}F^{rt}+2F_{\theta\phi}F^{\theta\phi}\right)\right)=-\frac{q^{2}+p^{2}}{r^{4}}$
and get this answer.

###  7.2 Quadratic polynomial $f(T)$ gravity

Substituting the energy-momentum tensor to the right-hand side of the
equations with $f(T)=-\frac{1}{12\beta}+T+\beta T^{2}$, we obtain the torsion
scalar

$T=-{1\over 6\beta}\pm\sqrt{\frac{2}{3\beta}}\cdot{\sqrt{p^{2}+q^{2}}\over
r^{2}}$ (88)

and the solution in the form

$A=C\cdot\left({r^{2}\over 36\beta}-{2m\over
r}+{3\left(p^{2}+q^{2}\right)\over
2r^{2}}+{\sqrt{6\beta\left(p^{2}+q^{2}\right)^{3}}\over 3r^{4}}\right),\quad
W=C\cdot\left(1+{\sqrt{6\beta\left(p^{2}+q^{2}\right)}\over r^{2}}\right)^{2}$
(89)

where we omitted the plus-minus sign assuming an arbitrary sign in the
definition of $\sqrt{p^{2}+q^{2}}$ as was mentioned above. And, needless to
say, we can put $C=1$. In absence of magnetic and electric charges the
teleparallel torsion scalar is constant $T_{c}=2T_{0}=-\frac{1}{6\beta}$.

Regarding the EM potentials, we have ${\cal A}_{4}=p\theta$ as before. For the
electric part, we know that ${\cal
A}^{\prime}_{0}=-q\frac{\sqrt{W(r)}}{r^{2}}$. Setting $C=1$ and using the
gauge freedom to omit an arbitrary constant additive term, we have

${\mathcal{A}}_{0}=\frac{q}{r}\cdot\left(1+{\sqrt{6\beta\left(p^{2}+q^{2}\right)}\over
3r^{2}}\right).$ (90)

Let us mention again that this behaviour is different from classical
electrodynamics due to non-constant nature of $W(r)$, and it was ignored in
the paper [49] leading to wrong solutions.

Obviously in absence of the electric charge $q$ the solution reduces to the
magnetically charge solution. Also, by setting $q=p=0$, one gets a constant
torsion solution. It is worth noticing here that if we try here to generate
another solution from this dyonic one, through electric-magnetic duality
transformation we end up with the same solution. One can see that by taking
$q\rightarrow p$ and $p\rightarrow-q$, it interchanges $q$ and $p$ in the
dyonic solution. But since the solution is the same as we interchange $q$ and
$p$, no new solution is produced from this symmetry transformations. In the
following we discuss the main physical features which characterize the
obtained solution.

###  7.3 Physical features of the Black Hole

BH horizon is one of the most important features which characterizes BHs. We
restrict the discussion to the feasible case $\beta>0$. Recalling the metric
potentials (89), we see that the polynomial $A(r)r^{4}$ has only one term with
a negative coefficient. Therefore, by the virtue of Descartes’ sign rule, it
can have at most two positive real roots. Then the solution can produce at
most two horizons where $A(r)=0$, see Fig. 10(a), the inner, $r_{in}$, and the
outer, $r_{out}$, horizons. Note that the function $W(r)$ does not produce new
horizons since it is always non-zero positive function.

Note however that, for the regions of $A(r)<0$, the tetrad fields (71) become
complex, though the measurable quantities (metric) and the action stay real.
This happens as well for many tetrad solutions with an event horizon, and is
unavoidable in our approach. Indeed, we use coordinates in which some
components of the diagonal metric flip their signs, and we assume a tetrad
which is diagonal in these coordinates. For example, in TEGR or in GR with
tetrad fields, one can construct Schwarzschild solution where the tetrad is
complex inside the horizon. Since the tetrad is a dynamical variable, in the
teleparallel framework, it still can be viewed as a problem from the
foundational point of view. The nice point is that our tetrad is always real
outside the horizon, i.e. in the potentially observable region. The properties
of the horizon and beyond should become a subject of a separate investigation.
Even though the metric geometry is perfectly smooth across the horizon, the
same can hardly be said about the tetradic structure, for it has a square root
of a function whose value passes through zero, and some scalar quantities such
as $\accentset{\circ}{\nabla}_{\mu}e^{\mu}_{0}$ appear to be singular there.
One more interesting point to mention is that the region in between the
horizons can also be done with a real tetrad by changing the Minkowski metric
there from $\mathrm{diag}(-1,+1,+1,+1)$ to $\mathrm{diag}(+1,-1,+1,+1)$.

When the two roots of $A(r)=0$ merge, this case of the extremal BH is
characterized by $A=0$ and $A^{\prime}=0$. We find that those two horizons are
possible as long as the BH mass exceeds a minimal value

$m_{\text{min}}=\frac{7}{36}\sqrt[4]{54(q^{2}+p^{2})^{3}\over\beta}.$ (91)

(a) Possible horizons

(b) Horizon mass-radius relation

Figure 1: Schematic plots of possible horizons: 0(a) Typical behaviour of the
function $A(r)$ given by (89). 0(b) The horizon mass-radius relation (93) with
the minimum mass (91) at which the two horizons coincide at the degenerate
horizon (92), where $m>m_{\text{min}}$ the BH has two horizons ($r_{in}$ and
$r_{out}$) at most, while for $m<m_{min}$ case the BH is naked. We take
$\beta=1/4$ and $q=p=1$.

This defines the minimum horizon mass-radius relation. At $m=m_{\text{min}}$,
the two horizons coincide forming a degenerate horizon
($r_{in}=r_{out}=r_{dg}$), where

$r_{dg}=\sqrt[4]{24\beta(q^{2}+p^{2})}.$ (92)

In this case the solution is called extremal solution.

For the case $m<m_{\text{min}}$ the solution possesses a naked singularity,
i.e., no BH solution. By setting $A(r_{h})=0$, one gets a relation between the
mass parameter, $m$ and the horizon radius $r_{h}$

$m_{h}\equiv m(r_{h})={r_{h}^{3}\over 72\beta}+{3(q^{2}+p^{2})\over
4r_{h}}+{[6\beta(q^{2}+p^{2})]^{3/2}\over 36\beta r_{h}^{3}}.$ (93)

This relation is depicted by Fig. 10(b) for certain values of the parameters.
Notice that another way of obtaining the degenerate horizon $r_{h}=r_{dg}$ is
to solve $dm_{h}/dr_{h}=0$ for $r_{h}$, which gives the same result as in
(92). Then, by substituting (92) into (93), the extremal mass (91) is re-
obtained.

####  7.3.1 Singularities

One of the interesting features of the magnetic and dyonic solutions presented
here is that they have milder singularities compared to that of constant
torsion solutions. Calculating the leading term of the torsion scalar one
obtains

$T={\sqrt{6\beta(q^{2}+p^{2})}\over 3\beta r^{2}}+O(r^{0}),$ (94)

which has a milder singularity as $r\rightarrow 0$ compared to the GR/TEGR
magnetic solution presented above. Furthermore, Ricci scalar, Ricci tensor
square and Kretschmann scalar have milder behavior near $r=0$ too,

$\displaystyle R=-{\sqrt{6(q^{2}+p^{2})/\beta}\over
3r^{2}}+O(r^{0}),\,\,\,\,R^{\mu\nu}R_{\mu\nu}={7(q^{2}+p^{2})\over 9\beta
r^{4}}+O(r^{-2}),$ $\displaystyle
R^{\mu\nu\alpha\beta}R_{\mu\nu\alpha\beta}={10(q^{2}+p^{2})\over 3\beta
r^{4}}+O(r^{-2}),$ (95)

where we allow ourselves the freedom of not putting circles over the Levi-
Civita quantities. To compare with the GR/TEGR solutions, let us list the
above invariants as $r\rightarrow 0$

$\displaystyle T={2(q^{2}+p^{2})\over r^{4}}+O(r^{0}),\hskip
72.26999ptR=2\,T_{0},$ $\displaystyle
R^{\mu\nu}R_{\mu\nu}={4(q^{2}+p^{2})^{2}\over
r^{8}}+O(r^{0}),\,\,\,\,R^{\mu\nu\alpha\beta}R_{\mu\nu\alpha\beta}={56(q^{2}+p^{2})^{2}\over
r^{8}}+O(r^{-6}).$ (96)

Notice that Ricci scalar is regular in the constant-torsion cases but diverges
for our solution. This shows that singularities within the quadratic
polynomial teleparallel gravity could be milder than the corresponding GR
solution. But one might ask, can we extend a test particle trajectory beyond
the singular point? Is it going to be less singular than that of the GR/TEGR
case? Let us check a simple trajectory, namely, a null-radial trajectory for
this case. It has the following equation

${dr\over dt}=\pm{A(r)\over\sqrt{W(r)}}.$ (97)

In the local region around $r=0$ we have

${dr\over dt}=\pm{(q^{2}+p^{2})\over 3r^{2}}+O(r^{0}).$ (98)

These trajectories can not be extended beyond the singular point at $r=0$,
since the right hand side is neither continues nor differentiable at $r=0$.
Therefore, the local solution around $r=0$, is neither guaranteed to exist nor
it is unique. Our simple calculation shows that in spite of the milder form of
curvature and torsion invariants the null-radial trajectories are still in-
extendable beyond the singular point as in the GR/TEGR case.

####  7.3.2 Conserved Quantities

Defining a conserved charge in GR, TEGR or their extensions is always a
nontrivial task, especially if we want to link it with the symmetries of the
underlying field theory. The authors in [50] used a Hamiltonian analysis to
find the energy and momentum of any stationary gravitational solution in TEGR
gravity. Later this expression was extended to $f(T)$ gravity by the authors
in [51]. Asymptotically, these charges produce the conserved 4-momentum in
Minkowski space. This 4-vector, $P^{a}\equiv(E,\mathbf{P})$ in $f(T)$ gravity
[51] is given by

$P^{a}=\frac{1}{8\pi}\int_{V}d^{3}x\,\partial_{\nu}\Pi^{a\nu}=\frac{1}{8\pi}\oint_{\Sigma}d\Sigma_{\nu}\,\Pi^{a\nu},$
(99)

where $\Pi^{a\nu}=eS^{a0\nu}f_{T}(T)$. The above expression represents the
total energy-momentum contained in a three-dimensional volume of space $V$
bounded by the surface $\Sigma$. In order to get the temporal component
$P^{0}$, we evaluate

$S^{(0)(0)r}\equiv e{{}^{0}}{{}_{\mu}}e{{}^{0}}{{}_{\nu}}S^{\mu\nu r}=-{A\over
rW}.$ (100)

Then we use the regularized expression of the energy–momentum (99), which
guarantees its vanishing for the flat spacetime

$P^{a}_{reg}=\frac{1}{8\pi}\oint_{\Sigma}d\Sigma_{\nu}\,[\Pi^{a\nu}-\Pi^{a\nu}_{dS}],$
(101)

where $\Pi^{a\nu}_{dS}$ denotes the contribution of pure de Sitter spacetime
by requiring the physical parameters ($m,\,q,\,p$) to vanish. Substituting
(100) into (101), for the solution set (89), we get

$E_{reg}={m\sigma\over 6\pi}-{(q^{2}+p^{2})\sigma\over 8\pi
r}-{\sqrt{6\beta}(q^{2}+p^{2})^{3/2}\,\sigma\over 36\pi
r^{3}}+O\left(\frac{1}{r^{5}}\right).$ (102)

By taking the relation $m=\tilde{G}M$ where $M$ is the BH physical mass,
$\tilde{G}=G/f_{T}(T_{c})$; at spatial infinity $r\to\infty$ one obtains the
total energy $E={M\sigma\over 4\pi}$ as measured by a stationary observer at
infinity. Here $\sigma$ is the area of the flat surface, or $\sigma=\int
d\theta\int d\phi$. This mass or total energy is the same as the one obtained
for the flat horizon BH in GR/TEGR case.

##  8 Conclusions

BH solutions represent an intricate topic in $f(T)$ gravity theories. The
standardly used spherical coordinates do not allow for a diagonal tetrad
solution (in pure tetrad gauge) with non-constant torsion scalar, and the non-
diagonal ”good” tetrads lead to very complicated equations. It prevents many
researchers from finding new non-trivial exact BH solutions in this framework.

In this work, we give a detailed review of the issue of diagonal tetrads for
the spherically symmetric class of spacetimes, with an overview of the
constant $T$ options. Most importantly, we have shown that in the case of flat
horizons one can use Cartesian coordinates on the horizon which allow for a
simple diagonal tetrad which solves the equations of motion, without requiring
a constant torsion scalar. Furthermore, we have generalised this interesting
result by proving the following statement: if all the components of a diagonal
tetrad depend on only one of the coordinates in which it is written, then this
tetrad automatically solves the antisymmetric part of equations. Based on
that, we presented non-trivial exact topological BH solutions of magnetic and
dyonic type. Usually the problem with simple tetrad choices in $f(T)$ is in
the anti-symmetric equations not being satisfied. However, for BHs with flat
horizon, these equations are satisfied if we used Cartesian coordinates on the
horizon because of the above Theorem. It is the $rr$ component what makes the
equations demand constant $T$. And very interestingly, the addition of matter
leads to the possibility of non-constant $T$ solutions.

By extending our investigation to the time-dependent cases, we find that
diagonal tetrads with spherical and hyperbolic symmetries lead to trivial
solutions with constant torsion scalar just as the static case. Interestingly,
for the time dependent tetrad with a planar symmetry (or a flat horizon), the
anti-symmetric part of the field equations becomes non-trivial and generically
does not lead to a constant torsion scalar when Cartesian coordinates on the
horizon are used. In this case, the presence of matter is required for
existence of non-trivial consistent solutions. We leave this case to future
work.

The new solutions presented here are characterised by mass, electric and
magnetic charges as well as their flat horizons which could be a torus or a
cylinder, depending on the global identifications. These solutions possess two
horizons, inner and outer ones, that coincide in the extremal case, i.e.,
$m=m_{\text{min}}$. Below the extremal mass the solutions form a naked
singularity. These solutions possess milder singularities at $r=0$ (compared
with GR solutions) as it is clear form the behavior of scalar invariants,
apart from Ricci scalar which is divergent in our case. It would be
interesting to study, in more details, the physical features of these
solutions, especially, their geometries and causal structures, as well as the
null-like and time-like trajectories in such space-times. It would be
interesting to find more of these exact solutions since they are important
Laboratories to study the $f(T)$ theories. We hope that we will be able to
report on this in our coming works.

## Acknowledgments

MJG has been supported by the European Regional Development Fund CoE program
TK133 “The Dark Side of the Universe” and by the Estonian Research Council
grant MOBJD622.

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