hep-th/yymmnnn

Black hole solutions in double field theory

Shuxuan Ying

Department of Physics

Chongqing University

Chongqing, 401331, China

<EMAIL_ADDRESS>
Abstract

In this paper, we study black hole solutions in double field theory. In the
first part, we introduce a solution-generating method and classify black hole
solutions into three categories in standard double field theory. To solve the
problem of double time, we utilize space/time split double field theory in the
second part to derive black hole solutions. By introducing a cosmological
constant and imposing the strong constraint, our findings indicate that when
considering the entire doubled spacetime, the curvature signifies an AdS
vacuum. However, in the subspace of the entire spacetime, black holes and
curvature singularities emerge.

###### Contents

  1. 1 Introduction
  2. 2 Standard double field theory
    1. 2.1 Equations of motion for the generalized metric and dilaton
    2. 2.2 Some notations and definitions
    3. 2.3 Classification of black hole solutions
  3. 3 Space/time split double field theory
    1. 3.1 Black hole solutions
  4. 4 Conclusion and discussion

##  1 Introduction

Double field theory (DFT) is a recent progress in closed string theory. This
approach doubles all spacetime coordinates and elevates T-duality to an
intrinsic symmetry of the action [1, 2, 3, 4, 5]. In closed string theory,
when a string moves in compactified spacetime, the momentum of its center of
mass can be quantized and is conjugate to the coordinates $x^{i}$.
Additionally, if the string wraps around the compactified background multiple
times, it introduces another quantum number, the winding number. Consequently,
the physical observables of the string depend on both the momentum and the
winding number. T-duality, which exchanges momentum and winding numbers
between two strings wrapped on different compactified radii, requires treating
these two quantum numbers equally within a field theory. Thus, it is
reasonable to introduce extra coordinates $\tilde{x}_{i}$ conjugate to the
winding number. DFT suggests that the usual spacetime coordinates
$x^{i}\left(i=1,2,\cdots,D\right)$ can be doubled to
$X^{M}=\left(\tilde{x}_{\mu},x^{\mu}\right)$, where $M=1,2,\cdots,2D$.
Consequently, all spacetime component fields, including the gravitational
field $g_{\mu\nu}$, the dilaton $\phi$, and the anti-symmetric Kalb-Ramond
field $b_{\mu\nu}$, depend on $X^{M}$. In this framework, $g_{\mu\nu}$ and
$b_{\mu\nu}$ combine into a generalized metric that transforms linearly under
continuous $O\left(D,D,\mathbb{R}\right)$ transformations. The generalized
coordinate transformations are also well-defined on this $2D$ manifold. Based
on these features, the DFT action can be constructed to explicitly exhibit
$O\left(D,D,\mathbb{R}\right)$ symmetry. Moreover, DFT must satisfy the strong
constraint, meaning the theory lives in a $D$-dimensional subspace of the $2D$
manifold. This constraint, derived from the level-matching condition of closed
string theory, requires that all fields depend on only one set of the double
coordinates $\left(\tilde{x},x\right)$.

In previous works [6, 7, 8, 9, 10, 11, 12], researchers have explored the
cosmological solutions of DFT. These studies have provided a consistent method
to group T-dual cosmological solutions from standard string cosmology [13, 14,
15]. These solutions unify pre- and post-big bang evolutions in a single line
element. Additionally, recent investigations have examined specific black hole
solutions under certain constraints [16, 17, 18, 19]. However, two key issues
remain unresolved:

  1. 1.

The presence of two time coordinates in the generalized metric lacks a
reasonable physical explanation. Directly imposing the strong constraint
eliminates the double coordinates of the solutions, reducing them to the
standard string theory solutions.

  2. 2.

The solutions appear to unify previous T-dual solutions without introducing
new physics.

Solving these two problems could make studying black hole solutions in DFT
highly valuable. To tackle the first issue, we can utilize recent advances in
the canonical formulation of DFT [20, 21, 22]. In this formulation, the ADM
(Arnowitt-Deser-Misner)-like decomposition splits the $2D$ manifold into
temporal and spatial parts $X^{M}=\left(\tilde{t},t,X^{\hat{M}}\right)$,
requiring all spacetime component fields to be independent of the dual time
coordinate $\tilde{t}$. Consequently, only the time coordinate remains
undoubled, reducing the $2D$-dimensional DFT to $2D-1$ dimensions. We refer to
this approach as space/time split DFT for the remainder of this paper. To
illustrate how this decomposition can generate new solutions, we must clarify
the differences among three types of $O\left(D,D,\mathbb{R}\right)$ invariant
theories as follows:

  * •

Low-energy effective action with $O\left(m,m,\mathbb{R}\right)$ duality: In
this action, spacetime coordinates are not doubled. The
$O\left(m,m,\mathbb{R}\right)$ dduality imposes significant constraints on the
form of the spacetime metric. Specifically, when the background metric is
independent of $m$ coordinates, the action exhibits
$O\left(m,m,\mathbb{R}\right)$ symmetry. For example, FLRW backgrounds display
$O\left(D-1,D-1,\mathbb{R}\right)$ symmetry, also known as "scale-factor
duality," where the action remains invariant under the scale factor
transformation $a\left(t\right)\rightarrow\frac{1}{a\left(t\right)}$. Recent
studies have explored the classification of all orders of $\alpha^{\prime}$
corrections based on this duality [23, 24, 25]. Non-perturbative and non-
singular solutions can be found in references[26, 27, 28, 29, 30, 31, 32, 33].
Solutions involving matter sources are discussed in references [34, 35, 36,
37].

  * •

DFT with $O\left(D,D,\mathbb{R}\right)$ symmetry: In this formulation, all
spacetime coordinates are doubled, and massless closed string fields depend on
these double coordinates without needing to be independent of $m$ coordinates.
The level-matching condition of closed strings imposes the strong constraint,
requiring the theory to live on a $D-1$ subspace of the doubled spacetime.
Vacuum cosmological solutions of DFT have been provided in [6, 7, 8] and later
extended to include matter sources [9, 10, 11, 12]. Additionally, interesting
solutions such as pp-waves [38], monopoles [39], and a summary of these
solutions can be found in [40].

  * •

Space/time split DFT with $O\left(D-1,D-1,\mathbb{R}\right)$ symmetry: In this
version, only the spatial coordinates are doubled, and fields are not required
to be independent of $m$ coordinates. Since a metric component in the time
direction is absent, solutions benefit from the double coordinates while also
extending beyond simple combinations. This approach holds potential for
discovering new solutions. There have been some efforts to study black hole
thermodynamics in space/time split DFT [41] and cosmology [22, 42].

Moreover, it is important to note that compact spacetime is not required from
the perspective of field theory. For example, in the low-energy effective
action with $O\left(D,D,\mathbb{R}\right)$ duality, the spacetime coordinates
are not doubled. If the background is compactified, even in field theory, the
continuous $O\left(D,D,\mathbb{R}\right)$ symmetry becomes a discrete
$O\left(D,D,\mathbb{Z}\right)$ symmetry. Throughout the rest of this paper, we
will abbreviate $O\left(D,D,\mathbb{R}\right)$ as $O\left(D,D\right)$.

The aim of this paper is to calculate the black hole solutions of DFT. To
preserve more non-trivial properties from $O\left(D,D\right)$ symmetry, we
adhere to the following calculation principles: 1) Since the strong constraint
erases all features of DFT, we initially calculate solutions that violate this
constraint. Although these solutions currently lack clear physical
interpretations, we hope future studies will solve this issue. 2) When
imposing the strong constraint to obtain physical solutions, we assume the
generalized metric $\mathcal{H}_{MN}$ represents the entire spacetime metric,
not just the usual metric $g_{ij}$. Based on these principles, we first
calculate the black hole solutions of standard DFT, which includes double time
coordinates. These solutions can be derived using the following three methods.
Let us recall the traditional low-energy effective action with a vanishing
Kalb-Ramond field for simplicity:

$S=\int d^{4}x\sqrt{-g}e^{-2\phi}\left[R+4\left(\nabla\phi\right)^{2}\right].$
(1.1)

where $g_{ij}$denotes the string metric and $\phi$ is the dilaton. The black
hole solutions of DFT can be constructed through the solutions of this action
(1.1):

1st kind of solutions: If $g_{ij}\left(x\right)$ and $\phi\left(x\right)$ are
solutions of the effective action (1.1), the solutions of DFT that satisfy the
strong constraint can be given as:

$ds^{2}=g^{ij}\left(x\right)d\tilde{x}_{i}d\tilde{x}_{j}+g_{ij}\left(x\right)dx^{i}dx^{j},\qquad
or\qquad
ds^{2}=\left(g^{-1}\right)^{ij}\left(\tilde{x}\right)d\tilde{x}_{i}d\tilde{x}_{j}+\left(g^{-1}\right)_{ij}\left(\tilde{x}\right)dx^{i}dx^{j}.$
(1.2)

2nd kind of solutions: If $g_{ij}\left(x\right)$ and
$\left(g^{-1}\right)_{ij}\left(x\right)$ are self-dual solutions of the
effective action (1.1), the constraint-satisfying solutions of DFT can be
given as:

$ds^{2}=g^{ij}\left(x\right)d\tilde{x}_{i}d\tilde{x}_{j}+g_{ij}\left(x\right)dx^{i}dx^{j},\qquad
or\qquad
ds^{2}=\left(g^{-1}\right)^{ij}\left(x\right)d\tilde{x}_{i}d\tilde{x}_{j}+\left(g^{-1}\right)_{ij}\left(x\right)dx^{i}dx^{j},$
(1.3)

or

$ds^{2}=g^{ij}\left(\tilde{x}\right)d\tilde{x}_{i}d\tilde{x}_{j}+g_{ij}\left(\tilde{x}\right)dx^{i}dx^{j},\qquad
or\qquad
ds^{2}=\left(g^{-1}\right)^{ij}\left(\tilde{x}\right)d\tilde{x}_{i}d\tilde{x}_{j}+\left(g^{-1}\right)_{ij}\left(\tilde{x}\right)dx^{i}dx^{j}.$
(1.4)

3rd kind of solutions: If $g_{ij}\left(x\right)$ and
$\left(g^{-1}\right)_{ij}\left(x\right)$ are self-dual solutions of the
effective action (1.1), we can construct constraint-violating solutions of DFT
such that
$g\left(\tilde{x},x\right)=g\left(\tilde{x}\right)\times\left(g^{-1}\right)\left(x\right)$.
These solutions depend on double coordinates simultaneously and take the
forms:

$ds^{2}=g^{ij}\left(x,\tilde{x}\right)d\tilde{x}_{i}d\tilde{x}_{j}+g_{ij}\left(x,\tilde{x}\right)dx^{i}dx^{j},\qquad
or\qquad
ds^{2}=\left(g^{-1}\right)^{ij}\left(x,\tilde{x}\right)d\tilde{x}_{i}d\tilde{x}_{j}+\left(g^{-1}\right)_{ij}\left(x,\tilde{x}\right)dx^{i}dx^{j}.$
(1.5)

The problem with these solutions is the presence of double timelike directions
$dt$ and $d\tilde{t}$, which lack explicit physical interpretations. To solve
this issue, we consider the $2D-1$ dimensional space/time split DFT, which is
independent of the dual time coordinate. To obtain non-trivial solutions, we
consider non-critical strings, which introduce the cosmological constant. We
then calculate the equations of motion (EOM) for the corresponding action and
use the generated methods mentioned above to derive the black hole solutions.
The new insights of these solutions arise from the curvature definition in
doubled spacetime. In DFT, the Ricci curvature is generalized to include the
doubled coordinates and dilatonic dynamics, known as the generalized Ricci
scalar $\mathcal{R}$. It satisfies the EOM for the dilaton and always vanishes
($\mathcal{R}=0$). Since we consider the cosmological constant, the
generalized Ricci scalar for the whole doubled spacetime is:

$\mathcal{R}=-\lambda^{2}.$ (1.6)

This implies that, regardless of the obtained solution, the entire doubled
spacetime is an AdS vacuum. On the other hand, since the geometry of the
subspace $x$ of the doubled spacetime can be completely determined by the low-
energy effective action, we can construct the black hole solution in this
subspace. In other words, considering the entire spacetime results in an AdS
vacuum, whereas black holes and curvature singularities emerge in the
subspace.

The remainder of this paper is outlined as follows. In Section 2, we review
the dilatonic and gravitational EOM of standard DFT, classify the solutions
into three kinds, and present the main examples. In Section 3, we introduce
the space/time split DFT and calculate the black hole solutions of its EOM.
Section 4 contains our discussion and conclusion.

##  2 Standard double field theory

###  2.1 Equations of motion for the generalized metric and dilaton

The EOM for the dilatonic and gravitational fields in DFT have been
extensively reviewed in the reference [4]. Here, we briefly summarize these
equations. To construct the spacetime action of DFT, we introduce the
$2D\times 2D$ “generalized metric,” which incorporates the gravitational field
$g_{ij}$ and the antisymmetric Kalb-Ramond field $b_{ij}$:

$\mathcal{H}_{MN}=\left(\begin{array}[]{cc}g^{ij}&-g^{ik}b_{kj}\\\
b_{ik}g^{kj}&g_{ij}-b_{ik}g^{kl}b_{lj}\end{array}\right),\quad\mathcal{H}^{MN}=\left(\begin{array}[]{cc}g_{ij}-b_{ik}g^{kl}b_{lj}&b_{ik}g^{kj}\\\
-g^{ik}b_{kj}&g^{ij}\end{array}\right),$ (2.7)

where $\mathcal{H}^{MP}\mathcal{H}_{PN}=\delta_{\;N}^{M}$,
$\mathcal{H}^{MN}\equiv\eta^{MP}\mathcal{H}_{PQ}\eta^{QN}=\mathcal{H}_{PQ}\eta^{PM}\eta^{QN}$,
and $\mathcal{H}^{MN}=\mathcal{H}^{NM}$ ($M,N,\ldots=1,2,\ldots,2D$). The
indices can be raised and lowered by the $O\left(D,D\right)$-invariant metric:

$\eta_{MN}=\left(\begin{array}[]{cc}0&I\\\
I&0\end{array}\right),\qquad\eta^{MN}=\eta_{MN}.$ (2.8)

The gauge-invariant DFT action can thus be constructed from
$O\left(D,D\right)$ scalars:

$\displaystyle S$ $\displaystyle=$ $\displaystyle\int
d^{D}xd^{D}\tilde{x}\mathcal{L}_{\mathrm{DFT}}\left(d,\mathcal{H}_{MN}\right)$
(2.9) $\displaystyle=$ $\displaystyle\int
d^{D}xd^{D}\tilde{x}e^{-2d}\left(\frac{1}{8}\mathcal{H}^{MN}\partial_{M}\mathcal{H}^{KL}\partial_{N}\mathcal{H}_{KL}-\frac{1}{2}\mathcal{H}^{MN}\partial_{N}\mathcal{H}^{KL}\partial_{L}\mathcal{H}_{MK}\right.$
$\displaystyle\left.-2\partial_{M}d\partial_{N}\mathcal{H}^{MN}+4\mathcal{H}^{MN}\partial_{M}d\partial_{N}d\right),$

where the derivatives are
$\partial_{M}=\left(\tilde{\partial}^{i},\partial_{i}\right)$ and the
$O\left(D,D\right)$-invariant dilaton $d$ is defined in terms of the physical
dilaton $\phi$ as:

$e^{-2d}=\sqrt{g}e^{-2\phi}.$ (2.10)

Since each term in the action is a scalar, the $O\left(D,D\right)$ symmetry is
manifest. Moreover, the action (2.9) can also be rewritten in terms of the
generalized scalar curvature $\mathcal{R}\left(d,\mathcal{H}_{MN}\right)$ with
some additional boundary terms:

$\mathcal{L}_{\mathrm{DFT}}\left(d,\mathcal{H}_{MN}\right)=e^{-2d}\mathcal{R}\left(d,\mathcal{H}_{MN}\right)+\partial_{M}\left(e^{-2d}\left[\partial_{N}\mathcal{H}^{MN}-4\mathcal{H}^{MN}\partial_{N}d\right]\right),$
(2.11)

where the generalized curvature scalar is defined as:

$\displaystyle\mathcal{R}\left(d,\mathcal{H}_{MN}\right)$ $\displaystyle=$
$\displaystyle\frac{1}{8}\mathcal{H}^{MN}\partial_{M}\mathcal{H}^{KL}\partial_{N}\mathcal{H}_{KL}-\frac{1}{2}\mathcal{H}^{MN}\partial_{N}\mathcal{H}^{KL}\partial_{L}\mathcal{H}_{MK}+4\mathcal{H}^{MN}\partial_{M}\partial_{N}d$
(2.12)
$\displaystyle+4\partial_{M}\mathcal{H}^{MN}\partial_{N}d-4\mathcal{H}^{MN}\partial_{M}d\partial_{N}d-\partial_{M}\partial_{N}\mathcal{H}^{MN}.$

Additionally, all fields must satisfy the strong constraint
$\eta^{MN}\partial_{M}\partial_{N}f=0$ (or equivalently
$\partial^{M}f\partial_{M}g=0$)due to the level-matching condition of closed
string theory. This constraint requires all fields to live only on a subspace
of the doubled coordinates. The gauge transformations of the DFT action (2.9)
are given by:

$\displaystyle\delta_{\xi}\mathcal{H}^{MN}$ $\displaystyle=$
$\displaystyle\hat{\mathcal{L}}_{\xi}\mathcal{H}^{MN}\equiv\xi^{P}\partial_{P}\mathcal{H}^{MN}+\left(\partial^{M}\xi_{P}-\partial_{P}\xi^{M}\right)\mathcal{H}^{PN}+\left(\partial^{N}\xi_{P}-\partial_{P}\xi^{N}\right)\mathcal{H}^{MP},$
$\displaystyle\delta d$ $\displaystyle=$
$\displaystyle\xi^{M}\partial_{M}d-\frac{1}{2}\partial_{M}\xi^{M},$ (2.13)

with $\xi^{M}=\left(\hat{\xi}_{i},\xi^{i}\right)$ and “generalized Lie
derivatives” $\hat{\mathcal{L}}_{\xi}$. The dilatonic EOM can be derived from
the variation of the DFT action: (2.9):

$\displaystyle\delta_{d}S$ $\displaystyle=$ $\displaystyle\int
dxd\tilde{x}e^{-2d}\left(-\frac{1}{4}\mathcal{H}^{MN}\partial_{M}\mathcal{H}^{KL}\partial_{N}\mathcal{H}_{KL}+\mathcal{H}^{MN}\partial_{N}\mathcal{H}^{KL}\partial_{L}\mathcal{H}_{MK}\right.$
(2.14)
$\displaystyle\left.+2\partial_{M}\partial_{N}\mathcal{H}^{MN}+8\mathcal{H}^{MN}\partial_{M}d\partial_{N}d-8\partial_{M}\mathcal{H}^{MN}\partial_{N}d-8\mathcal{H}^{MN}\partial_{M}\partial_{N}d\right)\delta
d.$

This gives:

$\displaystyle\frac{1}{8}\mathcal{H}^{MN}\partial_{M}\mathcal{H}^{KL}\partial_{N}\mathcal{H}_{KL}-\frac{1}{2}\mathcal{H}^{MN}\partial_{M}\mathcal{H}^{KL}\partial_{K}\mathcal{H}_{NL}$
$\displaystyle-\partial_{M}\partial_{N}\mathcal{H}^{MN}-4\mathcal{H}^{MN}\partial_{M}d\partial_{N}d+4\partial_{M}\mathcal{H}^{MN}\partial_{N}d+4\mathcal{H}^{MN}\partial_{M}\partial_{N}d=0.$
(2.15)

To obtain the gravitational field equation for the generalized metric
$\mathcal{H}$, we consider the variation of $\mathcal{H}^{MN}$:

$\delta_{\mathcal{H}}S=\int
dxd\tilde{x}e^{-2d}\delta\mathcal{H}^{MN}\mathcal{K}_{MN},$ (2.16)

where

$\displaystyle\mathcal{K}_{MN}$ $\displaystyle\equiv$
$\displaystyle\frac{1}{8}\partial_{M}\mathcal{H}^{KL}\partial_{N}\mathcal{H}_{KL}-\frac{1}{4}\left(\partial_{L}-2\partial_{L}d\right)\left(\mathcal{H}^{LK}\partial_{K}\mathcal{H}_{MN}\right)+2\partial_{M}\partial_{N}d$
(2.17)
$\displaystyle-\frac{1}{2}\partial_{\left(N\right.}\mathcal{H}^{KL}\partial_{L}\mathcal{H}_{\left.M\right)K}+\frac{1}{2}\left(\partial_{L}-2\partial_{L}d\right)\left(\mathcal{H}^{KL}\partial_{\left(N\right.}\mathcal{H}_{\left.M\right)K}+\mathcal{H}_{\quad\left(M\right.}^{K}\partial_{K}\mathcal{H}_{\quad\left.N\right)}^{L}\right).$

However, $\mathcal{K}_{MN}=0$ is not the gravitational field equation, as this
equation is not an $O\left(D,D\right)$-invariant field equation. To obtain the
$O\left(D,D\right)$-invariant EOM, we first utilize the notations
$\mathcal{H}\equiv\mathcal{H}^{\bullet\bullet}$ and
$\eta\equiv\mathcal{\eta}_{\bullet\bullet}$. Since the $O\left(D,D\right)$
symmetry requires $\mathcal{H}$ to satisfy
$\mathcal{H}\eta\mathcal{H}=\eta^{-1}$, its varied field
$\mathcal{H}^{\prime}=\mathcal{H}+\delta\mathcal{H}$ also needs to satisfy the
symmetric condition $\mathcal{H}^{\prime}\eta\mathcal{H}^{\prime}=\eta^{-1}$.
Then we have:

$\delta\mathcal{H}\eta\mathcal{H}+\mathcal{H}\eta\delta\mathcal{H}=0.$ (2.18)

To simplify our calculations, we define the following:

$S_{\quad N}^{M}\equiv\mathcal{H}_{\quad
N}^{M}=\eta^{MP}\mathcal{H}_{PN}=\mathcal{H}^{MP}\eta_{PN},\quad S^{2}=1.$
(2.19)

Using the notation $S\equiv S_{\quad\bullet}^{\bullet}=\mathcal{H}\eta$, we
derive the variation condition for

$\delta\mathcal{H}S^{t}+S\delta\mathcal{H}=0.$ (2.20)

Given that $S^{2}=1$, we can rewrite the variation as:

$\delta\mathcal{H}=-S\delta\mathcal{H}S^{t}.$ (2.21)

This can be further expressed as:

$\delta\mathcal{H}=\frac{1}{4}\left(1+S\right)\mathcal{M}\left(1-S^{t}\right)+\frac{1}{4}\left(1-S\right)\mathcal{M}\left(1+S^{t}\right),$
(2.22)

where $\mathcal{M}$ is an arbitrary symmetric matrix ensuring the symmetry of
$\delta\mathcal{H}$. Substituting this back into the variation of the action
(2.16), we get:

$\frac{1}{4}\left(1-S^{t}\right)\mathcal{K}\left(1+S\right)+\frac{1}{4}\left(1+S^{t}\right)\mathcal{K}\left(1-S\right)=0.$
(2.23)

Thus, the $O\left(D,D\right)$-invariant gravitational field equation is
obtained:

$\displaystyle\mathcal{R}_{MN}$ $\displaystyle=$
$\displaystyle\frac{1}{4}\left(\delta_{M}^{\quad P}-S_{\quad
M}^{P}\right)\mathcal{K}_{PQ}\left(\delta_{\quad N}^{Q}+S_{\quad
N}^{Q}\right)+\frac{1}{4}\left(\delta_{M}^{\quad P}+S_{\quad
M}^{P}\right)\mathcal{K}_{PQ}\left(\delta_{\quad N}^{Q}-S_{\quad
N}^{Q}\right)$ (2.24) $\displaystyle=$
$\displaystyle\frac{1}{2}\mathcal{K}_{MN}-\frac{1}{2}S_{\quad
M}^{P}\mathcal{K}_{PQ}S_{\quad N}^{Q}.$

In summary, the dilatonic and gravitational EOM of DFT are presented as
follows:

$\displaystyle\frac{1}{8}\mathcal{H}^{MN}\partial_{M}\mathcal{H}^{KL}\partial_{N}\mathcal{H}_{KL}-\frac{1}{2}\mathcal{H}^{MN}\partial_{M}\mathcal{H}^{KL}\partial_{K}\mathcal{H}_{NL}-\partial_{M}\partial_{N}\mathcal{H}^{MN}$
$\displaystyle-4\mathcal{H}^{MN}\partial_{M}d\partial_{N}d+4\partial_{M}\mathcal{H}^{MN}\partial_{N}d+4\mathcal{H}^{MN}\partial_{M}\partial_{N}d$
$\displaystyle=$ $\displaystyle 0,$ $\displaystyle\mathcal{K}_{MN}-S_{\quad
M}^{P}\mathcal{K}_{PQ}S_{\quad N}^{Q}$ $\displaystyle=$ $\displaystyle 0.$
(2.25)

###  2.2 Some notations and definitions

To calculate the EOM (2.25) in DFT, we need to define several notations and
calculation rules. First, we clarify the notations for the generalized metric.
The spatial vector in DFT is defined as:

$\partial_{M}\equiv\left(\begin{array}[]{c}\tilde{\partial}^{i}\\\
\partial_{i}\end{array}\right),$ (2.26)

where $i$ denotes the $D-1$ dimensional spatial coordinates. The corresponding
dual vector is given by:

$dX^{M}\equiv\left(\begin{array}[]{c}d\tilde{x}_{i}\\\
dx^{i}\end{array}\right).$ (2.27)

Using the dual vector, we can define the generalized line element in terms of
the generalized metric:

$ds^{2}=\mathcal{H}_{MN}dX^{M}dX^{N}.$ (2.28)

For simplicity, we use block matrix notation to rewrite the vector, the dual
vector, and the generalized metric:

$\partial_{M}=\left(\begin{array}[]{c}\partial_{\mathbf{1}}\\\
\partial_{\mathbf{2}}\end{array}\right),\quad
dX^{M}=\left(\begin{array}[]{c}dX^{\mathbf{\mathbf{1}}}\\\
dX^{\mathbf{2}}\end{array}\right),\quad\mathcal{H}_{MN}=\left(\begin{array}[]{cc}\mathcal{H}_{\mathbf{1}\mathbf{1}}&\mathcal{H}_{\mathbf{1}\mathbf{2}}\\\
\mathcal{H}_{\mathbf{2}\mathbf{1}}&\mathcal{H}_{\mathbf{2}\mathbf{2}}\end{array}\right),\quad\mathcal{H}^{MN}=\left(\begin{array}[]{cc}\mathcal{H}^{\mathbf{1}\mathbf{1}}&\mathcal{H}^{\mathbf{1}\mathbf{2}}\\\
\mathcal{H}^{\mathbf{2}\mathbf{1}}&\mathcal{H}^{\mathbf{2}\mathbf{2}}\end{array}\right),$
(2.29)

where $\mathbf{1}$ represents the set of coordinates $\tilde{x}_{i}$ and
$\mathbf{2}$ denotes the coordinates $x^{i}$.Using these notations, the
components of the generalized metric can be divided into four blocks:

$\displaystyle\mathcal{H}_{\mathbf{1}\mathbf{1}}\left(\frac{\partial}{\partial\tilde{x}_{i}},\frac{\partial}{\partial\tilde{x}_{j}}\right)=g^{ij},$
$\displaystyle\mathcal{H}_{\mathbf{1}\mathbf{2}}\left(\frac{\partial}{\partial\tilde{x}_{i}},\frac{\partial}{\partial
x^{j}}\right)=-g^{ik}b_{kj},$
$\displaystyle\mathcal{H}_{\mathbf{2}\mathbf{1}}\left(\frac{\partial}{\partial
x^{i}},\frac{\partial}{\partial\tilde{x}_{j}}\right)=b_{ik}g^{kj},$
$\displaystyle\mathcal{H}_{\mathbf{2}\mathbf{2}}\left(\frac{\partial}{\partial
x^{i}},\frac{\partial}{\partial x^{j}}\right)=g_{ij}-b_{ik}g^{kl}b_{lj}.$
(2.30)

It is worth noting that $g\left(x,\tilde{x}\right)$,
$b\left(x,\tilde{x}\right)$, and the dilaton $d\left(x,\tilde{x}\right)$
depend on two sets of coordinates. The generalized line element becomes:

$\displaystyle dS^{2}$ $\displaystyle=$
$\displaystyle\mathcal{H}_{\mathbf{1}\mathbf{1}}dX^{\mathbf{1}}dX^{\mathbf{1}}+\mathcal{H}_{\mathbf{1}\mathbf{2}}dX^{\mathbf{1}}dX^{\mathbf{2}}+\mathcal{H}_{\mathbf{2}\mathbf{1}}dX^{\mathbf{2}}dX^{\mathbf{1}}+\mathcal{H}_{\mathbf{2}\mathbf{2}}dX^{\mathbf{2}}dX^{\mathbf{2}}$
(2.31) $\displaystyle=$ $\displaystyle
g^{ij}d\tilde{x}_{i}d\tilde{x}_{j}-g^{ik}b_{kj}d\tilde{x}_{i}dx^{j}+b_{ik}g^{kj}dx^{i}d\tilde{x}_{j}+\left(g_{ij}-b_{ik}g^{kl}b_{lj}\right)dx^{i}dx^{j}.$

For simplicity in the rest of the paper, we will set the Kalb-Ramond field to
be zero, so the line element becomes:

$dS^{2}=\mathcal{H}_{\mathbf{1}\mathbf{1}}dX^{\mathbf{1}}dX^{\mathbf{1}}+\mathcal{H}_{\mathbf{2}\mathbf{2}}dX^{\mathbf{2}}dX^{\mathbf{2}}=g^{ij}d\tilde{x}_{i}d\tilde{x}_{j}+g_{ij}dx^{i}dx^{j}.$
(2.32)

Then, the blocks of the generalized metric are:

$\mathcal{H}_{\mathbf{1}\mathbf{1}}=g^{ij},\qquad\mathcal{H}^{\mathbf{1}\mathbf{1}}=g_{ij},\qquad\mathcal{H}^{\mathbf{2}\mathbf{2}}=g^{ij},\qquad\mathcal{H}_{\mathbf{\mathbf{2}}\mathbf{2}}=g_{ij}.$
(2.33)

To calculate the contraction of $O\left(D,D\right)$ indices, we introduce
extra indices $\mathcal{H}^{\mathbf{1}\left(i\right)\mathbf{1}\left(j\right)}$
to represent the contractions of the elements of the block matrix. Therefore,
the generalized metric can be rewritten by adding the extra indices:

$\mathcal{H}_{M\left(i\right)N\left(j\right)}=\left(\begin{array}[]{cc}\mathcal{H}_{\mathbf{1}\left(i\right)\mathbf{1}\left(j\right)}&0\\\
0&\mathcal{H}_{\mathbf{2}\left(i\right)\mathbf{2}\left(j\right)}\end{array}\right)=\left(\begin{array}[]{cc}g^{ij}&0\\\
0&g_{ij}\end{array}\right),$ (2.34)

where $b_{ij}=0$. Now, there exist two sets of indices: the block matrix
notations $M$, $N$ and the indices of the components of the block matrix $i$,
$j$. Based on these new notations, we give the following examples that we will
use in the calculations:

$\displaystyle\mathcal{H}^{M\left(i\right)N\left(j\right)}\partial_{M\left(i\right)}d\partial_{N\left(j\right)}d$
$\displaystyle=$
$\displaystyle\mathcal{H}^{\mathbf{1}\left(i\right)\mathbf{1}\left(j\right)}\partial_{\mathbf{1}\left(i\right)}d\partial_{\mathbf{1}\left(j\right)}d+\mathcal{H}^{\mathbf{2}\left(i\right)\mathbf{2}\left(j\right)}\partial_{\mathbf{2}\left(i\right)}d\partial_{\mathbf{2}\left(j\right)}d$
(2.35) $\displaystyle=$ $\displaystyle
g_{ij}\tilde{\partial}^{i}d\tilde{\partial}^{j}d+g^{ij}\partial_{i}d\partial_{j}d.$

Moreover, for the derivatives, we have:

$\displaystyle\partial_{\mathbf{1}\left(k\right)}\mathcal{H}^{\mathbf{2}\left(i\right)\mathbf{2}\left(j\right)}$
$\displaystyle=$ $\displaystyle\tilde{\partial}^{k}g^{ij}$
$\displaystyle\partial_{\mathbf{2}\left(k\right)}\mathcal{H}^{\mathbf{1}\left(i\right)\mathbf{1}\left(j\right)}$
$\displaystyle=$ $\displaystyle\partial_{k}g_{ij}$
$\displaystyle\partial_{\mathbf{1}\left(p\right)}\mathcal{H}^{\mathbf{1}\left(i\right)\mathbf{1}\left(j\right)}\partial_{\mathbf{2}\left(q\right)}\mathcal{H}_{\mathbf{1}\left(i\right)\mathbf{1}\left(j\right)}$
$\displaystyle=$ $\displaystyle\tilde{\partial}^{p}g_{ij}\partial_{q}g^{ij}$
(2.36)

On the other hand, the rules for $S_{\quad N}^{M}$ are

$\mathcal{H}_{\quad\mathbf{1}\left(j\right)}^{\mathbf{2}\left(i\right)}=g^{ij},\quad\mathcal{H}_{\quad\mathbf{2}\left(j\right)}^{\mathbf{1}\left(i\right)}=g_{ij}.$
(2.37)

and

$\partial_{\mathbf{2}\left(k\right)}\mathcal{H}_{\quad\mathbf{2}\left(j\right)}^{\mathbf{1}\left(i\right)}=\partial_{k}g_{ij},\qquad\partial_{\mathbf{1}\left(k\right)}\mathcal{H}_{\quad\mathbf{1}\left(j\right)}^{\mathbf{2}\left(i\right)}=\tilde{\partial}^{k}g^{ij}.$
(2.38)

Note that the raising and lowering of indices are given by:

$g_{ij}\tilde{\partial}^{i}=\tilde{\partial}_{j},\quad
g^{ij}\partial_{i}=\partial^{j}.$ (2.39)

###  2.3 Classification of black hole solutions

In this subsection, our goal is to outline the methodology for generating
typical DFT black hole solutions. These solutions are categorized into three
types, each accompanied by specific examples. However, due to the inclusion of
dual time coordinates, the physical interpretation of these solutions is not
explicitly defined.

Firstly, let us revisit the EOM for the graviton and dilaton fields:

$\displaystyle\frac{1}{8}\mathcal{H}^{MN}\partial_{M}\mathcal{H}^{KL}\partial_{N}\mathcal{H}_{KL}-\frac{1}{2}\mathcal{H}^{MN}\partial_{M}\mathcal{H}^{KL}\partial_{K}\mathcal{H}_{NL}-\partial_{M}\partial_{N}\mathcal{H}^{MN}$
$\displaystyle-4\mathcal{H}^{MN}\partial_{M}d\partial_{N}d+4\partial_{M}\mathcal{H}^{MN}\partial_{N}d+4\mathcal{H}^{MN}\partial_{M}\partial_{N}d$
$\displaystyle=$ $\displaystyle 0,$ $\displaystyle
2\mathcal{R}_{MN}=\mathcal{K}_{MN}-S_{\quad M}^{P}\mathcal{K}_{PQ}S_{\quad
N}^{Q}$ $\displaystyle=$ $\displaystyle 0.$ (2.40)

Assuming the Kalb-Ramond field is zero and substituting the generalized metric
formulation:

$\mathcal{H}_{M\left(i\right)N\left(j\right)}=\left(\begin{array}[]{cc}\mathcal{H}_{\mathbf{1}\left(i\right)\mathbf{1}\left(j\right)}&0\\\
0&\mathcal{H}_{\mathbf{2}\left(i\right)\mathbf{2}\left(j\right)}\end{array}\right)=\left(\begin{array}[]{cc}g^{ij}&0\\\
0&g_{ij}\end{array}\right),$ (2.41)

the EOM can be expanded using the notation and rules established in the
previous section. Specifically for the graviton field, we obtain:

$\displaystyle\mathcal{R}_{\mathbf{1}\left(p\right)\mathbf{1}\left(q\right)}$
$\displaystyle=$
$\displaystyle\frac{1}{8}\tilde{\partial}^{p}g_{ij}\tilde{\partial}^{q}g^{ij}+\frac{1}{8}\tilde{\partial}^{p}g^{ij}\tilde{\partial}^{q}g_{ij}-\frac{1}{4}\tilde{\partial}^{i}\left(g_{ij}\tilde{\partial}^{j}g^{pq}\right)-\frac{1}{4}\partial_{i}\left(g^{ij}\partial_{j}g^{pq}\right)$
$\displaystyle-\frac{1}{4}\tilde{\partial}^{q}g_{ij}\tilde{\partial}^{j}g^{pi}-\frac{1}{4}\tilde{\partial}^{p}g_{ij}\tilde{\partial}^{j}g^{qi}+\frac{1}{4}\tilde{\partial}^{i}\left(g_{ji}\tilde{\partial}^{q}g^{pj}\right)+\frac{1}{4}\tilde{\partial}^{i}\left(g_{ji}\tilde{\partial}^{p}g^{qj}\right)$
$\displaystyle+\frac{1}{4}\partial_{i}\left(g^{jp}\partial_{j}g^{iq}\right)+\frac{1}{4}\partial_{i}\left(g^{jq}\partial_{j}g^{ip}\right)-\frac{1}{8}g^{ip}\partial_{i}g_{kl}\partial_{j}g^{kl}g^{jq}-\frac{1}{8}g^{ip}\partial_{i}g^{kl}\partial_{j}g_{kl}g^{jq}$
$\displaystyle+\frac{1}{4}g^{ip}\tilde{\partial}^{l}\left(g_{lk}\tilde{\partial}^{k}g_{ij}\right)g^{jq}+\frac{1}{4}g^{ip}\partial_{l}\left(g^{lk}\partial_{k}g_{ij}\right)g^{jq}+\frac{1}{4}g^{ip}\partial_{j}g^{kl}\partial_{l}g_{ik}g^{jq}+\frac{1}{4}g^{ip}\partial_{i}g^{kl}\partial_{l}g_{jk}g^{jq}$
$\displaystyle-\frac{1}{4}g^{ip}\partial_{l}\left(g^{kl}\partial_{j}g_{ik}\right)g^{jq}-\frac{1}{4}g^{ip}\partial_{l}\left(g^{kl}\partial_{i}g_{jk}\right)g^{jq}-\frac{1}{4}g^{ip}\tilde{\partial}^{l}\left(g_{ki}\tilde{\partial}^{k}g_{lj}\right)g^{jq}-\frac{1}{4}g^{ip}\tilde{\partial}^{l}\left(g_{kj}\tilde{\partial}^{k}g_{li}\right)g^{jq}$
$\displaystyle+\frac{1}{2}\tilde{\partial}^{i}d\left(g_{ij}\tilde{\partial}^{j}g^{pq}\right)+\frac{1}{2}\partial_{i}d\left(g^{ij}\partial_{j}g^{pq}\right)+2\tilde{\partial}^{p}\tilde{\partial}^{q}d$
$\displaystyle-\frac{1}{2}\partial_{i}d\left(g^{jp}\partial_{j}g^{iq}\right)-\frac{1}{2}\partial_{i}d\left(g^{jq}\partial_{j}g^{ip}\right)-\frac{1}{2}\tilde{\partial}^{i}d\left(g_{ji}\tilde{\partial}^{q}g^{pj}\right)-\frac{1}{2}\tilde{\partial}^{i}d\left(g_{ji}\tilde{\partial}^{p}g^{qj}\right)$
$\displaystyle-\frac{1}{2}g^{ip}\tilde{\partial}^{l}d\left(g_{lk}\tilde{\partial}^{k}g_{ij}\right)g^{jq}-\frac{1}{2}g^{ip}\partial_{l}d\left(g^{lk}\partial_{k}g_{ij}\right)g^{jq}-2g^{ip}\partial_{i}\partial_{j}dg^{jq}$
$\displaystyle+\frac{1}{2}g^{ip}\partial_{l}d\left(g^{kl}\partial_{j}g_{ik}\right)g^{jq}+\frac{1}{2}g^{ip}\partial_{l}d\left(g^{kl}\partial_{i}g_{jk}\right)g^{jq}$
$\displaystyle+\frac{1}{2}g^{ip}\tilde{\partial}^{l}d\left(g_{ki}\tilde{\partial}^{k}g_{lj}\right)g^{jq}+\frac{1}{2}g^{ip}\tilde{\partial}^{l}d\left(g_{kj}\tilde{\partial}^{k}g_{li}\right)g^{jq}.$
$\displaystyle\mathcal{R}_{\mathbf{2}\left(p\right)\mathbf{2}\left(q\right)}$
$\displaystyle=$
$\displaystyle\frac{1}{8}\partial_{p}g_{ij}\partial_{q}g^{ij}+\frac{1}{8}\partial_{p}g^{ij}\partial_{q}g_{ij}-\frac{1}{4}\tilde{\partial}^{i}\left(g_{ij}\tilde{\partial}^{j}g_{pq}\right)-\frac{1}{4}\partial_{i}\left(g^{ij}\partial_{j}g_{pq}\right)$
$\displaystyle-\frac{1}{4}\partial_{q}g^{ij}\partial_{j}g_{pi}-\frac{1}{4}\partial_{p}g^{ij}\partial_{j}g_{qi}+\frac{1}{4}\partial_{i}\left(g^{ji}\partial_{q}g_{pj}\right)+\frac{1}{4}\partial_{i}\left(g^{ji}\partial_{p}g_{qj}\right)$
$\displaystyle+\frac{1}{4}\tilde{\partial}^{i}\left(g_{jp}\tilde{\partial}^{j}g_{iq}\right)+\frac{1}{4}\tilde{\partial}^{i}\left(g_{jq}\tilde{\partial}^{j}g_{ip}\right)-\frac{1}{8}g_{ip}\tilde{\partial}^{i}g_{kl}\tilde{\partial}^{j}g^{kl}g_{jq}-\frac{1}{8}g_{ip}\tilde{\partial}^{i}g^{kl}\tilde{\partial}^{j}g_{kl}g_{jq}$
$\displaystyle+\frac{1}{4}g_{ip}\tilde{\partial}^{l}\left(g_{lk}\tilde{\partial}^{k}g^{ij}\right)g_{jq}+\frac{1}{4}g_{ip}\partial_{l}\left(g^{lk}\partial_{k}g^{ij}\right)g_{jq}+\frac{1}{4}g_{ip}\tilde{\partial}^{j}g_{kl}\tilde{\partial}^{l}g^{ik}g_{jq}+\frac{1}{4}g_{ip}\tilde{\partial}^{i}g_{kl}\tilde{\partial}^{l}g^{jk}g_{jq}$
$\displaystyle-\frac{1}{4}g_{ip}\tilde{\partial}^{l}\left(g_{kl}\tilde{\partial}^{j}g^{ik}\right)g_{jq}-\frac{1}{4}g_{ip}\tilde{\partial}^{l}\left(g_{kl}\tilde{\partial}^{i}g\right)g_{jq}-\frac{1}{4}g_{ip}\partial_{l}\left(g^{ki}\partial_{k}g^{lj}\right)g_{jq}-\frac{1}{4}g_{ip}\partial_{l}\left(g^{kj}\partial_{k}g^{li}\right)g_{jq}$
$\displaystyle+\frac{1}{2}\tilde{\partial}^{i}d\left(g_{ij}\tilde{\partial}^{j}g_{pq}\right)+\frac{1}{2}\partial_{i}d\left(g^{ij}\partial_{j}g_{pq}\right)+2\partial_{p}\partial_{q}d$
$\displaystyle-\frac{1}{2}\tilde{\partial}^{i}d\left(g_{jp}\tilde{\partial}^{j}g_{iq}\right)-\frac{1}{2}\tilde{\partial}^{i}d\left(g_{jq}\tilde{\partial}^{j}g_{ip}\right)-\frac{1}{2}\partial_{i}d\left(g^{ji}\partial_{q}g_{pj}\right)-\frac{1}{2}\partial_{i}d\left(g^{ji}\partial_{p}g_{qj}\right)$
$\displaystyle-\frac{1}{2}g_{ip}\tilde{\partial}^{l}d\left(g_{lk}\tilde{\partial}^{k}\tilde{g}^{ij}\right)g_{jq}-\frac{1}{2}g_{ip}\partial_{l}d\left(g^{lk}\partial_{k}g^{ij}\right)g_{jq}-2g_{ip}\tilde{\partial}^{i}\tilde{\partial}^{j}dg_{jq}$
$\displaystyle+\frac{1}{2}g_{ip}\tilde{\partial}^{l}d\left(g_{kl}\tilde{\partial}^{j}g^{ik}\right)g_{jq}+\frac{1}{2}g_{ip}\tilde{\partial}^{l}d\left(g_{kl}\tilde{\partial}^{i}g^{jk}\right)g_{jq}$
$\displaystyle+\frac{1}{2}g_{ip}\partial_{l}d\left(g^{ki}\partial_{k}g^{lj}\right)g_{jq}+\frac{1}{2}g_{ip}\partial_{l}d\left(g^{kj}\partial_{k}g^{li}\right)g_{jq},$
$\displaystyle\mathcal{R}_{\mathbf{1}\left(p\right)\mathbf{2}\left(q\right)}$
$\displaystyle=$
$\displaystyle\frac{1}{8}\tilde{\partial}^{p}g_{ij}\partial_{q}g^{ij}+\frac{1}{8}\tilde{\partial}^{p}g^{ij}\partial_{q}g_{ij}-\frac{1}{4}\partial_{q}g_{ij}\tilde{\partial}^{j}g^{pi}-\frac{1}{4}\tilde{\partial}^{p}g^{ij}\partial_{j}g_{qi}$
$\displaystyle+\frac{1}{4}\tilde{\partial}^{i}\left(g_{ji}\partial_{q}g^{pj}\right)+\frac{1}{4}\partial_{i}\left(g^{ji}\tilde{\partial}^{p}g_{qj}\right)+\frac{1}{4}\tilde{\partial}^{i}\left(g^{jp}\partial_{j}g_{iq}\right)+\frac{1}{4}\partial_{i}\left(g_{jq}\tilde{\partial}^{j}g^{ip}\right)$
$\displaystyle-\frac{1}{8}g^{ip}\partial_{i}g_{kl}\tilde{\partial}^{j}g^{kl}g_{jq}-\frac{1}{8}g^{ip}\partial_{i}g^{kl}\tilde{\partial}^{j}g_{kl}g_{jq}+\frac{1}{4}g^{ip}\partial_{i}g_{kl}\tilde{\partial}^{l}g^{jk}g_{jq}+\frac{1}{4}g^{ip}\tilde{\partial}^{j}g^{kl}\partial_{l}g_{ik}g_{jq}$
$\displaystyle-\frac{1}{4}g^{ip}\tilde{\partial}^{l}\left(g_{kl}\partial_{i}g^{jk}\right)g_{jq}-\frac{1}{4}g^{ip}\partial_{l}\left(g^{kl}\tilde{\partial}^{j}g_{ik}\right)g_{jq}-\frac{1}{4}g^{ip}\partial_{l}\left(g_{ki}\tilde{\partial}^{k}g^{lj}\right)g_{jq}-\frac{1}{4}g^{ip}\tilde{\partial}^{l}\left(g^{kj}\partial_{k}g_{li}\right)g_{jq}$
$\displaystyle+2\tilde{\partial}^{p}\partial_{q}d-\tilde{\partial}^{i}d\left(g^{jp}\partial_{j}g_{iq}\right)-\partial_{i}d\left(g_{jq}\tilde{\partial}^{j}g^{ip}\right)$
$\displaystyle-\tilde{\partial}^{i}d\left(g_{ji}\partial_{q}g^{pj}\right)-\partial_{i}d\left(g^{ji}\tilde{\partial}^{p}g_{qj}\right)-2g^{ip}\partial_{i}\tilde{\partial}^{j}dg_{jq}$
$\displaystyle+\frac{1}{2}g^{ip}\tilde{\partial}^{l}d\left(g_{kl}\partial_{i}g^{jk}\right)g_{jq}+\frac{1}{2}g^{ip}\partial_{l}d\left(g^{kl}\tilde{\partial}^{j}g_{ik}\right)g_{jq}$
$\displaystyle+\frac{1}{2}g^{ip}\tilde{\partial}^{l}d\left(g^{kj}\partial_{k}g_{li}\right)g_{jq}+\frac{1}{2}g^{ip}\partial_{l}d\left(g_{ki}\tilde{\partial}^{k}g^{lj}\right)g_{jq},$
$\displaystyle\mathcal{R}_{\mathbf{2}\left(p\right)\mathbf{1}\left(q\right)}$
$\displaystyle=$
$\displaystyle\frac{1}{8}\partial_{p}g_{ij}\tilde{\partial}^{q}g^{ij}+\frac{1}{8}\partial_{p}g^{ij}\tilde{\partial}^{q}g_{ij}-\frac{1}{4}\partial_{p}g_{ij}\tilde{\partial}^{j}g^{qi}-\frac{1}{4}\tilde{\partial}^{q}g^{ij}\partial_{j}g_{pi}$
$\displaystyle+\frac{1}{4}\tilde{\partial}^{i}\left(g_{ji}\partial_{p}g^{qj}\right)+\frac{1}{4}\partial_{i}\left(g^{ji}\tilde{\partial}^{q}g_{pj}\right)+\frac{1}{4}\tilde{\partial}^{i}\left(g^{jq}\partial_{j}g_{ip}\right)+\frac{1}{4}\partial_{i}\left(g_{jp}\tilde{\partial}^{j}g^{iq}\right)$
$\displaystyle-\frac{1}{8}g_{ip}\tilde{\partial}^{i}g_{kl}\partial_{j}g^{kl}g^{jq}-\frac{1}{8}g_{ip}\tilde{\partial}^{i}g^{kl}\partial_{j}g_{kl}g^{jq}+\frac{1}{4}g_{ip}\partial_{j}g_{kl}\tilde{\partial}^{l}g^{ik}g^{jq}+\frac{1}{4}g_{ip}\tilde{\partial}^{i}g^{kl}\partial_{l}g_{jk}g^{jq}$
$\displaystyle-\frac{1}{4}g_{ip}\tilde{\partial}^{l}\left(g_{kl}\partial_{j}g^{ik}\right)g^{jq}-\frac{1}{4}g_{ip}\partial_{l}\left(g^{kl}\tilde{\partial}^{i}g_{jk}\right)g^{jq}-\frac{1}{4}g_{ip}\tilde{\partial}^{l}\left(g^{ki}\partial_{k}g_{lj}\right)g^{jq}-\frac{1}{4}g_{ip}\partial_{l}\left(g_{kj}\tilde{\partial}^{k}g^{li}\right)g^{jq}$
$\displaystyle
2\partial_{p}\tilde{\partial}^{q}d-\tilde{\partial}^{i}d\left(g^{jq}\partial_{j}g_{ip}\right)-\partial_{i}d\left(g_{jp}\tilde{\partial}^{j}g^{iq}\right)$
$\displaystyle-\tilde{\partial}^{i}d\left(g_{ji}\partial_{p}g^{qj}\right)-\partial_{i}d\left(g^{ji}\tilde{\partial}^{q}g_{pj}\right)-2g_{ip}\tilde{\partial}^{i}\partial_{j}dg^{jq}$
$\displaystyle+\frac{1}{2}g_{ip}\tilde{\partial}^{l}d\left(g_{kl}\partial_{j}g^{ik}\right)g^{jq}+\frac{1}{2}g_{ip}\partial_{l}d\left(g^{kl}\tilde{\partial}^{i}g_{jk}\right)g^{jq}$
$\displaystyle+\frac{1}{2}g_{ip}\tilde{\partial}^{l}d\left(g^{ki}\partial_{k}g_{lj}\right)g^{jq}+\frac{1}{2}g_{ip}\partial_{l}d\left(g_{kj}\tilde{\partial}^{k}g^{li}\right)g^{jq}.$

On the other hand, the EOM for the dilaton is given by:

$\displaystyle\frac{1}{8}g_{ij}\tilde{\partial}^{i}g_{kl}\tilde{\partial}^{j}g^{kl}+\frac{1}{8}g_{ij}\tilde{\partial}^{i}g^{kl}\tilde{\partial}^{j}g_{kl}+\frac{1}{8}g^{ij}\partial_{i}g_{kl}\partial_{j}g^{kl}+\frac{1}{8}g^{ij}\partial_{i}g^{kl}\partial_{j}g_{kl}$
$\displaystyle-\frac{1}{2}g_{ij}\tilde{\partial}^{i}g_{kl}\tilde{\partial}^{k}g^{jl}-\frac{1}{2}g^{ij}\partial_{i}g^{kl}\partial_{k}g_{jl}-\tilde{\partial}^{i}\tilde{\partial}^{j}g_{ij}-\partial_{i}\partial_{j}g^{ij}-4g_{ij}\tilde{\partial}^{i}d\tilde{\partial}^{j}d-4g^{ij}\partial_{i}d\partial_{j}d$
$\displaystyle+4\tilde{\partial}^{i}g_{ij}\tilde{\partial}^{j}d+4\partial_{i}g^{ij}\partial_{j}d+4g_{ij}\tilde{\partial}^{i}\tilde{\partial}^{j}d+4g^{ij}\partial_{i}\partial_{j}d=0.$
(2.46)

Given the symmetries between the generalized Ricci scalars
$\mathcal{R}_{M\left(p\right)N\left(q\right)}$:

$\displaystyle\mathcal{R}_{\mathbf{1}\left(p\right)\mathbf{1}\left(q\right)}$
$\displaystyle\underleftrightarrow{g^{\bullet\bullet}\leftrightarrow
g_{\bullet\bullet},\quad\tilde{\partial}^{\bullet}\leftrightarrow\partial_{\bullet}}$
$\displaystyle\mathcal{R}_{\mathbf{2}\left(p\right)\mathbf{2}\left(q\right)},$
$\displaystyle\mathcal{R}_{\mathbf{1}\left(p\right)\mathbf{2}\left(q\right)}$
$\displaystyle\underleftrightarrow{g^{\bullet\bullet}\leftrightarrow
g_{\bullet\bullet},\quad\tilde{\partial}^{\bullet}\leftrightarrow\partial_{\bullet}}$
$\displaystyle\mathcal{R}_{\mathbf{2}\left(p\right)\mathbf{1}\left(q\right)},$
(2.47)

we need only compute one set. In this paper, we focus on calculating
$\mathcal{R}_{\mathbf{2}\left(p\right)\mathbf{2}\left(q\right)}$ and
$\mathcal{R}_{\mathbf{2}\left(p\right)\mathbf{1}\left(q\right)}$.

Next, we present three types of black hole solutions for the EOM
(LABEL:eq:_GRA_EOM) and (2.46). To begin, we revisit the low-energy effective
action of closed string theory:

$S=\int
d^{4}x\sqrt{-g}e^{-2\phi}\left[R+4\left(\nabla\phi\right)^{2}-\frac{1}{12}H_{\mu\nu\rho}H^{\mu\nu\rho}\right].$
(2.48)

The solutions of standard DFT black holes are categorized by the solutions
$g_{ij}\left(x\right)$ and $\phi\left(x\right)$ from the action (2.48), as
follows:

#### 1st kind of black hole solutions

If $g_{ij}\left(x\right)$ and $\phi\left(x\right)$ satisfy the action (2.48),
they automatically become solutions of DFT that satisfy the strong constraint.
The corresponding line elements can take two forms:

$ds^{2}=g^{ij}\left(x\right)d\tilde{x}_{i}d\tilde{x}_{j}+g_{ij}\left(x\right)dx^{i}dx^{j},\qquad
or\qquad
ds^{2}=\left(g^{-1}\right)^{ij}\left(\tilde{x}\right)d\tilde{x}_{i}d\tilde{x}_{j}+\left(g^{-1}\right)_{ij}\left(\tilde{x}\right)dx^{i}dx^{j}.$
(2.49)

Moreover, when $\phi\left(x\right)=0$, the low-energy effective action (2.48)
reduces to the Einstein-Hilbert action, implying that DFT also includes the
black hole solutions of Einstein’s gravity. There is a notable exception: the
BTZ black hole. The BTZ black hole cannot be directly obtained by setting
$\phi\left(x\right)=0$; it also requires the inclusion of the Kalb-Ramond
field ($H_{ijk}\neq 0$) to precisely reduce to Einstein’s equations with a
negative cosmological constant [43]. Therefore, it is necessary to consider a
non-vanishing $b_{ij}$ in this case.

Example:

To illustrate, consider the following ansatz:

$\displaystyle dS^{2}$ $\displaystyle=$
$\displaystyle-A\left(\tilde{r},r\right)^{-1}d\tilde{t}^{2}+B\left(\tilde{r},r\right)^{-1}d\tilde{r}^{2}+C\left(\tilde{r},r\right)^{-1}\left(d\tilde{\theta}^{2}+D\left(\tilde{\theta},\theta\right)^{-1}d\tilde{\phi}^{2}\right)$
(2.50)
$\displaystyle-A\left(\tilde{r},r\right)dt^{2}+B\left(\tilde{r},r\right)dr^{2}+C\left(\tilde{r},r\right)\left(d\theta^{2}+D\left(\tilde{\theta},\theta\right)d\phi^{2}\right).$

Substituting this ansatz into the EOM (LABEL:eq:_GRA_EOM) and (2.46), we
obtain the EOM for the graviton:

$\displaystyle
B\left[\left(\frac{\tilde{\partial}^{r}B}{B}-\frac{\tilde{\partial}^{r}A}{A}\right)\frac{\tilde{\partial}^{r}A}{A}+\frac{\tilde{\partial}^{r}\tilde{\partial}^{r}A}{A}-2\tilde{\partial}^{r}d\frac{\tilde{\partial}^{r}A}{A}\right]-\frac{1}{B}\left[\left(\frac{\partial_{r}B}{B}+\frac{\partial_{r}A}{A}\right)\frac{\partial_{r}A}{A}-\frac{\partial_{r}\partial_{r}A}{A}+2\partial_{r}d\frac{\partial_{r}A}{A}\right]$
$\displaystyle=$ $\displaystyle 0,$ $\displaystyle
B\left[\frac{1}{2}\left(\frac{\tilde{\partial}^{r}A}{A}\right)^{2}-\frac{1}{2}\left(\frac{\tilde{\partial}^{r}B}{B}\right)^{2}+\left(\frac{\tilde{\partial}^{r}C}{C}\right)^{2}+\frac{\tilde{\partial}^{r}\tilde{\partial}^{r}B}{B}-2\tilde{\partial}^{r}d\frac{\tilde{\partial}^{r}B}{B}-4\tilde{\partial}^{r}\tilde{\partial}^{r}d\right]$
$\displaystyle-\frac{1}{B}\left[\frac{1}{2}\left(\frac{\partial_{r}A}{A}\right)^{2}+\frac{3}{2}\left(\frac{\partial_{r}B}{B}\right)^{2}+\left(\frac{\partial_{r}C}{C}\right)^{2}-\frac{\partial_{r}\partial_{r}B}{B}+2\partial_{r}d\frac{\partial_{r}B}{B}-4\partial_{r}\partial_{r}d\right]$
$\displaystyle=$ $\displaystyle 0,$
$\displaystyle\frac{1}{2}B\left[-\frac{\tilde{\partial}^{r}B}{B}\frac{\tilde{\partial}^{r}C}{C}-\frac{\tilde{\partial}^{r}\tilde{\partial}^{r}C}{C}+\left(\frac{\tilde{\partial}^{r}C}{C}\right)^{2}+2\tilde{\partial}^{r}d\frac{\tilde{\partial}^{r}C}{C}\right]-C\left[2\tilde{\partial}^{\theta}\tilde{\partial}^{\theta}d-\frac{1}{4}\left(\frac{\tilde{\partial}^{\theta}D}{D}\right)^{2}\right]$
$\displaystyle-\frac{1}{2}\frac{1}{B}\left[-\frac{\partial_{r}B}{B}\frac{\partial_{r}C}{C}+\frac{\partial_{r}\partial_{r}C}{C}-\left(\frac{\partial_{r}C}{C}\right)^{2}-2\partial_{r}d\frac{\partial_{r}C}{C}\right]-\frac{1}{C}\left[-2\partial_{\theta}\partial_{\theta}d+\frac{1}{4}\left(\frac{\partial_{\theta}D}{D}\right)^{2}\right]$
$\displaystyle=$ $\displaystyle 0,$
$\displaystyle\frac{1}{2}B\left[-\frac{\tilde{\partial}^{r}B}{B}\frac{\tilde{\partial}^{r}C}{C}-\frac{\tilde{\partial}^{r}\tilde{\partial}^{r}C}{C}+\left(\frac{\tilde{\partial}^{r}C}{C}\right)^{2}+2\tilde{\partial}^{r}d\frac{\tilde{\partial}^{r}C}{C}\right]-C\left[\frac{1}{2}\frac{\tilde{\partial}^{\theta}\tilde{\partial}^{\theta}D}{D}-\frac{1}{2}\left(\frac{\tilde{\partial}^{\theta}D}{D}\right)^{2}-\tilde{\partial}^{\theta}d\frac{\tilde{\partial}^{\theta}D}{D}\right]$
$\displaystyle-\frac{1}{2}\frac{1}{B}\left[-\frac{\partial_{r}B}{B}\frac{\partial_{r}C}{C}+\frac{\partial_{r}\partial_{r}C}{C}-\left(\frac{\partial_{r}C}{C}\right)^{2}-2\partial_{r}d\frac{\partial_{r}C}{C}\right]-\frac{1}{C}\left[\frac{1}{2}\frac{\partial_{\theta}\partial_{\theta}D}{D}-\frac{1}{2}\left(\frac{\partial_{\theta}D}{D}\right)^{2}-\partial_{\theta}d\frac{\partial_{\theta}D}{D}\right]$
$\displaystyle=$ $\displaystyle 0,$

and the EOM for the dilaton:

$\displaystyle\frac{1}{2}B\left(-\frac{1}{2}\left(\frac{\tilde{\partial}^{r}A}{A}\right)^{2}+\frac{1}{2}\left(\frac{\tilde{\partial}^{r}B}{B}\right)^{2}-\left(\frac{\tilde{\partial}^{r}C}{C}\right)^{2}-2\frac{\tilde{\partial}^{r}\tilde{\partial}^{r}B}{B}+8\frac{\tilde{\partial}^{r}B}{B}\tilde{\partial}^{r}d+8\tilde{\partial}^{r}\tilde{\partial}^{r}d-8\tilde{\partial}^{r}d\tilde{\partial}^{r}d\right)$
$\displaystyle+C\left[4\tilde{\partial}^{\theta}\tilde{\partial}^{\theta}d-4\tilde{\partial}^{\theta}d\tilde{\partial}^{\theta}d-\frac{1}{4}\left(\frac{\tilde{\partial}^{\theta}D}{D}\right)^{2}\right]$
$\displaystyle+\frac{1}{2}\frac{1}{B}\left(-\frac{1}{2}\left(\frac{\partial_{r}A}{A}\right)^{2}-\frac{7}{2}\left(\frac{\partial_{r}B}{B}\right)^{2}-\left(\frac{\partial_{r}C}{C}\right)^{2}+2\frac{\partial_{r}\partial_{r}B}{B}-8\frac{\partial_{r}B}{B}\partial_{r}d+8\partial_{r}\partial_{r}d-8\partial_{r}d\partial_{r}d\right)$
$\displaystyle+\frac{1}{C}\left[4\partial_{\theta}\partial_{\theta}d-4\partial_{\theta}d\partial_{\theta}d-\frac{1}{4}\left(\frac{\partial_{\theta}D}{D}\right)^{2}\right]=0.$
(2.52)

Keep in mind that
$\tilde{\partial}^{r}\equiv\frac{\partial}{\partial\tilde{r}}$ and
$\tilde{\partial}^{\theta}\equiv\frac{\partial}{\partial\tilde{\theta}}$.
After imposing the strong constraint, the solutions are organized as follows.
The first solution is:

$\displaystyle dS^{2}$ $\displaystyle=$
$\displaystyle-A\left(r\right)^{-1}d\tilde{t}^{2}+B\left(r\right)^{-1}d\tilde{r}^{2}+C\left(r\right)^{-1}\left(d\tilde{\theta}^{2}+D\left(\theta\right)^{-1}d\tilde{\phi}^{2}\right)$
(2.53)
$\displaystyle-A\left(r\right)dt^{2}+B\left(r\right)dr^{2}+C\left(r\right)\left(d\theta^{2}+D\left(\theta\right)d\phi^{2}\right),$

where

$\displaystyle A\left(r\right)$ $\displaystyle=$
$\displaystyle\left(1-\frac{2\eta}{r}\right)^{\frac{m+\sigma}{\eta}},$
$\displaystyle B\left(r\right)$ $\displaystyle=$
$\displaystyle\left(1-\frac{2\eta}{r}\right)^{\frac{\sigma-m}{\eta}},$
$\displaystyle C\left(r\right)$ $\displaystyle=$
$\displaystyle\left(1-\frac{2\eta}{r}\right)^{1+\frac{\sigma-m}{\eta}}r^{2},$
$\displaystyle D\left(\theta\right)$ $\displaystyle=$
$\displaystyle\sin^{2}\theta,$ $\displaystyle d\left(r,\theta\right)$
$\displaystyle=$
$\displaystyle\frac{\sigma}{2\eta}\ln\left(1-\frac{2\eta}{r}\right)-\frac{1}{4}\log\left[r^{4}\left(1-\frac{2\eta}{r}\right)^{2+3\frac{\sigma-m}{\eta}+\frac{m+\sigma}{\eta}}\sin^{2}\theta\right].$
(2.54)

Its dual solution can be obtained similarly using the following ansatz:

$\displaystyle dS^{2}$ $\displaystyle=$
$\displaystyle-A\left(\tilde{r}\right)d\tilde{t}^{2}+B\left(\tilde{r}\right)d\tilde{r}^{2}+C\left(\tilde{r}\right)\left(d\tilde{\theta}^{2}+D\left(\tilde{\theta}\right)d\tilde{\phi}^{2}\right)$
(2.55)
$\displaystyle-A\left(\tilde{r}\right)^{-1}dt^{2}+B\left(\tilde{r}\right)^{-1}dr^{2}+C\left(\tilde{r}\right)^{-1}\left(d\theta^{2}+D\left(\tilde{\theta}\right)^{-1}d\phi^{2}\right),$

and its solution is

$\displaystyle A\left(\tilde{r}\right)$ $\displaystyle=$
$\displaystyle\left(1-\frac{2\eta}{\tilde{r}}\right)^{\frac{m+\sigma}{\eta}},$
$\displaystyle B\left(\tilde{r}\right)$ $\displaystyle=$
$\displaystyle\left(1-\frac{2\eta}{\tilde{r}}\right)^{\frac{\sigma-m}{\eta}},$
$\displaystyle C\left(\tilde{r}\right)$ $\displaystyle=$
$\displaystyle\left(1-\frac{2\eta}{\tilde{r}}\right)^{1+\frac{\sigma-m}{\eta}}\tilde{r}^{2},$
$\displaystyle D\left(\tilde{\theta}\right)$ $\displaystyle=$
$\displaystyle\sin^{2}\tilde{\theta},$ $\displaystyle
d\left(\tilde{r},\tilde{\theta}\right)$ $\displaystyle=$
$\displaystyle\frac{\sigma}{2\eta}\ln\left(1-\frac{2\eta}{\tilde{r}}\right)-\frac{1}{4}\log\left[\tilde{r}^{4}\left(1-\frac{2\eta}{\tilde{r}}\right)^{2+3\frac{\sigma-m}{\eta}+\frac{m+\sigma}{\eta}}\sin^{2}\tilde{\theta}\right],$
(2.56)

where $m$ is the mass of the black hole, $\sigma$ is the scalar charge, and
$\eta^{2}=m^{2}+\sigma^{2}$. When $\sigma=0$, these reduce to doubly
Schwarzschild black holes. A similar solution in the low-energy effective
action was found in [44].

#### 2nd kind of black hole solutions

If $g_{ij}\left(x\right)$ and $\phi\left(x\right)$ are solutions of the action
(2.48), and $\left(g^{-1}\right)_{ij}\left(x\right)$ is also a solution for
the same action, these configurations are termed self-dual solutions. They
satisfy the strong constraint and encompass four distinct types, expressed as:

$ds^{2}=g^{ij}\left(x\right)d\tilde{x}_{i}d\tilde{x}_{j}+g_{ij}\left(x\right)dx^{i}dx^{j},\qquad
or\qquad
ds^{2}=\left(g^{-1}\right)^{ij}\left(x\right)d\tilde{x}_{i}d\tilde{x}_{j}+\left(g^{-1}\right)_{ij}\left(x\right)dx^{i}dx^{j}{\color[rgb]{.75,.5,.25}\definecolor[named]{pgfstrokecolor}{rgb}{.75,.5,.25},}$
(2.57)

or

$ds^{2}=g^{ij}\left(\tilde{x}\right)d\tilde{x}_{i}d\tilde{x}_{j}+g_{ij}\left(\tilde{x}\right)dx^{i}dx^{j},\qquad
or\qquad
ds^{2}=\left(g^{-1}\right)^{ij}\left(\tilde{x}\right)d\tilde{x}_{i}d\tilde{x}_{j}+\left(g^{-1}\right)_{ij}\left(\tilde{x}\right)dx^{i}dx^{j}.$
(2.58)

Example:

To illustrate this self-dual property, consider the following ansatz:

$\displaystyle dS^{2}$ $\displaystyle=$
$\displaystyle-A\left(\tilde{r},r\right)^{-1}d\tilde{t}^{2}+d\tilde{r}^{2}+A\left(\tilde{r},r\right)^{-1}\left(d\tilde{x}_{i}d\tilde{x}^{i}\right)$
(2.59)
$\displaystyle-A\left(\tilde{r},r\right)dt^{2}+dr^{2}+A\left(\tilde{r},r\right)\left(dx_{i}dx^{i}\right).$

The EOM (LABEL:eq:_GRA_EOM) and (2.46) for the graviton are then:

$\displaystyle\left[-\left(\frac{\tilde{\partial}^{r}A}{A}\right)^{2}+\frac{\tilde{\partial}^{r}\tilde{\partial}^{r}A}{A}-2\tilde{\partial}^{r}d\frac{\tilde{\partial}^{r}A}{A}\right]-\left[\left(\frac{\partial_{r}A}{A}\right)^{2}-\frac{\partial_{r}\partial_{r}A}{A}+2\partial_{r}d\frac{\partial_{r}A}{A}\right]$
$\displaystyle=$ $\displaystyle 0,$
$\displaystyle\left[\frac{D-1}{2}\left(\frac{\tilde{\partial}^{r}A}{A}\right)^{2}-4\tilde{\partial}^{r}\tilde{\partial}^{r}d\right]-\left[\frac{D-1}{2}\left(\frac{\partial_{r}A}{A}\right)^{2}-4\partial_{r}\partial_{r}d\right]$
$\displaystyle=$ $\displaystyle 0,$ (2.60)

while the EOM for the dilaton is:

$\left(-\frac{D-1}{2}\left(\frac{\tilde{\partial}^{r}A}{A}\right)^{2}+8\tilde{\partial}^{r}\tilde{\partial}^{r}d-8\tilde{\partial}^{r}d\tilde{\partial}^{r}d\right)+\left(-\frac{D-1}{2}\left(\frac{\partial_{r}A}{A}\right)^{2}+8\partial_{r}\partial_{r}d-8\partial_{r}d\partial_{r}d\right)=0.$
(2.61)

This formulation yields two types of solutions, given by:

$dS^{2}=-\left(\frac{r}{r_{0}}\right)^{\frac{\pm
2}{\sqrt{D-1}}}d\tilde{t}^{2}+d\tilde{r}^{2}+\left(\frac{r}{r_{0}}\right)^{\frac{\pm
2}{\sqrt{D-1}}}\left(d\tilde{x}_{i}d\tilde{x}^{i}\right)-\left(\frac{r}{r_{0}}\right)^{\frac{\mp
2}{\sqrt{D-1}}}dt^{2}+dr^{2}+\left(\frac{r}{r_{0}}\right)^{\frac{\mp
2}{\sqrt{D-1}}}\left(dx_{i}dx^{i}\right),$ (2.62)

which depend on the usual coordinate $r$, and two additional types of
solutions that depend on the dual coordinate $\tilde{r}$:

$dS^{2}=-\left(\frac{\tilde{r}}{\tilde{r_{0}}}\right)^{\frac{\pm
2}{\sqrt{D-1}}}d\tilde{t}^{2}+d\tilde{r}^{2}+\left(\frac{\tilde{r}}{\tilde{r_{0}}}\right)^{\frac{\pm
2}{\sqrt{D-1}}}\left(d\tilde{x}_{i}d\tilde{x}^{i}\right)-\left(\frac{\tilde{r}}{\tilde{r_{0}}}\right)^{\frac{\mp
2}{\sqrt{D-1}}}dt^{2}+dr^{2}+\left(\frac{\tilde{r}}{\tilde{r_{0}}}\right)^{\frac{\mp
2}{\sqrt{D-1}}}\left(dx_{i}dx^{i}\right).$ (2.63)

In cosmology, such solutions exhibit scale-factor duality.

#### 3rd kind of black hole solutions

This kind of solution in DFT involves double coordinates that are treated
simultaneously, leading to a violation of the strong constraint. These
solutions can be constructed using the product of self-dual solutions:
$g\left(\tilde{x},x\right)=g\left(\tilde{x}\right)\times\left(g^{-1}\right)\left(x\right)$.
The line element is expressed as:

$ds^{2}=g^{ij}\left(x,\tilde{x}\right)d\tilde{x}_{i}d\tilde{x}_{j}+g_{ij}\left(x,\tilde{x}\right)dx^{i}dx^{j},\qquad
or\qquad
ds^{2}=\left(g^{-1}\right)^{ij}\left(x,\tilde{x}\right)d\tilde{x}_{i}d\tilde{x}_{j}+\left(g^{-1}\right)_{ij}\left(x,\tilde{x}\right)dx^{i}dx^{j}$
(2.64)

Example:

Consider the following ansatz:

$\displaystyle dS^{2}$ $\displaystyle=$
$\displaystyle-A\left(\tilde{r},r\right)^{-1}d\tilde{t}^{2}+B\left(\tilde{r},r\right)^{-1}d\tilde{r}^{2}+C\left(\tilde{r},r\right)^{-1}\left(d\tilde{x}_{i}d\tilde{x}^{i}\right)-A\left(\tilde{r},r\right)dt^{2}+B\left(\tilde{r},r\right)dr^{2}+C\left(\tilde{r},r\right)\left(dx_{i}dx^{i}\right).$

Using this ansatz, the corresponding EOM for the graviton (LABEL:eq:_GRA_EOM)
are:

$\displaystyle
B\left[\left(\frac{\tilde{\partial}^{r}B}{B}-\frac{\tilde{\partial}^{r}A}{A}\right)\frac{\tilde{\partial}^{r}A}{A}+\frac{\tilde{\partial}^{r}\tilde{\partial}^{r}A}{A}-2\tilde{\partial}^{r}d\frac{\tilde{\partial}^{r}A}{A}\right]-\frac{1}{B}\left[\left(\frac{\partial_{r}B}{B}+\frac{\partial_{r}A}{A}\right)\frac{\partial_{r}A}{A}-\frac{\partial_{r}\partial_{r}A}{A}+2\partial_{r}d\frac{\partial_{r}A}{A}\right]$
$\displaystyle=$ $\displaystyle 0,$ $\displaystyle
B\left[\frac{1}{2}\left(\frac{\tilde{\partial}^{r}A}{A}\right)^{2}-\frac{1}{2}\left(\frac{\tilde{\partial}^{r}B}{B}\right)^{2}+\frac{D-2}{2}\left(\frac{\tilde{\partial}^{r}C}{C}\right)^{2}+\frac{\tilde{\partial}^{r}\tilde{\partial}^{r}B}{B}-2\tilde{\partial}^{r}d\frac{\tilde{\partial}^{r}B}{B}-4\tilde{\partial}^{r}\tilde{\partial}^{r}d\right]$
$\displaystyle-\frac{1}{B}\left[\frac{1}{2}\left(\frac{\partial_{r}A}{A}\right)^{2}+\frac{3}{2}\left(\frac{\partial_{r}B}{B}\right)^{2}+\frac{D-2}{2}\left(\frac{\partial_{r}C}{C}\right)^{2}-\frac{\partial_{r}\partial_{r}B}{B}+2\partial_{r}d\frac{\partial_{r}B}{B}-4\partial_{r}\partial_{r}d\right]$
$\displaystyle=$ $\displaystyle 0,$ $\displaystyle
B\left[-\frac{\tilde{\partial}^{r}B}{B}\frac{\tilde{\partial}^{r}C}{C}-\frac{\tilde{\partial}^{r}\tilde{\partial}^{r}C}{C}+\left(\frac{\tilde{\partial}^{r}C}{C}\right)^{2}+2\tilde{\partial}^{r}d\frac{\tilde{\partial}^{r}C}{C}\right]-\frac{1}{B}\left[-\frac{\partial_{r}B}{B}\frac{\partial_{r}C}{C}+\frac{\partial_{r}\partial_{r}C}{C}-\left(\frac{\partial_{r}C}{C}\right)^{2}-2\partial_{r}d\frac{\partial_{r}C}{C}\right]$
$\displaystyle=$ $\displaystyle 0,$

Similarily, we derive the EOM for the dilaton (2.46):

$\displaystyle
B\left(-\frac{1}{2}\left(\frac{\tilde{\partial}^{r}A}{A}\right)^{2}+\frac{1}{2}\left(\frac{\tilde{\partial}^{r}B}{B}\right)^{2}-\frac{D-2}{2}\left(\frac{\tilde{\partial}^{r}C}{C}\right)^{2}-2\frac{\tilde{\partial}^{r}\tilde{\partial}^{r}B}{B}+8\frac{\tilde{\partial}^{r}B}{B}\tilde{\partial}^{r}d+8\tilde{\partial}^{r}\tilde{\partial}^{r}d-8\tilde{\partial}^{r}d\tilde{\partial}^{r}d\right)$
$\displaystyle+\frac{1}{B}\left(-\frac{1}{2}\left(\frac{\partial_{r}A}{A}\right)^{2}-\frac{7}{2}\left(\frac{\partial_{r}B}{B}\right)^{2}-\frac{D-2}{2}\left(\frac{\partial_{r}C}{C}\right)^{2}+2\frac{\partial_{r}\partial_{r}B}{B}-8\frac{\partial_{r}B}{B}\partial_{r}d+8\partial_{r}\partial_{r}d-8\partial_{r}d\partial_{r}d\right)$
$\displaystyle=$ $\displaystyle 0.$

The solutions are expressed as:

$\displaystyle dS^{2}$ $\displaystyle=$
$\displaystyle-A\left(\tilde{r},r\right)^{\mp
1}d\tilde{t}^{2}+B\left(\tilde{r},r\right)^{\mp
1}d\tilde{r}^{2}+C\left(\tilde{r},r\right)^{\mp
1}\left(d\tilde{x}_{i}d\tilde{x}^{i}\right)$ (2.68)
$\displaystyle-A\left(\tilde{r},r\right)^{\pm}dt^{2}+B\left(\tilde{r},r\right)^{\pm}dr^{2}+C\left(\tilde{r},r\right)^{\pm}\left(dx_{i}dx^{i}\right),$

where

$\displaystyle A\left(\tilde{r},r\right)$ $\displaystyle=$
$\displaystyle\left(r+a\right)^{\frac{1}{\sqrt{3}}}\left(\tilde{r}+a\right)^{-\frac{1}{\sqrt{3}}},$
$\displaystyle B\left(\tilde{r},r\right)$ $\displaystyle=$
$\displaystyle\left(r+a\right)^{-1}\left(\tilde{r}+a\right),$ $\displaystyle
C\left(\tilde{r},r\right)$ $\displaystyle=$
$\displaystyle\left(r+a\right)^{\sqrt{\frac{2}{3\left(D-2\right)}}}\left(\tilde{r}+a\right)^{-\sqrt{\frac{2}{3\left(D-2\right)}}},$
$\displaystyle d\left(\tilde{r},\tilde{\theta},r,\theta\right)$
$\displaystyle=$ $\displaystyle const.$ (2.69)

##  3 Space/time split double field theory

In the standard DFT, all spacetime coordinates are doubled from $D$ to $2D$,
resulting in the presence of double time coordinates. This feature complicates
the physical interpretation of DFT solutions. A natural resolution to this
issue is to undouble the time coordinate, meaning that all fields in DFT
depend on a single time coordinate while the spatial coordinates are doubled.
The action for this $1+2\left(D-1\right)$-dimensional spacetime can be derived
through an ADM-like decomposition of the generalized metric and the dilaton
[20, 21, 22]. To perform this decomposition, we introduce the following
notation: the double coordinates $X^{M}$ can be rewritten as

$X^{M}=\left(\tilde{t},t,X^{\hat{M}}\right),$ (3.70)

where the hat indices ’$\hat{M}$’ denote the spatial part of the $2D$ manifold
and the tilde indices ’$\tilde{t}$’ represent the dual coordinates. Similarly,
the generalized metric $\mathcal{H}^{MN}$ and the invariant metric $\eta^{MN}$
can be decomposed as

$\mathcal{H}^{MN}=\left(\begin{array}[]{ccc}\mathcal{H}^{\tilde{t}\tilde{t}}&0&0\\\
0&\mathcal{H}^{tt}&0\\\
0&0&\mathcal{H}^{\hat{M}\hat{N}}\end{array}\right),\qquad\eta^{MN}=\left(\begin{array}[]{ccc}0&1&0\\\
1&0&0\\\ 0&0&\eta^{\hat{M}\hat{N}}\end{array}\right),$ (3.71)

where we set the generalized shift vector to zero, thus
$\mathcal{H}^{tM}$=$\mathcal{H}^{\tilde{t}M}$=0, $\mathcal{H}^{tt}=-n^{-2}$,
$\mathcal{H}^{\tilde{t}\tilde{t}}=-n^{2}$ and $n$ is the lapse function. The
dilaton can then be rewritten as

$e^{-2d}=ne^{-2\hat{d}}.$ (3.72)

Substituting the decompositions (3.71) and (3.72) back into the DFT action, we
obtain:

$S_{\mathrm{DFT}}=\int
d^{D}xd^{D}\tilde{x}\mathcal{L}_{\mathrm{DFT}}\left(d,\mathcal{H}_{MN}\right),$
(3.73)

where

$\displaystyle\mathcal{L}_{\mathrm{DFT}}\left(d,\mathcal{H}_{MN}\right)$
$\displaystyle=$ $\displaystyle
e^{-2d}\left(\frac{1}{8}\mathcal{H}^{MN}\partial_{M}\mathcal{H}^{KL}\partial_{N}\mathcal{H}_{KL}-\frac{1}{2}\mathcal{H}^{MN}\partial_{N}\mathcal{H}^{KL}\partial_{L}\mathcal{H}_{MK}\right.$
(3.74)
$\displaystyle\left.-2\partial_{M}d\partial_{N}\mathcal{H}^{MN}+4\mathcal{H}^{MN}\partial_{M}d\partial_{N}d\right).$

By removing the dual time-like coordinate through the constraint
$\frac{\partial}{\partial\tilde{t}}\left(\cdots\right)=0$, we obtain:

$S_{\mathrm{DFT}}=\int dt\int
d^{D-1}xd^{D-1}\tilde{x}\mathcal{\hat{L}}_{\mathrm{DFT}},$ (3.75)

where

$\displaystyle\mathcal{\hat{L}}_{\mathrm{DFT}}$ $\displaystyle=$
$\displaystyle\mathcal{L}_{\mathrm{DFT}}\left(\hat{d}-\frac{1}{2}\log
n,\mathcal{H}_{\hat{M}\hat{N}}\right)-\frac{1}{n}e^{-2\hat{d}}\left(4\left(\partial_{t}\hat{d}\right)^{2}+\frac{1}{8}\partial_{t}\mathcal{H}^{\hat{M}\hat{N}}\partial_{t}\mathcal{H}_{\hat{M}\hat{N}}+\mathcal{H}^{\hat{M}\hat{N}}\partial_{\hat{M}}n\partial_{\hat{N}}n\right).$

This implies that we replace $d$ as $\hat{d}-\frac{1}{2}\log n$ and
$\mathcal{H}_{MN}$ with $\mathcal{H}_{\hat{M}\hat{N}}$ in the Lagrangian
(3.74). This action is consistent with [21]. After a brief calculation, it
also shows that

$\mathcal{L}_{\mathrm{DFT}}\left(\hat{d}-\frac{1}{2}\log
n,\mathcal{H}_{\hat{M}\hat{N}}\right)-n^{-1}e^{-2d}\mathcal{H}^{\hat{M}\hat{N}}\partial_{\hat{M}}n\partial_{\hat{N}}n=ne^{-2d}\mathcal{R}\left(\hat{d},\mathcal{H}_{\hat{M}\hat{N}}\right)-b_{1},$
(3.77)

where $b_{1}$ is a total derivative term (boundary term):

$b_{1}=-\partial_{\hat{M}}\left(ne^{-2\hat{d}}\left[\partial_{\hat{N}}\mathcal{H}^{\hat{M}\hat{N}}-4\mathcal{H}^{\hat{M}\hat{N}}\partial_{\hat{N}}\hat{d}\right]\right).$
(3.78)

Finally, the $1+2\left(D-1\right)$-dimensional DFT action is given by

$\displaystyle S_{\mathrm{DFT}}$ $\displaystyle=$ $\displaystyle\int
dtd^{D-1}xd^{D-1}\tilde{x}\mathcal{\hat{L}}_{\mathrm{DFT}}$ (3.79)
$\displaystyle=$ $\displaystyle-\int
dtd^{D-1}xd^{D-1}\tilde{x}\left[\frac{1}{n}e^{-2\hat{d}}\left(4\left(\partial_{t}\hat{d}\right)^{2}+\frac{1}{8}\partial_{t}\mathcal{H}^{\hat{M}\hat{N}}\partial_{t}\mathcal{H}_{\hat{M}\hat{N}}\right)-ne^{-2d}\mathcal{R}\left(\hat{d},\mathcal{H}_{\hat{M}\hat{N}}\right)\right].$

Since all decomposed fields of DFT only depend on
$\left(t,X^{\hat{M}}\right)$:

$\mathcal{H}^{\hat{M}\hat{N}}\left(t,X^{\hat{M}}\right),\qquad\hat{d}\left(t,X^{\hat{M}}\right),\qquad
n\left(t,X^{\hat{M}}\right).$ (3.80)

and the spacetime action is an integral over a
$1+2\left(D-1\right)$-dimensional spacetime, the dual timelike direction
$\tilde{t}$ has no physical meaning. This implies that we can safely ignore
the dual timelike direction $\tilde{t}$, and the action (3.79) can be
effectively considered as the action for a $1+2\left(D-1\right)$ spacetime
with the following ansatz:

$ds^{2}=-n^{2}dt^{2}+\mathcal{H}_{\hat{M}\hat{N}}dX^{\hat{M}}dX^{\hat{N}},$
(3.81)

which can be expanded as

$\displaystyle ds^{2}$ $\displaystyle=$
$\displaystyle-n^{2}dt^{2}+\mathcal{H}_{\hat{\mathbf{1}}\hat{\mathbf{1}}}dX^{\hat{\mathbf{1}}}dX^{\hat{\mathbf{1}}}+\mathcal{H}_{\hat{\mathbf{1}}\hat{\mathbf{2}}}dX^{\hat{\mathbf{1}}}dX^{\hat{\mathbf{2}}}+\mathcal{H}_{\hat{\mathbf{2}}\hat{\mathbf{1}}}dX^{\hat{\mathbf{2}}}dX^{\hat{\mathbf{1}}}+\mathcal{H}_{\hat{\mathbf{2}}\hat{\mathbf{2}}}dX^{\hat{\mathbf{2}}}dX^{\hat{\mathbf{2}}}$
(3.82) $\displaystyle=$
$\displaystyle-n^{2}dt^{2}+g^{ij}d\tilde{x}_{i}d\tilde{x}_{j}-g^{ik}b_{kj}d\tilde{x}_{i}dx^{j}+b_{ik}g^{kj}dx^{i}d\tilde{x}_{j}+\left(g_{ij}-b_{ik}g^{kl}b_{lj}\right)dx^{i}dx^{j}.$

For simplicity in the rest of the paper, we set the Kalb-Ramond field to zero,
so the line element becomes:

$ds^{2}=-n^{2}dt^{2}+\mathcal{H}_{\hat{\mathbf{1}}\hat{\mathbf{1}}}dX^{\hat{\mathbf{1}}}dX^{\hat{\mathbf{1}}}+\mathcal{H}_{\hat{\mathbf{2}}\hat{\mathbf{2}}}dX^{\hat{\mathbf{2}}}dX^{\hat{\mathbf{2}}}=-n^{2}dt^{2}+g^{ij}d\tilde{x}_{i}d\tilde{x}_{j}+g_{ij}dx^{i}dx^{j}.$
(3.83)

Thus, our aim is to calculate the black hole solutions for this ansatz in the
rest of the paper.

To obtain the EOM, we can also substitute the field decompositions (3.71) and
(3.72) into the equations (2.25). The dilatonic EOM (2.46) then becomes:

$\displaystyle\frac{1}{8}\mathcal{H}^{\hat{M}\hat{N}}\partial_{\hat{M}}\mathcal{H}^{\hat{K}\hat{L}}\partial_{\hat{N}}\mathcal{H}_{\hat{K}\hat{L}}-\frac{1}{2}\mathcal{H}^{\hat{M}\hat{N}}\partial_{\hat{M}}\mathcal{H}^{\hat{K}\hat{L}}\partial_{\hat{K}}\mathcal{H}_{\hat{N}\hat{L}}-\partial_{\hat{M}}\partial_{\hat{N}}\mathcal{H}^{\hat{M}\hat{N}}$
$\displaystyle-4\mathcal{H}^{\hat{M}\hat{N}}\partial_{\hat{M}}d\partial_{\hat{N}}d+4\partial_{\hat{M}}\mathcal{H}^{\hat{M}\hat{N}}\partial_{\hat{N}}d+4\mathcal{H}^{\hat{M}\hat{N}}\partial_{\hat{M}}\partial_{\hat{N}}d$
$\displaystyle+\frac{1}{4}\frac{1}{n^{2}}\partial_{t}n^{-2}\partial_{t}n^{2}+\partial_{t}\partial_{t}n^{-2}-4\partial_{t}n^{-2}\partial_{t}d-\frac{1}{8}\frac{1}{n^{2}}\partial_{t}\mathcal{H}^{\hat{K}\hat{L}}\partial_{t}\mathcal{H}_{\hat{K}\hat{L}}+\frac{1}{4}\mathcal{H}^{\hat{M}\hat{N}}\partial_{\hat{M}}n^{-2}\partial_{\hat{N}}n^{2}$
$\displaystyle=$ $\displaystyle 0.$ (3.84)

The gravitational EOM (LABEL:eq:_GRA_EOM) are given by:

$\displaystyle\mathcal{R}_{tt}=\mathcal{K}_{tt}-S_{\quad
t}^{P}\mathcal{K}_{PQ}S_{\quad t}^{Q}$ $\displaystyle=$ $\displaystyle 0,$
$\displaystyle\mathcal{R}_{\hat{M}\hat{N}}=\mathcal{K}_{\hat{M}\hat{N}}-S_{\quad\hat{M}}^{P}\mathcal{K}_{PQ}S_{\quad\hat{N}}^{Q}$
$\displaystyle=$ $\displaystyle 0.$ (3.85)

For simplicity, we will continue to use $d$ instead of $\hat{d}$ in the rest
of the paper.

###  3.1 Black hole solutions

To calculate the black hole solutions of space/time split DFT, we utilize the
following ansatz:

$ds^{2}=-n^{2}dt^{2}+\mathcal{H}_{\hat{M}\hat{N}}dX^{\hat{M}}dX^{\hat{N}},$
(3.86)

where we assume the generalized metric with the vanishing Kalb-Ramond field
takes the form:

$\mathcal{H}_{\hat{M}\left(i\right)\hat{N}\left(j\right)}\equiv\left(\begin{array}[]{cc}\mathcal{H}_{\hat{\mathbf{1}}\left(i\right)\hat{\mathbf{1}}\left(j\right)}&0\\\
0&\mathcal{H}_{\hat{\mathbf{2}}\left(i\right)\hat{\mathbf{2}}\left(j\right)}\end{array}\right)=\left(\begin{array}[]{cc}g^{ij}&0\\\
0&g_{ij}\end{array}\right),$ (3.87)

where $g_{ij}$ denotes the spatial section of spacetime. Substituting this
ansatz into the gravitational EOM (3.85), the non-vanishing generalized Ricci
scalar can be given by
$\mathcal{R}_{\hat{M}\left(i\right)\hat{N}\left(j\right)}$ and
$\mathcal{R}_{tt}$ which are given as follows:

$\mathcal{R}_{\hat{M}\left(i\right)\hat{N}\left(j\right)}=\mathcal{K}_{\hat{M}\hat{N}}-S_{\quad\hat{M}}^{P}\mathcal{K}_{PQ}S_{\quad\hat{N}}^{Q}\equiv\left(\begin{array}[]{cc}\mathcal{R}_{\hat{\mathbf{1}}\left(i\right)\hat{\mathbf{1}}\left(j\right)}&\mathcal{R}_{\hat{\mathbf{1}}\left(i\right)\hat{\mathbf{2}}\left(j\right)}\\\
\mathcal{R}_{\hat{\mathbf{2}}\left(i\right)\hat{\mathbf{1}}\left(j\right)}&\mathcal{R}_{\hat{\mathbf{2}}\left(i\right)\hat{\mathbf{2}}\left(j\right)}\end{array}\right),$
(3.88)

and

$\displaystyle\mathcal{R}_{tt}$ $\displaystyle=$
$\displaystyle\frac{1}{4}\partial_{t}g_{ij}\partial_{t}g^{ij}-\frac{1}{4}\partial_{t}n^{2}\partial_{t}n^{-2}+\frac{1}{4}\partial_{t}\left(n^{-2}\partial_{t}n^{2}\right)$
(3.89)
$\displaystyle-\frac{1}{4}n^{4}\partial_{t}\left(n^{-2}\partial_{t}n^{-2}\right)-\frac{1}{2}\partial_{t}d\left(n^{-2}\partial_{t}n^{2}\right)+2\partial_{t}\partial_{t}d$
$\displaystyle+\frac{1}{4}\tilde{\partial}^{i}\left(g_{ij}\tilde{\partial}^{j}n^{2}\right)-\frac{1}{4}n^{4}\tilde{\partial}^{i}\left(g_{ij}\tilde{\partial}^{j}n^{-2}\right)-\frac{1}{2}\tilde{\partial}^{i}d\left(g_{ij}\tilde{\partial}^{j}n^{2}\right)+\frac{1}{2}n^{4}\tilde{\partial}^{i}d\left(g_{ij}\tilde{\partial}^{j}n^{-2}\right)$
$\displaystyle+\frac{1}{4}\partial_{i}\left(g^{ij}\partial_{j}n^{2}\right)-\frac{1}{4}n^{4}\partial_{i}\left(g^{ij}\partial_{j}n^{-2}\right)-\frac{1}{2}\partial_{i}d\left(g^{ij}\partial_{j}n^{2}\right)+\frac{1}{2}n^{4}\partial_{i}d\left(g^{ij}\partial_{j}n^{-2}\right),$

Then, $\mathcal{R}_{\hat{M}\left(i\right)\hat{N}\left(j\right)}$ can be
expanded as:

$\displaystyle\mathcal{R}_{\hat{\mathbf{1}}\left(p\right)\hat{\mathbf{1}}\left(q\right)}$
$\displaystyle=$
$\displaystyle\frac{1}{4}\tilde{\partial}^{p}n^{-2}\tilde{\partial}^{q}n^{2}-\frac{1}{4}g^{ip}\partial_{i}n^{2}\partial_{j}n^{-2}g^{jq}+\frac{1}{4}\partial_{t}\left(n^{-2}\partial_{t}g^{pq}\right)+\frac{1}{4}g^{ip}\partial_{t}\left(n^{-2}\partial_{t}n^{2}\right)g^{jq}$
(3.90)
$\displaystyle-\frac{1}{2}\partial_{t}d\left(n^{-2}\partial_{t}g^{pq}\right)+\frac{1}{2}g^{ip}\partial_{t}d\left(n^{-2}\partial_{t}g_{ij}\right)g^{jq}$
$\displaystyle+\frac{1}{4}\tilde{\partial}^{p}g^{ij}\tilde{\partial}^{q}g_{ij}-\frac{1}{4}\tilde{\partial}^{i}\left(g_{ij}\tilde{\partial}^{j}g^{pq}\right)-\frac{1}{4}\partial_{i}\left(g^{ij}\partial_{j}g^{pq}\right)$
$\displaystyle-\frac{1}{4}\tilde{\partial}^{q}g_{ij}\tilde{\partial}^{j}g^{pi}-\frac{1}{4}\tilde{\partial}^{p}g_{ij}\tilde{\partial}^{j}g^{qi}+\frac{1}{4}\tilde{\partial}^{i}\left(g_{ji}\tilde{\partial}^{q}g^{pj}\right)+\frac{1}{4}\tilde{\partial}^{i}\left(g_{ji}\tilde{\partial}^{p}g^{qj}\right)$
$\displaystyle+\frac{1}{4}\partial_{i}\left(g^{jp}\partial_{j}g^{iq}\right)+\frac{1}{4}\partial_{i}\left(g^{jq}\partial_{j}g^{ip}\right)-\frac{1}{8}g^{ip}\partial_{i}g_{kl}\partial_{j}g^{kl}g^{jq}-\frac{1}{8}g^{ip}\partial_{i}g^{kl}\partial_{j}g_{kl}g^{jq}$
$\displaystyle+\frac{1}{4}g^{ip}\tilde{\partial}^{l}\left(g_{lk}\tilde{\partial}^{k}g_{ij}\right)g^{jq}+\frac{1}{4}g^{ip}\partial_{l}\left(g^{lk}\partial_{k}g_{ij}\right)g^{jq}+\frac{1}{4}g^{ip}\partial_{j}g^{kl}\partial_{l}g_{ik}g^{jq}+\frac{1}{4}g^{ip}\partial_{i}g^{kl}\partial_{l}g_{jk}g^{jq}$
$\displaystyle-\frac{1}{4}g^{ip}\partial_{l}\left(g^{kl}\partial_{j}g_{ik}\right)g^{jq}-\frac{1}{4}g^{ip}\partial_{l}\left(g^{kl}\partial_{i}g_{jk}\right)g^{jq}-\frac{1}{4}g^{ip}\tilde{\partial}^{l}\left(g_{ki}\tilde{\partial}^{k}g_{lj}\right)g^{jq}-\frac{1}{4}g^{ip}\tilde{\partial}^{l}\left(g_{kj}\tilde{\partial}^{k}g_{li}\right)g^{jq}$
$\displaystyle+\frac{1}{2}\tilde{\partial}^{i}d\left(g_{ij}\tilde{\partial}^{j}g^{pq}\right)+\frac{1}{2}\partial_{i}d\left(g^{ij}\partial_{j}g^{pq}\right)+2\tilde{\partial}^{p}\tilde{\partial}^{q}d$
$\displaystyle-\frac{1}{2}\partial_{i}d\left(g^{jp}\partial_{j}g^{iq}\right)-\frac{1}{2}\partial_{i}d\left(g^{jq}\partial_{j}g^{ip}\right)-\frac{1}{2}\tilde{\partial}^{i}d\left(g_{ji}\tilde{\partial}^{q}g^{pj}\right)-\frac{1}{2}\tilde{\partial}^{i}d\left(g_{ji}\tilde{\partial}^{p}g^{qj}\right)$
$\displaystyle-\frac{1}{2}g^{ip}\tilde{\partial}^{l}d\left(g_{lk}\tilde{\partial}^{k}g_{ij}\right)g^{jq}-\frac{1}{2}g^{ip}\partial_{l}d\left(g^{lk}\partial_{k}g_{ij}\right)g^{jq}-2g^{ip}\partial_{i}\partial_{j}dg^{jq}$
$\displaystyle+\frac{1}{2}g^{ip}\partial_{l}d\left(g^{kl}\partial_{j}g_{ik}\right)g^{jq}+\frac{1}{2}g^{ip}\partial_{l}d\left(g^{kl}\partial_{i}g_{jk}\right)g^{jq}$
$\displaystyle+\frac{1}{2}g^{ip}\tilde{\partial}^{l}d\left(g_{ki}\tilde{\partial}^{k}g_{lj}\right)g^{jq}+\frac{1}{2}g^{ip}\tilde{\partial}^{l}d\left(g_{kj}\tilde{\partial}^{k}g_{li}\right)g^{jq},$
$\displaystyle\mathcal{R}_{\hat{\mathbf{2}}\left(p\right)\hat{\mathbf{2}}\left(q\right)}$
$\displaystyle=$
$\displaystyle\frac{1}{4}\partial_{p}n^{2}\partial_{q}n^{-2}-\frac{1}{4}g_{pp}\tilde{\partial}^{p}n^{2}\tilde{\partial}^{q}n^{-2}g_{qq}+\frac{1}{4}\partial_{t}\left(n^{-2}\partial_{t}g_{pq}\right)+\frac{1}{4}g_{ip}\partial_{t}\left(n^{-2}\partial_{t}g^{ij}\right)g_{jq}$
(3.91)
$\displaystyle-\frac{1}{2}g_{ip}\partial_{t}d\left(n^{-2}\partial_{t}g^{ij}\right)g_{jq}+\frac{1}{2}\partial_{t}d\left(n^{-2}\partial_{t}g_{pq}\right)$
$\displaystyle+\frac{1}{4}\partial_{p}g^{ij}\partial_{q}g_{ij}-\frac{1}{4}\tilde{\partial}^{i}\left(g_{ij}\tilde{\partial}^{j}g_{pq}\right)-\frac{1}{4}\partial_{i}\left(g^{ij}\partial_{j}g_{pq}\right)$
$\displaystyle-\frac{1}{4}\partial_{q}g^{ij}\partial_{j}g_{pi}-\frac{1}{4}\partial_{p}g^{ij}\partial_{j}g_{qi}+\frac{1}{4}\partial_{i}\left(g^{ji}\partial_{q}g_{pj}\right)+\frac{1}{4}\partial_{i}\left(g^{ji}\partial_{p}g_{qj}\right)$
$\displaystyle+\frac{1}{4}\tilde{\partial}^{i}\left(g_{jp}\tilde{\partial}^{j}g_{iq}\right)+\frac{1}{4}\tilde{\partial}^{i}\left(g_{jq}\tilde{\partial}^{j}g_{ip}\right)-\frac{1}{8}g_{ip}\tilde{\partial}^{i}g_{kl}\tilde{\partial}^{j}g^{kl}g_{jq}-\frac{1}{8}g_{ip}\tilde{\partial}^{i}g^{kl}\tilde{\partial}^{j}g_{kl}g_{jq}$
$\displaystyle+\frac{1}{4}g_{ip}\tilde{\partial}^{l}\left(g_{lk}\tilde{\partial}^{k}g^{ij}\right)g_{jq}+\frac{1}{4}g_{ip}\partial_{l}\left(g^{lk}\partial_{k}g^{ij}\right)g_{jq}+\frac{1}{4}g_{ip}\tilde{\partial}^{j}g_{kl}\tilde{\partial}^{l}g^{ik}g_{jq}+\frac{1}{4}g_{ip}\tilde{\partial}^{i}g_{kl}\tilde{\partial}^{l}g^{jk}g_{jq}$
$\displaystyle-\frac{1}{4}g_{ip}\tilde{\partial}^{l}\left(g_{kl}\tilde{\partial}^{j}g^{ik}\right)g_{jq}-\frac{1}{4}g_{ip}\tilde{\partial}^{l}\left(g_{kl}\tilde{\partial}^{i}g^{jk}\right)g_{jq}-\frac{1}{4}g_{ip}\partial_{l}\left(g^{ki}\partial_{k}g^{lj}\right)g_{jq}-\frac{1}{4}g_{ip}\partial_{l}\left(g^{kj}\partial_{k}g^{li}\right)g_{jq}$
$\displaystyle+\frac{1}{2}\tilde{\partial}^{i}d\left(g_{ij}\tilde{\partial}^{j}g_{pq}\right)+\frac{1}{2}\partial_{i}d\left(g^{ij}\partial_{j}g_{pq}\right)+2\partial_{p}\partial_{q}d$
$\displaystyle-\frac{1}{2}\tilde{\partial}^{i}d\left(g_{jp}\tilde{\partial}^{j}g_{iq}\right)-\frac{1}{2}\tilde{\partial}^{i}d\left(g_{jq}\tilde{\partial}^{j}g_{ip}\right)-\frac{1}{2}\partial_{i}d\left(g^{ji}\partial_{q}g_{pj}\right)-\frac{1}{2}\partial_{i}d\left(g^{ji}\partial_{p}g_{qj}\right)$
$\displaystyle-\frac{1}{2}g_{ip}\tilde{\partial}^{l}d\left(g_{lk}\tilde{\partial}^{k}\tilde{g}^{ij}\right)g_{jq}-\frac{1}{2}g_{ip}\partial_{l}d\left(g^{lk}\partial_{k}g^{ij}\right)g_{jq}-2g_{ip}\tilde{\partial}^{i}\tilde{\partial}^{j}dg_{jq}$
$\displaystyle+\frac{1}{2}g_{ip}\tilde{\partial}^{l}d\left(g_{kl}\tilde{\partial}^{j}g^{ik}\right)g_{jq}+\frac{1}{2}g_{ip}\tilde{\partial}^{l}d\left(g_{kl}\tilde{\partial}^{i}g^{jk}\right)g_{jq}$
$\displaystyle+\frac{1}{2}g_{ip}\partial_{l}d\left(g^{ki}\partial_{k}g^{lj}\right)g_{jq}+\frac{1}{2}g_{ip}\partial_{l}d\left(g^{kj}\partial_{k}g^{li}\right)g_{jq},$
$\displaystyle\mathcal{R}_{\hat{\mathbf{1}}\left(p\right)\hat{\mathbf{2}}\left(q\right)}$
$\displaystyle=$
$\displaystyle\frac{1}{8}\tilde{\partial}^{p}n^{2}\partial_{q}n^{-2}+\frac{1}{8}\tilde{\partial}^{p}n^{-2}\partial_{q}n^{2}$
(3.92)
$\displaystyle+\frac{1}{8}\tilde{\partial}^{p}g_{ij}\partial_{q}g^{ij}+\frac{1}{8}\tilde{\partial}^{p}g^{ij}\partial_{q}g_{ij}-\frac{1}{4}\partial_{q}g_{ij}\tilde{\partial}^{j}g^{pi}-\frac{1}{4}\tilde{\partial}^{p}g^{ij}\partial_{j}g_{qi}$
$\displaystyle+\frac{1}{4}\tilde{\partial}^{i}\left(g_{ji}\partial_{q}g^{pj}\right)+\frac{1}{4}\partial_{i}\left(g^{ji}\tilde{\partial}^{p}g_{qj}\right)+\frac{1}{4}\tilde{\partial}^{i}\left(g^{jp}\partial_{j}g_{iq}\right)+\frac{1}{4}\partial_{i}\left(g_{jq}\tilde{\partial}^{j}g^{ip}\right)$
$\displaystyle-\frac{1}{8}g^{ip}\partial_{i}g_{kl}\tilde{\partial}^{j}g^{kl}g_{jq}-\frac{1}{8}g^{ip}\partial_{i}g^{kl}\tilde{\partial}^{j}g_{kl}g_{jq}+\frac{1}{4}g^{ip}\partial_{i}g_{kl}\tilde{\partial}^{l}g^{jk}g_{jq}+\frac{1}{4}g^{ip}\tilde{\partial}^{j}g^{kl}\partial_{l}g_{ik}g_{jq}$
$\displaystyle-\frac{1}{4}g^{ip}\tilde{\partial}^{l}\left(g_{kl}\partial_{i}g^{jk}\right)g_{jq}-\frac{1}{4}g^{ip}\partial_{l}\left(g^{kl}\tilde{\partial}^{j}g_{ik}\right)g_{jq}-\frac{1}{4}g^{ip}\partial_{l}\left(g_{ki}\tilde{\partial}^{k}g^{lj}\right)g_{jq}-\frac{1}{4}g^{ip}\tilde{\partial}^{l}\left(g^{kj}\partial_{k}g_{li}\right)g_{jq}$
$\displaystyle+2\tilde{\partial}^{p}\partial_{q}d-\tilde{\partial}^{i}d\left(g^{jp}\partial_{j}g_{iq}\right)-\partial_{i}d\left(g_{jq}\tilde{\partial}^{j}g^{ip}\right)$
$\displaystyle-\tilde{\partial}^{i}d\left(g_{ji}\partial_{q}g^{pj}\right)-\partial_{i}d\left(g^{ji}\tilde{\partial}^{p}g_{qj}\right)-2g^{ip}\partial_{i}\tilde{\partial}^{j}dg_{jq}$
$\displaystyle+\frac{1}{2}g^{ip}\tilde{\partial}^{l}d\left(g_{kl}\partial_{i}g^{jk}\right)g_{jq}+\frac{1}{2}g^{ip}\partial_{l}d\left(g^{kl}\tilde{\partial}^{j}g_{ik}\right)g_{jq}$
$\displaystyle+\frac{1}{2}g^{ip}\tilde{\partial}^{l}d\left(g^{kj}\partial_{k}g_{li}\right)g_{jq}+\frac{1}{2}g^{ip}\partial_{l}d\left(g_{ki}\tilde{\partial}^{k}g^{lj}\right)g_{jq},$
$\displaystyle\mathcal{R}_{\hat{\mathbf{2}}\left(p\right)\hat{\mathbf{1}}\left(q\right)}$
$\displaystyle=$
$\displaystyle\frac{1}{8}\partial_{p}n^{2}\tilde{\partial}^{q}n^{-2}+\frac{1}{8}\partial_{p}n^{-2}\tilde{\partial}^{q}n^{2}$
(3.93)
$\displaystyle+\frac{1}{8}\partial_{p}g_{ij}\tilde{\partial}^{q}g^{ij}+\frac{1}{8}\partial_{p}g^{ij}\tilde{\partial}^{q}g_{ij}-\frac{1}{4}\partial_{p}g_{ij}\tilde{\partial}^{j}g^{qi}-\frac{1}{4}\tilde{\partial}^{q}g^{ij}\partial_{j}g_{pi}$
$\displaystyle+\frac{1}{4}\tilde{\partial}^{i}\left(g_{ji}\partial_{p}g^{qj}\right)+\frac{1}{4}\partial_{i}\left(g^{ji}\tilde{\partial}^{q}g_{pj}\right)+\frac{1}{4}\tilde{\partial}^{i}\left(g^{jq}\partial_{j}g_{ip}\right)+\frac{1}{4}\partial_{i}\left(g_{jp}\tilde{\partial}^{j}g^{iq}\right)$
$\displaystyle-\frac{1}{8}g_{ip}\tilde{\partial}^{i}g_{kl}\partial_{j}g^{kl}g^{jq}-\frac{1}{8}g_{ip}\tilde{\partial}^{i}g^{kl}\partial_{j}g_{kl}g^{jq}+\frac{1}{4}g_{ip}\partial_{j}g_{kl}\tilde{\partial}^{l}g^{ik}g^{jq}+\frac{1}{4}g_{ip}\tilde{\partial}^{i}g^{kl}\partial_{l}g_{jk}g^{jq}$
$\displaystyle-\frac{1}{4}g_{ip}\tilde{\partial}^{l}\left(g_{kl}\partial_{j}g^{ik}\right)g^{jq}-\frac{1}{4}g_{ip}\partial_{l}\left(g^{kl}\tilde{\partial}^{i}g_{jk}\right)g^{jq}-\frac{1}{4}g_{ip}\tilde{\partial}^{l}\left(g^{ki}\partial_{k}g_{lj}\right)g^{jq}-\frac{1}{4}g_{ip}\partial_{l}\left(g_{kj}\tilde{\partial}^{k}g^{li}\right)g^{jq}$
$\displaystyle+2\partial_{p}\tilde{\partial}^{q}d-\tilde{\partial}^{i}d\left(g^{jq}\partial_{j}g_{ip}\right)-\partial_{i}d\left(g_{jp}\tilde{\partial}^{j}g^{iq}\right)$
$\displaystyle-\tilde{\partial}^{i}d\left(g_{ji}\partial_{p}g^{qj}\right)-\partial_{i}d\left(g^{ji}\tilde{\partial}^{q}g_{pj}\right)-2g_{ip}\tilde{\partial}^{i}\partial_{j}dg^{jq}$
$\displaystyle+\frac{1}{2}g_{ip}\tilde{\partial}^{l}d\left(g_{kl}\partial_{j}g^{ik}\right)g^{jq}+\frac{1}{2}g_{ip}\partial_{l}d\left(g^{kl}\tilde{\partial}^{i}g_{jk}\right)g^{jq}$
$\displaystyle+\frac{1}{2}g_{ip}\tilde{\partial}^{l}d\left(g^{ki}\partial_{k}g_{lj}\right)g^{jq}+\frac{1}{2}g_{ip}\partial_{l}d\left(g_{kj}\tilde{\partial}^{k}g^{li}\right)g^{jq}.$

On the other hand, the EOM for the dilaton (3.84) is expressed as:

$\displaystyle\frac{1}{8}g_{ij}\tilde{\partial}^{i}g_{kl}\tilde{\partial}^{j}g^{kl}+\frac{1}{8}g_{ij}\tilde{\partial}^{i}g^{kl}\tilde{\partial}^{j}g_{kl}+\frac{1}{8}g^{ij}\partial_{i}g_{kl}\partial_{j}g^{kl}+\frac{1}{8}g^{ij}\partial_{i}g^{kl}\partial_{j}g_{kl}$
$\displaystyle-\frac{1}{2}g_{ij}\tilde{\partial}^{i}g_{kl}\tilde{\partial}^{k}g^{jl}-\frac{1}{2}g^{ij}\partial_{i}g^{kl}\partial_{k}g_{jl}-\tilde{\partial}^{i}\tilde{\partial}^{j}g_{ij}-\partial_{i}\partial_{j}g^{ij}-4g_{ij}\tilde{\partial}^{i}d\tilde{\partial}^{j}d-4g^{ij}\partial_{i}d\partial_{j}d$
$\displaystyle+4\tilde{\partial}^{i}g_{ij}\tilde{\partial}^{j}d+4\partial_{i}g^{ij}\partial_{j}d+4g_{ij}\tilde{\partial}^{i}\tilde{\partial}^{j}d+4g^{ij}\partial_{i}\partial_{j}d+\frac{1}{4}g_{ij}\tilde{\partial}^{i}n^{2}\tilde{\partial}^{j}n^{-2}+\frac{1}{4}g^{ij}\partial_{i}n^{2}\partial_{j}n^{-2}$
$\displaystyle-\frac{1}{4}n^{-2}\partial_{t}g_{kl}\partial_{t}g^{kl}+\frac{1}{4}n^{-2}\partial_{t}n^{2}\partial_{t}n^{-2}+\partial_{t}\partial_{t}n^{-2}-4\partial_{t}n^{-2}\partial_{t}d=0.$
(3.94)

Next, we consider the $D=1+\left(2+2\right)$ dimensional black hole line
element:

$dS^{2}=-n^{2}dt^{2}+\mathcal{H}_{\hat{M}\hat{N}}dX^{\hat{M}}dX^{\hat{N}}=-n\left(r\right)^{2}dt^{2}+g^{ij}d\tilde{x}_{i}d\tilde{x}_{j}+g_{ij}dx^{i}dx^{j},$
(3.95)

where the strong constraint
$\eta_{\hat{M}\hat{N}}\partial^{\hat{M}}\partial^{\hat{N}}\left(\cdots\right)=0$
is imposed, and the metric assumes the form:

$g_{ij}\left(r\right)=\left(\begin{array}[]{cc}B\left(r\right)&0\\\
0&C\left(r\right)\end{array}\right).$ (3.96)

Furthermore, due to symmetries among generalized Ricci scalars
$\mathcal{R}_{M\left(p\right)N\left(q\right)}$,

$\displaystyle\mathcal{R}_{\hat{\mathbf{1}}\left(p\right)\mathbf{1}\left(q\right)}$
$\displaystyle\underleftrightarrow{g^{\bullet\bullet}\leftrightarrow
g_{\bullet\bullet},\quad\tilde{\partial}^{\bullet}\leftrightarrow\partial_{\bullet}}$
$\displaystyle\mathcal{R}_{\hat{\mathbf{2}}\left(p\right)\hat{\mathbf{2}}\left(q\right)},$
$\displaystyle\mathcal{R}_{\hat{\mathbf{1}}\left(p\right)\hat{\mathbf{2}}\left(q\right)}$
$\displaystyle\underleftrightarrow{g^{\bullet\bullet}\leftrightarrow
g_{\bullet\bullet},\quad\tilde{\partial}^{\bullet}\leftrightarrow\partial_{\bullet}}$
$\displaystyle\mathcal{R}_{\hat{\mathbf{2}}\left(p\right)\hat{\mathbf{1}}\left(q\right)},$
(3.97)

we only need to calculate one set of them. Here, we compute
$\mathcal{R}_{\hat{\mathbf{2}}\left(p\right)\hat{\mathbf{2}}\left(q\right)}$
and
$\mathcal{R}_{\hat{\mathbf{2}}\left(p\right)\hat{\mathbf{1}}\left(q\right)}$.
Consequently, the gravitational EOM (3.85) become:

$\displaystyle\mathcal{R}_{tt}$ $\displaystyle=$ $\displaystyle
nB\left[\left(\frac{\tilde{\partial}^{r}B}{B}-\frac{\tilde{\partial}^{r}n}{n}\right)\tilde{\partial}^{r}n+\tilde{\partial}^{r}\tilde{\partial}^{r}n-2\tilde{\partial}^{r}d\tilde{\partial}^{r}n\right]$
$\displaystyle-n\frac{1}{B}\left[\left(\frac{\partial_{r}B}{B}+\frac{\partial_{r}n}{n}\right)\partial_{r}n-\partial_{r}\partial_{r}n+2\partial_{r}d\partial_{r}n\right]=0,$
$\displaystyle\mathcal{R}_{\hat{\mathbf{2}}\left(r\right)\hat{\mathbf{2}}\left(r\right)}$
$\displaystyle=$ $\displaystyle
B\left[B\left(\frac{\tilde{\partial}^{r}n}{n}\right)^{2}+\frac{1}{4}B\left(\frac{\tilde{\partial}^{r}C}{C}\right)^{2}+\frac{1}{2}\tilde{\partial}^{r}\tilde{\partial}^{r}B-\frac{1}{4}\frac{\left(\tilde{\partial}^{r}B\right)^{2}}{B}-\tilde{\partial}^{r}d\tilde{\partial}^{r}B-2B\tilde{\partial}^{r}\tilde{\partial}^{r}d\right]$
$\displaystyle-\frac{1}{B}\left[B\left(\frac{\partial_{r}n}{n}\right)^{2}+\frac{1}{4}B\left(\frac{\partial_{r}C}{C}\right)^{2}-\frac{1}{2}\partial_{r}\partial_{r}B+\frac{3}{4}\frac{\left(\partial_{r}B\right)^{2}}{B}+\partial_{r}d\partial_{r}B-2B\partial_{r}\partial_{r}d\right]=0,$
$\displaystyle\mathcal{R}_{\hat{\mathbf{2}}\left(\varphi\right)\hat{\mathbf{2}}\left(\varphi\right)}$
$\displaystyle=$
$\displaystyle\frac{1}{2}B\left[-\frac{\tilde{\partial}^{r}B}{B}\tilde{\partial}^{r}C-\tilde{\partial}^{r}\tilde{\partial}^{r}C+\frac{\left(\tilde{\partial}^{r}C\right)^{2}}{C}+2\tilde{\partial}^{r}d\tilde{\partial}^{r}C\right]$
(3.98)
$\displaystyle-\frac{1}{2}\frac{1}{B}\left[-\frac{\partial_{r}B}{B}\partial_{r}C+\partial_{r}\partial_{r}C-\frac{\left(\partial_{r}C\right)^{2}}{C}-2\partial_{r}d\partial_{r}C\right]=0.$

Moreover, to compute the black hole solution in non-critical string theory, we
must incorporate the cosmological constant into the action. This constant does
not break the $O\left(d,d\right)$ symmetry and the diffeomorphism of the
action:

$S_{\mathrm{m}}=\int dt\int d^{D-1}xd^{D-1}\tilde{x}e^{-2d}\lambda^{2}.$
(3.99)

Therefore, the EOM for the dilaton (3.84) is expressed as:

$\displaystyle
B\left[-\left(\frac{\tilde{\partial}^{r}n}{n}\right)^{2}+\frac{1}{4}\left(\frac{\tilde{\partial}^{r}B}{B}\right)^{2}-\frac{1}{4}\left(\frac{\tilde{\partial}^{r}C}{C}\right)^{2}-\frac{\tilde{\partial}^{r}\tilde{\partial}^{r}B}{B}+4\frac{\tilde{\partial}^{r}B}{B}\tilde{\partial}^{r}d-4\tilde{\partial}^{r}d\tilde{\partial}^{r}d+4\tilde{\partial}^{r}\tilde{\partial}^{r}d\right]$
$\displaystyle+\frac{1}{B}\left[-\left(\frac{\partial_{r}n}{n}\right)^{2}-\frac{7}{4}\left(\frac{\partial_{r}B}{B}\right)^{2}-\frac{1}{4}\left(\frac{\partial_{r}C}{C}\right)^{2}+\frac{\partial_{r}\partial_{r}B}{B}-4\frac{\partial_{r}B}{B}\partial_{r}d-4\partial_{r}d\partial_{r}d+4\partial_{r}\partial_{r}d\right]$
$\displaystyle=$ $\displaystyle-\lambda^{2}.$

At first, we attempt to derive a solution corresponding to the 3rd kind of
solution found in standard DFT as discussed in the previous section. The
solution, which violates constraints, is given by:

$ds^{2}=\left[-n\left(r,\tilde{r}\right)^{2}dt^{2}+dr^{2}+dx^{2}\right]+\left[d\tilde{r}^{2}+d\tilde{x}^{2}\right],$
(3.101)

where

$\displaystyle n\left(r,\tilde{r}\right)$ $\displaystyle=$
$\displaystyle\left(\tanh\left(\frac{\sqrt{2}\lambda}{4}\tilde{r}\right)\right)^{\mp
1}\left(\tanh\left(\frac{\sqrt{2}\lambda}{4}r\right)\right)^{\pm 1},$
$\displaystyle d\left(r,\tilde{r}\right)$ $\displaystyle=$
$\displaystyle-\frac{1}{2}\log\left(\sinh\left(\frac{\sqrt{2}\lambda}{2}\tilde{r}\right)\sinh\left(\frac{\sqrt{2}\lambda}{2}r\right)\right).$
(3.102)

This configuration represents a double black string solution. However, it
violates the strong constraint due to its dependence on dual coordinates
$\left(r,\tilde{r}\right)$. Further investigation into this solution is
postponed until a compelling reason to relax the strong constraint and
interpret dual coordinates emerges. Alternatively, we impose the strong
constraint and derive two types of solutions for the EOM (3.98) and
(LABEL:eq:dilaton_eom), corresponding to the 1st kind of solution discussed in
the previous section. The solution is

$ds^{2}=\left[-n\left(r\right)^{2}dt^{2}+B\left(r\right)dr^{2}+C\left(r\right)d\varphi^{2}\right]+\left[B\left(r\right)^{-1}d\tilde{r}^{2}+C\left(r\right)^{-1}d\tilde{\varphi}^{2}\right],$
(3.103)

where

$\displaystyle
n\left(r\right)=\left(\sqrt{\left(1-\frac{M}{r}\right)^{a}}\right)^{\pm
1},\qquad
B\left(r\right)=\left(1-\frac{M}{r}\right)^{-c}\frac{1}{\lambda^{2}r^{2}},\qquad
C\left(r\right)=\left(\left(1-\frac{M}{r}\right)^{b}\frac{1}{\lambda^{2}}\right)^{\pm
1},$ $\displaystyle d=\mathrm{const},\qquad a^{2}+b^{2}=1,\qquad c=1.$ (3.104)

The choices of $\pm$ for $n\left(r\right)$ and $C\left(r\right)$ do not need
to be the same simultaneously. The dual solution is given by:

$ds^{2}=\left[-n\left(\tilde{r}\right)^{2}dt^{2}+B\left(\tilde{r}\right)^{-1}d\tilde{r}^{2}+C\left(\tilde{r}\right)^{-1}d\tilde{\varphi}^{2}\right]+\left[B\left(\tilde{r}\right)dr^{2}+C\left(\tilde{r}\right)d\varphi^{2}\right]$
(3.105)

where

$\displaystyle
n\left(\tilde{r}\right)=\left(\sqrt{\left(1-\frac{M}{\tilde{r}}\right)^{a}}\right)^{\mp
1},\qquad
B\left(\tilde{r}\right)=\left(1-\frac{M}{\tilde{r}}\right)^{c}\lambda^{2}\tilde{r}^{2},\qquad
C\left(\tilde{r}\right)=\left(\left(1-\frac{M}{\tilde{r}}\right)^{-b}\lambda^{2}\right)^{\mp
1},$ $\displaystyle d=\mathrm{const},\qquad a^{2}+b^{2}=1,\qquad c=1.$ (3.106)

A similar solution in the low-energy effective action was studied in reference
[45]. Note that our solutions apply to both compact and noncompact $\varphi$.

For simplicity, we analyze only one of these solutions, specifically:

$\displaystyle ds^{2}$ $\displaystyle=$
$\displaystyle\left[-\left(1-\frac{M}{r}\right)^{a}dt^{2}+\left(1-\frac{M}{r}\right)^{-c}\frac{1}{\lambda^{2}r^{2}}dr^{2}+\left(1-\frac{M}{r}\right)^{b}\frac{1}{\lambda^{2}}d\varphi^{2}\right]$
(3.107)
$\displaystyle+\left[\left(1-\frac{M}{r}\right)^{c}\lambda^{2}r^{2}d\tilde{r}^{2}+\left(1-\frac{M}{r}\right)^{-b}\lambda^{2}d\tilde{\varphi}^{2}\right].$

To determine its physical interpretation, we introduce two key results:

#### A. Generalized scalar curvature

To investigate the singularities of this metric, we need to consider the
scalar curvature within the framework of this theory. The geometry of DFT is
structured around generalized diffeomorphisms. Therefore, the connection must
incorporate covariant derivatives that respect these symmetries. This
connection can be defined analogously to the Levi-Civita connection in
Riemannian geometry, ensuring compatibility with the generalized metric and
the $O\left(d,d\right)$ structure. Using this connection, the generalized
Ricci curvature is derived under the condition of vanishing torsion. However,
the generalized Ricci scalar $\mathcal{R}$ and tensor $\mathcal{R}_{MN}$ are
constrained to vanish due to satisfying the equations of motion for the
graviton and dilaton: $\mathcal{R}=0$ and $\mathcal{R}_{MN}=0$. This property
complicates the measurement of curvature, even in spacetime singularities
within DFT [41]. In our solutions, incorporating the cosmological constant in
DFT leads to the generalized Ricci scalar being

$\mathcal{R}=-\lambda^{2},$ (3.108)

indicating that our solutions represent AdS vacuum solutions in double
coordinates. Importantly, DFT possesses AdS vacuum solutions due to the
generalized Ricci scalar includes dilatonic effects.

#### B. Reduction to low-energy effective action

To impose the strong constraint $\tilde{\partial}\left(\ldots\right)=0$,
requiring all fields to be independent of the dual coordinates, the DFT action
simplifies to the traditional low-energy effective action:

$\left.S_{\mathrm{DFT}}\right|_{\tilde{\partial}=0}=\int dt\int
d^{D-1}xd^{D-1}\tilde{x}\sqrt{-g}e^{-2\phi}\left[R+4\left(\nabla\phi\right)^{2}-\frac{1}{12}H_{\mu\nu\rho}H^{\mu\nu\rho}\right].$
(3.109)

Alternatively, this also can be expressed as:

$\left.\mathcal{R}\right|_{\tilde{\partial}=0}=\sqrt{-g}e^{-2\phi}\left[R+4\left(\nabla\phi\right)^{2}-\frac{1}{12}H_{\mu\nu\rho}H^{\mu\nu\rho}\right].$
(3.110)

While integrating out the dual coordinates $\tilde{x}$ only yields an overall
constant in the action, it may seem negligible. However, to retain the novel
insights of DFT, we preserve this term and interpret the generalized metric as
the complete metric of spacetime. Moreover, this reduction implies that the
undoubled geometry described by $x$ can be entirely governed by the
traditional low-energy effective action, without the need for any additional
theories. This indicates that the traditional interpretation of black holes in
the low-energy effective action also applies to DFT black holes. Thus, the
Kretschmann scalar for the undoubled spacetime (spanned by $x$) of the
generalized metric (3.107) is given by

$R_{\mu\nu\rho\sigma}R^{\mu\nu\rho\sigma}=\frac{\lambda^{4}M^{2}\left(1-\frac{M}{r}\right)^{2c}\left(a^{4}M^{2}+2a^{3}M\left(cM-2r\right)+a^{2}\left(b^{2}M^{2}+\left(cM-2r\right)^{2}\right)+b^{2}\left(bM+cM-2r\right)^{2}\right)}{4\left(M-r\right)^{4}}.$
(3.111)

Based on these results, we aim to understand the physical significance of the
black hole solution (3.107) in DFT. In the framework of DFT, the entire
doubled metric of spacetime is given by:

$ds^{2}=\underset{\mathrm{AdS\;vacuum\;in\;DFT}}{\underbrace{\underset{\mathrm{Black\;hole\;in\;ordinary\;string\;theory}}{\underbrace{{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\left[-n\left(r\right)^{2}dt^{2}+B\left(r\right)dr^{2}+C\left(r\right)d\varphi^{2}\right]}}}+{\color[rgb]{0,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,1,1}\pgfsys@color@cmyk@stroke{1}{0}{0}{0}\pgfsys@color@cmyk@fill{1}{0}{0}{0}\left[B\left(r\right)^{-1}d\tilde{r}^{2}+C\left(r\right)^{-1}d\tilde{\varphi}^{2}\right]}}},$
(3.112)

This entire metric describes the AdS vacuum in double coordinates. The first
bracket corresponds to the black hole solution known from ordinary string
theory, while the second bracket represents the additional dimensions
introduced by DFT. Since the first bracket is fully determined by the low-
energy effective action, the usual interpretations in this framework can be
directly applied. This suggests that this conventional, undoubled spacetime
$x$ is self-contained and exhibits its own dynamics. If we consider the
additional part $\left(\tilde{r},\tilde{\varphi}\right)$, there is no action
to describe its geometry independently. Therefore, we must consider the entire
doubled spacetime. DFT governs its dynamics, and it manifests as an AdS
vacuum. In other words, if we obscure the doubled coordinates and focus only
on the conventional region, the black hole and its singularity emerge as
solutions of the low-energy effective action, as depicted in Figure. (1).

Doubled spacetime $\displaystyle\left(x,\tilde{x}\right)$ AdS vacuum Ordinary
spacetime $\displaystyle(x)$ Black holes completely described by the low
energy effective action Figure 1: The dark blue region represents the
conventional undoubled spacetime described by the low-energy effective action,
which contains the black holes. When the additional region from the double
coordinates is included, only DFT can describe the entire doubled spacetime,
consistently characterizing it as an AdS vacuum.

To specify, we examine the following double black string solutions with
specific values for $a$, $b$ and $c$:

##### 1\. When $a=1$, $b=0$, $c=1$, we obtain:

$ds^{2}=\left[-\left(1-\frac{M}{r}\right)dt^{2}+\frac{1}{r\left(r-M\right)}\frac{1}{\lambda^{2}}dr^{2}+\frac{1}{\lambda^{2}}d\varphi^{2}\right]+\left[r\left(r-M\right)\lambda^{2}d\tilde{r}^{2}+\lambda^{2}d\tilde{\varphi}^{2}\right]$
(3.113)

If we only consider the first square bracket, the corresponding Kretschmann
scalar is
$R_{\mu\nu\rho\sigma}R^{\mu\nu\rho\sigma}=\frac{\lambda^{4}M^{2}}{r^{2}}$,
indicating this black hole solution possesses a spacetime singularity: $r=0$
where the event horizon is located at $r=M$. Considering the whole metric,
there are two time-like directions $dr^{2}$ and $d\tilde{r}^{2}$ acrossing the
event horizon. Additionally, accounting for the coordinates
$\left(\tilde{r},\tilde{\varphi}\right)$, there is a coordinate singularity as
$r\rightarrow\infty$. However, it is impossible to measure the curvature in
the region $\left(\tilde{r},\tilde{\varphi}\right)$ independently. When
considering the doubled coordinates, the metric corresponds to the AdS vacuum.

##### 2\. When $a=-1$, $b=0$, $c=1$, we have:

$ds^{2}=\left[-\left(1-\frac{M}{r}\right)^{-1}dt^{2}+\frac{1}{r\left(r-M\right)}\frac{1}{\lambda^{2}}dr^{2}+\frac{1}{\lambda^{2}}d\varphi^{2}\right]+\left[r\left(r-M\right)\lambda^{2}d\tilde{r}^{2}+\lambda^{2}d\tilde{\varphi}^{2}\right],\qquad
M\leq 0,$ (3.114)

Considering the first square bracket, it is the solution of low-energy
effective action and the corresponding Kretschmann scalar is
$R_{\mu\nu\rho\sigma}R^{\mu\nu\rho\sigma}=\frac{\lambda^{4}M^{2}}{\left(M-r\right)^{2}}$.
This solution has a single curvature singularity at $r=M$, with a coordinate
singularity as $r\rightarrow\infty$. And for complete metric, the metric still
corresponds to the AdS vacuum.

##### 3\. When $a=0$, $b=\pm 1$, $c=1$, we have:

$\displaystyle ds^{2}$ $\displaystyle=$
$\displaystyle\left[-dt^{2}+\left(1-\frac{M}{r}\right)^{-1}\frac{1}{\lambda^{2}r^{2}}dr^{2}+\left(1-\frac{M}{r}\right)^{\pm
1}\frac{1}{\lambda^{2}}d\varphi^{2}\right]$ (3.115)
$\displaystyle+\left[\left(1-\frac{M}{r}\right)\lambda^{2}r^{2}d\tilde{r}^{2}+\left(1-\frac{M}{r}\right)^{\mp
1}\lambda^{2}d\tilde{\varphi}^{2}\right],\qquad M\geq 0,$

Here, $b=1$ for $M\geq 0$ and $b=-1$ for $M\leq 0$. When $b=-1$, there is a
conical singularity at $r=M$ due to
$R_{\mu\nu\rho\sigma}R^{\mu\nu\rho\sigma}=\frac{\lambda^{4}M^{2}}{\left(M-r\right)^{2}}$.
When $b=1$, the curvature singularity is at $r=0$ because
$R_{\mu\nu\rho\sigma}R^{\mu\nu\rho\sigma}=\frac{\lambda^{4}M^{2}}{r^{2}}$.
Furthermore, when the doubled coordinates are considered, the metric
corresponds to the AdS vacuum.

##### 4\. When $a=b=c=0$, the solution is given as follows:

$\displaystyle ds^{2}$ $\displaystyle=$
$\displaystyle\left[-dt^{2}+\frac{1}{\lambda^{2}r^{2}}dr^{2}+\lambda^{\pm
2}d\varphi^{2}\right]+\left[\lambda^{2}r^{2}d\tilde{r}^{2}+\lambda^{\mp
2}d\tilde{\varphi}^{2}\right].$ (3.116)

The first square bracket of this metric describe flat spacetime, as
$R_{\mu\nu\rho\sigma}R^{\mu\nu\rho\sigma}=0$. This implies that the half-
geometry of the AdS vacuum in the doubled geometry can also manifest as flat
spacetime when the doubled coordinates are ignored.

##  4 Conclusion and discussion

In this paper, we studied black hole solutions in DFT across two main
sections. Initially, we directly calculated black hole solutions within
standard DFT, where metrics involve two time-like directions. These solutions
were categorized into three classes: two imposing to the strong constraint and
one violating it by depending on dual coordinates. Due to the ambiguity in
interpreting solutions with two time-like directions, further investigation is
warranted for these cases. In the second part, we explored the space/time
split DFT, where the theory only depends on conventional time-like directions,
manifesting in a $1+2\left(D-1\right)$ dimensional spacetime
$\left(t,x,\tilde{x}\right)$. In this framework, we investigated non-critical
string theory incorporating a cosmological constant, also deriving solutions
that violate the constraint. These solutions, similarly, are left for future
investigation. Furthermore, we derived double black string solutions
satisfying the strong constraint. In the context of the entire doubled
spacetime, the generalized Ricci scalar consistently manifests as a negative
constant, indicating AdS vacuum solutions. Upon hidden the dual coordinates
$x$, the standard geometry described by $\tilde{x}$ reveals black hole
solutions and curvature singularities.

Finally, we present some valuable study points for the near future:

#### Cosmology

Using our method, the cosmology of DFT can be revisited. By incorporating the
cosmological constant, we can derive cosmological solutions. The corresponding
generalized FLRW line element is given by:

$ds^{2}=-dt^{2}+a^{2}\left(t\right)dx^{i}dx_{i}+a^{-2}\left(t\right)d\tilde{x}^{i}d\tilde{x}_{i}.$
(4.117)

This solution addresses the double time problem discussed in reference [6].
Similar to the black hole solution, when considering the entire doubled
spacetime, the generalized curvature implies that this solution corresponds to
a dS vacuum.

#### Black hole soluions in Type II DFT

Considering the DFT action of type II theories [46, 47, 48]:

$S=\int
dxd\tilde{x}\left(e^{-2d}\mathcal{R}\left(\mathcal{H},d\right)+\frac{1}{4}\left(\cancel{\partial}\chi\right)^{\dagger}\mathbb{S}\cancel{\partial}\chi\right),$
(4.118)

where the first term represents the standard DFT action describing the NS-NS
sector, and the second term is the DFT action for the massless RR fields. The
RR sector introduces two independent fields. The first field is
$\mathbb{S}=\mathbb{S}^{\dagger}$,
$\mathbb{S}\in\mathrm{Spin}\left(10,10\right)$. The second field is a spinor
$\chi$, which can be represented by a Majorana-Weyl spinor of
$O\left(10,10\right)$. Using our generating method, we can also easily obtain
the double black $p$-brane solution of this action. It would be intriguing to
explore whether this approach introduces new insights into the AdS/CFT
correspondence.

Acknowledgements We are deeply indebt to Houwen Wu for many discussions and
suggestions. This work was supported by NSFC Grant No.12105031, No.12347101
and PSFC Grant No. cstc2021jcyj-bshX0227.

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