11institutetext: Department of mathematicas and applied mathematics, Beijing
University of Chemical Technology, Beijing 100029, China22institutetext:
School of Physics and Astronomy, University of Nottingham, University Park,
Nottingham NG7 2RD, United Kingdom33institutetext: Nottingham Centre of
Gravity, University of Nottingham, Nottingham NG7 2RD, UK

# Kerr-like metric in 4D Double Field Theory

Shunrui Li 2,3 and Yang Liu<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

Double Field Theory suggests that people can view the whole massless NS-NS
sector as the gravitational unity. The $O(D,D)$ covariance and the doubled
diffeomorphisms determine precisely how the Standard Model as well as a
relativistic point particle should couple to the NS-NS sector. The theory also
refines the notion of singularity. In KPS2017 , the authors derive
analytically the most general, spherically symmetric, asymptotically flat,
static vacuum solution to $D=4$ Double Field Theory. The solution contains
three free parameters and consequently generalizes the Schwarzschild geometry.
In this paper, we generalize the metric in KPS2017  to obtain the ’Kerr-like’
metric in $D=4$ double field theory (DFT) in the Einstein frame and string
frame. Then we apply ’covariant phase space’ approach to study the
thermodynamic properties of the metric we have obtained. We explore the first
law of black hole thermodynamics and Hawking radiation for this metric
carefully. As a special case, the ’Schwarzschild-like’ metric in $4D$ double
field theory in KPS2017  can be recovered.

##  1 Introduction

In recent years, one of developments in string theory is double field theory
(DFT) Siegel19931 ; Siegel19932 ; D1990 , which describes supergravity in a
manifestly duality invariant manner. DFT introduces the dual coordiantes
$\tilde{x}_{\mu}$ such that all fields and gauge parameters may depend on the
$2D$ coordinates $X^{M}=(x^{\mu},\tilde{x}_{\mu})$, which form an $O(D,D)$
vector. In order for the action of generalised diffeomorphisms to give a
closed algebra, people need to impose further conditions on the fields and
gauge parameters of double field theory. The easiest way is to require that
the fields and gauge parameters only depend on at most half of the
coordinates, in which case the DFT action can be reduced back to the following
action:

$S_{NS-NS}=\int d^{D}x\sqrt{\det
g}e^{-2\phi}\left(R+4(\nabla\phi)^{2}-\frac{1}{12}H^{2}\right),\qquad
H_{\mu\nu\rho}\equiv 3\partial_{[\mu}B_{\nu\rho]},$ (1.0.1)

where $g_{\mu\nu}$ is the spacetime metric, $B_{\mu\nu}$ 2-form and $\phi$
dilaton. The constraint is called the $section\ condition$ or $strong\
constraint$. Therefore, double field theory provides a natural framework to
study possibilities suggested by T-duality which go beyond supergravity.
Imposing the section condition implies that all supergravity backgrounds can
be viewed as solutions of DFT ASA and CDAB 2017 .

Black hole (BH) solutions are an excellent setup to test string theory, which
is the only consistent theory of quantum gravity so far. The matching between
the microscopic states counting for the 5-dimensional BPS black hole solution
of the string effective action and the Bekenstein-Hawking entropy is still one
of the main successes of superstring theory SW1996 ; Z2023 . Furthermore, the
non-extremal version of such BH has been studied and some limits of the
Bekenstein-Hawking entropy have been matched with the microscopic states
counting using U-duality arguments HMS1996 . Since then, a lot of following
work has been done to extend the results of black hole in string theory to
other solutions and at higher order in $\alpha^{\prime}$. The research has two
sides. On the one hand, from the macroscopic side, a major development is due
to the introduction of the entropy function formalism S2008 , whose ideas have
been used to compute the corrections to the entropy of asymptotically-flat
extremal BHs using only their near horizon limit CDKL20071 ; CDKL20072 ;
CDKL2008 . On the other hand, from the microscopic side, the corrections to
the Cardy formula have been computed in CM2009 ; KLL1999 ; KL2005  finding a
perfect match.

Now we discuss the black hole solutions in double field theory. If we treat
the whole closed string massless sector as stringy graviton fields, then
double field theory may evolve into Stringy Gravity, i.e. the stringy
augmentation of General Relativity ACP2018 . In ACP2018 , equipped with an
$O(D,D)$ covariant differential geometry beyond Riemann, the authors spell out
the definition of the energy–momentum tensor in Stringy Gravity and derive the
corresponding on-shell conservation law from doubled general covariance.
Equating it with the recently identified stringy Einstein curvature tensor,
all the equations of motion of the closed string massless sector can be
unified into a single expression, $G_{AB}=8\pi GT_{AB}$, which is the so-
called Einstein Double Field Equations. In KPS2017 ; ACP2018 , the authors
study the most general $D=4$ static, asymptotically flat, spherically
symmetric, ‘regular’ solution, sourced by the stringy energy–momentum tensor
which is nontrivial only up to a finite radius from the center. Outside this
radius, the solution matches the known vacuum geometry which has four constant
parameters. Furthermore, the authors express these as volume integrals of the
interior stringy energy–momentum tensor and discuss the related energy
conditions of the black hole metric ACP2018 .

The metric in KPS2017 ; ACP2018  is ’Schwarzschild-like’ metric in $4D$ DFT.
In this paper, we will generalize this metric to rotating case, namely, ’Kerr-
like’ metric. The main result of our investigation has two parts. First, we
obtain the Kerr-like metric in $4D$ double field theory in the Einstein frame
and string frame. We mimic the derivation method of Kerr metric in General
Relativity and obtain the Kerr-like metric in $D=4$ DFT. We do the
complexification for the physical quantities in this metric. We discuss the
conditions which dilaton and $H$-flux should satisfy as well. Second, we study
the black hole theormodynamics and Hawking radiation of the Kerr-like metric
we have obtained. We can make use of the ’covariant phase space’ approach V.L
and R.M.W 1994 ; R.M.W 1993 ; J.L and R.M.W 1990 . In this approach the first
law of black hole thermodynamics is re-expressed in a ’differential’ form, as
the vanishing of the exterior derivative of a certain $(D-2)$-form constructed
out of the fields and their variations. Then Stokes’ theorem sets the integral
of this form on a horizon cross-section equal to the integral on a sphere at
infinity, recovering the usual, integrated form of the first law of black hole
thermodynamics ASA and CDAB 2017 .

The rest of this paper is organised as follows. In section 2, we review the
main results of background we use in this paper. In section 3, we derive the
Kerr-like metric in $4D$ double field theory. In section 4, we discuss the
thermodynamic properties of the Kerr-like metric which is obtained in section
3. In section 5, we conclude our results and discuss some remaining concerns.

##  2 Basics

###  2.1 Review of double field theory

Double field theory (DFT) generalizes the spacetime action, which can possess
T-duality on the level of component fields manifestly WY2014 . Early works can
see T1990 ; D1990 . Due to the equivalence of spacetime momenta and winding
numbers in the string spectra WY2014 , a set of conjugated coordinates
$\tilde{x}^{i}$, which is conjugated to winding numbers, can be introduced
naturally WY2014 . These conjugated coordinates are treated on the same
footing as the usual coordinates $x_{i}$. Then the dimension of spacetime is
doubled from $D$ to $D+D$ WY2014 . The action of DFT unifies the metric
$g_{ij}$, the two-form $b_{ij}$ and the dilaton $\phi$ by rewriting these
fields in an $O(D,D)$ covariant way. The group $O(D,D)$ itself is defined to
the set of transformations perserving the $O(D,D)$ structure

$\eta_{MN}=\left[\begin{array}[]{cc}0&I_{D}\\\ I_{D}&0\end{array}\right],$
(2.1.1)

while the generalized metric $\mathcal{H}_{MN}$ is given by

$\mathcal{H}_{MN}=\begin{bmatrix}g^{ij}&-g^{ik}B_{kj}\\\
B_{ik}g^{kj}&g_{ij}-B_{ik}g^{kl}B_{lj}\end{bmatrix}.$ (2.1.2)

The generalised metric is symmetric and constrained to satisfy
$\mathcal{H}_{M}^{N}\mathcal{H}_{N}^{P}=\delta_{M}^{P}$, which implies that it
parameterises the coset $O(D,D)/O(1,D-1)\times O(1,D-1)$.
If there is no dependence on the conjugated coordinates, the DFT can be
reduced to the supergravity action B2019 . The action is given by

$S=\frac{1}{16\pi G_{DFT}}\int dxd\tilde{x}e^{-2d}\mathcal{R},$ (2.1.3)

where $d$ is the shifted dilaton, which contains the determinant of the metric
and the usual dilaton $\phi$ both B2019 ; HZ20101 , i.e.,

$e^{-2d}=\sqrt{-g}e^{-2\phi},$ (2.1.4)

and generalised Ricci scalar $\mathcal{R}$ is

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

The gauge fields $A_{\mu}^{M}$ and $B_{\mu\nu}$ constitute the ”tensor
hierarchy” of the split theory. The field strengths are

$\mathcal{F}^{M}_{\mu\nu}=\partial_{\mu}A^{M}_{\nu}-\partial_{\nu}A^{M}_{\mu}-[A_{\mu},A_{\nu}]^{M}_{C}+\partial^{M}B_{\mu\nu},$
(2.1.6)
$\mathcal{H}_{\mu\nu\rho}=3D_{[\mu}B_{\nu\rho]}-3A_{[\mu}^{N}\partial_{\nu}A_{\rho]N}+A_{[\mu|N|}[A_{\nu},A_{\rho]}]_{C}^{N},$
(2.1.7)

and the double field internal index $C_{\mu\nu}$ can be defined by

$C_{\mu\nu}\equiv-B_{\mu\nu}+\frac{1}{2}A_{\mu}^{M}A_{\nu M}.$ (2.1.8)

In the above, we have introduced the derivative
$D_{\mu}=\partial_{\mu}-\mathcal{L}_{A_{\mu}}$, which is covariant under
generalised diffeomorphisms, as explained in OH and HS 2013 .

###  2.2 Spherical solutions in D=4 double field theory

The massless spectrum of any of closed string theories has a common sector of
the NS-NS field: the spacetime metric $g_{\mu\nu}$, 2-form $B_{\mu\nu}$ amd
dilaton $\phi$, the low energy of these field is

$S_{NS-NS}=\int
d^{D}x\sqrt{g}e^{-2\phi}(R+4(\nabla\phi)^{2}-\frac{1}{12}H_{\mu\nu\omega}H^{\mu\nu\omega}),$
(2.2.1)

where $H_{\mu\nu\omega}\equiv 3\partial_{[\mu}B_{\nu\omega]}$. We can obtain
generalised Einstein equations for pure double field theory by variational
method,

$R_{\mu\nu}+2\nabla_{\mu}(\partial_{\nu}\phi)-\frac{1}{4}H_{\mu\rho\sigma}H_{\nu}^{\rho\sigma}=0,$
(2.2.2) $d\star(e^{2\phi}H_{(3)})=0,$ (2.2.3)
$R+4\Box\phi-4\partial_{\mu}\phi\partial^{\mu}\phi-\frac{1}{12}H_{\mu\rho\sigma}H^{\nu\rho\sigma}=0.$
(2.2.4)

More details can be found in JHP2019 ; KPS2017 .

We review the most general form of the static, asymptotically flat and
spherically symmetric vacuum solutions to $D=4$ double field theory briefly
KPS2017 ; ACP2018 . Without loss of generality, the metric for the string
frame can be assumed as

$ds^{2}=e^{2\phi(r)}[-A(r)dt^{2}+A(r)^{-1}dr^{2}+A(r)^{-1}C(r)d\Omega^{2}],$
(2.2.5)

where

$d\Omega^{2}=d\theta^{2}+\sin^{2}\theta d\varphi^{2}.$ (2.2.6)

It is worthwhile to note that our string frame metric ansatz takes the product
form of the dilaton factor, $e^{2\phi}$, times the Einstein frame metric.
If the spacetime is asymptotically ’flat’, then the metric should satisfy the
following three boundary conditions KPS2017 :

$\lim_{r\rightarrow\infty}A(r)=1,$ (2.2.7)
$\lim_{r\rightarrow\infty}r^{-2}C(r)=1,$ (2.2.8)
$\lim_{r\rightarrow\infty}\phi(r)=0.$ (2.2.9)

From the asymptotic ‘smoothness’, the metric should satisfy:

$\lim_{r\rightarrow\infty}A^{\prime}(r)=\lim_{r\rightarrow\infty}A^{\prime\prime}(r)=0,$
(2.2.10)
$\lim_{r\rightarrow\infty}r^{-1}C^{\prime}(r)=\lim_{r\rightarrow\infty}C^{\prime\prime}(r)=2,$
(2.2.11)
$\lim_{r\rightarrow\infty}\phi^{\prime}(r)=\lim_{r\rightarrow\infty}\phi^{\prime\prime}(r)=0.$
(2.2.12)

We write the $B$-field using the form notation KPS2017

$B_{(2)}=\frac{1}{2}B_{\mu\nu}dx^{\mu}\wedge dx^{\nu}=B(r)\cos\theta dr\wedge
d\varphi+h\cos\theta dt\wedge d\varphi.$ (2.2.13)

The $H$-flux, which takes the most general spherically symmetric form, can be
written as KPS2017

$H_{(3)}=\frac{1}{3!}H_{\lambda\mu\nu}dx^{\lambda}\wedge dx^{\mu}\wedge
dx^{\nu}=B(r)\sin\theta dr\wedge d\theta\wedge d\varphi+h\sin\theta dt\wedge
d\theta\wedge d\varphi,$ (2.2.14)

which is closed for constant $h$ KPS2017 . As a result, with four constants
$a$, $b$, $c$, $h$ and KPS2017

$c_{+}=c+\frac{1}{2}\sqrt{a^{2}+b^{2}},$ (2.2.15)
$c_{-}=c-\frac{1}{2}\sqrt{a^{2}+b^{2}},$ (2.2.16)
$\gamma_{\pm}=\frac{1}{2}(1\pm\sqrt{1-h^{2}/b^{2}}),$ (2.2.17)

the metric can be given KPS2017 ; ACP2018 :

$e^{2\phi}=\gamma_{+}\left(\frac{r-\alpha}{r+\beta}\right)^{\frac{b}{\sqrt{a^{2}+b^{2}}}}+\gamma_{-}\left(\frac{r+\beta}{r-\alpha}\right)^{\frac{b}{\sqrt{a^{2}+b^{2}}}}.$
(2.2.18)

It can be proved that the asymptotic flatness is inconsistent with the
magnetic $H$-flux, therefore we should set $B(r)=0$ KPS2017 . Then we have

$B_{(2)}=h\cos\theta dt\wedge d\varphi,$ (2.2.19) $H_{(3)}=h\sin\theta
dt\wedge d\theta\wedge d\varphi.$ (2.2.20)
$ds^{2}=e^{2\phi}[-\left(\frac{r-\alpha}{r+\beta}\right)^{\frac{a}{\sqrt{a^{2}+b^{2}}}}dt^{2}+\left(\frac{r+\beta}{r-\alpha}\right)^{\frac{a}{\sqrt{a^{2}+b^{2}}}}\times\\{dr^{2}+(r-\alpha)(r+\beta)d\Omega^{2}\\}],$
(2.2.21)

where

$\alpha=\frac{a}{a+b}\sqrt{a^{2}+b^{2}},$ (2.2.22)

and

$\beta=\frac{b}{a+b}\sqrt{a^{2}+b^{2}}.$ (2.2.23)

###  2.3 Noether charge and the first law of black hole in double field theory

For the black hole in general relativity, Robert M.Wald, Joohan Lee and Vivek
Lyer verified that black hole entropy is Noether charge by covariant phase
space formalism R.M.W 1993 ; J.L and R.M.W 1990 ; V.L and R.M.W 1994 .
The Noether charge $(n-2)$-from can always be expressed in the form

$\mathcal{Q}=\textbf{W}_{c}(\phi)\xi^{c}+\textbf{X}^{cd}\nabla_{[c}\xi_{d]}+d\textbf{Z}(\phi,\xi),$
(2.3.1)

where $\textbf{W}_{c},\textbf{X}^{ab},\textbf{Y}$ and Z are covariant
quantities which are locally constructed from the indicated fields and their
derivatives (with Y linear in $\mathcal{L}_{\xi}\phi$ and Z linear in $\xi$).
For DFT, it is more convenient to use the variation of action

$\delta S_{DFT}=\int e^{-2d}(-2\delta d\mathcal{R}+\delta
H^{MN}\mathcal{R}_{MN})+\int\partial_{M}(e^{-2d}\Theta^{M}[H,dt;\delta
H,\delta d],$ (2.3.2)

where

$\displaystyle\Theta^{M}=$ $\displaystyle\delta
H^{QP}(\frac{1}{4}H^{MR}\partial_{R}H_{QP}-\frac{1}{2}H^{MR}\partial_{Q}H_{RP}-\frac{1}{2}H^{R}_{Q}\partial_{R}H^{M}_{P})+$
(2.3.3) $\displaystyle 2\delta H^{MP}\partial_{P}d-\partial_{P}\delta
H^{MP}+4\partial_{P}(\delta d)H^{MP}.$

We consider the varition of the DFT action under a generalised diffeomorphism
with parameter $\Lambda^{M}$,

$\delta_{\Lambda}S_{DFT}=\int\partial_{M}(\Lambda^{M}e^{-2d}\mathcal{R}).$
(2.3.4)

By comparing the two variations (2.3.2) and (2.3.4), we can get a boundary
term

$\mathcal{J}=e^{-2d}(\Theta[H,d;\mathcal{L}_{\Lambda}H,\mathcal{L}_{\Lambda}d]-\Lambda^{M}\mathcal{R}),$
(2.3.5)

which is divergence-free whenever $(d,H)$ are on-shell

$\mathcal{R}=0=\mathcal{R}_{MN}\rightarrow\partial_{M}\mathcal{J}^{M}=0,$
(2.3.6)

and an anstisymmetric $\mathcal{J}^{MN}$ satisfies

$\mathcal{J}^{M}=\partial_{N}\mathcal{J}^{MN}.$ (2.3.7)

In DFT we have

$\not{\delta}Q_{\Lambda}=\Omega[d,H;(\delta d,\delta
H),(\mathcal{L}_{\Lambda}d,\mathcal{L}_{\Lambda}H)].$ (2.3.8)

We will consider it as a definition of the infinitesimal Noether charge
$\not{\delta}Q_{\Lambda}$ associated to $\Lambda^{M}$ and more details can be
found in R.M.W 1993 ; J.L and R.M.W 1990 ; V.L and R.M.W 1994 .
Additionally, we can write symplectic form on the right hand of (2.3.8) as

$\Omega[d,H;(\delta_{1}d,\delta_{1}H),(\delta_{2}d,\delta_{2}H)]=\delta_{1}[e^{-2d}\Theta^{M}(H,d;\delta_{2}H,\delta_{2}d)]-\delta_{2}[e^{-2d}\Theta^{M}(H,d;\delta_{1}H,\delta_{1}d)].$
(2.3.9)

For double field theory, we also have

$\mathcal{J}^{\mu\nu}=e^{-2\hat{d}}(-2g^{\rho[\mu}\nabla_{\rho}(\xi^{\prime})^{\nu]}-\widetilde{\lambda}_{p}H^{\mu\nu\rho}-\widetilde{\Lambda}^{M}H_{MN}\mathcal{F}^{\mu\nu
N})\epsilon_{\mu\nu},$ (2.3.10)

where $\xi^{\prime}$ is unit surface gravity Killing vector, which is defined
from Killing vector $\xi$

$\xi^{\prime}=\frac{1}{\kappa}\xi.$ (2.3.11)

Moreover, in this equation, $\kappa$ is also defined by $\xi$

$\partial_{\mu}(\xi^{\sigma}\xi^{\nu}g_{\sigma\nu})=(-2\kappa)\xi_{\mu},$
(2.3.12)

where the Killing vector is defined by

$\xi^{\mu}=\frac{\partial}{\partial t}.$ (2.3.13)

Furthermore, on the bifurcation, in (2.3.10), we have

$\widetilde{\lambda_{p}}\equiv-(\xi^{\prime})^{\rho}C_{\rho\mu}\qquad\widetilde{\Lambda}^{M}\equiv(\xi^{\prime})^{\rho}A_{\rho}^{M}.$
(2.3.14)

The existence of a Noether charge $Q_{\Lambda}$ whose variation equals the
right hand of (2.3.9) is equivalent to the existence of the ’boundary term’
$B^{M}$ such that

$\delta\int_{\partial
C(\infty)}e^{-2d}(2B^{M}\Lambda^{P})\epsilon_{MP}=\int_{\partial
C(\infty)}e^{-2d}(2\Theta^{[M}\Lambda^{P]})\epsilon_{MP}=\int_{C}\partial_{P}(2e^{-2d}\Theta^{[M}\Lambda^{P]})\epsilon_{MP},$
(2.3.15)

where the $\epsilon$ are the normal and binormal to the codimension 1 “Cauchy
surface” $C$ and its boundary at infinity $\partial C(\infty)$ respectively.
The Noether charge is

$Q_{\Lambda}=\int_{C}\mathcal{J}^{M}\epsilon_{M}-\int_{\partial
C(\infty)}e^{-2d}(2B^{M}\Lambda^{P})\epsilon_{MP}=\int_{\partial
C(\infty)}(\mathcal{J}^{M}-e^{-2d}2B^{M}\Lambda^{P})\epsilon_{MP}.$ (2.3.16)

Let us assume that we have a background for which there exists a horizon
specified by $R=R_{0}$ for a radial coordinate $R$. Then if we integrate
against the codimension 1 “Cauchy surface” $C$ given by $t=t_{0}$, so we can
define mass as the Noether charge

$M\equiv Q_{\xi}=\frac{1}{16\pi G_{DFT}}\int_{\partial
C(\infty)}d^{d-2}xd^{2n}X(\mathcal{J}^{\mu\nu}-e^{-2d}2B^{\mu}\xi^{\nu})\epsilon_{\mu\nu},$
(2.3.17)

where $n=D-d$. As a result, we can get the first law of black hole
thermdynamics

$\delta M=\frac{\kappa}{2\pi}\delta S+\not{\delta}W,$ (2.3.18)

where $S$ is entropy,

$S\equiv 2\pi
Q_{hor,\xi^{\prime}}=\frac{1}{4G_{DFT}}\int_{t=t_{0},R=R_{0}}d^{d-2}xd^{2n}Xe^{-2d}\sqrt{|detg_{d-2}|},$
(2.3.19)

and $\not{\delta}W$ is $O(n,n)$ invariant thermodynamic work contribution,

$\not{\delta}W\equiv-\frac{1}{16\pi
G_{DFT}}\int_{t=t_{0},R=R_{0}}d^{d-2}xd^{2n}X[\tilde{\lambda}_{\rho}\delta(e^{-2d}\mathcal{H}^{\mu\nu\rho}+\tilde{\Lambda}^{M}\delta(e^{-2d}H_{MN}\mathcal{F}^{\mu\nu
N})]\epsilon_{\mu\nu}.$ (2.3.20)

Extending (2.3.18) to the case where the external metric $g_{\mu\nu}$ is only
stationary, rather than static, is a straightforward task. However, such an
extension necessitates assuming — or proving — a form of horizon rigidity
theorem, relevant within the context of the split parametrization elucidated
in OH and HS 2013 . This theorem would ensure the existence of a certain
number of commuting Killing vector fields $\partial/\partial\varphi^{I}$ so
that $\xi=\partial/\partial t+\Omega^{I}\partial/\partial\varphi^{I}$; the
left-hand side of (2.3.18) would then be replaced by $\delta
M-\Omega^{I}\delta J_{I}$. Therefore, the first law of black hole
thermodynamics for stationary metric is

$\delta M^{\prime}=\frac{\kappa}{2\pi}\delta S+\not{\delta}W,$ (2.3.21)

where

$\delta M^{\prime}=\delta M-\Omega^{I}\delta J_{I}.$ (2.3.22)

More details of Noether charge and black hole thermodynamics in DFT can be
found in ASA and CDAB 2017 .

##  3 Derivation of Kerr-like metric

In this section, we present the Kerr-like metric in the context of DFT.
Initially, our focus lies on deriving the metric for the Einstein frame,
excluding considerations related to dilation. We propose a more generalized
approach for deriving the metric, utilizing the Schwarzschild-like metric to
derive the Kerr-like metric. Subsequently, the metric under this specified
condition is acquired through a sequence of transformations. Finally, we
discuss the conditions which dilaton and $H$-flux should satisfy.

###  3.1 General derivation method

In this subsection, we present a more general derivation method for the Kerr-
like metric within the Einstein frame. Initially, we employ transformations
aimed at eliminating the $g_{rr}$ term from the metric, while simultaneously
transitioning the coordinate system from $(t,r,\theta,\varphi)$ to
$(u,r,\theta,\varphi)$. Subsequently, leveraging contravariant coordinates, we
embark on a series of transformations, some of which need to do
complexification. Upon acquiring the inverse metric, we proceed to obtain the
metric in $(u,r,\theta,\varphi)$ coordinates by computing the inverse matrix.
Finally, we revert to the original coordinates $(t,r,\theta,\varphi)$ to
deduce the Kerr-like metric within the Einstein frame.
The Schwarzschild-like metric in $4D$ DFT is given by (2.2.5). Now,we
generalize the spherically symmetric metric to Kerr-like metric. In the
following section we will obtain the Kerr-like metric in the Einstein metric.
Therefore, $e^{2\phi(r)}$ term should be neglected. First, we can rewrite the
angular part of metric (2.2.6) in a general form

$d\Omega^{2}=f_{1}^{2}(\theta)d\theta^{2}+f_{2}^{2}(\theta)d\varphi^{2}.$
(3.1.1)

We introduce a new variable $u$, which is given by

$du=dt-A^{-1}(r)dr.$ (3.1.2)

Then the metric can be rewritten as

$ds^{2}=-A(r)du^{2}+2dudr+A^{-1}(r)C(r)d{\Omega}^{2}.$ (3.1.3)

According to Appendix A, the metric can now be expressed in terms of a null
tetrad $\\{l^{\mu},n^{\mu},m^{\mu},\bar{m}^{\mu}\\}$. The vectors
$l^{\mu},n^{\mu}$ are real while $m^{\mu},\bar{m}^{\mu}$ are complex
conjugates; the only non-vanishing scalar products between tetrad vectors are:
$l^{\mu}n_{\mu}=1$ and $n^{\mu}n_{\mu}=0$.
As a result, we have:

$g^{\mu\nu}=-l^{\mu}n^{\nu}-n^{\mu}l^{\nu}+m^{\mu}\bar{m}^{\nu}+\bar{m}^{\mu}m^{\nu}.$
(3.1.4)

and

$l^{\mu}=\delta_{1}^{\mu},$ (3.1.5)
$n^{\mu}=\delta_{0}^{\mu}+\frac{1}{2}A(r)\delta_{1}^{\mu},$ (3.1.6)
$m^{\mu}=\frac{1}{\sqrt{2A^{-1}(r)C(r)}}\left(\frac{1}{f_{1}}\delta_{2}^{\mu}+\frac{i}{f_{2}}\delta_{3}^{\mu}\right),$
(3.1.7)
$\bar{m}^{\mu}=\frac{1}{\sqrt{2A^{-1}(r)C(r)}}\left(\frac{1}{f_{1}}\delta_{2}^{\mu}-\frac{i}{f_{2}}\delta_{2}^{\mu}\right),$
(3.1.8)

where the $\delta$’s are the unit vectors of cotangent space.
Then we introduce two complex transformations

$r^{\prime}=r+i\psi P(\theta)\>\qquad\text{and}\qquad\>u^{\prime}=u+i\psi
W(\theta),$ (3.1.9)

where we have defined rotation parameter

$\psi=J/M,$ (3.1.10)

where $J$ is angular momentum.
Taking the derivative of (3.1.9), we can get:

$dr^{\prime}=dr+i\psi
P^{\prime}d\theta\>\qquad\text{and}\qquad\>du^{\prime}=du+i\psi
W^{\prime}d\theta,$ (3.1.11)

where $P^{\prime}$ and $W^{\prime}$ are the derivatives with respect to
$\theta$.
In fact, (3.1.11) can be rewritten as a matrix

$\left[\begin{array}[]{c}du^{\prime}\\\ dr^{\prime}\\\ d\theta^{\prime}\\\
d\varphi^{\prime}\end{array}\right]=\left[\begin{array}[]{cccc}1&0&i\psi
P^{\prime}&0\\\ 0&1&i\psi W^{\prime}&0\\\ 0&0&1&0\\\
0&0&0&1\end{array}\right]\left[\begin{array}[]{c}du\\\ dr\\\ d\theta\\\
d\varphi\end{array}\right].$ (3.1.12)

After applying transformation (3.1.12), these unit vectors can be written as:

$\delta_{0}^{\mu}=\delta^{\prime\mu}_{0},$ (3.1.13)
$\delta_{1}^{\mu}=\delta^{\prime\mu}_{1},$ (3.1.14)
$\delta_{2}^{\mu}=\delta^{\prime\mu}_{2}+i\psi
P^{\prime}\delta^{\prime\mu}_{1}+i\psi W^{\prime}\delta^{\prime\mu}_{0},$
(3.1.15) $\delta_{3}^{\mu}=\delta^{\prime\mu}_{3}.$ (3.1.16)

Then we can rewrite $l^{\mu},n^{\mu},m^{\mu}$ as:

$l^{\mu}=\delta_{1}^{\mu},$ (3.1.17)
$n^{\mu}=\delta^{\prime\mu}_{0}+\frac{1}{2}\hat{A}(r)\delta^{\prime\mu}_{1},$
(3.1.18)
$m^{\mu}=\frac{1}{\sqrt{2\widehat{(A^{-1}C)}(r^{\prime},\bar{r^{\prime}})}}\left(\frac{1}{f_{1}}\delta^{\prime\mu}_{2}+\frac{i\psi
P^{\prime}}{f_{1}}\delta^{\prime\mu}_{1}+\frac{i\psi
W^{\prime}}{f_{1}}\delta^{\prime\mu}_{0}+\frac{i}{f_{2}}\delta^{\prime\mu}_{3}\right),$
(3.1.19)

where $\hat{A}(r)$ and $\widehat{A^{-1}C}(r^{\prime},\bar{r^{\prime}})$ are
the complexification of $A$ and $A^{-1}(r)C(r)$. In subsection 3.2, we will
focus on complexification. The function $\hat{A}(r)$ is obtained by preserving
the norm complex continuation for $A(r)$, namely,

$\hat{A}(r)=A(r^{\prime},\bar{r^{\prime}}),$ (3.1.20)

In Kerr-Newman metric, $A=1-\frac{2M-Q^{2}}{r^{2}}$ and
$\hat{A}=1-\frac{2Mr-Q^{2}}{r^{2}+a^{2}\cos^{2}\theta}$. More details of the
derivation of the Kerr-Newman metric in general relativity can be found in the
Appendix A.
From eqs.(3.1.17) to (3.1.19), we obtain that:

$l^{\mu}n^{\nu}+l^{\nu}n^{\mu}=\left[\begin{array}[]{cccc}0&1&0&0\\\
1&\hat{A}&0&0\\\ 0&0&0&0\\\ 0&0&0&0\end{array}\right],$ (3.1.21)

and

$m^{\mu}\bar{m}^{\nu}+m^{\nu}\bar{m}^{\mu}=\left[\begin{array}[]{cccc}\frac{1}{\widehat{(A^{-1}C)}}\frac{\psi^{2}W^{\prime
2}}{f_{1}^{2}}&\frac{1}{\widehat{(A^{-1}C)}}\frac{\psi^{2}W^{\prime}P^{\prime}}{f_{1}^{2}}&0&\frac{1}{\widehat{(A^{-1}C)}}\frac{\psi
W^{\prime}}{f_{1}f_{2}}\\\
\frac{1}{\widehat{(A^{-1}C)}}\frac{\psi^{2}W^{\prime}P^{\prime}}{f_{1}^{2}}&\frac{1}{\widehat{(A^{-1}C)}}\frac{\psi^{2}P^{\prime
2}}{f_{1}^{2}}&0&\frac{1}{\widehat{(A^{-1}C)}}\frac{\psi
P^{\prime}}{f_{1}f_{2}}\\\ 0&0&\frac{1}{\widehat{(A^{-1}C)}f_{1}^{2}}&0\\\
\frac{1}{\widehat{(A^{-1}C)}}\frac{\psi
W^{\prime}}{f_{1}f_{2}}&\frac{1}{\widehat{(A^{-1}C)}}\frac{\psi
P^{\prime}}{f_{1}f_{2}}&0&\frac{1}{\widehat{(A^{-1}C)}f_{2}^{2}}\end{array}\right],$
(3.1.22)

Combining (3.1.4), (3.1.21) and (3.1.22)

$g^{\mu\nu}=\left[\begin{array}[]{cccc}\frac{1}{\widehat{(A^{-1}C)}}\frac{(\psi
W^{\prime})^{2}}{f^{2}_{1}}&-1+\frac{1}{\widehat{(A^{-1}C)}}\frac{\psi^{2}P^{\prime}W^{\prime}}{f^{2}_{1}}&0&\frac{1}{\widehat{(A^{-1}C)}}\frac{\psi
W^{\prime}}{f_{1}f_{2}}\\\
-1+\frac{1}{\widehat{(A^{-1}C)}}\frac{\psi^{2}P^{\prime}W^{\prime}}{f^{2}_{1}}&-\hat{A}+\frac{1}{\widehat{(A^{-1}C)}}\frac{(\psi
P^{\prime})^{2}}{f^{2}_{1}}&0&\frac{1}{\widehat{(A^{-1}C)}}\frac{\psi
P^{\prime}}{f_{1}f_{2}}\\\ 0&0&\frac{1}{\widehat{(A^{-1}C)}f^{2}_{1}}&0\\\
\frac{1}{\widehat{(A^{-1}C)}}\frac{\psi
W^{\prime}}{f_{1}f_{2}}&\frac{1}{\widehat{(A^{-1}C)}}\frac{\psi
P^{\prime}}{f_{1}f_{2}}&0&\frac{1}{\widehat{(A^{-1}C)}}\frac{1}{f^{2}_{2}}\end{array}\right].$
(3.1.23)

Since the metric should be written in terms of line elements, we need to find
the inverse of the matrix (3.1.23), which is given by:

$g_{\mu\nu}=\left[\begin{array}[]{cccc}-\hat{A}&-1&0&\hat{A}aW^{\prime}\frac{f_{2}}{f_{1}}+\psi
P^{\prime}\frac{f_{2}}{f_{1}}\\\ -1&0&0&\psi W^{\prime}\frac{f_{2}}{f_{1}}\\\
0&0&\widehat{(A^{-1}C)}f_{1}^{2}&0\\\ \hat{A}\psi
W^{\prime}\frac{f_{2}}{f_{1}}+\psi P^{\prime}\frac{f_{2}}{f_{1}}&\psi
W^{\prime}\frac{f_{2}}{f_{1}}&0&-\hat{A}\psi^{2}W^{\prime
2}\frac{f_{2}^{2}}{f_{1}^{2}}+\widehat{(A^{-1}C)}f_{2}^{2}-2\psi^{2}P^{\prime}W^{\prime}\frac{f_{2}^{2}}{f_{1}^{2}}\end{array}\right].$
(3.1.24)

In order to cancel the (0,1),(1,0),(1,3),(3,1) components of matrix (3.1.24),
we need to introduce the following transformation:

$\left[\begin{array}[]{c}dt\\\ dr\\\ d\theta\\\
d\varphi\end{array}\right]=\left[\begin{array}[]{cccc}1&-M&0&0\\\ 0&1&0&0\\\
0&0&1&0\\\ 0&-N&0&1\end{array}\right]\left[\begin{array}[]{c}du\\\ dr\\\
d\theta\\\ d\varphi\end{array}\right],$ (3.1.25)

where

$M=\frac{-\psi^{2}P^{\prime}W^{\prime}+\widehat{(A^{-1}C)}f_{1}^{2}}{\hat{A}\widehat{(A^{-1}C)}+\psi^{2}P^{\prime
2}},$ (3.1.26) $N=\frac{\psi
P^{\prime}}{(\hat{A}\widehat{(A^{-1}C)}+\psi^{2}P^{\prime
2})\left(-\frac{f_{2}}{f_{1}}\right)}.$ (3.1.27)

Applying transformation (3.1.25), we can obtain

$g_{\mu\nu}=\left[\begin{array}[]{cccc}-\hat{A}&0&0&\hat{A}\psi
W^{\prime}\frac{f_{2}}{f_{1}}+\psi P^{\prime}\frac{f_{2}}{f_{1}}\\\
0&\frac{\widehat{(A^{-1}C)}f_{1}^{2}}{\psi^{2}P^{\prime
2}+\hat{A}(\widehat{A^{-1}C})}&0&0\\\ 0&0&\widehat{(A^{-1}C)}f_{1}^{2}&0\\\
\hat{A}\alpha W^{\prime}\frac{f_{2}}{f_{1}}+\psi
P^{\prime}\frac{f_{2}}{f_{1}}&0&0&-\hat{A}\psi^{2}W^{\prime
2}\frac{f_{2}^{2}}{f_{1}^{2}}+\widehat{(A^{-1}C)}f_{2}^{2}-2\psi^{2}P^{\prime}W^{\prime}\frac{f_{2}^{2}}{f_{1}^{2}}\end{array}\right].$
(3.1.28)

Finally, the metric in term of line elements is given by:

$\begin{split}ds^{2}=&-\hat{A}dt^{2}+\frac{\widehat{(A^{-1}C)}f_{1}^{2}}{\psi^{2}P^{\prime
2}+\hat{C}}dr^{2}+\widehat{(A^{-1}C)}f_{1}^{2}d\theta^{2}+2\left(\hat{A}\alpha
W^{\prime}\frac{f_{2}}{f_{1}}+\psi
P^{\prime}\frac{f_{2}}{f_{1}}\right)dtd\varphi\\\
&+\left(-\hat{A}\psi^{2}W^{\prime
2}\frac{f_{2}^{2}}{f_{1}^{2}}+\widehat{(A^{-1}C)}f_{2}^{2}-2\psi^{2}P^{\prime}W^{\prime}\frac{f_{2}^{2}}{f_{1}^{2}}\right)d\varphi^{2},\end{split}$
(3.1.29)

where $\psi$ has been introduced as a rotation parameter in (3.1.10).

###  3.2 Kerr-like metric in DFT

In this section, we first apply the metric derivation method from the previous
section to DFT metric. Then we give the complexification of $A$, $A^{-1}C$ and
$C$.
From (2.2.21), we can get

$A(r)=\left(\frac{r-c_{+}}{r-c_{-}}\right)^{\frac{a}{c_{+}-c_{-}}}\qquad
C(r)=(r-c_{+})(r-c_{-}),$ (3.2.1)

For the Kerr-like metric in DFT, we set:

$r^{\prime}=r+i\psi\cos{\theta}\qquad u^{\prime}=u-i\psi\cos{\theta}.$ (3.2.2)

Taking the derivative of (3.2.2), we can get

$dr^{\prime}=dr-i\psi\sin{\theta}\qquad du^{\prime}=u+i\psi\sin{\theta},$
(3.2.3)

so we have

$f_{1}=1,\quad f_{2}=\sin\theta\quad\text{and}\quad
P^{\prime}=-\sin\theta,\quad W^{\prime}=\sin\theta.$ (3.2.4)

Insert (3.2.4) into (3.1.28), then we have

$g_{\mu\nu}=\left[\begin{array}[]{cccc}-\hat{A}&0&0&\psi\sin^{2}{\theta}(\hat{A}-1)\\\
0&\frac{\widehat{A^{-1}C}}{\hat{A}\widehat{(A^{-1}C)}+\psi^{2}\sin^{2}\theta}&0&0\\\
0&0&\widehat{(A^{-1}C)}&0\\\
\psi\sin^{2}{\theta}(\hat{A}-1)&0&0&-\hat{A}\psi^{2}\sin^{4}{\theta}+\widehat{(A^{-1}C)}\sin^{2}{\theta}+2\psi^{2}\sin^{4}{\theta}\end{array}\right].$
(3.2.5)

where $\hat{A}$, $\widehat{(A^{-1}C)}$ will be given by (3.2.10) and (3.2.11).
Moreover, considering (3.1.26) and (3.1.27), we can obtain the corresponding
$M$ and $N$ in ’Kerr-like’ metric directly,

$M=\frac{\widehat{(A^{-1}C)}+\psi^{2}\sin^{2}\theta}{\hat{A}\widehat{(A^{-1}C)}+\psi^{2}\sin^{2}\theta},$
(3.2.6) $N=\frac{\psi}{\hat{A}\widehat{(A^{-1}C)}+\psi^{2}\sin^{2}\theta}.$
(3.2.7)

In order to find the metric without dilaton, we also need to do
complexification to get $\hat{A}$ and $\widehat{A^{-1}C}$. For easier
calculation, we first consider the special condition with $c_{-}=0$:

$A(r)|_{c_{-}=0}=\left(1-\frac{\sqrt{a^{2}+b^{2}}}{r}\right)^{\frac{a}{\sqrt{a^{+}b^{2}}}}.$
(3.2.8)

The complexification of $A(r)|_{c_{-}=0}$ is:

$\hat{A}(r)|_{c_{-}=0}=\left(\frac{r^{2}+\psi^{2}\cos^{2}{\theta}-r\sqrt{a^{2}+b^{2}}}{r^{2}+\psi^{2}\cos^{2}{\theta}}\right)^{\frac{a}{\sqrt{a^{+}b^{2}}}}.$
(3.2.9)

We replace $r$ by $r-c_{-}$, which also satisfy (3.2.3), so we can get:

$\hat{A}(r)=\left(\frac{(r-c_{-})^{2}+\psi^{2}\cos^{2}{\theta}-(r-c_{-})\sqrt{a^{2}+b^{2}}}{(r-c_{-})^{2}+\psi^{2}\cos^{2}{\theta}}\right)^{\frac{a}{\sqrt{a^{2}+b^{2}}}},$
(3.2.10)

Similarly, we can get the complexification of $A^{-1}C$ and $C$:

$\widehat{(A^{-1}C)}=((r-c_{-})^{2}+\psi^{2}\cos^{2}{\theta}-(r-c_{-})\sqrt{a^{2}+b^{2}})^{1-\frac{a}{\sqrt{a^{2}+b^{2}}}}((r-c_{-})^{2}+\psi^{2}\cos^{2}{\theta})^{\frac{a}{\sqrt{a^{2}+b^{2}}}},$
(3.2.11)
$\hat{C}=(r-c_{-})^{2}+\psi^{2}\cos^{2}{\theta}-(r-c_{-})\sqrt{a^{2}+b^{2}}.$
(3.2.12)

When we consider the complete Schwarzschild metric in string frame KPS2017

$\begin{split}ds^{2}&=-(\gamma_{+}(\frac{r-c_{+}}{r-c_{-}})^{\frac{a+b}{\sqrt{a^{2}+b^{2}}}}+\gamma_{-}(\frac{r-c_{+}}{r-c_{-}})^{\frac{a-b}{\sqrt{a^{2}+b^{2}}}})dt^{2}\\\
&+(\gamma_{+}(\frac{r-c_{+}}{r-c_{-}})^{\frac{-a+b}{\sqrt{a^{2}+b^{2}}}}+\gamma_{-}(\frac{r-c_{+}}{r-c_{-}})^{-\frac{a+b}{\sqrt{a^{2}+b^{2}}}})dr^{2}\\\
&+(\gamma_{+}(r-c_{+})^{1+\frac{-a+b}{\sqrt{a^{2}+b^{2}}}}(r-c_{-})^{1+\frac{a-b}{a^{2}+b^{2}}}+\gamma_{-}(r-c_{-})^{1-\frac{a+b}{\sqrt{a^{2}+b^{2}}}}(r-c_{+})^{1+\frac{a+b}{\sqrt{a^{2}+b^{2}}}}d\theta^{2}\\\
&+(\gamma_{+}(r-c_{+})^{1+\frac{-a+b}{\sqrt{a^{2}+b^{2}}}}(r-c_{-})^{1+\frac{a-b}{a^{2}+b^{2}}}+\gamma_{-}(r-c_{-})^{1-\frac{a+b}{\sqrt{a^{2}+b^{2}}}}(r-c_{+})^{1+\frac{a+b}{\sqrt{a^{2}+b^{2}}}}\sin^{2}\theta
d\varphi^{2},\end{split}$ (3.2.13)

we cannot directly give a complete metric, that is, we cannot give the
specific form of the dilaton corresponding to this metric. Now we denote the
corresponding dilaton as $e^{2\hat{\phi}}$.
Now, based on EA and JC 2001 , we can write the general Kerr-like metric in
DFT in string frame is

$\begin{split}ds^{2}&=e^{2\hat{\phi}}(-\hat{A}dt^{2}+\left(\frac{\widehat{(A^{-1}C)}}{{\hat{A}}\widehat{(A^{-1}C)}+\psi^{2}\sin^{2}\theta}\right)dr^{2}+\widehat{(A^{-1}C)}d\theta^{2}+(-\hat{A}\psi^{2}\sin^{4}{\theta}+\widehat{(A^{-1}C)}\sin^{2}{\theta}\\\
&+2\psi^{2}\sin^{4}{\theta})d\varphi^{2}+\psi\sin^{2}{\theta}(\hat{A}-1)2dtd\varphi).\end{split}$
(3.2.14)

###  3.3 Discussion about dilaton and $H$-flux

In this subsection, we consider the conditions which $H$-flux and dilaton
should satisfy. For Kerr-like metric, we can set the general form of
$B_{(2)}$:

$B_{(2)}=Udt\wedge dr+Vdt\wedge d\theta+Wdt\wedge d\varphi+Xdr\wedge
d\theta+Ydr\wedge d\varphi+Zd\theta\wedge d\varphi$ (3.3.1)

where $U$,$V$,$W$,$X$,$Y$,$Z$ are all the functions of $r$ and $\theta$. The
explicit expressions for these functions should be determined from string
theory KPS2017 ; K.S. STELLE 1997 , but we will not discuss it in this paper.
From (3.3.1) we give $H_{(3)}$:

$H_{(3)}=\left(\frac{\partial U}{\partial\theta}-\frac{\partial V}{\partial
r}\right)dt\wedge dr\wedge d\theta-\frac{\partial W}{\partial r}dt\wedge
dr\wedge d\varphi-\frac{\partial W}{\partial\theta}dt\wedge d\theta\wedge
d\varphi+\left(-\frac{\partial Y}{\partial\theta}+\frac{\partial Z}{\partial
r}\right)dr\wedge d\theta\wedge d\varphi,$ (3.3.2)

For subsequent calculations, we assume that the dilaton $\hat{\phi}$ is a
function of $r$ and $\theta$ due to Kerr-like metric. Now we calculate (2.2.3)

$\begin{split}d\star(e^{2\hat{\phi}}H_{(3)})&=d(e^{2\hat{\phi}}\left(\frac{\partial
U}{\partial\theta}-\frac{\partial V}{\partial
r}\right)(-\sqrt{|g|}g^{\varphi\varphi})d\varphi-e^{2\hat{\phi}}\frac{\partial
W}{\partial r}\sqrt{|g|}g^{\theta\theta}d\theta\\\
&-e^{2\hat{\phi}}\frac{\partial
W}{\partial\theta}\sqrt{|g|}g^{rr}dr+e^{2\hat{\phi}}\left(-\frac{\partial
Y}{\partial\theta}+\frac{\partial Z}{\partial r}\right)\sqrt{|g|}g^{tt}dt)\\\
&=-\frac{\partial}{\partial r}\left(e^{2\hat{\phi}}\left(-\frac{\partial
Y}{\partial\theta}+\frac{\partial Z}{\partial
r}\right)\sqrt{|g|}g^{tt}\right)dt\wedge
dr-\frac{\partial}{\partial\theta}\left(e^{2\hat{\phi}}\left(-\frac{\partial
Y}{\partial\theta}+\frac{\partial Z}{\partial
r}\right)\sqrt{|g|}g^{tt}\right)dt\wedge d\theta\\\
&+\left(-\frac{\partial}{\partial\theta}\left(-e^{2\hat{\phi}}\frac{\partial
W}{\partial\theta}\sqrt{|g|}g^{rr}\right)+\frac{\partial}{\partial
r}\left(-e^{2\hat{\phi}}\frac{\partial W}{\partial
r}\sqrt{|g|}g^{\theta\theta}\right)\right)dr\wedge d\theta\\\
&+\frac{\partial}{\partial r}\left(e^{2\hat{\phi}}\left(\frac{\partial
U}{\partial\theta}-\frac{\partial V}{\partial
r}\right)(-\sqrt{|g|}g^{\varphi\varphi})\right)dr\wedge
d\varphi+\frac{\partial}{\partial\theta}\left(e^{2\hat{\phi}}\left(\frac{\partial
U}{\partial\theta}-\frac{\partial V}{\partial
r}\right)(-\sqrt{|g|}g^{\varphi\varphi})\right)d\theta\wedge
d\varphi.\end{split}$ (3.3.3)

We calculate some components in metric:

$\begin{split}g^{tt}&=e^{-2\hat{\phi}}\frac{-\hat{C}+\psi^{2}(-2+\hat{A})\sin^{2}\theta}{\hat{A}\hat{C}+\psi^{2}\sin^{2}\theta}\\\
g^{rr}&=e^{-2\hat{\phi}}\frac{\hat{A}(\hat{C}+\psi^{2}\sin^{2}\theta)}{\hat{C}}\\\
g^{\theta\theta}&=e^{-2\hat{\phi}}\frac{\hat{A}}{\hat{C}}\\\
g^{\varphi\varphi}&=e^{-2\hat{\phi}}\frac{\hat{A}\csc^{4}\theta}{\psi^{2}+\hat{A}\csc^{2}\theta\hat{C}}\\\
\sqrt{|g|}&=e^{4\hat{\phi}}\frac{\hat{C}\sin^{2}\theta}{\hat{A}}\sqrt{\frac{\hat{A}\hat{C}+\psi^{2}\sin^{2}\theta}{\hat{C}+\psi^{2}\sin^{2}\theta}}\end{split}$
(3.3.4)

and some terms in $d\star(e^{2\hat{\phi}}H_{(3)})=0$:

$\begin{split}e^{2\hat{\phi}}\left(-\frac{\partial
Y}{\partial\theta}+\frac{\partial Z}{\partial
r}\right)\sqrt{|g|}g^{tt}&=\left(-\frac{\partial
Y}{\partial\theta}+\frac{\partial Z}{\partial r}\right)P_{1}\\\
e^{2\hat{\phi}}\frac{\partial
W}{\partial\theta}\sqrt{|g|}g^{rr}&=-\frac{\partial W}{\partial\theta}P_{2}\\\
e^{2\hat{\phi}}\frac{\partial W}{\partial
r}\sqrt{|g|}g^{\theta\theta}&=\frac{\partial W}{\partial r}P_{3}\\\
e^{2\hat{\phi}}\left(\frac{\partial U}{\partial\theta}-\frac{\partial
V}{\partial r}\right)(-\sqrt{|g|}g^{\varphi\varphi})&=\left(\frac{\partial
U}{\partial\theta}-\frac{\partial V}{\partial r}\right)P_{4}\end{split}$
(3.3.5)

where

$\begin{split}P_{1}&=\left(\frac{-\hat{C}+\psi^{2}(-2+\hat{A})\sin^{2}\theta}{\hat{A}\hat{C}+\psi^{2}\sin^{2}\theta}\right)\left(\frac{\hat{C}\sin^{2}\theta}{\hat{A}}\right)\sqrt{\frac{\hat{A}\hat{C}+\psi^{2}\sin^{2}\theta}{\hat{C}+\psi^{2}\sin^{2}\theta}}\\\
P_{2}&=\left(\frac{\hat{A}(\hat{C}+\psi^{2}\sin^{2}\theta)}{\hat{C}}\right)\left(\frac{\hat{C}\sin^{2}\theta}{\hat{A}}\right)\sqrt{\frac{\hat{A}\hat{C}+\psi^{2}\sin^{2}\theta}{\hat{C}+\psi^{2}\sin^{2}\theta}}\\\
P_{3}&=\left(\frac{\hat{A}}{\hat{C}}\right)\left(\frac{\hat{C}\sin^{2}\theta}{\hat{A}}\right)\sqrt{\frac{\hat{A}\hat{C}+\psi^{2}\sin^{2}\theta}{\hat{C}+\psi^{2}\sin^{2}\theta}}\\\
P_{4}&=\left(\frac{\hat{A}\csc^{4}\theta}{\psi^{2}+\hat{A}\csc^{2}\theta\hat{C}}\right)\left(\frac{\hat{C}\sin^{2}\theta}{\hat{A}}\right)\sqrt{\frac{\hat{A}\hat{C}+\psi^{2}\sin^{2}\theta}{\hat{C}+\psi^{2}\sin^{2}\theta}}\end{split}$
(3.3.6)

Then we need to solve a series of equations from (3.3.2)

$\begin{split}\left(-\frac{\partial Y}{\partial\theta}+\frac{\partial
Z}{\partial r}\right)P_{1}&=c_{1}\\\ \left(\frac{\partial
U}{\partial\theta}-\frac{\partial V}{\partial r}\right)P_{4}&=c_{2}\\\
\frac{\partial}{\partial\theta}\left(\frac{\partial
W}{\partial\theta}\right)P_{2}&=\frac{\partial}{\partial
r}\left(\frac{\partial W}{\partial r}\right)P_{3}\end{split}$ (3.3.7)

where $c_{1}$,$c_{2}$ are the constants. We can also calculate the components
of $H_{\mu\rho\sigma}H_{\nu}^{\ \rho\sigma}$:

$\begin{split}H_{t\rho\sigma}H_{t}^{\ \rho\sigma}&=2\left(\left(\frac{\partial
U}{\partial\theta}-\frac{\partial V}{\partial
r}\right)^{2}g^{\theta\theta}g^{rr}+\left(\frac{\partial W}{\partial
r}\right)^{2}g^{rr}g^{\varphi\varphi}+\left(\frac{\partial
W}{\partial\theta}\right)^{2}g^{\theta\theta}g^{\varphi\varphi}\right)\\\
H_{t\rho\sigma}H_{r}^{\ \rho\sigma}&=-2\frac{\partial
W}{\partial\theta}\left(-\frac{\partial Y}{\partial\theta}+\frac{\partial
Z}{\partial r}\right)g^{\theta\theta}g^{\varphi\varphi}\\\
H_{t\rho\sigma}H_{\theta}^{\ \rho\sigma}&=-2\frac{\partial
W}{\partial\theta}\left(-\frac{\partial Y}{\partial\theta}+\frac{\partial
Z}{\partial r}\right)g^{rr}g^{\varphi\varphi}\\\ H_{t\rho\sigma}H_{\varphi}^{\
\rho\sigma}&=2\left(\frac{\partial U}{\partial\theta}-\frac{\partial
V}{\partial r}\right)\left(-\frac{\partial Y}{\partial\theta}+\frac{\partial
Z}{\partial r}\right)g^{rr}g^{\varphi\varphi}\\\ H_{r\rho\sigma}H_{r}^{\
\rho\sigma}&=2\left(\left(\frac{\partial U}{\partial\theta}-\frac{\partial
V}{\partial r}\right)^{2}g^{tt}g^{\theta\theta}+\left(\frac{\partial
W}{\partial\theta}\right)^{2}g^{tt}g^{\varphi\varphi}+\left(-\frac{\partial
Y}{\partial\theta}+\frac{\partial Z}{\partial
r}\right)^{2}g^{\theta\theta}g^{\ \varphi\varphi}\right)\\\
H_{r\rho\sigma}H_{\theta}^{\ \rho\sigma}&=2\frac{\partial
W}{\partial\theta}\frac{\partial W}{\partial r}g^{tt}g^{\varphi\varphi}\\\
H_{r\rho\sigma}H_{\varphi}^{\ \rho\sigma}&=2(\left(\frac{\partial
U}{\partial\theta}-\frac{\partial V}{\partial r}\right)\left(\frac{\partial
W}{\partial r}\right)g^{tt}g^{\theta\theta}\\\
H_{\theta\rho\sigma}H_{\theta}^{\ \rho\sigma}&=2\left(\left(\frac{\partial
U}{\partial\theta}-\frac{\partial V}{\partial
r}\right)^{2}g^{tt}g^{rr}+\left(\frac{\partial
W}{\partial\theta}\right)^{2}g^{tt}g^{\varphi\varphi}+\left(-\frac{\partial
Y}{\partial\theta}+\frac{\partial Z}{\partial
r}\right)^{2}g^{rr}g^{\varphi\varphi}\right)\\\
H_{\theta\rho\sigma}H_{\varphi}^{\ \rho\sigma}&=-2\left(\frac{\partial
U}{\partial\theta}-\frac{\partial V}{\partial r}\right)\frac{\partial
W}{\partial r}g^{tt}g^{rr}\\\ H_{\varphi\rho\sigma}H_{\varphi}^{\
\rho\sigma}&=2\left(\left(\frac{\partial W}{\partial
r}\right)^{2}g^{tt}g^{rr}+\left(\frac{\partial
W}{\partial\theta}\right)^{2}g^{tt}g^{\theta\theta}+\left(-\frac{\partial
Y}{\partial\theta}+\frac{\partial Z}{\partial
r}\right)^{2}g^{rr}g^{\theta\theta}\right)\end{split}$ (3.3.8)

Now we deal with $\nabla_{\mu}(\partial_{\nu}\phi)$:

$\left[\begin{array}[]{cccc}-(\Gamma^{r}_{tt}\partial_{r}\phi+\Gamma^{\theta}_{tt}\partial_{\theta}\phi)&0&0&-(\Gamma^{r}_{t\varphi}\partial_{r}\phi+\Gamma^{\theta}_{t\varphi}\partial_{\theta}\phi)\\\
0&\partial_{r}\partial_{r}\phi-\Gamma^{r}_{rr}\partial_{r}\phi-\Gamma^{\theta}_{rr}\partial_{\theta}\phi&\partial_{\theta}\partial_{r}\phi-\Gamma^{r}_{r\theta}\partial_{r}\phi-\Gamma^{\theta}_{r\theta}\partial_{\theta}\phi&0\\\
0&\partial_{r}\partial_{\theta}\phi-\Gamma^{r}_{r\theta}\partial_{r}\phi-\Gamma^{\theta}_{r\theta}\partial_{\theta}\phi&\partial_{r}\partial_{r}\phi-\Gamma^{r}_{\theta\theta}\partial_{r}\phi-\Gamma^{\theta}_{\theta\theta}\partial_{\theta}\phi&0\\\
-(\Gamma^{r}_{t\varphi}\partial_{r}\phi+\Gamma^{\theta}_{t\varphi}\partial_{\theta}\phi)&0&0&-(\Gamma^{r}_{\varphi\varphi}\partial_{r}\phi+\Gamma^{\theta}_{\varphi\varphi}\partial_{\theta}\phi)\end{array}\right],$
(3.3.9)

where $\Gamma$ is Christoffel symbol. We will list the non-vanishing
components of the Christoffel symbol in Appendix B. Regarding the curvature
tensor, due to its complexity, we will not list explicit results, but we can
piont out that $R_{tt}$, $R_{rr}$, $R_{\theta\theta}$, $R_{\varphi\varphi}$,
$R_{t\varphi}=R_{\varphi t}$ and $R_{r\theta}=R_{\theta r}$ are the non-
vanishing components of Ricci curvature. Based on (3.3.8) and (3.3.9),we have:

$\begin{split}\frac{\partial W}{\partial\theta}\left(-\frac{\partial
Y}{\partial\theta}+\frac{\partial Z}{\partial r}\right)&=0\\\
\left(\frac{\partial U}{\partial\theta}-\frac{\partial V}{\partial
r}\right)\frac{\partial W}{\partial r}&=0\end{split}$ (3.3.10)

We have $\frac{\partial W}{\partial\theta}=0$ or $-\frac{\partial
Y}{\partial\theta}+\frac{\partial Z}{\partial r}=0$ and $\frac{\partial
U}{\partial\theta}-\frac{\partial V}{\partial r}=0$ or $\frac{\partial
W}{\partial r}=0$. Since this metric can degenerate to the solution of
spherically symmetric ansatz (under the condition of $\psi=0$), we have:

$H_{(3)}|_{\psi=0}=h\sin\theta dt\wedge d\theta\wedge d\varphi$ (3.3.11)

##  4 Black hole thermodynamics and Hawking radiation

In this section, we elaborate on the first law of thermodynamics of black
holes in DFT based on subsection 2.3. Then we give the explicit form of the
first law of thermodynamics of the metric we have obtained in subsection 3.2.
Afterwards, we also calculate the Hawking radiation for the $4D$ DFT Kerr-like
metric in detail.

###  4.1 The first law of the black-hole thermodynamics of the DFT Kerr-like
metric

Let us consider the thermodynamics of black holes of the $4D$ DFT Kerr-like
metric, which is given by (3.1.29).
Part 1. Entropy term:
To find the entropy term, we need to obtain the event horizon, which
corresponds to ”Cauchy surface” mentioned in subsection 2.3. The first step is
to find its radius, namely, the pole of (3.1.26) and (3.1.27):

$\hat{A}\widehat{(A^{-1}C)}+\psi^{2}\sin^{2}\theta=0.$ (4.1.1)

By substituting the two terms $\hat{A}$ (3.2.10) and $\widehat{A^{-1}C}$
(3.2.11) into (4.1.1), we can obtain that

$\left((r-c_{-})^{2}+\psi^{2}\cos^{2}{\theta}-(r-c_{-})\sqrt{a^{2}+b^{2}}\right)+\psi^{2}\sin^{2}\theta=0.$
(4.1.2)

Similar to the Kerr metric in general relativity, this equation has a maximum
solution, denoted as $r=r_{+}$, which can lead to an event horizon, on which
thermodynamics exists. Now, we give the event horizon in the following
equation:

$\mathcal{A}=\int_{r_{+}}\sqrt{g_{\theta\theta}g_{\varphi\varphi}}d\theta
d\varphi=\int_{r_{+}}e^{2\hat{\phi}}\sqrt{{\widehat{(A^{-1}C)}}(-\hat{A}\psi^{2}\sin^{4}{\theta}+\widehat{(A^{-1}C)}\sin^{2}{\theta}+2\psi^{2}\sin^{4}{\theta})}$
(4.1.3)

From Bekenstein-Hawking entropy formula, we can get the entropy

$S=\frac{1}{4G_{DFT}}\mathcal{A}$ (4.1.4)

Part 2. $\delta M^{\prime}$ and $\not{\delta}W$:
As discussed earlier, we can give the remaining two items $\delta M^{\prime}$
and $\not{\delta}W$. Before that, we need to get $\Omega^{I}$ and $J_{I}$.
Similar to general relativity R.M.Wald 1984 , we can directly have

$J_{I}\equiv(\frac{1}{16\pi
G_{DFT}})\int_{t=t_{0},r=r_{+}}e^{2\hat{\phi}}\epsilon_{abcd}\nabla^{c}\xi^{d}.$
(4.1.5)

Moverover, as a Killing field, $\xi^{d}$ is defined as

$\xi^{d}=\frac{\partial}{\partial
t}+\Omega^{I}\frac{\partial}{\partial\varphi^{I}}$ (4.1.6)

To give $\Omega^{I}$, we need to first define $\Omega$. By taking the limit
$r\rightarrow r_{+}$, $\Omega$ approaches $\Omega^{I}$, namely,

$\begin{split}\Omega^{I}&=\lim_{r\rightarrow r_{+}}\Omega=\lim_{r\rightarrow
r_{+}}\frac{d\varphi}{dt}=\lim_{r\rightarrow
r_{+}}\left(-\frac{g_{t\varphi}}{g_{\varphi\varphi}}\right)\\\
&=\lim_{r\rightarrow
r_{+}}\left(-\frac{\psi\sin^{2}\theta(\hat{A}-1)}{-\hat{A}\psi^{2}\sin^{4}\theta+\widehat{(A^{-1}C)}\sin^{2}\theta+2\psi^{2}\sin^{4}\theta}\right).\end{split}$
(4.1.7)

and based on (2.3.20), we can define $\not{\delta}W$, which is $O(n,n)$
invariant thermodynamic work contribution corresponding to metric (3.2.5)

$\begin{split}\not{\delta}W=&-\frac{1}{16\pi
G_{DFT}}\int_{t=t_{0},r=r_{+}}d^{2}x[\tilde{\lambda}_{\rho}\delta(e^{2\hat{\phi}}\mathcal{H}^{\mu\nu\rho}+\tilde{\Lambda}^{M}\delta(e^{2\hat{\phi}}H_{MN}\mathcal{F}^{\mu\nu
N})]\epsilon_{\mu\nu}\\\ &-\Omega^{I}\left(\frac{1}{16\pi
G_{DFT}}\right)\delta\int_{t=t_{0},r=r_{+}}e^{2\hat{\phi}}\epsilon_{abcd}\nabla^{c}\xi^{d}.\end{split}$
(4.1.8)

Then we can get $\delta M^{\prime}$, which is

$\begin{split}\delta M^{\prime}=&\delta M-\Omega^{I}\delta J_{I}=\delta
M-\Omega^{I}\left(\frac{1}{16\pi
G_{DFT}}\right)\delta\int_{t=t_{0},r=r_{+}}\epsilon_{abcd}\nabla^{c}\xi^{d}\\\
&=\frac{1}{16\pi
G_{DFT}}\delta\int_{t=t_{0},r=r_{+}}d^{2}xe^{2\hat{\phi}}(-2g^{\rho[\mu}\nabla_{\rho}(\xi^{\prime})^{\nu]}-\widetilde{\lambda}_{p}H^{\mu\nu\rho}-\widetilde{\Lambda}^{M}H_{MN}\mathcal{F}^{\mu\nu
N})\epsilon_{\mu\nu}\\\ &-\Omega^{I}\left(\frac{1}{16\pi
G_{DFT}}\right)\delta\int_{t=t_{0},r=r_{+}}e^{2\hat{\phi}}\epsilon_{abcd}\nabla^{c}\xi^{d}.\end{split}$
(4.1.9)

Moreover, based on discussion about the ”Komar term” ASA and CDAB 2017 , we
can rewrite (4.1.9) as

$\begin{split}\delta M^{\prime}&=\frac{\kappa}{16\pi
G_{DFT}}\delta\int_{t=t_{0},r=r_{+}}e^{2\hat{\phi}}\sqrt{g_{\theta\theta}g_{\varphi\varphi}}\\\
&-\frac{1}{16\pi
G_{DFT}}\int_{t=t_{0},r=r_{+}}d^{2}x[\tilde{\lambda}_{\rho}\delta(e^{2\hat{\phi}}\mathcal{H}^{\mu\nu\rho}+\tilde{\Lambda}^{M}\delta(e^{2\hat{\phi}}\mathcal{F}^{\mu\nu
N})]\epsilon_{\mu\nu}\\\ &-\Omega^{I}\left(\frac{1}{16\pi
G_{DFT}}\right)\delta\int_{t=t_{0},r=r_{+}}e^{2\hat{\phi}}\epsilon_{abcd}\nabla^{c}\xi^{d},\\\
&=\frac{\kappa}{2\pi}\delta S-\frac{1}{16\pi
G_{DFT}}\int_{t=t_{0},r=r_{+}}d^{2}x[\tilde{\lambda}_{\rho}\delta(e^{2\hat{\phi}}\mathcal{H}^{\mu\nu\rho}+\tilde{\Lambda}^{M}\delta(e^{2\hat{\phi}}H_{MN}\mathcal{F}^{\mu\nu
N})]\epsilon_{\mu\nu}\\\ &-\Omega^{I}\left(\frac{1}{16\pi
G_{DFT}}\right)\delta\int_{t=t_{0},r=r_{+}}e^{2\hat{\phi}}\epsilon_{abcd}\nabla^{c}\xi^{d},\end{split}$
(4.1.10)

Combining (4.1.8) and (4.1.10), the first law of black-hole thermodynamics
(2.3.21) can be obtained.

###  4.2 Hawking radiation as tunneling

In this subsection, we study the Hawking radiation of the Kerr-metric we have
obtained in (3.1.29). In line with the analysis of Parikh&Wilczek:2000 ;
Johnson&March-Russell:2020 , we study the semiclassical tunnelling of
particles through the horizon of black holes in this section. Further
discussion can be seen in Kraus&Wilczek:1995 ; Parikh:2004 ; Parikh:2003 ;
Vanzo&Acquaviva&Criscienzo:2011 . On the one hand, the rate of emission
$\Gamma$ will have exponential part given by Johnson&March-Russell:2020

$\Gamma\sim\exp(-2ImS),$ (4.2.1)

where $S$ is the tunnelling action of particles and $ImS$ is the imaginary
part of the tunnelling action $S$. On the other hand, according to the Planck
radiation law, the emitted rate $\Gamma$ of particles with frequency $\omega$
can be written as Johnson&March-Russell:2020

$\Gamma\sim\exp(-\omega/T_{BH}).$ (4.2.2)

Therefore, combining eqs.(4.2.1) and (4.2.2), the temperature at which the
black hole radiates can be read off Johnson&March-Russell:2020 :

$T_{BH}=\frac{\omega}{2ImS}.$ (4.2.3)

In this section, we study Hawking radiation process in Painlevé–Gullstrand
coordinates. Firstly, we will rewrite the metric (3.2.5) in
Painlevé–Gullstrand coordinates. We define

$t_{s}=t-F(r,\rho),$ (4.2.4)

where $F(r,\rho)$ is a function of $r$ and $\rho$. Then we have

$dt=dt_{s}+dF(r,\rho)=dt_{s}+\frac{\partial F(r,\rho)}{\partial
r}dr+\frac{\partial F(r,\rho)}{\partial\rho}d\rho,$ (4.2.5)

and

$\begin{split}dt^{2}=dt^{2}_{s}+\left(\frac{\partial F(r,\rho)}{\partial
r}\right)^{2}dr^{2}+\left(\frac{\partial
F(r,\rho)}{\partial\rho}\right)^{2}d\rho^{2}\\\ +2\frac{\partial
F(r,\rho)}{\partial r}dt_{s}dr+2\frac{\partial
F(r,\rho)}{\partial\rho}dt_{s}d\rho+2\frac{\partial F(r,\rho)}{\partial
r}\frac{\partial F(r,\rho)}{\partial\rho}drd\rho.\\\ \end{split}$ (4.2.6)

The metric (3.2.5) can be written as

$ds^{2}=g_{00}dt^{2}+g_{11}dr^{2}+g_{22}d\rho^{2}+g_{33}d\theta^{2}+2g_{03}dtd\theta.$
(4.2.7)

In order to obtain the metric in Painlevé–Gullstrand coordinates, we can set
Jiang&Wu&Cai:2006

$d\theta=-\frac{g_{03}}{g_{33}}dt.$ (4.2.8)

Then consider (4.2.5) and (4.2.6) and ignore $\rho$ sector for simplicity, we
have

$ds^{2}=\left(g_{00}-\frac{g^{2}_{03}}{g_{33}}\right)dt^{2}_{s}+\left[\left(g_{00}-\frac{g^{2}_{03}}{g_{33}}\right)\left(\frac{\partial
F(r,\rho)}{\partial
r}\right)^{2}+g_{11}\right]dr^{2}+2\left(g_{00}-\frac{g^{2}_{03}}{g_{33}}\right)\frac{\partial
F(r,\rho)}{\partial r}dt_{s}dr.$ (4.2.9)

If we set

$\left(g_{00}-\frac{g^{2}_{03}}{g_{33}}\right)\left(\frac{\partial
F(r,\rho)}{\partial r}\right)^{2}+g_{11}=1,$ (4.2.10)

then we have

$\frac{\partial F(r,\rho)}{\partial
r}=\sqrt{\frac{1-g_{11}}{g_{00}-\frac{g^{2}_{03}}{g_{33}}}}.$ (4.2.11)

Then the metric (4.2.9) can be written as

$ds^{2}=-\left(\frac{g^{2}_{03}}{g_{33}}-g_{00}\right)dt^{2}_{s}+2\sqrt{\left(g_{00}-\frac{g^{2}_{03}}{g_{33}}\right)\left(1-g_{11}\right)}dt_{s}dr+dr^{2}.$
(4.2.12)

In the following, we take $dt^{2}_{s}=dt^{2}$ for convenience, namely,

$ds^{2}=-\left(\frac{g^{2}_{03}}{g_{33}}-g_{00}\right)dt^{2}+2\sqrt{\left(g_{00}-\frac{g^{2}_{03}}{g_{33}}\right)\left(1-g_{11}\right)}dtdr+dr^{2}.$
(4.2.13)

The action for a particle moving freely in a curved background can be written
as:

$S=\int p_{\mu}dx^{\mu},$ (4.2.14)

with

$p_{\mu}=mg_{\mu\nu}\frac{dx^{\nu}}{d\sigma},$ (4.2.15)

where $\sigma$ is an affine parameter along the worldline of the particle,
chosen so that $p_{\mu}$ coincides with the physical 4-momentum of the
particle. For a massive particle, this requires that $d\sigma=d\tau/m$, with
$\tau$ the proper time Johnson&March-Russell:2020 ; YL2022 . For simplicity,
in this paper we only consider the case of massless scalar field $\Phi$ and
ignore the angular directions. Following the spirit of Johnson&March-
Russell:2020 , we will obtain the formula of Hawking temperature for the
’Kerr-like’ solution in $D=4$ double field theory.
The radial dynamics of massless particles in 4d spacetime are determined by
the equations:

$\left(\frac{g^{2}_{03}}{g_{33}}-g_{00}\right)\dot{t}^{2}-2\sqrt{\left(g_{00}-\frac{g^{2}_{03}}{g_{33}}\right)\left(1-g_{11}\right)}\dot{t}\dot{r}-\dot{r}^{2}=0,$
(4.2.16)
$\left(\frac{g^{2}_{03}}{g_{33}}-g_{00}\right)\dot{t}-\sqrt{\left(g_{00}-\frac{g^{2}_{03}}{g_{33}}\right)\left(1-g_{11}\right)}\dot{r}=\omega.$
(4.2.17)

The second equation is the geodesic equation corresponding to the time-
independence of the metric; in terms of the momentum defined in eq.(4.2.15),
it can be written $p_{t}=-\omega$, and so $\omega$ has the interpretation of
the energy of the particle as measured at infinity Johnson&March-Russell:2020
; YL2022 .

In the following, we shall investigate the tunneling behavior of massless
particles from the horizon. For this purpose, let us first evaluate the
radial, null geodesics. Since the tunneling processes take place near the
event horizon, we may consider a particle tunneling across the event horizon
as an ellipsoid shell and hold that the particle should still be an ellipsoid
shell during the tunneling process. In other words, the particle does not have
motion in the $\rho$-direction Jiang&Wu&Cai:2006 . Therefore, under these
assumptions ($ds^{2}=0=d\rho$), the radial, null geodesics followed by
massless particles are given by the following.

According to (4.2.16) and (4.2.17), for an outgoing massless particle, we have

$\frac{dr}{dt}=\frac{\dot{r}}{\dot{t}}=-\sqrt{\left(g_{00}-\frac{g^{2}_{03}}{g_{33}}\right)(1-g_{11})}+\sqrt{-g_{11}\left(g_{00}-\frac{g^{2}_{03}}{g_{33}}\right)},$
(4.2.18)
$\dot{t}=\frac{\omega}{\left(\frac{g^{2}_{03}}{g_{33}}-g_{00}\right)[\sqrt{-g_{11}(1-g_{11})}+g_{11}]},$
(4.2.19)

and

$\dot{r}=\frac{-\omega}{\sqrt{g_{00}-\frac{g^{2}_{03}}{g_{33}}}}\frac{\sqrt{-g_{11}}-2g_{11}\sqrt{1-g_{11}}}{g_{11}}.$
(4.2.20)

For the case we are considering, we have

$p_{r}=g_{rt}\dot{t}+g_{rr}\dot{r}=\frac{\omega}{\sqrt{g_{00}-\frac{g^{2}_{03}}{g_{33}}}}\left(\frac{\sqrt{1-g_{11}}}{g_{11}+\sqrt{-g_{11}(1-g_{11})}}-\frac{\sqrt{-g_{11}}-2g_{11}\sqrt{1-g_{11}}}{g_{11}}\right).$
(4.2.21)

Then according to (4.2.14), we have

$ImS=Im\int p_{r}dr.$ (4.2.22)

Insert the components of metric (3.2.14) into (4.2.21) and (4.2.22), we can
obtain the explicit formula for $ImS$. Since it is too lengthy, we do not list
the result. Finally, from (4.2.3), we can obtain the Hawking temperature for
the Kerr-like metric in $4D$ DFT, which is obtained in section 3. In fact, if
we consider the non-rotating black hole, the result for the ’Schwarzschild-
like’ metric obtained in YL2022  can be obtained as a special case.

##  5 Conclusions and Discussions

In recent years, a significant advancement in string theory is the development
of double field theory (DFT) Siegel19931 ; Siegel19932 ; D1990 . DFT aims to
provide a manifestly duality-invariant description of supergravity. By
introducing dual coordinates $\tilde{x}_{\mu}$, DFT allows fields and gauge
parameters to depend on the $2D$ coordinates
$X^{M}=(x^{\mu},\tilde{x}_{\mu})$, forming an $O(D,D)$ vector. If we impose
the ”section condition” or ”strong constraint,” restricting fields and gauge
parameters to depend on at most half of the coordinates, the action $S_{NSNS}$
can be derived from the action of DFT. Thus, DFT offers a natural framework
for exploring T-duality-inspired possibilities beyond supergravity.

The study of black hole (BH) solutions in string theory is a dynamic area of
research. A notable achievement of superstring theory is the correspondence
between the microscopic state counting for the 5-dimensional BPS black hole
solution of the string effective action and the Bekenstein-Hawking entropy
SW1996 ; Z2023 . Subsequently, numerous efforts have been made to extend these
results to other solutions in string theory HMS1996 ; S2008 ; CDKL20071 ;
CDKL20072 ; CDKL2008 ; CM2009 ; KLL1999 ; KL2005 . The first black hole
solution within double field theory was presented in KPS2017 , where the
authors investigated a general static, asymptotically flat, spherically
symmetric solution in $D=4$, sourced by the stringy energy–momentum tensor.
This solution matches the known vacuum geometry outside a finite radius from
the center, with parameters expressed as volume integrals of the interior
stringy energy–momentum tensor. Additionally, the authors discuss the energy
conditions of the black hole metric ACP2018 .

However, the metric proposed in KPS2017 ; ACP2018  is a ’Schwarzschild-like’
metric in $D=4$ DFT. In this paper, we extend this metric to the rotating
case, i.e., a ’Kerr-like’ metric. Our investigation yields two main results.
First, we derive the Kerr-like metric in $4D$ double field theory in both the
Einstein frame and the string frame, following a methodology akin to deriving
the Kerr metric in General Relativity. We complexify the relevant physical
quantities in this metric. We discuss the conditions which $H$-flux and
dilaton should satisfy as well. Second, we explore the thermodynamics and
Hawking radiation of the Kerr-like metric obtained. We employ the ’covariant
phase space’ approach V.L and R.M.W 1994 ; R.M.W 1993 ; J.L and R.M.W 1990 ,
which reformulates the first law of black hole thermodynamics into a
’differential’ form. We recover the usual integrated form of the first law of
black hole thermodynamics ASA and CDAB 2017 . Moreover, we provide explicit
formulas for the first law of black hole thermodynamics and the Hawking
radiation temperature for the ’Kerr-like’ metric in $4D$ DFT. Notably, the
results for the ’Schwarzschild-like’ metric are regained in the non-rotating
case.

In conclusion, we offer a brief outlook on potential future developments.
Firstly, we aim to derive additional black hole metrics within $4D$ double
field theory. Secondly, we will explore whether the metrics in $D=4$ DFT
satisfy the CA-duality (’complexity=action’). Lastly, we will investigate the
disparities between solutions in $4D$ double field theory and General
Relativity, with hopes of identifying observable differences in future
astronomical observations.

#### Acknowledgements

YL was supported by an STFC studentship. For the purpose of open access, the
authors have applied a CC BY public copyright licence to any Author Accepted
Manuscript version arising.

##  Appendix A Kerr-Newman metric in General Relativity

In this appendix, we will review how to derive the Kerr-Newman metric in
general relativity. We first write Reissner-Nordstroem metric for an
electrically charged, static, and spherically symmetric body

$ds^{2}_{RN}=-\left(1-\frac{2M}{r}+\frac{Q^{2}}{r^{2}}\right)dt^{2}+\left(1-\frac{2M}{r}+\frac{Q^{2}}{r^{2}}\right)^{-1}dr^{2}+r^{2}d\Omega^{2},$
(A.0.1)

where $d\Omega^{2}=d\theta^{2}+\sin{\theta}^{2}d\varphi^{2}$. We set:

$du=-dt+\left(1-\frac{2M}{r}+\frac{Q^{2}}{r^{2}}\right)^{-1}dr.$ (A.0.2)

Then this metric can be rewritten as:

$ds^{2}=-\left(1-\frac{2M}{r}+\frac{Q^{2}}{r^{2}}\right)du^{2}+2dudr+r^{2}d\Omega^{2}.$
(A.0.3)

Because $dr^{2}$ is eliminated, the metric can now be expressed in terms of a
null tetrad $\\{l^{\mu},n^{\mu},m^{\mu},\bar{m}^{\mu}\\}$. The vectors
$l^{\mu},n^{\mu}$ are real while $m^{\mu},\bar{m}^{\mu}$ are complex
conjugates; the only non-vanishing scalar products between tetrad vectors are:

$l^{\mu}n_{\mu}=1,n^{\mu}n_{\mu}=0,$ (A.0.4)

and we can set

$l^{\mu}=\delta_{1}^{\mu}$ (A.0.5)
$n^{\mu}=\delta_{0}^{\mu}+C\delta_{1}^{\mu},$ (A.0.6)

where the $\delta$’s are the unit vector of cotangent space, and $C$ is an
unknown function. From (A.0.4), (A.0.5) and (A.0.6), we have

$C=-\frac{1}{2}\left(1-\frac{2M}{r}+\frac{Q^{2}}{r^{2}}\right).$ (A.0.7)

Combining (3.1.4) and $m^{\mu}\bar{m}_{\mu}=-1$, we have

$m^{\mu}=\frac{1}{\sqrt{2}r}\left(\delta_{2}^{\mu}+\frac{i}{\sin{\theta}}\delta_{3}^{\mu}\right),$
(A.0.8)
$\bar{m}_{\mu}=\frac{1}{\sqrt{2}r}\left(\delta_{2}^{\mu}-\frac{i}{\sin{\theta}}\delta_{3}^{\mu}\right).$
(A.0.9)

We need to do complexification and write the expression as:

$dr^{\prime}=dr-ia\sin\theta d\theta,$ (A.0.10) $du^{\prime}=du+ia\sin\theta
d\theta,$ (A.0.11)

Then based on transformations (A.0.10) and (A.0.11), we can obtain

$1-\frac{2M}{r}+\frac{Q^{2}}{r^{2}}\rightarrow 1-\frac{2Mr-Q^{2}}{r^{\prime
2}+a^{2}\cos^{2}\theta},$ (A.0.12) $r^{2}\rightarrow
r^{2}+a^{2}\cos^{2}\theta,$ (A.0.13)

so

$l^{\mu}=\delta_{1}^{\mu},$ (A.0.14)
$n^{\mu}=\delta_{0}^{\mu}-\frac{1}{2}\left(1-\frac{2Mr-Q^{2}}{r^{\prime
2}+a^{2}\cos^{2}\theta}\right)\delta_{1}^{\mu},$ (A.0.15)
$m^{\mu}=\frac{1}{\sqrt{2}(r^{\prime
2}+a^{2}\cos^{2}\theta)}\left(\delta_{2}^{\mu}-ia\sin\theta\delta_{1}^{\mu}+ia\sin\theta\delta_{0}^{\mu}+\frac{i}{\sin\theta}\theta_{3}^{\mu}\right).$
(A.0.16)

According to (3.1.4), we can get:

$g^{\mu\nu}=\left[\begin{array}[]{cccc}\frac{a^{2}\sin^{2}\theta}{\rho^{2}}&-\frac{r^{\prime
2}+a^{2}}{\rho^{2}}&0&\frac{a}{\rho^{2}}\\\ -\frac{r^{\prime
2}+a^{2}}{\rho^{2}}&\frac{r^{\prime
2}+a^{2}-2Mr+Q^{2}}{\rho^{2}}&0&-\frac{a}{\rho^{2}}\\\
0&0&\frac{1}{\rho^{2}}&0\\\
\frac{a}{\rho^{2}}&-\frac{a}{\rho^{2}}&0&\frac{1}{\rho^{2}\sin^{2}\theta}\end{array}\right],$
(A.0.17)

and $\rho^{2}=r^{\prime 2}+a^{2}\cos^{2}\theta$. The inverse of (A.0.17) is

$g_{\mu\nu}=\left[\begin{array}[]{cccc}-\left(1-\frac{2Mr-Q^{2}}{\rho^{2}}\right)&-1&0&-\frac{a\sin^{2}\theta(2Mr-Q^{2})}{\rho^{2}}\\\
-1&0&0&a\sin^{2}\theta\\\ 0&0&\rho^{2}&0\\\
-\frac{a\sin^{2}\theta(2Mr-Q^{2})}{\rho^{2}}&a\sin^{2}\theta&0&\sin^{2}\theta\left(r^{2}+a^{2}+a^{2}\sin^{2}\theta\frac{2Mr-Q^{2}}{\rho^{2}}\right)\end{array}\right].$
(A.0.18)

We can use the following transformation to turn back to coordinates
$(t,r,\theta,\phi)$

$\left[\begin{array}[]{c}dt\\\ dr\\\ d\theta\\\
d\varphi\end{array}\right]=\left[\begin{array}[]{cccc}1&-\frac{r^{2}+a^{2}}{r^{2}+a^{2}-2Mr+Q^{2}}&0&0\\\
0&1&0&0\\\ 0&0&1&0\\\
0&-\frac{a}{r^{2}+a^{2}-2Mr+Q^{2}}&0&1\end{array}\right]\left[\begin{array}[]{c}du\\\
dr\\\ d\theta\\\ d\varphi\end{array}\right],$ (A.0.19)

After transforming the coordinates, we will find that $(0,1)$,$(1,0)$,$(1,3)$
and $(3,1)$ terms in following metric is vanishing carter 1968 . Then we can
get the metric:

$g_{\mu\nu}=\left[\begin{array}[]{cccc}-\left(1-\frac{2Mr-Q^{2}}{\rho^{2}}\right)&0&0&-\frac{a\sin^{2}\theta(2Mr-Q^{2})}{\rho^{2}}\\\
0&\frac{\rho^{2}}{r^{2}+a^{2}-2Mr+Q^{2}}&0&0\\\ 0&0&\rho^{2}&0\\\
-\frac{a\sin^{2}\theta(2Mr-Q^{2})}{\rho^{2}}&0&0&\sin^{2}\theta\left(r^{2}+a^{2}+a^{2}\sin^{2}\theta\frac{2Mr-Q^{2}}{\rho^{2}}\right)\end{array}\right].$
(A.0.20)

More details can be found in T.A and E.T.N 2016 .

##  Appendix B Christoffel symbols of new metric

We give the non-vanishing Christoffel symbols

$\begin{split}\Gamma_{tt}^{r}&=\frac{\hat{A}(\hat{C}+\psi^{2}\sin^{2}\theta)}{2\hat{C}}\left(\frac{\partial\hat{A}}{\partial
r}+2\hat{A}\frac{\partial\hat{\phi}}{\partial r}\right)\\\
\Gamma^{\theta}_{tt}&=\frac{\hat{A}}{2\hat{C}}\left(\frac{\partial\hat{A}}{\partial
r}+2\hat{A}\frac{\partial\hat{\phi}}{\partial r}\right)\\\
\Gamma^{t}_{tr}=\Gamma^{t}_{rt}&=\frac{\psi^{2}\sin^{2}\theta(\frac{\partial\hat{A}}{\partial
r}+2\frac{\partial\hat{\phi}}{\partial
r})+\hat{C}(\frac{\partial\hat{A}}{\partial
r}+2\hat{A}\frac{\partial\hat{\phi}}{\partial
r})}{2(\hat{A}\hat{C}+\psi^{2}\sin^{2}\theta)}\\\
\Gamma^{\varphi}_{tr}=\Gamma^{\varphi}_{rt}&=\frac{\psi\frac{\partial
A}{\partial r}}{2(\hat{A}\hat{C}+\psi^{2}\sin^{2}\theta)}\\\
\Gamma^{t}_{t\theta}=\Gamma^{t}_{\theta
t}&=\frac{\psi^{2}\hat{A}^{2}\sin(2\theta)+\hat{C}\frac{\partial\hat{A}}{\partial\theta}+\hat{A}(-4\psi^{2}\cos\theta\sin\theta+2\hat{C}\frac{\partial\hat{\phi}}{\partial\theta})}{2(\hat{A}\hat{C}+\psi^{2}\sin\theta)}\\\
&+\frac{\psi^{2}\sin\theta(2\cos\theta+\sin\theta\frac{\partial\hat{A}}{\partial\theta}+2\sin\theta\frac{\partial\hat{\phi}}{\partial\theta})}{2(\hat{A}\hat{C}+\psi^{2}\sin\theta)}\\\
\Gamma^{\varphi}_{t\theta}=\Gamma^{\varphi}_{\theta
t}&=\frac{\psi\csc^{2}\theta(-2\hat{A}\cot\theta+2\hat{A}^{2}\cot\theta+\frac{\partial\hat{A}}{\partial\theta})}{2(\psi^{2}+\hat{A}\csc^{2}\theta\hat{C})}\\\
\Gamma^{\theta}_{t\varphi}=\Gamma^{\theta}_{\varphi
t}&=-\frac{\psi\hat{A}\sin^{2}\theta(\hat{C}+\psi^{2}\sin^{2}\theta)(\frac{\partial\hat{A}}{\partial
r}+2(-1+\hat{A})\frac{\partial\hat{\phi}}{\partial r})}{2\hat{C}}\\\
\Gamma^{r}_{rr}&=-\frac{\psi^{2}\hat{A}\sin^{2}\theta\frac{\partial\hat{C}}{\partial
r}+\hat{C}^{2}(\frac{\partial\hat{A}}{\partial
r}-2\hat{A}\frac{\partial\hat{\phi}}{\partial
r})+\psi^{2}\hat{C}\sin^{2}\theta(\frac{\partial\hat{A}}{\partial
r}-2\hat{A}\frac{\partial\hat{\phi}}{\partial
r})}{2\hat{A}\hat{C}(\hat{C}+\psi^{2}\sin^{2}\theta)}\\\
\Gamma^{\theta}_{rr}&=\frac{\hat{C}(\hat{C}+\psi^{2}\sin^{2}\theta)\frac{\partial\hat{A}}{\partial\theta}-\hat{A}\sin^{2}\theta\frac{\partial\hat{C}}{\partial\theta}-2\hat{C}^{2}\hat{A}\frac{\partial\hat{\phi}}{\partial\theta}}{2\hat{A}\hat{C}(\hat{C}+\psi^{2}\sin^{2}\theta)^{2}}+\frac{\psi^{2}(\sin
2\theta-2\sin^{2}\theta\frac{\partial\hat{\phi}}{\partial\theta})00}{2\hat{A}\hat{C}(\hat{C}+\psi^{2}\sin^{2}\theta)^{2}}\\\
\Gamma^{r}_{r\theta}=\Gamma^{r}_{\theta
r}&=\frac{1}{2}\left(-\frac{\partial\hat{A}}{\partial\theta}\frac{1}{\hat{A}}+\frac{\psi^{2}\sin^{2}\theta\frac{\partial\hat{C}}{\partial\theta}+2\hat{C}^{2}\frac{\partial\hat{\phi}}{\partial\theta}+2\psi^{2}\hat{C}\sin\theta(-\cos\theta+\sin\theta\frac{\partial\hat{\phi}}{\partial\theta})}{\hat{C}(\hat{C}+\psi^{2}\sin^{2}\theta)}\right)\\\
\Gamma^{\theta}_{r\theta}=\Gamma^{\theta}_{\theta
r}&=-\frac{1}{2}\frac{\partial\hat{A}}{\partial
r}\frac{1}{\hat{A}}+\frac{1}{2}\frac{\partial\hat{C}}{\partial
r}\frac{1}{\hat{C}}+\frac{\partial\hat{\phi}}{\partial r}\\\
\Gamma^{t}_{t\varphi}=\Gamma^{t}_{\varphi
t}&=-\frac{\psi\sin^{2}\theta(\hat{C}\frac{\partial\hat{A}}{\partial
r}+\psi^{2}\sin^{2}\theta\frac{\partial\hat{A}}{\partial
r}-(-1+\hat{A}\frac{\partial\hat{C}}{\partial
r}))}{2\hat{A}\hat{C}+\psi^{2}\sin^{2}\theta}\\\
\Gamma^{\varphi}_{r\varphi}=\Gamma^{\varphi}_{\varphi
t}&=\frac{-\psi^{2}\sin^{2}\theta\frac{\partial\hat{A}}{\partial
r}+2\psi^{2}\sin^{2}\theta\frac{\partial\hat{\phi}}{\partial
r}+\hat{A}(\frac{\partial\hat{C}}{\partial
r}+2\hat{C}\frac{\partial\hat{\phi}}{\partial
r})}{2\hat{A}\hat{C}+\psi^{2}\sin^{2}\theta}\\\
\Gamma^{r}_{\theta\theta}&=\frac{(\hat{C}+\psi^{2}\sin^{2}\theta)(-\hat{A}\frac{\partial\hat{C}}{\partial
r}+\hat{C}(\frac{\partial\hat{A}}{\partial
r}-2\hat{A}\frac{\partial\hat{\phi}}{\partial r}))}{2\hat{A}\hat{C}}\\\
\Gamma^{\theta}_{\theta\theta}&=-\frac{1}{2}\frac{\partial\hat{A}}{\partial\theta}\frac{1}{\hat{A}}+\frac{1}{2}\frac{\partial\hat{C}}{\partial\theta}\frac{1}{\hat{C}}+\frac{\partial\hat{\phi}}{\partial\theta}\\\
\end{split}$ (B.0.1)
$\begin{split}\Gamma^{t}_{\theta\varphi}=\Gamma^{t}_{\varphi\theta}&=-\frac{\psi\sin^{2}\theta(2\psi^{2}\sin
2\theta+\psi^{2}\hat{A}\sin
2\theta+\hat{C}\frac{\partial\hat{A}}{\partial\theta}+\psi^{2}\sin^{2}\theta\frac{\partial\hat{A}}{\partial\theta}+\frac{\partial\hat{C}}{\partial\theta})}{2(\hat{A}\hat{C}+\psi^{2}\sin^{2}\theta)}\\\
&+\frac{\psi\sin^{2}\hat{A}(3\psi^{2}\sin
2\theta+\frac{\partial\hat{C}}{\partial\theta})}{2(\hat{A}\hat{C}+\psi^{2}\sin^{2}\theta)}\\\
\Gamma^{\varphi}_{\theta\varphi}=\Gamma^{\varphi}_{\varphi\theta}&=\frac{-2\psi^{2}\hat{A}^{2}\cot\theta+\psi^{2}(-\frac{\partial\hat{A}}{\partial\theta}+2(\cot\theta+\frac{\partial\hat{\phi}}{\partial\theta}))}{2(\psi^{2}+\hat{A}\csc^{2}\theta\hat{C})}\\\
&+\frac{\hat{A}(4\psi^{2}\cot\theta+\csc^{2}\theta\frac{\partial\hat{C}}{\partial\theta}+2\csc^{2}\theta\hat{C}\frac{\partial\hat{\phi}}{\partial\theta})}{2(\psi^{2}+\hat{A}\csc^{2}\theta\hat{C})}\\\
\Gamma^{r}_{\varphi\varphi}&=\frac{\hat{A}\sin^{2}\theta(\hat{C}+\psi^{2}\sin^{2}\theta)(\psi^{2}\sin^{2}\theta\frac{\partial\hat{A}}{\partial
r}-\frac{\partial\hat{C}}{\partial
r}-2(\hat{C}-\psi^{2}(-2+\hat{A})\sin^{2}\theta)\frac{\partial\hat{\phi}}{\partial
r})}{2\hat{C}}\\\
\Gamma^{\theta}_{\varphi\varphi}&=\frac{\hat{A}\sin^{2}\theta(-8\psi^{2}\cos\theta\sin\theta+\psi^{2}\sin^{2}\theta\frac{\partial\hat{A}}{\partial\theta}-\frac{\partial\hat{C}}{\partial\theta}-4\psi^{2}\sin^{2}\frac{\partial\hat{\phi}}{\partial\theta}+2\psi^{2}\hat{A}\sin\theta(2\cos\theta+\sin\theta\frac{\partial\hat{\phi}}{\partial\theta}))}{2\hat{C}}\\\
&+\frac{\hat{A}\sin\theta(-2\hat{C}(\cos\theta+\sin\theta\frac{\partial\hat{\phi}}{\partial\theta}))}{2\hat{C}}\end{split}$
(B.0.2)

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