# Holographic entanglement entropy of the double Wick rotated BTZ black hole

Mitsutoshi Fujita<EMAIL_ADDRESS>Jun Zhang
<EMAIL_ADDRESS>
###### Abstract

In this paper, we analyze the holographic covariant entanglement entropy in
the double Wick rotated version of a rotating BTZ black hole (3 dimensional
Kerr-AdS solution), where the periodicity of Euclidean time and spatial
direction are changed. The dual field theory has negative energy in the
Lorentzian signature. The holographic entanglement entropy agrees with its CFT
counterpart, which is obtained by a conformal transformation of the
correlation functions of twisted operators.

##  1 Introduction

In contrast to correlation functions in quantum field theories, the
entanglement entropy of a subsystem becomes a non-local quantity [1, 2].
Dividing the space at a fixed time into two parts $A$ and $B$, the
entanglement entropy is defined as

$\displaystyle S_{A}=-\mbox{Tr}_{A}(\rho_{A}\log\rho_{A}),$ (1)

where the reduced density matrix $\rho_{A}=\mbox{Tr}_{B}\rho$ is the trace of
the total density matrix over the subregion $B$. For quantum critical phase
transitions, the entanglement entropy diverges at the critical point and
becomes an order parameter [3]. It captures geometric discernment of field
theories such as an area law [4]: the entanglement entropy defined in a
subregion is dependent on properties of a shared boundary and resembles the
black hole entropy. An area law also implies that the most entangled degrees
of freedom come from the high energy ones located around an infinitesimal
neighborhood of the entangling surface.

An interesting discovery was made by changing the periodicity of spacetime or
adding boundaries [5, 6]. When we express the entanglement entropy in the path
integral formalism, the reduced density matrix and the entanglement entropy
can be defined using the partition function on the $n$ copies of the manifold
glued along $A$ at the fixed time. We consider Euclidean time and space
direction. When the subsystem $A$ is a single interval along a space direction
with a periodic boundary condition, the entanglement entropy in 1+1
dimensional CFT becomes [7]

$\displaystyle S_{A}=\dfrac{c}{3}\log\Big{(}\dfrac{R}{\pi
a}\sin\Big{(}\dfrac{\pi l}{R}\Big{)}\Big{)}+a_{1},$ (2)

where $R$ is the circumference of the circle and $a_{1}$ is a constant, which
is not universal. On the other hand, when the space direction is infinitely
long, entanglement entropy of a thermal mixed state at a finite temperature
becomes

$\displaystyle S_{A}=\dfrac{c}{3}\log\Big{(}\dfrac{R}{\pi
a}\sinh\Big{(}\dfrac{\pi l}{R}\Big{)}\Big{)}+a_{1}.$ (3)

If two point functions of twisted operators are considered, this can be done
via a conformal transformation. Originally, the cut was made into a circle.
After other conformal transformations, the cut is along the axis of a
cylinder. The entanglement entropy at a semi-infinite line or a finite $1d$
system with open boundaries has a constant term called boundary entropy, which
does not depend on the length at zero temperature limit. The boundary entropy
has the expression $S_{0}=\log g$, where $g$ is the ground state degeneracy.

The Riemann surface can also have a general periodicity $z\sim z+2\pi\tau$,
where $z$ is a Riemann surface complex coordinate. The density matrix then
depends on the Hamiltonian and momentum of the CFT as follows:

$\displaystyle\rho=e^{-2\pi\tau_{2}H+2\pi i\tau_{1}P}.$ (4)

Entanglement entropy of CFT was computed with general periodicity [8], where
the cut is along a spatial direction. The gravity dual of this CFT has angular
momentum and becomes a rotating BTZ black hole, which is stationary but not
static. The metric does not have a curvature singularity at a surface in the
origin without matter couplings. This surface is a singularity in the causal
structure. A rotating BTZ black hole has its Euclidean version with an
analytic continuation of the angular momentum.

Periodicity can be changed by using a coordinate transformation (the double
Wick rotation) $z^{\prime}=iz\equiv\sigma_{1}^{\prime}+i\sigma_{2}^{\prime}$
as follows:

$\displaystyle z^{\prime}\sim z^{\prime}+2\pi(-\tau_{2}+i\tau_{1}).$ (5)

The motivation of this paper is to compute entanglement entropy with changed
periodicity on both the CFT side and the gravity dual. The gravity dual is the
double Wick rotated version of the rotating BTZ metric. It is interesting to
see changes in the gravity dual because the rotating BTZ black hole is not
static. It is necessary to analyze whether the metric is a black hole. Degrees
of freedom in time-dependent backgrounds are also open problems. The
entanglement entropy in $2d$ CFT is proportional to the central charge
(degrees of freedom) and is useful.

In this work, we compute the holographic entanglement entropy of the double
Wick rotated spacetime and the CFT counterpart. The Ryu-Takayanagi formula
proposes the holographic dual of the entanglement entropy [10, 11], which is
the area of minimal surfaces (see also the review [12, 13]). Entanglement
entropy is a powerful tool to analyze strongly coupled systems. The
hologrpahic entanglement entropy has been the order parameter of the
confinement/deconfinement phase transition in a confining gauge theory [14,
15, 16, 17]. A relation similar to the first law of thermodynamics is analyzed
in [18, 19, 20]. This relation was rewritten in terms of the relative entropy
[21]. [8] proposed a covariant generalization of the holographic entanglement
entropy, which is powerful to analyze the time-dependence in the dual field
theory side. The area of the holographic covariant entanglement entropy (an
extremal surface) can probe the interior of the black hole horizon, ending on
a boundary [9]. The double Wick rotated version of the entanglement entropy is
called geometric entropy. Holographic models are useful for computing
geometric entropy. Using geometric entropy, [30] analyzed the
confinement/deconfinement phase transition of the Yang-Mills theory on compact
space at finite temperature on both sides of duality. 111Third order phase
transitions were captured by geometric entropy in free QCD-like theory with
flavor on $S^{1}\times S^{3}$ [31].  The geometric entropy probed $AdS$
Schwarzschild black holes [32] and the Reissner-Nordstrom $AdS$ background
(effects of background charges) [33, 34] by using a minimal surface. However,
the standard modular Hamiltonian is unusual in geometric entropy. A space
direction is considered as the time in geometric entropy. By making use of the
analytic continuation, one can define $\rho(t)$, which is the density matrix
of geometric entropy. One usually relies on the path integral formulation to
compute geometric entropy. Thus, the entanglement entropy is a good starting
point to analyze spacetime with general periodicity.

The rest of this paper is structured as follows. In Section 2, we review the
rotating BTZ black hole. We derive the periodicity of Euclidean coordinates
and the thermodynamics of the rotating BTZ black hole. In Section 3, we
analyze the double Wick rotated version of a rotating BTZ black hole. We
obtain the periodicity of Euclidean coordinates via the double Wick rotation.
In Section 4, we derive the holographic covariant entanglement entropy of the
double Wick rotated spacetime. In Section 5, we derive the entanglement
entropy in the dual CFT.

##  2 The rotating BTZ black hole

This section is the review of [22, 23]. We derive the periodicity of Euclidean
coordinates in the rotating BTZ black hole by using coordinate
transformations. We derive the stress energy tensor and free energy by using
holographic renormalization. We analyze the thermodynamic properties of the
rotating BTZ black hole, which describes theory at high temperature.

The rotating BTZ metric becomes a solution of the $3d$ Einstein-Hilbert action
as follows:

$\displaystyle
ds^{2}=r^{2}\Big{(}dx+\dfrac{r_{+}r_{-}}{r^{2}L}dt\Big{)}^{2}+\dfrac{L^{2}r^{2}dr^{2}}{(r^{2}-r_{+}^{2})(r^{2}-r_{-}^{2})}-\dfrac{(r^{2}-r_{-}^{2})(r^{2}-r_{+}^{2})}{r^{2}L^{2}}dt^{2},$
$\displaystyle 8G_{3}{M}=\dfrac{r_{+}^{2}+r_{-}^{2}}{L^{2}},\quad
J=\dfrac{r_{+}r_{-}}{4G_{3}L},$ (6)

where $x\sim x+2\pi$ and $L$ denotes the AdS radius. Two integration constants
are the mass $M$ and angular momentum $J$, which are conserved charges. $J$ is
related to rotational invariance.

After the coordinate transformation $(r,t)\to L(r,t)$ and $(r_{+},r_{-})\to
L(r_{+},r_{-})$, coordinates become dimensionless. The metric is rewritten as

$\displaystyle
ds^{2}=L^{2}\Big{[}r^{2}\Big{(}dx+\dfrac{r_{+}r_{-}}{r^{2}}dt\Big{)}^{2}+\dfrac{r^{2}dr^{2}}{(r^{2}-r_{+}^{2})(r^{2}-r_{-}^{2})}-\dfrac{(r^{2}-r_{-}^{2})(r^{2}-r_{+}^{2})}{r^{2}}dt^{2}\Big{]},$
(7) $\displaystyle 8G_{3}M=r_{+}^{2}+r_{-}^{2},\quad
J=\dfrac{Lr_{+}r_{-}}{4G_{3}},$ (8)

Recall that the dimensions of $M,\ G_{3}^{-1},\ J/L$ are all 1. 222However, we
keep $G_{3}$ because it is a dimensional quantity. The blackening factor (or
the lapse function) vanishes for two roots as follows:

$\begin{gathered}r_{+}^{2}=4{G_{3}}\left({M+\sqrt{{M^{2}}-\frac{{{J^{2}}}}{{{L^{2}}}}}}\right),\hfill\\\
r_{-}^{2}=4{G_{3}}\left({M-\sqrt{{M^{2}}-\frac{{{J^{2}}}}{{{L^{2}}}}}}\right).\hfill\\\
\end{gathered}$ (9)

The Euclidean version $t\to-i\tau_{E}$ and $r_{-}\to-i\tilde{r}_{-}$ $(J\to-
iJ_{E})$ becomes

$\displaystyle
ds_{E}^{2}=L^{2}\Big{[}r^{2}\Big{(}dx-\dfrac{r_{+}\tilde{r}_{-}}{r^{2}}{d\tau_{E}}\Big{)}^{2}+\dfrac{r^{2}dr^{2}}{(r^{2}-r_{+}^{2})(r^{2}+\tilde{r}_{-}^{2})}+\dfrac{(r^{2}+\tilde{r}_{-}^{2})(r^{2}-r_{+}^{2})}{r^{2}}d\tau_{E}^{2}\Big{]}.$
(10)

The blackening factor vanishes when

$\displaystyle
r_{+}=2\Big{[}G_{3}M\Big{(}1+\Big{(}1+\dfrac{J_{E}^{2}}{M^{2}L^{2}}\Big{)}^{\frac{1}{2}}\Big{)}\Big{]}^{\frac{1}{2}},$
$\displaystyle\tilde{r}_{-}=2\Big{[}G_{3}M\Big{(}-1+\Big{(}1+\dfrac{J_{E}^{2}}{M^{2}L^{2}}\Big{)}^{\frac{1}{2}}\Big{)}\Big{]}^{\frac{1}{2}}.$
(11)

These two values don’t agree when $M$ and $J_{E}$ are finite. This difference
shows that there are no extreme limits, unlike Lorentzian black holes. 333In a
Lorentzian rotating black hole, $J_{E}=iJ$ and $\tilde{r}_{-}=ir_{-}$. The
extreme limit means $r_{+}=r_{-}$ when $J=ML$. Moreover, the black hole mass
should be larger than the angular momentum $ML\geq J$ when the black hole
horizon exists.

The rotating BTZ metric in the Euclidean signature has constant negative
curvature and the spacetime becomes locally equivalent to hyperbolic three-
space. The following Euclidean coordinate transformation maps to a Poincare
AdS metric

$\displaystyle
X=r_{1}\cos\theta=\sqrt{\dfrac{r^{2}-r_{+}^{2}}{r^{2}+\tilde{r}_{-}^{2}}}\cos\Big{(}r_{+}\tau_{E}+\tilde{r}_{-}x\Big{)}\exp\Big{(}r_{+}x-\tilde{r}_{-}\tau_{E}\Big{)},$
$\displaystyle
Y=r_{1}\sin\theta=\sqrt{\dfrac{r^{2}-r_{+}^{2}}{r^{2}+\tilde{r}_{-}^{2}}}\sin\Big{(}{r_{+}}\tau_{E}+\tilde{r}_{-}x\Big{)}\exp\Big{(}{r_{+}}x-{\tilde{r}_{-}}\tau_{E}\Big{)},$
$\displaystyle
z=\sqrt{\dfrac{r_{+}^{2}+\tilde{r}_{-}^{2}}{r^{2}+\tilde{r}_{-}^{2}}}\exp\Big{(}{r_{+}}x-{\tilde{r}_{-}}\tau_{E}\Big{)},$
(12)

where a spherical coordinate is represented by
$\theta=r_{+}\tau_{E}+\tilde{r}_{-}x$ and
$r_{1}=\sqrt{\frac{r^{2}-r_{+}^{2}}{r^{2}+\tilde{r}_{-}^{2}}}\exp\Big{(}r_{+}x-\tilde{r}_{-}\tau_{E}\Big{)}$.
When $r\to\infty$, coordinates mentioned above describe $(X,Y)$ plane at
$z=0$. When $r\to r_{+}$, coordinates mentioned above describe the $z$ axis at
$r_{1}=0$. The Euclidean metric is rewritten as

$\displaystyle
ds_{E}^{2}=\dfrac{L^{2}}{z^{2}}(dX^{2}+dY^{2}+dz^{2})=\dfrac{L^{2}}{z^{2}}(dr_{1}^{2}+r_{1}^{2}d\theta^{2}+dz^{2}).$
(13)

The periodicity of $x$ means

$\displaystyle(r_{1},\theta)\sim(r_{1}e^{2\pi
r_{+}},\theta+{2\pi\tilde{r}_{-}}),$ (14)

where a translation is along a radial direction and a $2\pi\tilde{r}_{-}$
rotation is a rotation around the $z$ axis.

The periodicity of the trigonometric functions $(\theta\sim\theta+2\pi)$ and
nonsingular transformation at the $z$ axis ($r=r_{+}$ and $r_{1}=0$) require

$\displaystyle(x,\tau_{E})\sim(x+\phi_{0},\tau_{E}+\beta_{0}),$ (15)
$\displaystyle\beta_{0}=\dfrac{1}{TL}=\dfrac{2\pi
r_{+}}{r_{+}^{2}+\tilde{r}_{-}^{2}},\quad\phi_{0}=\dfrac{2\pi\tilde{r}_{-}}{r_{+}^{2}+\tilde{r}_{-}^{2}}.$
(16)

If we don’t require the periodicity mentioned above, the Euclidean black hole
receives a conical singularity at the black hole horizon.

We define the chemical potential as the shift of the rotating BTZ black hole
$\Omega_{E}=\frac{4G_{3}J_{E}}{r_{+}^{2}L^{2}}=\frac{\tilde{r}_{-}}{r_{+}L}$.
We then find the relationship

$\displaystyle\phi_{0}=\beta_{0}\Omega_{E}L.$ (17)

We use a complex coordinate $z=z+i2\pi\tau_{E}$. Periodicity of $z$ is given
by

$\displaystyle z\sim
z+\beta_{0}\Omega_{E}L+i\beta,\quad\bar{z}\sim\bar{z}+\beta_{0}\Omega_{E}L-i\beta.$
(18)

Following the analytic continuation $\Omega=-i\Omega_{E}$, left and right
temperature are as follows:

$\displaystyle z\sim
z+i\beta_{0}(1+L\Omega),\bar{z}\sim\bar{z}-i\beta_{0}(1-L\Omega).$ (19)

We then compute the holographic stress energy tensor. To compute the
holographic stress energy tensor, we perform the coordinate transformation and
use the FG coordinate with $g_{\mu\rho}=0$ as follows:

$\displaystyle
ds^{2}=\frac{L^{2}}{\rho^{2}}(d\rho^{2}+\bar{g}_{ij}dx^{i}dx^{j}),$ (20)

where

$\displaystyle\bar{g}_{ij}=\bar{g}_{ij}^{(0)}+\bar{g}_{ij}^{(2)}\rho^{2}+\bar{g}_{ij}^{(4)}\rho^{4}+\dots$
(21)

The radial coordinate $\rho$ can be perturbatively represented by the large
$r$ expansion. Metric for (7) becomes

$\displaystyle\bar{g}_{tt}^{(0)}=-1,\ \bar{g}_{xx}^{(0)}=1,\
\bar{g}^{(0)}_{tx}=0,\
\bar{g}_{tt}^{(2)}=\bar{g}_{xx}^{(2)}=\dfrac{r_{+}^{2}+r_{-}^{2}}{2},\
\bar{g}_{tx}^{(2)}=r_{+}r_{-}.$ (22)

The expectation value of the renormalized stress tensor of $2d$ CFT can be
computed by the variation of the action, a Gibbons-Hawking term and the
counter-terms in terms of the metric at the $AdS$ boundary as follows [28,
29]:

$\displaystyle\langle\tilde{T}_{\mu\nu}(x)\rangle=\dfrac{L}{8\pi
G_{3}}(\bar{g}_{\mu\nu}^{(2)}-\bar{g}_{\mu\nu}^{(0)}\bar{g}^{(2)\alpha}{}_{\alpha}).$
(23)

Substituting (22) into (23),

$\displaystyle\langle\tilde{T}_{tt}\rangle=\langle\tilde{T}_{xx}\rangle=\dfrac{ML}{2\pi},\
\langle\tilde{T}_{tx}(x)\rangle=\dfrac{J}{2\pi}.$ (24)

The stress tensor mentioned above is written in terms of dimensionless
parameters. By recovering dimensions $\langle
T_{\mu\nu}\rangle=\langle\tilde{T}_{\mu\nu}\rangle/L^{2}$, we find

$\displaystyle\langle T_{tt}\rangle=\langle T_{xx}\rangle=\dfrac{M}{2\pi L},\
\langle T_{tx}\rangle=\dfrac{J}{2\pi L^{2}}.$ (25)

Energy is obtained as the integration of energy density (25) as follows:

$\displaystyle E=\int dx\langle T_{tt}\rangle=M,$ (26) $\displaystyle P=\int
dx\langle T_{t\tilde{x}}\rangle=\dfrac{J}{L},$ (27)

where $x$ has a periodicity of $x$. Note that the zero point energy is zero
when the mass vanishes. The normalization of energy shows that the anti-de
Sitter space with the Poincare coordinate has zero mass.

We must evaluate the grand canonical partition function to obtain the
Euclidean path integral

$\displaystyle Z=\int[dg]e^{-I_{E}[g]},$ (28)

where $I_{E}[g]$ becomes the Euclidean action. Because the classical limit is
taken into account, the classical approximation is defined as the steepest
descent approximation $Z\sim e^{-I_{E}[g]}$.

We add Gibbons-Hawking terms and counter-terms to renormalize the action
$I_{E}[g]$ as follows:

$\displaystyle I_{\mathrm{ren}}=-\dfrac{1}{16\pi G_{3}}\int
d^{3}x\sqrt{g}(R-2\Lambda)+\dfrac{1}{8\pi G_{3}}\int_{\partial
M}d^{2}x\sqrt{\gamma}\Big{(}\Theta+\dfrac{1}{L}\Big{)},$ (29)

where the second term makes the variation principle well-defined. According to
the AdS/CFT correspondence, free energy is temperature times the value of the
Euclidean action on the Euclidean continuation of the black hole

$\displaystyle F$ $\displaystyle=TI_{\mathrm{ren}}[{g}],$ (30)

, where $1/T$ is the inverse temperature (the Euclidean Killing time). This
temperature is obtained as an analytic continuation $J_{E}\to iJ$ and
$\tilde{r}_{-}=ir_{-}$ of (15) as follows:

$\displaystyle\beta_{0}=\dfrac{1}{TL}=\dfrac{2\pi
r_{+}}{r_{+}^{2}-r_{-}^{2}}.$ (31)

Using the metric (7), free energy becomes

$\displaystyle\bar{F}=-L^{2}M\sqrt{1-\dfrac{J^{2}}{M^{2}L^{2}}}.$ (32)

We can recover dimensions after the coordinate transformation
$(t,x,r)\to(t/L,x/L,r/L)$ and $(r_{+},r_{-})\to(r_{+}/L,r_{-}/L)$ (see the
measure part). Free energy is represented by energy $M$ and a rotational
chemical potential $\Omega$,

$\displaystyle F$
$\displaystyle=-M\sqrt{1-\dfrac{J^{2}}{L^{2}M^{2}}}=M-Ts-\Omega J,$ (33)

where the spatial volume is $V_{x}=2\pi L$. According to the Euler relation
$M=Ts-PV_{x}+\mu Q$, free energy is related to pressure $F=-PV_{x}$. The
chemical potential is defined as the angular shift of a rotating BTZ black
hole at $r=r_{+}$, namely,
$\Omega=-\frac{4G_{3}J}{r_{+}^{2}}=-\frac{r_{-}}{r_{+}L}$. Note that $J$ and
$\Omega$ are thermodynamical conjugates. An entropy is given by Bekenstein-
Hawking formula $s=\frac{2\pi r_{+}}{4G_{3}}$. The first law of thermodynamics
then follows: $dE=Tds-PdV_{x}+\mu dQ$. The result of holographic
renormalization (33) is consistent with [22, 24].

Free energy (33) mentioned above is consistent with the form of the density
matrix $e^{-\beta H+i\beta\Omega_{E}LP}=e^{-\beta H+\beta\Omega LP}$ in the
CFT side, where $\Omega_{E}=i\Omega$ and momentum $P$ is defined in (26). That
is, the periodicity of a complex coordinate $z=x+i{\tau}_{E}$ is
$2\pi\tau_{E}=\beta\Omega_{E}L+i\beta$: $z\sim z+2\pi\tau$.

The rotating BTZ black hole has the following properties. When the black hole
horizon vanishes, temperature (31) becomes zero. The extreme rotating black
hole ($J=Ml$ or $r_{+}=r_{-}$) has zero temperature and non-zero entropy.
Moreover, the quantum correction to the path integral was computed in [25,
26]. The quantum corrected temperature and entropy were obtained [27].

##  3 The double Wick rotated version of a rotating BTZ black hole

In this section, we consider the double Wick rotated version of a rotating BTZ
black hole. We perform most of the computation in the Euclidean frame and
derive the periodicity of this metric. We argue that this spacetime does not
have Hawking temperature and entropy in the Lorentzian signature.

A rotating asymptotic $AdS_{3}$ background can be obtained by doing the double
Wick rotation(by taking $\tilde{t},\ \tilde{x}$ to $ix,\ it$) on the metric
(7), which is

$d{s^{2}}={L^{2}}[-{r^{2}}{(\frac{{{r_{-}}{r_{+}}}}{{{r^{2}}}}dx+dt)^{2}}+\frac{{({r^{2}}-r_{-}^{2})({r^{2}}-r_{+}^{2})}}{{{r^{2}}}}d{x^{2}}+\frac{{{r^{2}}d{r^{2}}}}{{({r^{2}}-r_{-}^{2})({r^{2}}-r_{+}^{2})}}].$
(34)

We perform a complete square once again to rewrite (34) in the ADM form as
follows:

$\displaystyle ds^{2}=-N^{2}dt^{2}+h_{ab}(dy^{a}+N^{a}dt)(dy^{b}+N^{b}dt)$
$\displaystyle=l^{2}\Big{[}-\dfrac{(r^{2}-r_{+}^{2})(r^{2}-r_{-}^{2})}{(r^{2}-r_{+}^{2}-r_{-}^{2})}dt^{2}+(r^{2}-r_{+}^{2}-r_{-}^{2})\Big{(}dx-\dfrac{r_{+}r_{-}}{(r^{2}-r_{+}^{2}-r_{-}^{2})}dt\Big{)}^{2}$
$\displaystyle+\dfrac{r^{2}dr^{2}}{(r^{2}-r_{+}^{2})(r^{2}-r_{-}^{2})}.\Big{]}$
(35)

The above formula shows that a hypersurface $\Sigma_{t}$ is spanned by $x$ and
$r$ at a constant $t$. The lapse and the shift are obtained as follows:

$\displaystyle
N=\sqrt{\dfrac{(r^{2}-r_{-}^{2})(r^{2}-r_{+}^{2})}{r^{2}-r_{-}^{2}-r_{+}^{2}}},\quad
N^{\mu}=(0,-\dfrac{r_{-}r_{+}}{r^{2}-r_{-}^{2}-r_{+}^{2}},0),$ (36)

where $N^{t}=0$ because it is perpendicular to a hypersurface.

We take the analytical continuation by letting $t\to-i\tau_{E}$ and $r_{-}\to
i\tilde{r}_{-}$ $(J\to iJ_{E})$

$d{s^{2}}={L^{2}}[{r^{2}}{(\frac{{{{\tilde{r}}_{-}}{r_{+}}}}{{{r^{2}}}}dx-d\tau_{E})^{2}}+\frac{{({r^{2}}+\tilde{r}_{-}^{2})({r^{2}}-r_{+}^{2})}}{{{r^{2}}}}d{x^{2}}+\frac{{{r^{2}}d{r^{2}}}}{{({r^{2}}+\tilde{r}_{-}^{2})({r^{2}}-r_{+}^{2})}}].$
(37)

Correspondingly, we have

$\begin{gathered}r_{+}^{2}=4{G_{3}}\left({M+\sqrt{{M^{2}}+\frac{{J_{E}^{2}}}{{{L^{2}}}}}}\right),\hfill\\\
r_{-}^{2}=4{G_{3}}\left({M-\sqrt{{M^{2}}{\text{ +
}}\frac{{J_{E}^{2}}}{{{L^{2}}}}}}\right).\hfill\\\ \end{gathered}$ (38)

Note that the metric in (37) becomes equivalent to (10) after interchanging
$x$ and $\tau_{E}$. We can obtain periodicity in spacetime by the double Wick
rotation of $x$ and $\tau_{E}$ directions in (10) and (15). Alternatively,
periodicity can be obtained by requiring the regularity of the metric on a
cone spanned by $(x,r)$. The black hole horizon is at $r={r_{+}}$, and the
metric near the black hole horizon can be approximated as

$d{s^{2}}={L^{2}}[r_{+}^{2}{(\frac{{{{\tilde{r}}_{-}}}}{{{r_{+}}}}dx-d\tau_{E})^{2}}+\frac{{2(r_{+}^{2}+\tilde{r}_{-}^{2})(r-{r_{+}})}}{{{r_{+}}}}d{x^{2}}+\frac{{{r_{+}}d{r^{2}}}}{{2(r_{+}^{2}+\tilde{r}_{-}^{2})(r-{r_{+}})}}].$
(39)

By taking

$\rho=L\sqrt{\frac{{2{r_{+}}\left({r-{r_{+}}}\right)}}{{r_{+}^{2}+\tilde{r}_{-}^{2}}}},$
(40)

we can have

$d{s^{2}}={L^{2}}r_{+}^{2}{(\frac{{{{\tilde{r}}_{-}}}}{{{r_{+}}}}dx-d\tau_{E})^{2}}+{\rho^{\text{2}}}d{\left({\frac{{r_{+}^{2}+\tilde{r}_{-}^{2}}}{{{r_{+}}}}x}\right)^{2}}+d{\rho^{2}}.$
(41)

To avoid a conical singularity at the black hole horizon, we need to make the
following identification

$\left({\tau_{E},x}\right)\sim\left({\tau_{E}+\eta,x+\zeta}\right)$ (42)

with

$\begin{gathered}\eta=\frac{{2\pi{{\tilde{r}}_{-}}}}{{r_{+}^{2}+\tilde{r}_{-}^{2}}},\hfill\\\
\zeta=\frac{{2\pi{r_{+}}}}{{r_{+}^{2}+\tilde{r}_{-}^{2}}}.\hfill\\\
\end{gathered}$ (43)

One can also see periodicity in the ADM decomposition of (39). When we require
the regularity of the metric on a cone spanned by $(\tau_{E},r)$, we obtain
periodicity in (43).

We also define the chemical potential, which is equal to the shift (36) as
follows:

$\displaystyle\mu_{1}=-\dfrac{N^{x}_{E}}{L}=\dfrac{r_{+}}{\tilde{r}_{-}L}.$
(44)

$\eta$ and $\zeta$ satisfy the following equation

$\displaystyle\mu_{1}L\eta=\zeta.$ (45)

Compared with the density matrix $\rho=e^{-2\pi\tau_{2}H+2\pi\tau_{1}iP}$ in
the dual CFT, we obtain the complex parameter as follows:

$\displaystyle\tau_{1}=\dfrac{\zeta}{2\pi}=\dfrac{\mu_{1}L\eta}{2\pi},\quad\tau_{2}=\dfrac{\eta}{2\pi}.$
(46)

We then consider an analytic continuation of the double Wick rotated metric
(3) to the Lorentz signature. We employ a complex coordinate system
$z=x+i\tau_{E}$. Periodicity of $z$ is given by

$\displaystyle z\sim z+\zeta+i\eta.$ (47)

After the analytic continuation $\tilde{r}_{-}\to-ir_{-}$
($\eta\to-i\eta_{L}$), only the periodicity of spatial directions is obtained
as follows:

$\displaystyle z\sim z+L_{1},\quad\bar{z}\sim\bar{z}+L_{2},$ (48)

where the periodicity is defined as

$\displaystyle
L_{1}\equiv\zeta+\eta_{L}=\dfrac{2\pi}{\Delta_{-}}=\dfrac{2\pi}{r_{+}-r_{-}},\quad
L_{2}\equiv\zeta-\eta_{L}=\dfrac{2\pi}{\Delta_{+}}=\dfrac{2\pi}{r_{+}+r_{-}}.$
(49)

Note that the two periodicities of holomorphic and anti-holomorphic sectors
are different (see also left and right temperature (19) of the rotating BTZ
black hole). The periodicity of $\tau_{E}$ is considered to be infinite in
(48). Thus, the temperature of (34) vanishes. Note that $t=$const and
$r=r_{+}$ surfaces become timelike $g_{xx}=-L^{2}r_{-}^{2}\leq 0$ after the
analytic continuation to the Lorentzian frame. It shows that the Bekenstein
entropy cannot be defined and the background (34) has a closed time-like
surface [22]. Using the stress energy tensor computed in appendix A and
changing into the Lorentzian frame $t\to-i\tau_{E}$ and $r_{-}\to
i\tilde{r}_{-}$, moreover, we obtain energy density and momentum density as
follows:

$\displaystyle\epsilon_{1}=\langle T_{tt}\rangle=-\dfrac{M}{2\pi
L},\quad\langle T_{tx}\rangle=-\dfrac{J}{2\pi L^{2}},\quad\langle
T_{xx}\rangle=-\dfrac{M}{2\pi L}.$ (50)

Energy density in (50) is negative and analogous to the Casimir energy of dual
field theory as described by [35]. By setting $r_{-}=0$, actually, the metric
becomes the $AdS_{3}$ soliton. It is dual to field theory at low temperature.
These results show that the Lorentzian background (34) is not a black hole.

##  4 Holographic entanglement entropy in the double Wick rotated metric

We derive covariant entanglement entropy in the double Wick rotated version of
the rotating BTZ black hole. This background is considered to be dual to CFT
with changed periodicity of Euclidean time and spatial direction
$(x,t_{E})\sim(x+\zeta,t_{E}+\eta)$, while the computation is performed in the
Lorentzian signature (34) because the time slice of a time dependent system is
well-defined.

We consider the subsystem $A$ as a strip with a length $\Delta l=x_{1}-x_{2}$.
The subsystem $B$ is the complement of $A$. The holographic formula is a
codimension 2 extremal surface $\gamma_{A}$ defined as the saddle point of the
area function in the Lorentzian spacetime as follows: [8]

$\displaystyle S_{A}=\dfrac{\mbox{Area}(\gamma_{A})}{4G_{N}},$ (51)

where $\gamma_{A}$ has the same boundary as the region $A$ and $\gamma_{A}$ is
homotopic to $A$. The extremal surface corresponds to the space-like geodesic
in the bulk connecting points of $\partial A$ when the $3d$ bulk spacetime is
considered. The extremal surface condition is equivalent to the vanishing of
null expansions.

The metric (34) is the double Wick rotated version of the rotating BTZ (by
taking $\tilde{t},\ \tilde{x}$ to $ix,\ it$). It is locally equivalent to the
pure $AdS_{3}$. The transformation from pure $AdS_{3}$ to the double Wick
rotated metric (34) becomes

$\displaystyle
w_{+}=\sqrt{\dfrac{r^{2}-r_{+}^{2}}{r^{2}-r_{-}^{2}}}e^{i(t+x)\Delta_{+}}=X+T,$
$\displaystyle
w_{-}=\sqrt{\dfrac{r^{2}-r_{+}^{2}}{r^{2}-r_{-}^{2}}}e^{i(t-x)\Delta_{-}}=X-T,$
$\displaystyle
z=\sqrt{\dfrac{r_{+}^{2}-r_{-}^{2}}{r^{2}-r_{-}^{2}}}e^{i(tr_{+}+xr_{-})},$
(52)

where $\Delta_{\pm}=r_{+}\pm r_{-}$. The pure $AdS$ is written as
$ds^{2}=\frac{L^{2}(dw_{+}dw_{-}+dz^{2})}{z^{2}}$. The spacelike geodesic in
pure $AdS_{3}$ becomes $(X-X_{*})^{2}+z^{2}=R^{2}$. The extremal surface is
obtained by mapping geodesics in pure $AdS_{3}$ into the one in the double
Wick rotated metric.

The extremal surface should be on the hypersurface

$\displaystyle\gamma w_{+}-\gamma^{-1}w_{-}=const,$ (53)

where $\gamma$ means boosts. The subsystem is on a constant time slice of
$t_{0}$. The value of $t$ should be the same. When the extremal surface has
two endpoints, $x_{1}$ and $x_{2}$, the constraint of the extremal surface
becomes

$\displaystyle\gamma^{2}e^{i(t_{0}+x_{1})\Delta_{+}}-e^{i(t_{0}-x_{1})\Delta_{-}}=\gamma^{2}e^{i(t_{0}+x_{2})\Delta_{+}}-e^{i(t_{0}-x_{2})\Delta_{-}}.$
(54)

The cut-off and the length of the interval near the $AdS$ boundary become

$\displaystyle\epsilon=\dfrac{\sqrt{r_{+}^{2}-r_{-}^{2}}}{r_{\infty}}e^{i(r_{+}t_{0}+r_{-}x_{1})},$
$\displaystyle(\Delta x)^{2}=\Delta w_{+}\Delta
w_{-}=(e^{i\Delta_{+}(t_{0}+x_{1})}-e^{i\Delta_{+}(t_{0}+x_{2})})(e^{i\Delta_{-}(t_{0}-x_{1})}-e^{i\Delta_{-}(t_{0}-x_{2})}).$
(55)

The holographic entanglement entropy becomes the geodesic in $AdS_{3}$.
Substituting the cut-off and the length into the formula of the geodesic, we
obtain the entanglement entropy in the double Wick rotated metric as follows:

$\displaystyle S_{A}$ $\displaystyle=\dfrac{c}{6}\log\Big{(}\dfrac{(\Delta
x)^{2}}{\epsilon_{1}\epsilon_{2}}\Big{)}=\dfrac{c}{6}\log\Big{(}\dfrac{4r_{\infty}^{2}}{\Delta_{+}\Delta_{-}}\sin\dfrac{\Delta_{+}\Delta
l}{2}\sin\dfrac{\Delta_{-}\Delta l}{2}\Big{)}$ (56)
$\displaystyle=\dfrac{c}{6}\log\Big{(}\dfrac{(\zeta^{2}-\eta_{L}^{2})}{\pi^{2}\epsilon^{2}}\sin\dfrac{\pi\Delta
l}{\zeta-\eta_{L}}\sin\dfrac{\pi\Delta l}{\zeta+\eta_{L}}\Big{)},$

where $\Delta l=x_{1}-x_{2}$ and $r_{\infty}=l^{2}/\epsilon$. The central
charge $c$ is defined as $c=3L/(2G_{3})$. The entanglement entropy factorizes
into the left and right sector

$\displaystyle S_{A}=S_{R}+S_{L},$ (57)

where

$\displaystyle
S_{L}=\dfrac{c}{6}\log\Big{(}\dfrac{L_{1}}{\pi\epsilon}\sin\dfrac{\pi\Delta
l}{L_{1}}\Big{)},$ $\displaystyle
S_{R}=\dfrac{c}{6}\log\Big{(}\dfrac{L_{2}}{\pi\epsilon}\sin\dfrac{\pi\Delta
l}{L_{2}}\Big{)}.$ (58)

Here, $L_{1}=\zeta+\eta_{L}$ and $L_{2}=\zeta-\eta_{L}$ are the periodicities
of the (anti-)holomorphic sector $z$(and $\bar{z}$), respectively (see eq.
(49)). The left sector will decouple from the right sector. $S_{L}$ and
$S_{R}$ are proportional to entanglement entropy of a single interval along a
spatial direction with a periodic boundary condition (periodicity $L_{1}$ and
$L_{2}$). When $r_{-}=0$ ($\eta_{L}=0$), $S_{L}=S_{R}$ and entanglement
entropy $S_{A}$ becomes (2).

##  5 The CFT side

We periodically identify the two dimensional Euclidean manifold. The partition
function of bosons becomes

$\displaystyle Z(\tau)=\mbox{tr}\Big{[}\exp(2\pi
i\tau_{1}P-2\pi\tau_{2}H)\Big{]}=\mbox{tr}(q^{L_{0}-\frac{c}{24}}\bar{q}^{\tilde{L}_{0}-\frac{\tilde{c}}{24}}),$
(59)

where we have used $q=e^{2\pi i\tau}$. Momentum generates the translation
along the $x$ direction $P=L_{0}-\tilde{L}_{0}$ and Hamiltonian generates the
translation along the time direction
$H=L_{0}+\tilde{L}_{0}-\frac{c+\tilde{c}}{24}$. To match the gravity dual, the
periodicity is chosen as

$\displaystyle\tau_{1}=\dfrac{\zeta}{2\pi},\quad\tau_{2}=\dfrac{\eta}{2\pi}=-\dfrac{\zeta\Omega_{E}}{2\pi},$
(60)

where $\Omega_{E}=1/(\mu_{1}L)$. Because the metric is invariant under the
complex conjugation and degenerate for a real $\tau$, we restrict to
$\mbox{Im}(\tau)>0$ ($\Omega_{E}<0$) or vice versa [36].

This is realized in terms of the following conformal map

$\displaystyle w=\exp\Big{(}\frac{iv}{\tau}\Big{)},$ (61)

where the new coordinate $v=x+it_{E}$ has the periodicity $v\sim v+2\pi\tau$
with modulus $\tau=\frac{\zeta(1-i\Omega_{E})}{2\pi}$. Coordinates are
transformed as follows:

$\displaystyle(x,t_{E})\sim(x+\zeta,t_{E}-\zeta\Omega_{E}).$ (62)

One can compare the above result with dual CFT to the rotating BTZ black hole.
Recall that it is obtained by the conformal map $w=\exp(\tilde{w}/\tau)$,
where $\tau$ is the same as the above case with $\zeta=\beta$. The new
coordinate $\tilde{w}$ has the periodicity $\tilde{w}\sim\tilde{w}+i2\pi\tau$
or

$\displaystyle(\tilde{x},\tilde{t}_{E})\sim(\tilde{x}+\beta\Omega_{E},\tilde{t}_{E}+\beta),$
(63)

where $i\tau$ becomes the new modulus. Thus, the periodicity is inversed when
it is compared with (62).

###  5.1 Entanglement entropy in the dual CFT

The value of $\mbox{tr}(\rho_{A}^{n})$ for the reduced density matrix is
equivalent to the correlation functions of twisted operators with conformal
weights $\delta_{n}=\bar{\delta}_{n}=\frac{c}{24}(n-\frac{1}{n})$.

When CFT lives on $2d$ Euclidean space and the region $A$ is $(u_{1}\leq x\leq
u_{2})$ at a constant time slice,

$\displaystyle\mbox{tr}(\rho_{A}^{n})=\langle\Phi(u_{1})\Phi(u_{2})\rangle=c_{n}\Big{(}\dfrac{|u_{1}-u_{2}|}{\epsilon}\Big{)}^{-\frac{c}{6}(n-\frac{1}{n})},$
(64)

where the UV cut-off is $\epsilon$ and $c_{n}$ is a constant.

Under the general conformal transformation, $\mbox{tr}\rho_{A}^{n}$ transforms
as a 2-point function of primary fields $\Phi$. This implies that one can
compute it in different geometries. Under the conformal transformation (61),
the twisted field is transformed as

$\displaystyle\langle\Phi(w_{1}^{\prime})\Phi(w_{2}^{\prime})\rangle=\Big{|}\dfrac{\partial
w_{1}^{\prime}}{\partial w_{1}}\Big{|}^{-2\delta_{n}}\Big{|}\dfrac{\partial
w_{2}^{\prime}}{\partial
w_{2}}\Big{|}^{-2\delta_{n}}\langle\Phi(w_{1})\Phi(w_{2})\rangle,$ (65)

where $\partial w^{\prime}/\partial w=\tau/(iw)$ and
$\partial\bar{w}^{\prime}/\partial\bar{w}=-\bar{\tau}/(i\bar{w})$.

We then have

$\displaystyle\mbox{tr}(\rho_{A}^{n})=\left(\dfrac{\epsilon^{2}e^{\frac{i}{2}\Big{(}\frac{w_{1}^{\prime}}{\tau}-\frac{\bar{w}_{1}^{\prime}}{\bar{\tau}}\Big{)}}e^{\frac{i}{2}\Big{(}\frac{w_{2}^{\prime}}{\tau}-\frac{\bar{w}_{2}^{\prime}}{\bar{\tau}}\Big{)}}}{|\tau|^{2}\Big{(}e^{i\frac{w_{1}^{\prime}}{\tau}}-e^{i\frac{w_{2}^{\prime}}{\tau}}\Big{)}\Big{(}e^{-i\frac{\bar{w}_{1}^{\prime}}{\bar{\tau}}}-e^{-i\frac{\bar{w}_{2}^{\prime}}{\bar{\tau}}}\Big{)}}\right)^{2\delta_{n}}=\left(\dfrac{\pi^{2}\epsilon^{2}}{\zeta^{2}(1+\Omega_{E}^{2})\sin\Big{(}\frac{\Delta
l}{2\tau}\Big{)}\sin\Big{(}\frac{\Delta\bar{l}}{2\bar{\tau}}\Big{)}}\right)^{2\delta_{n}},$
(66)

where the length of the interval becomes $\Delta
l=w_{1}^{\prime}-w_{2}^{\prime}=x_{1}-x_{2}$.

By differentiating in terms of $n$, the entanglement entropy is obtained as

$\displaystyle S_{A}$ $\displaystyle=-\dfrac{\partial}{\partial
n}\log\mbox{tr}(\rho_{A}^{n})\Big{|}_{n=1}=\dfrac{c}{6}\log\Big{(}\dfrac{\zeta^{2}(1+\Omega_{E}^{2})}{\pi^{2}\epsilon^{2}}\sin\Big{(}\dfrac{\Delta
l}{2\tau}\Big{)}\sin\Big{(}\dfrac{\Delta\bar{l}}{2\bar{\tau}}\Big{)}\Big{)}$
(67)
$\displaystyle=\dfrac{c}{6}\log\Big{(}\dfrac{(\zeta^{2}+\eta^{2})}{\pi^{2}\epsilon^{2}}\Big{|}\sin\dfrac{\pi\Delta
l}{\zeta+i\eta}\Big{|}^{2}\Big{)}.$

Note that the entanglement entropy factorizes into two modes, which will
decouple in $2d$ CFT. We perform the analytic continuation $\tilde{r}_{-}\to-
ir_{-}$ $(\eta\to-i\eta_{L})$ and then realize (56) obtained from the gravity
dual.

##  6 Discussion

In this paper, the holographic covariant entanglement entropy was analyzed in
the double Wick rotated version of a rotating BTZ black hole and agreed with
entanglement entropy in the CFT side (67). (56) in the Lorentzian signature
shows that entanglement entropy factorizes into the left and right sectors.
Each sector is proportional to entanglement entropy of a single interval along
a compact spatial direction with periodicity $L_{1}$ and $L_{2}$ (49). The
constraint of the causality annoys in the Lorentzian signature as mentioned
below. However, analysis on the CFT side implies that it will also work for
any places where the extremal surface probes the interior of the background
(34).

The entanglement entropy has also been computed on the CFT side. The
entanglement entropy was obtained by a conformal transformation (61) of
correlation functions of twisted operators in the analogue of [8]. The
entanglement entropy factorized into two modes, which would decouple like the
one in the gravity dual. Because a cylinder can be considered as the infinite
limit of a torus periodicity, our result should be realized from entanglement
entropy on the torus [37, 38]. The modular transformation of entanglement
entropy on the torus will also be interesting. When the chemical potential
conjugate to momentum is zero, $\tau\to-1/\tau$ transforms from zero
temperature to high temperature. The modular transformation analysis with
$\tau_{1}\neq 0$ should be applicable for ours.

The double Wick rotated geometry may have problems with causality in the
Lorentzian frame. We make use of a Killing vector to analyze the causal
structure of the background. A Killing vector $\partial_{x}$ has the norm
$\xi\cdot\xi=r^{2}-r_{-}^{2}-r_{+}^{2}$, which is spacelike for $r^{2}\geq
r_{+}^{2}+r_{-}^{2}$. However, it becomes timelike for $r^{2}\leq
r_{+}^{2}+r_{-}^{2}$. 444 In addition, $t=$const and $r=r_{+}$ surfaces become
timelike $g_{xx}=-L^{2}r_{-}^{2}\leq 0$.  It shows that the double Wick
rotated geometry has a closed time-like curve [22]. It will be possible to
explore how the law of physics allows a closed time-like curve [39]. Second,
the background is not a black hole in the Lorentzian signature because it does
not have Hawking temperature, which vanishes after the analytic continuation
$\tilde{r}_{-}\to-ir_{-}$ (see (43)). It also has negative energy, unlike
thermal field theory. This negative energy will correspond to Casimir energy
in the CFT side [35].

In the Euclidean signature, on the other hand, the gravitational background
becomes the double Wick rotated version of a rotating BTZ black hole, where
the periodicity of Euclidean time and spatial direction is changed as (43).
This background has properties of rotating $AdS$ black holes, such as Hawking
temperature by requiring regularity at the black hole horizon. The black hole
entropy will appear in this frame. It will be interesting to analyze the
thermodynamics of the double Wick rotated geometry. The entanglement entropy
(67) in the Euclidean signature will realize the results of dual CFT to black
holes.

## Acknowledgments

We would like to thank B. S. Kim for collaboration in the initial stage of
this project and for helpful discussions. MF would like to thank S. He, S.
Lin, and J. Sun for discussions and comments. We also thank T. Takayanagi for
valuable discussion and comments.

##  Appendix A Holographic stress energy tensor

In this appendix, we compute the holographic stress tensor for the double Wick
rotated metric. We compute the holographic stress energy tensor in the FG
coordinate
$ds^{2}=\frac{L^{2}}{\rho^{2}}(d\rho^{2}+\bar{g}_{ij}^{E}dx^{i}dx^{j})$, where
$\bar{g}^{E}_{ij}=\bar{g}^{E(0)}_{ij}+\bar{g}^{E(2)}_{ij}\rho^{2}+\dots$.
Metric for (3) becomes

$\displaystyle\bar{g}_{\tau_{E}\tau_{E}}^{E(0)}=1,\quad\bar{g}_{xx}^{E(0)}=1,\quad\bar{g}_{\tau_{E}x}^{E(0)}=0,$
$\displaystyle\bar{g}_{\tau_{E}\tau_{E}}^{E(2)}=\dfrac{r_{+}^{2}-\tilde{r}_{-}^{2}}{2},\quad\bar{g}_{xx}^{E(2)}=\dfrac{-r_{+}^{2}+\tilde{r}_{-}^{2}}{2},\quad\bar{g}^{E(2)}_{\tau_{E}x}=-r_{+}\tilde{r}_{-}.$
(68)

The boundary stress tensor becomes

$\displaystyle\langle\tilde{T}_{\mu\nu}^{E}(x)\rangle=\dfrac{L}{8\pi
G_{3}}(\bar{g}_{\mu\nu}^{(2)}-\bar{g}_{\mu\nu}^{(0)}\bar{g}^{2\alpha}{}_{\alpha}).$
(69)

Substituting (68) into (69) and performing the coordinate transformation into
$(\tau_{E}^{\prime},x^{\prime})=(L\tau_{E},Lx)$, we obtain the stress energy
tensor in the dual CFT as follows:

$\displaystyle\langle T^{E}_{\tau_{E}\tau_{E}}\rangle=\dfrac{M_{E}}{2\pi
L},\quad\langle T_{\tau_{E}x}\rangle=-\dfrac{J_{E}}{2\pi L^{2}},\quad\langle
T_{xx}\rangle=-\dfrac{M_{E}}{2\pi L}.$ (70)

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