Kinematic space and wormholes
Received: November
Kinematic space and wormholes
Jiandong Zhang 0 1 3 6
Bin Chen 0 1 2 3 4 5
Open Access 0 1 3
c The Authors. 0 1 3
0 5 Yiheyuan Rd , Beijing 100871 , P.R. China
1 Peking University , Beijing 100871 , P.R. China
2 Center for High Energy Physics, Peking University
3 Zhuhai 519082 , Guangdong , P.R. China
4 Department of Physics and State Key Laboratory of Nuclear Physics and Technology
5 Collaborative Innovation Center of Quantum Matter
6 TianQin Research Center for Gravitational Physics, Sun Yatsen University
The kinematic space could play a key role in constructing the bulk geometry from dual CFT. In this paper, we study the kinematic space from geometric points of view, without resorting to di erential entropy. We nd that the kinematic space could be intrinsically de ned in the embedding space. For each oriented geodesic in the Poincare disk, there is a corresponding point in the kinematic space. This point is the tip of the causal diamond of the disk whose intersection with the Poincare disk determines the geodesic. In this geometric construction, the causal structure in the kinematic space can be seen clearly.
wormholes; AdSCFT Correspondence; Black Holes

nd that every transformation in the SL(2; R) leads to a geodesic in the
kinematic space. In particular, for a hyperbolic transformation de ning a BTZ black hole,
it is a timelike geodesic in the kinematic space. We show that the horizon length of the
static BTZ black hole could be computed by the geodesic length of corresponding points
in the kinematic space. Furthermore, we discuss the fundamental regions in the kinematic
space for the BTZ blackhole and multiboundary wormholes.
1 Introduction 2 3 2.1
AdS3 and its kinematic space
H2 and dS2
Geodesics in H2
Symmetries on AdS3 and its quotients
Fuchsian group and its action
BTZ black hole as quotient
Multiboundary wormhole
Threeboundary wormhole
Torus wormhole
Kinematic space and wormhole
Geodesics in the kinematic space
Symmetry transformation and kinematic space geodesic
BTZ and kinematic space
Multiboundary wormhole in kinematic space
Conclusion and discussion
Introduction
One of recent developments in the AdS/CFT correspondence is on the emergence of
spacetime and di eomorphism. The key notion in the study of the emergent spacetime is the
entanglement and its holographic computation. The holographic entanglement entropy in
the Einstein gravity is proposed in [1, 2] to be
SRT =
where A is the area of the minimal surface which is homologous to the boundary region.
This formula, being reminiscent of the BekensteinHawking formula for the black hole
entropy [3], suggest a deep relation between quantum gravity and quantum information. It
has been widely suspected that the holographic entanglement entropy could play a pivotal
role in constructing bulk spacetime and even bulk physics.
There are several proposals to construct the bulk geometry from boundary CFT,
mainly based on the concept of the tensor network [4{10]. Among them, one
promising approach proposed by B. Czech et al. is to view the MERA (Multiscale
Entanglement Renormalization Ansantz) tensor network as a discrete version of vacuum kinematic
space [7, 8]. This proposal is inspired by the study of the hole entropy in the bulk from
dual CFT data, which suggests a way to de ne the bulk geometry from di erential
entropy [11{18]. To compute the length of a curve
in the hyperbolic plane, one could
apply integral geometry rather than di erential geometry. The length could be given by
the Crofton formula
Length of the curve
1 Z
label the oriented geodesic in the Poincare disk, n ( ; ) is the intersection
number of the geodesic with the curve
and K denotes the kinematic space. The most
interesting part is on the measure !( ; ) in the kinematic space, which has the form as
or in terms of the coordinates of the ending points of the geodesics on the disk boundary
the form of the measure becomes
This measure is more suggestive when being given by
!( ; ) =
u =
v =
!(u; v) =
!(u; v) =
@2S(u; v)
where S(u; v) is the entanglement entropy of the interval (u; v). In [19], the authors
furthermore suggest that the Crofton form should be interpretated as the conditional mutual
information.1 The basic picture on the kinematic space is that it is an auxiliary Lorentzian
geometry, whose metric is de ned in terms of conditional mutual information.
In this paper, we would like to study the kinematic space from geometric points of
We show that the kinematic space can be de ned in a geometric way. Simply
speaking, every geodesics in the Poincare disk could de ne a causal cone, whose tips are in
the kinematic space. The causal structure in the kinematic space can be seen clearly in this
geometric picture. Moreover, we discuss the static wormhole solution in the AdS3 gravity
and its representation in the kinematic space. We show that the timelike geodesic in the
kinematic space is closely related to the isometric transformation of hyperbolic type. For
the BTZ spacetime formed by the identi cation of the geodesics with respect to a hyperbolic
element in the Fuchsian group, its horizon length could be read from the timelike geodesic
distance in the kinematic space between the points corresponding to the geodesics in the
disk. Therefore for the eternal BTZ black hole formed by the identi cation of a pair of
1For the higher dimensional study of the Crofton form and its interpretation, see [20, 21].
2While we are preparing this manuscript, there appeared two works [22, 23], which partially overlap our
discussion in section 2.
geodesics, it could be described by two timelike separated points in the kinematic space.
These two points cannot be determined uniquely. As long as a pair of points lie in the
timelike geodesic determined by the transformation and the geodesic distance between
xed, they describe the same BTZ spacetime. On the other hand, the timelike
geodesic de ned by a hyperbolic transformation is unique. In this sense, the BTZ spacetime
could be related to a timelike geodesic in the kinematic space. In the similar spirit, we can
describe the multiboundary wormhole easily.
Another interesting issue is to consider the kinematic space for the BTZ wormhole and
other multiboundary wormhole background. The kinematic space can still be de ned by
the geodesics in these spacetime. We start from the kinematic space for AdS3, and take
into account of the quotient identi cation de ning the wormhole. We discuss carefully how
to classify the geodesics in the BTZ spacetime and propose a consistent rule to de ne the
fundamental region in the kinematic space for the BTZ spacetime. We furthermore show
that the fundamental region for the multiboundary wormhole could be de ned to be the
intersection of the fundamental regions for the BTZ spacetimes, each being de ned by the
fundamental elements of the Fuchsian group.
The remaining part of this article is organized as follows. In section 2, after giving
a brief review of AdS3 spacetime and its di erent coordinate systems, we show how to
describe the kinematic space. In section 3, we review the construction of the static BTZ
black hole and general multiboundary wormholes by using the Fuchsian group identi
cation. Especially we discuss the threeboundary wormhole and singleboundary torus
wormhole. In section 4, we discuss the properties of the kinematic space. We show that
a SL(2; R) transformation, being the isometric transformation of AdS3, de ne a geodesic
in the kinematic space. In particular, we study the three boundary wormhole to get its
fundamental regions in the kinematic space, and give a method to get the fundamental
region in kinematic space for general wormholes. We end with conclusions and discussions
in section 5.
AdS3 and its kinematic space
The AdS3 can be taken to be a hyperboloid space in the 2 + 2 dimensional at spacetime
R2;2 with the metric
ds2 =
dV 2 + dX2 + dY 2;
The AdS3 spacetime is de ned by the relation
V 2 + X2 + Y 2 =
U = cosh cos ;
X = sinh cos ;
V = cosh sin ;
Y = sinh sin ;
we can read the metric of AdS3 in the global AdS coordinates
ds2 =
cosh2 d 2 + d 2 + sinh2 d 2;
The classical solutions in the AdS3 gravity could be constructed by the quotient
identi cation by the discrete subgroup of the isometry group SL(2; R). If we focus on the
static solutions, the construction could be understood as the identi cation of the geodesics
pairwise in the constant time slice of AdS3. The constant time slice is a twodimensional
hyperboloid H2, the socalled Poincare upper half plane, which is of the metric
ds2 = d 2 + sinh2 d 2:
In fact, for simplicity we just take the
= 0 slice, this is equivalent to V = 0.
H2 and dS2
The relation between the constant time slice of AdS3 and the kinematic space is most easily
ds2 =
dU 2 + dX2 + dY 2:
Y 2 = 1 :
With the embedding coordinates
U = cosh ;
X = sinh cos ;
Y = sinh sin ;
we can project the upper sheet of the hyperboloid onto the unit disk
at U = 0 ;
which is usually called the Poincare disk. With the disk coordinates xD, yD, we can read
the relations between the points on the hyperboloid and the disk
We may introduce the polar coordinates on the disk
xD =
yD =
sinh cos
sinh sin
xD = r cos # ;
yD = r sin # :
ds2 = 4
dr2 + r2d#2
= 4
dx2D + dyD2
On the other hand, the twodimensional de Sitter spacetime can be embedded into the
same spacetime (2.6) as well. It is de ned by the relation
U 2 = 1 :
By de ning a new coordinate system with the following relation
U = sinh ;
X = cosh cos ;
Y = cosh sin ;
we can get the metric of dS2
If we make another coordinate transformation
x =
xD =
x2 + (1 + y)2
y =
yD =
x2 + (1 + y)2
then in terms of the coordinates ( ; ) we nd another metric form of the dS2 spacetime
And the point ( ; ) in this coordinate will correspond to
in (U; X; Y )
We can de ne H2 and dS2 on the Poincare upper half plane by introducing
x =
y =
Then the metrics of the hyperbolic space and de Sitter spacetime are respectively
Let us de ne
ones on the Poincare disk is
Or, more explicitly,
ds2 =
dx2 + dy2
ds2 =
zD = xD + iyD ;
z = x + iy ;
z =
zD =
then the transformation between the coordinates on the Poincare upper half plane and the
ds2 =
ds2 =
sheeted hyperboloid is the kinematic space. The unit disk in the center is the Poincare disk. In the
gure, the plane crossing the origin intersects H2 on a curve, which is a geodesic in H2. The
line orthogonal to the plane and crossing the origin intersects the kinematic hyperboloid with two
points, corresponding to the geodesics with di erent orientations. In the middle
gure, we show
the projection from the H2 hyperboloid to the Poincare disk. The geodesic is mapped to an arc
of a circle in the disk. In the right
gure, we show that the the future and the past domains of
dependence of the disk, whose boundary circle intersects the Poincare disk with the arc, form a
causal diamond with its tips being in the kinematic space. This gives another geometric construction
of the kinematic space.
Geodesics in H2
The geodesics in H2 are simple. On H2, the equation of a geodesic without orientation is
tanh cos(
0) = cos 0 ;
In the coordinates of R2;1, this is
sin 0Y = 0 :
This is a plane crossing the origin. So for any geodesic on H2 we can nd a corresponding
plane crossing the origin, and the intersection curve between this plane and hyperboloid
H2 is just the geodesic. The line normal to the plane and crossing the origin intersect the
dS2 spacetime (2.13) at two points, as shown in the left of gure 1. The coordinates of
these points in terms of (U; X; Y ) are
In terms of ( ; ) coordinate, these two points are at ( 0; 0) and ( + 0;
0) respectively,
corresponding to the geodesics with opposite orientations. The rst point correspond to
geodesic starting from
0 and ending on 0 +
0 on the boundary, and the second
point correspond to geodesic starting from 0 + 0 and ending on 0
0 on the boundary.
In the Poincare upper plane coordinate for H2, the geodesic equation corresponding
to (2.23) is
2 cos 0x + cos 0 + sin 0 = 0 :
It is either a semicircle or a straight line normal to the xaxis
(Semicircle centered at cos 0 sin 0 ; 0 with radius cos 0 sin 0
cos 0 sin 0
; cos 0 6= sin 0 ;
cos 0 = sin 0
In the Poincare disk, as shown in the middle of gure 1, the geodesic equation
corresponding to (2.23) is
2 sin 0yD = 0 ;
or a line crossing the origin when 0 = 2
The geometric meaning of 0 and 0 is clear: 0 is the opening angle of the arc of the
unit circle intersected by the geodesic, and 0 is the angular coordinate of the midpoint of
this arc. In the disk, we can also denote each geodesic by the angular coordinates ( ; ) of
its two endpoints on the unit circle, then we may have
= 0
= 0 + 0 ;
to de ne the kinematic space [19]. We should notice that in the kinematic space the points
0) denote the same geodesic but with di erent orientations.
Remarkably, the kinematic space is exactly the dS2 spacetime (2.17) with the coordinates
( ; ) and the metric (2.17) given above. Therefore we can conclude that the dS2 spacetime
de ned by (2.13) is exactly the kinematic space of H2.
In the above discussion, we have the picture that the corresponding points of a geodesic
in the kinematic space are the same as the points we get on dS2 in eq. (2.25) by the
intersection of the normal line to the plane (2.24). This picture shows explicitly the relation
between a hyperbolic space and its kinematic space.3 However, in the kinematic space, the
points could be timelike or spacelike separated, depending on whether the corresponding
geodesics have intersection or not [7]. It is not clear to see why there exist such a kind of
relations in the above construction.
There is another geometric construction to show the causal relation of the points in
the kinematic space more clearly. As we shown above, the geodesic (2.26) in the Poincare
disk is actually part of a circle. This circle is the boundary of a disk, which in general
is not of unit radius. The interesting point is that the future and the past domains of
dependence of this disk form a causal diamond with its tips being actually in the kinematic
space, as showed in the right gure of gure 1. In the embedding space, the coordinates of
the tips are
while in the kinematic space, their corresponding coordinates ( ~; ~) satisfy the relation
~ = 0 ;
~ =
3This has already been pointed out in gure 15 in [19].
They are slightly di erent from the points ( 0; 0) or ( 0 + ;
geodesic by using the normal line.
However, the di erence is just a constant translation. It is the same as the kinematic
space. Therefore, we can safely take the tips of the diamond as the points corresponding
to the geodesic.
This picture has the advantage to see the causal structure clearly. For example, in
the Poincare disk, if two geodesics have no intersection but have the same orientation,
then the casual diamond of the outer geodesics encloses the one of the inner geodesics,
such that the corresponding point of the inner geodesic is at the casual past of the one
of the outer geodesic. This shows that the causal relation can be seen directly from the
embedding picture by the relation of the corresponding light cone. If the causal diamond of
two geodesic has no intersection, the corresponding geodesics have no intersection as well.
And if two causal diamond have intersection, then the geodesics will also have intersection.
Moreover, in the rst picture, we must decide the embedded dS2 surface rst, then
we can get the corresponding point. But in the second picture, we do not need to know
the surface of kinematic space. We can directly get corresponding points of all geodesics,
which form the kinematic space. And then we can get the induced metric on this surface,
and this is exactly the metric of the kinematic space.
Symmetries on AdS3 and its quotients
Every classical solution in the AdS3 graviy is locally AdS3. They could be constructed by
the quotient identi cation of global AdS3. For example, the BTZ geometry is a quotient
of AdS3 by a discrete subgroup of PSL(2; R) [24, 25]. It is a twoboundary wormhole, or
an eternal black hole [26].
More interesting, there exist many di erent kinds of multiboundary wormholes with
di erent topology. For the static spacetime, one may identify the geodesics in the Poincare
disk to construct such multiboundary wormholes. The detailed construction could be
found in [27{32]. In this section, we will give a brief review of these solutions and discuss
three examples carefully, they are BTZ, threeboundary wormhole and singleboundary
torus wormhole.
Fuchsian group and its action
In this subsection, let us focus on the symmetry transformation on the constant time slice
of AdS3. For simplicity, we start from the Poincare upper plane. The symmetry group is
with j j = ad
de nes the Fuchsian group of the second kind. On the half plane, we have the complex
Mobius transformation
z0 =
Such a transformation leads to a Riemann surface
= H2= , where is a discrete subgroup
which is called the Fuchsian group and is generated by its fundamental element
= f njn 2 Zg.
For each transformation, we also have an oneparameter family of ow lines
f (x; y) = cx2 + (d
a)x + cy2 + ey
b = 0 ;
These ow lines are the integral curve of the transformation. Every ow line is a circle
which crosses the two
the geodesic ow line. For every point z, we can
nd a e such that the point locates on
the corresponding ow line, then the point z locates on the same ow line as well.
Under a transformation , a geodesic (x
x00)2 + y2 = r02 with the parameters being
x0)2 + y2 = r2 changes to another geodesic
x00 =
r02 =
r2) + (ad + bc)x0 + bd
(cx0 + d)2
(cx0 + d)2
Given two geodesics, the transformation relating them to each other is not unique. If
a geodesic is normal to every
ow lines of a transformation, then it is called a
normalgeodesic of the transformation. Given two geodesics without intersection, there exist many
transformation that can transform one to another. But there exist a unique transformation
such that both geodesics are the normalgeodesics of this transformation.
The discussion is similar in the Poincare disk. As shown in
gure 2, among the ow
lines intersecting with the geodesics, the geodesic ow line is special. Actually, the distance
between the identi ed points of two geodesics is the shortest along the geodesic ow line.
Such a distance is de ned to be the distance of two geodesics.
BTZ black hole as quotient
The BTZ black hole could be taken as the quotient of the global AdS3 by a discrete group
of PSL(2; R). The action could be seen most easily in the Poincare disk, if we are only
interested in the static con guration. In fact, if we want to get a BTZ black hole, we
should start from a Fuchsian group de ned by
is the fundamental element such that the group is denoted as
= f g
. On the disk,
we can choose a pair of nonintersecting geodesics to be identi ed by this element. Such
identi cation can be extended to AdS3 and leads to a static BTZ black hole. The horizon
length of the BTZ black hole LH can be computed directly by [33]
= f n
2 PSL(2; R); n 2 Zg, where j Tr j = 2 cosh
normalgeodesics of a PSL(2; R) transformation. The green and red arcs are the
ow lines of this
transformation, and they are normal to the two geodesics. The intersection point of one geodesic
with a ow line is transformed into the intersection point of the other geodesic with the same ow
line. Especially, the red arc is the geodesic ow line. The length of the arc between two intersection
points of the geodesics with the geodesic ow line is the distance between two geodesics.
The group f g and fM 1 M g represent the same BTZ black hole. And the ow line
of the fundamental element represents the angle direction of the black hole, while the
normalgeodesics represents the radius direction.
It is relatively more complex to read the horizon length from the geometric picture of
identi ed geodesics. Let us start from a diagonal transformation
+ y2 =
+ y2 =
LH =
= 2 ln ;
of the normalgeodesic under the transformation is
L: x2 + y2 =
4r2. Without losing
generality, we can assume
> 1. In this case, the distance between two geodesics is just
which is independent of the value of r and match with the computation from the trace of
the element (3.5). For a general element , we can always nd a transformation M such
that 0 = M 1 M is diagonal.
0 =
M =
If is hyperbolic, then we will have ja + dj > 2, and the above parameters are all real. Now
A and B are the coordinates of the two
xed points of
on the boundary for hyperbolic
. The normalgeodesic and its image are respectively
Then we can compute the distance of any two nonintersecting geodesics described by
x1)2 + y2 = r12 ;
x2)2 + y2 = r22 ;
we can assume that v1 < v2. For convenience, we introduce three parameters
A = (u1
B = (u1
C = (u1
Then the horizon length of the BTZ got by identifying the two geodesics is just
The above discussion can be translated into the language in the Poincare disk easily.
of the form
The points on the circle is parameterized by the angular coordinate . The point (x; 0) on
the boundary of the upper half plane is mapped to the point (1; ) with
= 2 arctan x
them is just
LH = 2 ln
Every geodesic in the disk can be characterized by the angular coordinates of its ending
points ( ; ),
< , or equivalently in terms of the coordinates in the kinematic space
. For two geodesics ( 1; 1), ( 2; 2) in the disk, the distance between
One subtle point is that each geodesic actually corresponds to two points in the
kinematic space, depending on the orientation. The points ( ; ) and ( + ;
to the same geodesic if we disregard its orientation. If two points ( 1; 1) and ( 2; 2) are
timelike separated, then the points ( 1; 1) and ( 2 + ;
2) must be spacelike separated.
The distance (3.15) between two geodesics is insensitive to the relative orientation of the
geodesics. However, in order to construct the BTZ black hole by identifying the geodesics
in pair, the geodesics should have the right orientations. Correspondingly the points in
the kinematic space must be timelike separated. If two points in the kinematic space are
timelike separated, then their corresponding geodesics contain each other, have no
intersection and have the same orientation. If they are null separated, then their corresponding
geodesics have one common endpoint. And if two points are spacelike separated, then
their corresponding geodesics either have intersection or have di erent orientation without
LH = 2 ln
ds2 = 4
For the multiboundary wormhole, the construction is similar. Now we need more pairs
of nonintersecting geodesics in the disk. Here for simplicity, we focus on the case with
two pairs of geodesics. With four geodesics, there exist two kinds of identi cation, leading
to a threeboundary wormhole and a singleboundary wormhole with the torus behind the
horizon respectively. There are two fundamental elements 1
, 2 for the Fuchsian group
Threeboundary wormhole
If the geodesic ow lines of the two fundamental elements do not intersect each other,
we obtain a threeboundary wormhole. This wormhole have three asymptotic boundaries,
each of which there exists a black hole. Outside every black hole's horizon, the spacetime
is described exactly by the BTZ metric. In other words, the observer at the asymptotic
in nity of each boundary sees a BTZ black hole. Inside the horizons, the three boundaries
are connected by a region with topology of a pair of pants.
The threeboundary wormhole could be characterized by the horizon lengths Li de ned
3 =
1 2 1. For simplicity, we can always choose the geodesic
on each boundary. The horizon lengths for the rst two boundaries Li are given by the i:
ow lines of both
transformations 1 and
2 to be symmetric about the xaxis on the disc. Moreover, we
can also choose the transformation matrix of 1 to be diagonal. Then we can assume the
transformation matrices are of the form
1 =
2 =
with ; > 1.
The horizon lengths of the black holes on the rst two boundaries are respectively
L1 = 2 ln ;
L2 = 2 ln :
And the horizon length of the black hole on the third boundary L3 is given by
cosh L3 =
= cosh L1 cosh L2
Obviously the length L3 depends on the real parameter , which is restricted by
The fundamental region of a threeboundary wormhole means that every point on the disc
outside this region can be mapped into it by an element of . The four geodesics are
gure on the left shows the fundamental region of threeboundary wormhole. The dashed
between these geodesics is a fundamental region of this wormhole. B1 and B2 denote the rst two
boundaries, corresponding to the transformation
. B31 and B32 denote the two parts of
the third boundary, which corresponds to the transformation
1. And the
slice of three boundary wormhole. The dashed lines on the right gure are the horizons of the black
holes at each boundary. The blue and red lines are the two pair of identi ed normalgeodesics.
as showed on the right of gure 3, we can get a surface which has the topology of a pair of
pants with three boundaries.
Moreover, by acting an element
, the fundamental region we chose above will
be mapped to another fundamental region. The two pairs of normalgeodesics Li;j will be
Li;j , and the corresponding two fundamental elements are
. Then for
any fundamental region, all of its images under the action of the elements in
the whole Poincare disc and have no intersection with each other.
Torus wormhole
For the torus wormhole, the construction is similar as the threeboundary case, and the only
di erence is that the two geodesic ow lines will intersect with each other. This wormhole
have just one asymptotic boundary with a black hole described by the BTZ metric outside
the horizon. Inside the horizon, the region has the topology of a torus with a boundary, or
a pair of pants with two legs being glued together.
The torus wormhole could also be characterized by three parameters, two of them
ow lines of two fundamental elements 1 and 2 have a intersection. The
gure on the left
shows the fundamental region of the torus wormhole, and the marks we use here are the same as the
threeboundary case. The identi cations 1 and 2 leads to two length L1 and L2, which determines
the shape of the torus. B11, B12, B13, B14 denote the four parts of the boundary. They correspond
to the transformations 1 2 1
1. These transformations
are similar to each other such that they give the asymptotic boundary of the wormhole and we can
The three dashed lines include the horizon of the black hole and two cycles with length L1, L2
that determine the region inside the horizon. The blue and red lines are the two pair of identi ed
For simplity, we can choose the geodesic ow lines of both transformations 1
yD = xD tan . Then we may set
1 =
2 =
The two lengths L1 = 2 ln
and L2 = 2 ln
characterize the torus inside the black hole.
The horizon of the black hole is determined by the element H , and the horizon length for
the torus wormhole is just
LH = 2 arccosh
meaning of the marks and the way of construction is similar to the threeboundary case.
In section 2, we introduced the kinematic space from a geometric point of view. In this
section, we study the properties of the kinematic space. We discuss the geodesics in the
kinematic space and show that the geodesic distance between two timelike point is equal
to the horizon length of corresponding BTZ black hole. We also show that the
normalgeodesics of a given SL(2; R) transformation form a geodesic in the kinematic space.
Furthermore we discuss the kinematic space of the wormholes, including the BTZ wormhole
and multiboundary wormholes.
Geodesics in the kinematic space
The kinematic space dS2 could be described by the upper half plane (x; y), y
0 with the
The geodesics in it are of three types
On the other hand, the kinematic space can be described in terms of the coordinates
( ; ) with the metric (2.17). Then the geodesics are described by
ds2 =
dy2 + dx2
y2 = 0 ;
y2 =
= A cos(
8>jAj > 1 ; timelike geodesic
jAj = 1 ; null geodesic
>>:jAj < 1 ; spacelike geodesic
= 0 ;
the kinematic space.
has the parameters
which represents a timelike geodesic. In gure 5, we have drawn the di erent geodesics in
For any two timelike separated points ( 1; 1), ( 2; 2), the geodesic connecting them
A2 =
A cos 0 =
2 cos 1 cos 2 cos( 1
Now the nature of the geodesic could be equivalently determined by the quantity
instead of A2. When this quantity is greater than 1, the geodesic is timelike, and when it
is less than 1 or equals 1, the corresponding geodesic is spacelike or null respectively.
2. The blue lines are the null geodesics with A =
1. The green lines are the spacelike geodesics with A =
The proper time between the two points along the timelike geodesic is
d =
= arctanh pA2
2 cos 1 cos 2 cos( 1
2 cos 1 cos 2 cos( 1
This is exactly the distance between two geodesics which should be identi ed to obtain the
BTZ black hole. Although it seems that (4.7) and (3.15) have very di erent form, they
can be proved to be equal, i.e.
Therefore we arrive the picture that the length of the horizon of the BTZ black hole can be
read from the geodesic distance of the two timelike separated points in the kinematic space,
where the two points correspond to the geodesics to be identi ed in the Poincare disk.
Symmetry transformation and kinematic space geodesic
More interestingly, a given isometric transformation de nes a geodesic in the kinematic
space. Let us consider a hyperbolic transformation
whose normalgeodesics in the Poincare upper half plane can be parameterized by r0 and
where A, B are elements in the matrix M . In the disk, the normalgeodesics are given by
are given by
xP )2 + y2 = rP2 ;
xP =
rP =
2yDy + 1 = 0 ;
xD =
yD =
The points in the kinematic space corresponding to the above oneparameter geodesics
> 0. The coordinates of these points are determined by the
Then we nd a curve in the kinematic space, which is determined by the relation
tan 0 =
This is a timelike geodesic in the kinematic space. On the contrary, if we require that this
curve is timelike, we should have
which is equivalent to
Then we have
In other words, the element
should be hyperbolic.
For hyperbolic and elliptic transformation , the normalgeodesics are
j Tr j = ja + dj > 2 :
xP )2 + y2 = x
xD =
yD =
In the kinematic space, the corresponding points form a geodesic, still described by
eq. (4.15). However, the geodesic is no longer timelike. Actually, for an elliptic
transformation the geodesic is spaclike, while for a parabolic transformation the geodesic is null.
We have showed that the horizon length LH in the BTZ spacetime equals the geodesic
in the kinematic space. In the kinematic space, for any pair of timelike
separated points, it corresponds to a BTZ spacetime. On the other hand, for a xed BTZ
spacetime obtained by the identi cation f g on a pair of geodesics in the Poincare disk,
it would be interesting to discuss its kinematic space. The kinematic space for the BTZ
spacetime is still de ned by the geodesics in the BTZ spacetime. We may start from the
geodesics in the Poincare disk, and take into account of the identi cation f g on all the
As showed in the left gure of gure 6, the BTZ spacetime is obtained by identifying
a pair of nonintersected geodesics L1, L2 =
L1. Between these two geodesics, there is a
fundamental region. On the boundary, the two geodesics divide the boundary of the disk
into four parts. We mark them as B1, B2, C1, C2, where Bi's are the boundaries of the
fundamental region, corresponding to the two boundaries of the BTZ wormhole, and Ci's
are the remaining parts on the circle. Then we can label any geodesic in H2 by the regions
where its two endpoints locate. For example, a geodesic with one endpoint in B1 and the
other in C2 is labelled by B1C2 or C2B1. Note that the order in the label represents the
orientation: B1C2 means the geodesics have a starting point in B1 and an ending point in
C2. Here we should notice that a geodesic with the parameter ( ; ) has the starting point
and the ending point at
As the BTZ spacetime is a quotient of AdS3 under the action of a Fuchsian group ,
its kinematic space cannot be the same as the one of AdS3. If two points in the kinematic
space of AdS3 can be transformed to each other under the action of an element in the
, they represent the same geodesic in the BTZ spacetime.
We would like to
fundamental region in the kinematic space of AdS3, corresponding to the BTZ black hole.
Here \fundamental" means that each geodesic with orientation in the BTZ spacetime has
and only has one corresponding point in that region. Since there are many di erent points
representing the same geodesic up to identi cation, there is ambiguity in choosing the
fundamental region. Here, we give a universal rule based on the two identi ed geodesics
de ning the fundamental region in the Poincare disk.
As shown in the right
gure of gure 6, the kinematic space can be separated into
20 regions by the geodesics with di erent ending points and orientations. Note that if we
reverse the orientations of the L1 and L2 simultaneously, the identi cation of them leads to
the same BTZ black hole. We label the points corresponding to the geodesics with opposite
orientation to L1 and L2 as L1 and L2.
There are two xed points under the action
= f g, as shown in the left gure of
gure 6 which are the intersection point between the orange geodesic and the boundary.
They lie on the boundaries C1 and C2, labelled by f1 and f2. They divides the boundaries
C1 and C2 into two parts respectively. The fundamental region for the BTZ spacetime in
the Poincare disk is the region between two geodesics L1 and L2 with two boundaries B1
and B2. Under the action of , the fundamental region is transformed to the region with the
boundaries in C1 next to B1 and B2. Similarly the action of
1 transforms the fundamental
by the two oriented red geodesics identi ed with each other. The orange line is the horizon, and
its two endpoints on the boundary is the xed points of the corresponding transformation. On the
right, we separate the whole kinematic space into 20 regions and marked them by the starting point
and ending point of the corresponding geodesics.
region to the region with the boundaries in C2 next to B1 and B2. Furthermore all the
regions under action of n; n 2 Z on the fundamental region cover the whole Poincare disk.
On the other hand, each geodesics in the Poincare disk can be related to the one in
the BTZ spacetime by the action of . If the ending point of the geodesic in the disc is
in Ci, it can always be mapped to the point in Bi. However, the resulting geodesic in the
BTZ spacetime may wind around the horizon. In order to classify the geodesics in the BTZ
spacetime, we need to consider the action of the Fuchsian group more carefully.
To discuss the action of the Fuchsian group on the geodesics in the disk, we start from
the geodesics with at least one ending point being the xed point and study the action of
on them. Actually, as any point on the boundary can be mapped to the xed point by
the continual action of the fundamental element , all the geodesics on the Poincare disk
can be related to the geodesics ending at the xed point. In other words, starting from the
geodesics ending at the xed point and consider its images under the action of the element
n; n 2 Z, these images constitutes a line in the kinematic space, starting and ending
xed points. Moreover, each
xed point with the angular coordinate
corresponds to two points with the coordinates ( ; 0) and ( + ; ) on the boundary of the
kinematic space. Therefore, as shown in
gure 7, there are various lines, connecting two
of the xed points. Among all the lines, there are a few special ones, which are drawn in
colored lines in gure 7.
The blue lines represent all the normalgeodesics of , and the red points on it represent
the geodesics L1, L2, L1 and L2 respectively. These geodesics represent the radial direction,
all the points on the same geodesic having the same angular coordinate. Moreover, the
blue lines themselves are also geodesic in the kinematic space.
The points on yellow lines represent all the geodesics with one endpoint being the xed
point and having in nite windings around the horizon. The two intersection points between
the yellow lines represent the geodesic connecting the two xed points on the boundaries
geodesics in H2 are the normalgeodesics of . The points on yellow lines represent the geodesics
with one endpoint being the xed point of , and having in nite windings around the horizon. The
orange intersection points of two yellow lines correspond to the geodesic covering the horizon of the
BTZ black hole. The points on the green lines represent the geodesics with two endpoints having
the same angular coordinate and winding the horizon once.
of H2. The geodesic actually covers the horizon of the BTZ wormhole. If the point in the
kinematic space is timelike separated from the intersection points, then the corresponding
geodesic does not intersect with the horizon and its endpoints are on the same boundary.
And if the point is spacelikely separated from the intersection point, then the corresponding
geodesic does intersect with the horizon and so its endpoints are on di erent boundaries.
Thus, the yellow lines separate all geodesics into the ones with two endpoints on the same
boundary and those on di erent boundaries.
The points on the green lines correspond to the geodesics for which one of its endpoint
can be mapped into the other by the fundamental transformation . Or in other words, such
geodesics wind around the horizon once. Therefore the green lines separate all geodesics
into the ones with or without the winding around the horizon. The points in the regions
between the two timelike green lines containing the blue lines correspond to the geodesics
without winding and with the endings on di erent boundaries. The points in the regions
between the spacelike green lines and the boundary of kinematic space correspond to the
geodesics without winding and with the endings on the same boundary. The points in the
regions between all the green lines containing the yellow lines correspond to the geodesics
with windings on the horizon, and the yellow lines divide them into the ones ending on
di erent boundaries or on the same boundary. Another important feature is that the causal
relation of two points will be invariant under the action of any transformation.
Now we can determine the fundamental region of the BTZ wormhole in the kinematic
space. The points in the regions B1B1 and B2B2 represent the geodesics ending on the same
boundary, and the ones in B1B2 represent the geodesics ending on di erent boundaries,
we include the regions BiC1 and C1Bi, and on the right we include the regions BiC2 and C2Bi
instead. Both of them includes the BiBj regions. We de ne the left
gure corresponding to the
and all of them correspond to the geodesics without winding. For a point in the region
C1C1, C2C2 or C1C2, we can always nd an element in
which transforms at least one
endpoint of the corresponding geodesic into Bi. So the regions CiCj will not be included
into the fundamental region. Then the main question is focused on the regions BiCj and
Cj Bi. Since the geodesics corresponding to the points in these two regions di er only on
orientation, we discuss only BiCj bellow. For every geodesic in BiC2 we can always nd
an element in
which transforms the endpoint in Bi to C1 and the other endpoint in C2
to Bi, we just need to choose the regions C1Bi and BiC1, or the regions C2Bi and BiC2, to
be part of the fundamental region. For the former choice, the corresponding fundamental
region is drawn in blue in the left
gure 8. These is similar as the choice in
gure 17 of [7], but with a little di erence because they ignore the orientation. For the
latter choice, the fundamental region is drawn in yellow in the right gure of gure 8.
Here we make a rule for the choice, which will be used in the discussion of
multiL2 =
L1, then we choose the fundamental region to include C1Bi and BiC1. But if
the Fuchsian group we choose is generated by
1, G = f
1g, with L1 =
1L2, then we
choose the fundamental region to include C2Bi and BiC2. Actually, both choices correspond
to the same wormhole with the same identi cation, we make this rule just for selfconsistent
discussion. It does not make any di erence if we choose an opposite rule. If we glue the
points on the boundaries that are identi ed and has the same orientation, then the topology
of the fundamental region is just two disconnected cylinders.
Multiboundary wormhole in kinematic space
For a threeboundary wormhole and a torus wormhole, both are de ned by the identi
cations of two pairs of geodesics. The identi cations are generated by two fundamental
elements 1
, 2 of the corresponding Fuchsian group
= f 1; 2g. We denote the four
geodesics as Li;j with Li;2 =
Poincare disk are shown in the left gure of gure 9. In this gure, we choose the identi
cation to get a threeboundary wormhole. The discussion for other identi cation is similar.
gure, the two pairs of identi ed geodesics are shown in Poincare disk, and they divided the
boundary into eight regions. In the right gure, the fundamental region in the kinematic space is
shown. The red and green points correspond to the identi ed geodesics, and the blue lines are the
timelike geodesics formed by normalgeodesics of 1, 2
Now the geodesics on the Poincare disk can be classi ed into 72 classes, depending on their
ending points. Correspondingly, the kinematic space is separated into 72 regions.
Now let us discuss the fundamental region for this threeboundary wormhole. Notice
that each pair of identi ed geodesics Li;1; Li;2 can de ne a BTZ spacetime corresponding
to i such that Li;2 =
iLi;1, and we can read the fundamental region for the resulting
BTZ according to the rule we de ned above. Since any points outside this fundamental
region can be mapped into it by an element i
n, then the regions including those points
must not be a part of the fundamental region for the wormhole. So the fundamental region
of a threeboundary wormhole, as shown in the right gure of gure 9, is the intersection
of the fundamental regions of all the BTZ de ned by each pair of geodesics. And this way
to choose the fundamental region works for all kinds of multiboundary wormhole.
As we mentioned above, for the same four geodesics a di erent kind of identi cation
leads to a singleboundary torus wormhole. As showed in the upper half of gure 10, the
geodesics in the same color are identi ed. We should notice that the fundamental region
in this identi cation is the same as the threeboundary wormhole in
gure 9. This is just
because we choose the group to be generated by 1; 2. If we choose the generators to
be 1; 2 1, the fundamental region is showed in the lower half of
gure 10. Although
the fundamental region may be the same for di erent kinds of wormhole, the same point
corresponds to di erent kinds of geodesics in these wormholes, since the identi cation is
To read the topology of the fundamental region, one should glue the identi ed points
with same orientation on the boundaries of the fundamental region. As all the boundaries
are parts of the boundary for the fundamental region of some fundamental element i, the
way to glue them is just the same as the BTZ case. But there is some slight di erence in
the wormhole cases. For the threeboundary wormhole, the naive guess that the topology
of its fundamental region in the kinematic space is just two pairs of pants is not correct.
In fact, the rectangle parts in the fundamental region a ects the topology. The red point
shown in Poincare disk, and they divided the boundary into eight regions. In the right gure, the
fundamental region in the kinematic space is shown. The red and green points correspond to the
identi ed geodesics, and the blue lines are the timelike geodesics formed by normalgeodesics of 1
2. On the lower half part, the same wormhole but the group is generated di erently
In the left gure, the direction of the red geodesics are changed, suggesting the element is 2 1.
representing L1;2 should be glued to L1;1, while the green point L2;2 should be glued to
L2;1. Then the lower triangle part and the upper triangle part which represents the same
geodesics with di erent orientation is connected by this rectangle part. The topology of
the fundamental region in the kinematic space turns out to be a surface with six
boundaries. This can be seen by cutting each pair of pants and gluing them together. For the
torus wormhole, the topology of the fundamental region is a surface of genus 2 with two
In order to distinguish di erent kinds of wormholes, it is not enough to consider only
the fundamental region, which is determined by the fundamental elements in the Fuchsian
group. We need to take the exact identi cation into account. One simple way to do this is
to draw the geodesics corresponding to the fundamental elements clearly. For example, the
fundamental region in yellow in the upper of gure 10 looks the same as the one in gure 9.
However, the geodesics characterizing the identi cations i are obviously di erent.
In this paper, we studied the properties of the kinematic space from geometric points of
view. First of all we showed how the kinematic space of AdS3 can be constructed
geometrically in the embedding space. As every geodesic on the Poincare disk is the boundary
of the intersection between the Poincare disk and another disk centered outside, the
kinematic space is actually formed by the tip points of the causal diamond of the other disk in
the embedding space. In this picture, the causal structure in the kinematic space is easily
understood. Moreover we discussed the Fuchsian group and its action on the geodesics to
get the multiboundary wormhole. We showed that for each SL(2; R) transformation in
the Fuchsian group its normalgeodesics make up a geodesic in the kinematic space. If the
transformation is hyperbolic, elliptic or parabolic, the corresponding geodesic in the
kinematic space is timelike, spacelike or null respectively. More surprisingly, the horizon length
of the BTZ wormhole can be read by the length of the corresponding timelike geodesic in the
kinematic space. Finally we discussed the kinematic space for the multiboundary
wormhole. We started from the kinematic space for global AdS3 and considered the identi cation
of the elements in the Fuchsian group. For the BTZ blackhole, we de ned consistently its
fundamental region in the kinematic space. For the threeboundary wormhole, we argued
that its fundamental region in the kinematic space is formed by the intersection of two
fundamental regionals of the BTZ wormhole constructed by two fundamental elements in
its Fuchsian group. For the singleboundary wormhole, its fundamental region could be
same as the one for the threeboundary wormhole, but the timelike geodesics corresponding
to the identi cation are di erent.
Our study on the kinematic space is purely geometrical, having nothing to do with the
di erential entropy. The discussion is quite di erent from the ones in the literature. Our
approach could be applied to the study of the holographic entanglement entropy and bit
threads [34]. We would like to leave them for future study [35].
Acknowledgments
The work was in part supported by NSFC Grant No. 11275010, No. 11335012 and
No. 11325522. We would like to thank B. Czech and M. Headrick for helpful discussions.
Open Access.
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any medium, provided the original author(s) and source are credited.
Phys. Rev. Lett. 96 (2006) 181602 [hepth/0603001] [INSPIRE].
JHEP 08 (2006) 045 [hepth/0605073] [INSPIRE].
[arXiv:0905.1317] [INSPIRE].
JHEP 07 (2016) 100 [arXiv:1512.01548] [INSPIRE].
correspondence, JHEP 08 (2016) 086 [arXiv:1606.00621] [INSPIRE].
spacetime, JHEP 10 (2013) 220 [arXiv:1305.0856] [INSPIRE].
[INSPIRE].
JHEP 06 (2014) 044 [arXiv:1403.3416] [INSPIRE].
Phys. Rev. D 90 (2014) 066012 [arXiv:1405.4684] [INSPIRE].
JHEP 11 (2014) 015 [arXiv:1406.4889] [INSPIRE].
JHEP 10 (2014) 149 [arXiv:1408.4770] [INSPIRE].
JHEP 10 (2015) 175 [arXiv:1505.05515] [INSPIRE].
Universitat Autonoma de Barcelona (2003).
JHEP 12 (2015) 081 [arXiv:1507.04633] [INSPIRE].
conformal eld theory, JHEP 09 (2016) 154 [arXiv:1604.02687] [INSPIRE].
[INSPIRE].
[hepth/0106112] [INSPIRE].
[1] S. Ryu and T. Takayanagi , Holographic derivation of entanglement entropy from AdS/CFT , [2] S. Ryu and T. Takayanagi , Aspects of holographic entanglement entropy , [3] A. Lewkowycz and J. Maldacena , Generalized gravitational entropy , JHEP 08 ( 2013 ) 090 [4] B. Swingle , Entanglement renormalization and holography , Phys. Rev. D 86 (2012) 065007 [5] M. Van Raamsdonk , Building up spacetime with quantum entanglement , Gen. Rel. Grav . 42 ( 2010 ) 2323 [arXiv:1005.3035] [INSPIRE].
[6] F. Pastawski , B. Yoshida , D. Harlow and J. Preskill , Holographic quantum errorcorrecting codes: toy models for the bulk/boundary correspondence , JHEP 06 ( 2015 ) 149 [7] B. Czech , L. Lamprou , S. McCandlish and J. Sully , Tensor networks from kinematic space , [8] B. Czech et al., Tensor network quotient takes the vacuum to the thermal state , [9] P. Hayden et al., Holographic duality from random tensor networks , JHEP 11 ( 2016 ) 009 [10] A. Bhattacharyya , Z.S. Gao , L.Y. Hung and S.N. Liu , Exploring the tensor networks /AdS [11] V. Balasubramanian , B. Czech , B.D. Chowdhury and J. de Boer, The entropy of a hole in [12] V. Balasubramanian , B.D. Chowdhury , B. Czech , J. de Boer and M.P. Heller , Bulk curves from boundary data in holography , Phys. Rev. D 89 ( 2014 ) 086004 [arXiv:1310.4204] [13] R.C. Myers , J. Rao and S. Sugishita , Holographic holes in higher dimensions , [14] B. Chen and J. Long , Strong subadditivity and emergent surface , [15] B. Czech , X. Dong and J. Sully , Holographic reconstruction of general bulk surfaces , [16] M. Headrick , R.C. Myers and J. Wien , Holographic holes and di erential entropy , [17] B. Czech and L. Lamprou , Holographic de nition of points and distances , Phys. Rev. D 90 ( 2014 ) 106005 [arXiv:1409.4473] [INSPIRE].
[18] B. Czech , P. Hayden , N. Lashkari and B. Swingle , The information theoretic interpretation of the length of a curve , JHEP 06 ( 2015 ) 157 [arXiv:1410.1540] [INSPIRE].
[19] B. Czech , L. Lamprou , S. McCandlish and J. Sully , Integral geometry and holography , [20] G. Solanes , Integral geometry and curvature integrals in hyperbolic space , Ph.D. Thesis , [21] X. Huang and F.L. Lin , Entanglement renormalization and integral geometry , [22] C.T. Asplund , N. Callebaut and C. Zukowski , Equivalence of emergent de Sitter spaces from [23] J. de Boer , F.M. Haehl , M.P. Heller and R.C. Myers , Entanglement, holography and causal diamonds, JHEP 08 ( 2016 ) 162 [arXiv:1606.03307] [INSPIRE].
Phys. Rev. Lett . 69 ( 1992 ) 1849 [hepth /9204099] [INSPIRE].
[24] M. Ban~ados , C. Teitelboim and J. Zanelli , Black hole in threedimensional spacetime , [25] M. Ban~ados , M. Henneaux , C. Teitelboim and J. Zanelli , Geometry of the (2+1) black hole , Phys. Rev. D 48 ( 1993 ) 1506 [Erratum ibid . D 88 ( 2013 ) 069902] [grqc/9302012] [26] J.M. Maldacena , Eternal black holes in antide Sitter , JHEP 04 ( 2003 ) 021 [27] S. Aminneborg , I. Bengtsson , D. Brill , S. Holst and P. Peldan , Black holes and wormholes in (2+1)dimensions, Class . Quant. Grav. 15 ( 1998 ) 627 [grqc /9707036] [INSPIRE].
[28] D. Brill , Black holes and wormholes in (2+1)dimensions , Lect. Notes Phys. 537 (2000) 143 [29] K. Krasnov , Holography and Riemann surfaces, Adv. Theor. Math. Phys. 4 ( 2000 ) 929 [30] K. Skenderis and B.C. van Rees, Holography and wormholes in 2+1 dimensions, Commun . Math. Phys. 301 ( 2011 ) 583 [arXiv:0912. 2090 ] [INSPIRE].
[31] V. Balasubramanian , P. Hayden , A. Maloney , D. Marolf and S.F. Ross , Multiboundary wormholes and holographic entanglement , Class. Quant. Grav . 31 ( 2014 ) 185015 [32] D. Marolf , H. Max eld , A. Peach and S.F. Ross , Hot multiboundary wormholes from bipartite [33] H. Max eld , Entanglement entropy in three dimensional gravity , JHEP 04 ( 2015 ) 031 [34] M. Freedman and M. Headrick , Bit threads and holographic entanglement , [35] J.d. Zhang and B. Chen , work in progress.