PHYSICAL REVIEW D 67, 124025 共2003兲
Black holes in a compactified spacetime
Andrei V. Frolov*
CITA, University of Toronto, Toronto, Ontario, Canada M5S 3H8
Valeri P. Frolov†
Theoretical Physics Institute, Department of Physics, University of Alberta, Edmonton, Alberta, Canada T6G 2J1
共Received 13 February 2003; published 25 June 2003兲
We discuss the properties of a 4-dimensional Schwarzschild black hole in a spacetime where one of the
spatial dimensions is compactified. As a result of the compactification the event horizon of the black hole is
distorted. We use Weyl coordinates to obtain the solution describing such a distorted black hole. This solution
is a special case of the Israel-Khan metric. We study the properties of the compactified Schwarzschild black
hole, and develop an approximation which allows one to find the size, shape, surface gravity, and other
characteristics of the distorted horizon with a very high accuracy in a simple analytical form. We also discuss
the possible instabilities of a black hole in compactified space.
DOI: 10.1103/PhysRevD.67.124025
PACS number共s兲: 04.50.⫹h, 04.70.Bw
I. INTRODUCTION
Black hole solutions in a compactified spacetime have
been studied in many publications. A lot of attention was
paid to Kaluza-Klein higher-dimensional black holes. By
compactifying black hole solutions along Killing directions
one obtains lower-dimensional solutions of Einstein equations with additional scalar, vector and other fields 共see, e.g.,
Ref. 关1兴, and references therein兲. The generation of black
hole and black string solutions by the Kaluza-Klein procedure was extensively used in string theory 共see, e.g., Ref. 关2兴,
and references therein兲.
The solution which we consider in this paper is of a different nature. We study a Schwarzschild black hole in a
spacetime with one compactified spatial dimension. This dimension does not coincide with any Killing vector; for this
reason the black hole metric is distorted as a result of compactification.
The recent interest in compactified spacetimes with black
holes is connected with brane-world models. The general
properties of black holes in the Randall-Sundrum model
were discussed in Refs. 关4,5兴. In the latter paper a
4-dimensional C-metric was used to obtain an exact
(3⫹1)-dimensional black hole solution in AdS spacetime
with the Randall-Sundrum brane. Black holes in RS braneworlds were discussed in a number of publications 共see, e.g.,
Ref. 关6兴, and references therein兲.
Black hole solutions in a spacetime with compactified dimensions are also interesting in connection with other types
of brane models, which were considered historically first in
Ref. 关7兴. In the Arkani-Hamed–Dimopoulos–Dvali– 共ADD兲
type of brane worlds the tension of the brane can be not very
large. If one neglects its action on the gravitational field of a
black hole, one obtains a black hole in a spacetime where
some of the dimensions are compactified. Compactification
of a special class of solutions, generalized Majumdar-
*Email address: frolov@cita.utoronto.ca
†
Email address: frolov@phys.ualberta.ca
0556-2821/2003/67共12兲/124025共11兲/$20.00
Papapetrou metrics, was discussed by Myers 关3兴. In this paper he also made some general remarks concerning compactification of the 4-dimensional Schwarzschild metric. Some
of the properties of compactified 4-dimensional Schwarzschild metrics were also considered in Ref. 关8兴. For a recent
discussion of higher-dimensional black holes on cylinders
see Ref. 关9兴.
In this paper we study a solution describing a
4-dimensional Schwarzschild black hole in a spacetime
where one of the dimensions is compactified. This solution is
a special case of the Israel-Khan metric 关10兴, where an infinite set of equal mass rods is placed along the axis of symmetry so that the distance between any two of the adjacent
rods is the same. Each of these rods is a source for a harmonic function, the Newtonian potential of the Schwarzschild black hole. The general properties of this solution were
discussed by Korotkin and Nicolai 关11兴. As a result of the
compactification, the event horizon of the black hole is distorted. In our paper we focus our attention on the properties
of the distorted horizon. We use Weyl coordinates to obtain a
solution describing such a distorted black hole. This approach to the study of axisymmetric static black holes is well
known and was developed long ago by Geroch and Hartle
关12兴 共see also Ref. 关18兴兲.1 In Weyl coordinates, the metric
describing a distorted 4-dimensional black hole contains 2
arbitrary functions. One of them, playing the role of gravitational potential, obeys a homogeneous linear equation. Because of the linearity, one can present the solution as a linear
superposition of the unperturbed Schwarzschild gravitational
potential and its perturbation. After this, the second function
which enters the solution can be obtained by simple integration. To find the gravitational potential one can either use the
Green’s function method or expand a solution into a series
1
For generalization of this approach to the case of electrically
charged distorted 4D black holes see Refs. 关13,14兴. A generalization
of the Weyl method to higher-dimensional spacetimes was discussed in Ref. 关15兴. An initial value problem for 5D black holes was
discussed in Refs. 关16,17兴.
67 124025-1
©2003 The American Physical Society
PHYSICAL REVIEW D 67, 124025 共2003兲
A. V. FROLOV AND V. P. FROLOV
over the eigenmodes. We discuss both of the methods since
they give two different convenient representations for the
solution. We develop an approximation which allows one to
find the size, shape, surface gravity and other characteristics
of the distorted horizon with very high accuracy in a simple
analytical form. We study properties of compactified
Schwarzschild black holes and discuss their possible instability.
The paper is organized as follows. We recall the main
properties of 4D distorted black holes in Sec. II. In Sec. III,
we obtain the solution for a static 4-dimensional black hole
in a spacetime with 1 compactified dimension. In Sec. IV, we
study this solution. In particular we discuss its asymptotic
form at large distances, and the size, form and shape of the
horizon. We conclude the paper by general remarks in Sec.
V.
⌬U⫽4 ␲ j,
A. Weyl form of the Schwarzschild metric
A static axisymmetric 4-dimensional metric in the canonical Weyl coordinates takes the form 关12,15,18兴
dS 2 ⫽⫺e 2U dT 2 ⫹e ⫺2U 关 e 2V 共 dR 2 ⫹dZ 2 兲 ⫹R 2 d ␾ 2 兴 , 共1兲
where
⌰共 x 兲⫽
V ,Z ⫽2RU ,R U ,Z .
U S 共 R,Z 兲 ⫽⫺
兩 x 兩 ⭐1,
0,
兩 x 兩 ⬎1.
共8兲
冕
dZ ⬘
M
⫺M
冋
冑R
2
⫹ 共 Z⫺Z ⬘ 兲 2
冑共 M ⫺Z 兲 2 ⫹R 2 ⫺Z⫹M
冑共 M ⫹Z 兲 2 ⫹R 2 ⫺Z⫺M
册
.
共9兲
G (3) 共 x,x⬘ 兲 ⫽
⫽
1
4 ␲ 兩 x⫺x⬘ 兩
1
4␲
1
冑R
2
⫹R ⬘ ⫺2RR ⬘ cos共 ␺ ⫺ ␺ ⬘ 兲 ⫹ 共 Z⫺Z ⬘ 兲 2
2
.
共10兲
Sometimes the solution 共9兲 is presented in another equivalent
form
共3兲
L⫺M
1
,
U S 共 R,Z 兲 ⫽ ln
2
L⫹M
冉 冊
1
L⫽ 共 L ⫹ ⫹L ⫺ 兲 ,
2
共4兲
R⫽ 冑r 共 r⫺2M 兲 sin ␪ ,
共12兲
Z⫽ 共 r⫺M 兲 cos ␪ .
共13兲
冊
共14兲
One has
冉
L 2 ⫺M 2
1
,
V S 共 R,Z 兲 ⫽ ln 2
2
L ⫺␩2
1
2
␩ ⫽ 共 L ⫹ ⫺L ⫺ 兲 .
共6兲
R→0
In fact, if V(0,Z 0 )⫽0 at any point Z 0 of the Z axis then Eq.
共3兲 implies that V(0,Z)⫽0 at any other point of the Z axis
which is connected with Z 0 .
For a four-dimensional Schwarzschild metric, the function
U is the potential of an infinitely thin finite rod of mass 1/2
per unit length located at ⫺M ⭐Z⭐M portion of the Z axis
L ⫾ ⫽ 冑R 2 ⫹ 共 Z⫾M 兲 2 .
共11兲
The function V S (R,Z) for the Schwarzschild metric can be
found either by solving Eq. 共3兲 or by direct change of the
coordinates
共5兲
where ⌬ is a flat Laplace operator in the metric 共4兲. It is easy
to check that Eq. 共2兲 plays the role of the integrability condition for the linear first order equations 共3兲. The regularity
condition implies that at regular points of the symmetry axis
R⫽0
lim V 共 R,Z 兲 ⫽0.
1,
共2兲
be an auxiliary 3-dimensional flat metric, then solutions of
Eq. 共2兲 coincide with axially symmetric solutions of the
3-dimensional Laplace equation
⌬U⫽0,
1
2
1
⫽⫺ ln
2
Let
dl 2 ⫽dR 2 ⫹R 2 d ␺ 2 ⫹dZ 2
再
The corresponding solution is
where U and V are functions of R and Z. This metric is a
solution of vacuum Einstein equations if and only if these
functions obey the equations
2
2
⫺U ,Z
V ,R ⫽R 共 U ,R
兲,
共7兲
The integral representation in the right hand side of Eq. 共9兲 is
obtained by using the 3-dimensional Green’s function for Eq.
共5兲, which is of the form
II. FOUR-DIMENSIONAL WEYL BLACK HOLES
⳵ 2U 1 ⳵ U ⳵ 2U
⫹
⫹
⫽0,
⳵R2 R ⳵R ⳵Z2
1 ␦共 ␳ 兲
⌰ 共 z/M 兲 ,
4␲ ␳
j⫽
共15兲
In the coordinates (R,Z) the black hole horizon H is the line
segment ⫺M ⭐Z⭐M of the R⫽0 axis.
B. A distorted black hole
General static axisymmetric distorted black holes were
studied in Ref. 关12兴. A distorted black hole is described by a
124025-2
PHYSICAL REVIEW D 67, 124025 共2003兲
BLACK HOLES IN A COMPACTIFIED SPACETIME
static axisymmetric Weyl metric with a regular Killing horizon. One can write the solution (U,V) for a distorted black
hole as
U⫽U S ⫹Û,
V⫽V S ⫹V̂,
It is a sphere deformed in an axisymmetric manner. The surface gravity ␬ is constant over the horizon surface
␬⫽
共16兲
where (U S ,V S ) is the Schwarzschild solution with mass M.
Since both V and V S vanish at the axis R⫽0 outside the
horizon, the function V̂ has the same property. The function
Û obeys the homogeneous equation 共2兲, while the equations
for V̂ follow from Eq. 共3兲. One of these equations is of the
form
eu
.
4M 0
共24兲
III. 4D COMPACTIFIED SCHWARZSCHILD
BLACK HOLE
A. Compactified Weyl metric
In what follows it is convenient to rewrite the Weyl metric
共1兲 in the dimensionless form dS 2 ⫽L 2 ds 2 ,
共17兲
ds 2 ⫽⫺e 2U dt 2 ⫹e ⫺2U 关 e 2V 共 d ␳ 2 ⫹dz 2 兲 ⫹ ␳ 2 d ␾ 2 兴 , 共25兲
Near the horizon Û is regular, while U S,R ⫽O(R ⫺1 ) and
U S,Z ⫽O(1). Thus near the horizon V̂ ,Z ⬃2Û ,Z . Integrating
this relation along the horizon from Z⫽⫺M to Z⫽M and
using the relations V̂(0,⫺M )⫽V̂(0,M )⫽0, we obtain that
Û has the same value u at both ends of the line segment H.
By integrating the same equation along the segment H from
the end point to an arbitrary point of H one obtains for
⫺M ⭐Z⭐M
where L is the scale parameter of the dimensionality of the
length and
V̂ ,Z ⫽2R 共 U S,R Û ,Z ⫹U S,Z Û ,R ⫹Û ,R Û ,Z 兲 .
V̂ 共 0,Z 兲 ⫽2 关 Û 共 0,Z 兲 ⫺u 兴 .
R⫽e u 冑r 共 r⫺2M 0 兲 sin ␪ ,
M 0 ⫽M e ⫺u ,
U S 共 ␳ ,z 兲 ⫽⫺
z⫽
Z
L
共26兲
冋
1
2
log
冑共 ␮ ⫺z 兲 2 ⫹ ␳ 2 ⫺z⫹ ␮
冑共 ␮ ⫹z 兲 2 ⫹ ␳ 2 ⫺z⫺ ␮
册
.
共27兲
For 兩 z 兩 ⬎ ␮ , the gravitational potential U S remains finite at
the symmetry axis
U S 共 0,z 兲 ⫽
共20兲
it is possible to recast the metric 共1兲 of a distorted black hole
into the form
冉
2M 0
2M 0
dT 2 ⫹e 2(V̂⫺Û⫹u) 1⫺
r
r
冊
⫹e 2(V̂⫺Û⫹u) r 2 共 d ␪ 2 ⫹e ⫺2V̂ sin2 ␪ d ␾ 2 兲 .
⫺1
dr 2
共21兲
In these coordinates, the event horizon is described by the
equation r⫽2M 0 , and the 2-dimensional metric on its surface is
d ␥ 2 ⫽4M 20 关 e 2(Û⫺u) d ␪ 2 ⫹e ⫺2(Û⫺u) sin2 ␪ d ␾ 2 兴 .
共22兲
The horizon surface has area
A⫽16␲ M 20 .
R
,
L
z⫺ ␮
1
ln
,
2
z⫹ ␮
兩z兩⬎␮.
共28兲
For 兩 z 兩 ⭐ ␮ , the gravitational potential U S is divergent at ␳
⫽0. The leading divergent term is
and defining
冊
␳⫽
are dimensionless coordinates. We shall also use instead of
mass M its dimensionless version ␮ ⫽M /L. The Schwarzschild solution 共9兲 can then be rewritten as
共19兲
Z⫽e u 共 r⫺M 0 兲 cos ␪ ,
冉
T
,
L
共18兲
Geroch and Hartle 关12兴 demonstrated that if Û is a regular
smooth solution of Eq. 共5兲 in any small open neighborhood
of H 共including H itself兲 which takes the same values u on
both ends of the segment H, then the solution is regular at the
horizon and describes a distorted black hole.
Using the coordinate transformation
dS 2 ⫽⫺e ⫺2Û 1⫺
t⫽
共23兲
U S 共 ␳ ,z 兲 ⬃
␳2
1
ln
, 兩z兩⭐␮.
2
4 共 ␮ 2 ⫺z 2 兲
共29兲
We will now obtain a new solution describing a
Schwarzschild black hole in a space in which the Z coordinate is compactified. We will call this solution a compactified
Schwarzschild 共CS兲 metric. For this purpose we assume that
the coordinate Z is periodic with a period 2 ␲ L. We shall use
the radius of compactification L as the scale factor.
Our space manifold M has topology S 1 ⫻R 2 and we are
looking for a solution of Eq. 共5兲 on M which is periodic in
z with the period 2 ␲ , z苸(⫺ ␲ , ␲ ). The source for this solution is an infinitely thin rod of the linear density 1/2 located
along z axis in the interval (⫺ ␮ , ␮ ), ␮ ⭐ ␲ . This problem
can be solved by two different methods, either by using
Green’s functions or by expanding a solution into a series
over the eigenmodes. We discuss both of the methods since
they give two different convenient representations for the
solution. We begin with the method of Green’s functions.
124025-3
PHYSICAL REVIEW D 67, 124025 共2003兲
A. V. FROLOV AND V. P. FROLOV
B. 3D Green’s function
To obtain this solution we proceed as follows. Our first
(3)
on
step is to obtain a 3-dimensional Green’s function G M
the manifold M. It can be done, for example, by the method
of images applied to the Green’s function for Eq. 共5兲 which
(3)
. It is more convegives the series representation for G M
nient to use another method which gives the integral representation. For this purpose we note that the flat
3-dimensional Green’s function can be obtained by the dimensional reduction from the 4-dimensional one. Namely, let
X⫽(X,Y ,Z,W)
dh 2 ⫽dX2 ⫽dX 2 ⫹dY 2 ⫹dZ 2 ⫹dW 2 ,
and hence it behaves as if the space had one dimension less.
It is obviously a result of compactification.
In the reduction procedure this creates a technical problem since the integral over w becomes divergent. It is easy to
deal with this problem as follows. Denote
(4,␣ )
(4,reg)
GM
共 X,X⬘ 兲 ⫽G M
共 X,X⬘ 兲 ⫹
(4,reg)
GM
共 X,X⬘ 兲 ⫽
共30兲
冋
1
sinh ␤
1
2 2
8 ␲ L ␤ cosh ␤ ⫺cos共 z⫺z ⬘ 兲
then
G (3) 共 x,x⬘ 兲 ⬅
1
4 ␲ 兩 x⫺x⬘ 兩
⫽
冕
⬁
⫺⬁
⫺
dWG (4) 共 X,X⬘ 兲 ,
共31兲
where x⫽(X,Y ,Z),
G (4) 共 X,X⬘ 兲 ⫽
and G
erator
(4)
1
1
,
2
4 ␲ 兩 X⫺X⬘ 兩 2
共32兲
1
冑␤
2
⫹b 2
册
共40兲
.
(4,␣ )
Here b is any positive number. For ␣ ⫽1, G M
does not
depend on b and coincides with Eq. 共31兲. At large ␤ the term
(4,reg)
has asymptotic behavior ⬃ ␤ ⫺2 .
GM
We also have
冕
(X,X ⬘ ) is the Green’s function for the Laplace op⌬ (4) G (4) 共 X,X⬘ 兲 ⫽⫺ ␦ 4 共 X⫺X⬘ 兲 .
1
1
,
2 2
8 ␲ L 共 ␤ 2 ⫹b 2 兲 ␣ /2
共39兲
⬁
⫺⬁
dw
共 ␤ 2 ⫹b 2 兲
⫽
␣ /2
⬃
共33兲
关 ␴ 2 ⫹b 2 兴 (1⫺ ␣ )/2⌫ 关共 ␣ ⫺1 兲 /2兴
⌫ 共 ␣ /2兲
2 冑␲
1
冋
1
1
1
⫹ln 2⫺ ln共 ␴ 2 ⫹b 2 兲
␲ ␣ ⫺1
2
册
⫹O 共 ␣ ⫺1 兲 .
Denote
(4)
GM
共 X,X⬘ 兲 ⫽
1
4␲2
⬁
兺
n⫽⫺⬁
1
共 Z⫺Z ⬘ ⫹2 ␲ Ln 兲 ⫹B
2
2
,
共34兲
where
B ⫽ 共 X⫺X ⬘ 兲 ⫹ 共 Y ⫺Y ⬘ 兲 ⫹ 共 W⫺W ⬘ 兲 .
2
2
2
2
共35兲
(4)
The function G M
is periodic in Z with the period 2 ␲ L and is
a Green’s function on the manifold M. The sum can be
calculated explicitly by using the relation
⬁
␲
1
sinh共 2 ␲ b 兲
.
兺
2
2⫽
b cosh共 2 ␲ b 兲 ⫺cos共 2 ␲ a 兲
⫺⬁ 共 a⫹n 兲 ⫹b
共36兲
Thus one has
(4)
GM
共 X,X⬘ 兲 ⫽
1
sinh ␤
,
8 ␲ L ␤ 关 cosh ␤ ⫺cos共 z⫺z ⬘ 兲兴
2 2
Here ␴ 2 ⫽(x⫺x ⬘ ) 2 ⫹(y⫺y ⬘ ) 2 . By omitting unimportant
共divergent兲 constant we regularize the expression for the integral. By using the reduction procedure 共31兲 we get
(3)
GM
共 x,x⬘ 兲 ⫽
1
,
8 ␲ 2L 2␤
冕
⬁
⫺⬁
(4,reg)
dWG M
共 X,X⬘ 兲 ⫺
1
ln共 ␴ 2 ⫹b 2 兲 .
16␲ 2 L
共42兲
C. Integral representation for the gravitational potential
To obtain the potential U( ␳ ,z) which determines the
(3)
(x,x⬘ ) with reblack hole metric we need to integrate G M
spect to x⬘ along the interval (⫺M ,M ) at R ⬘ ⫽0 axis. It is
convenient to use the representation 共42兲 and to change the
order of integrals. We use the integral (a⬎1,0⬍ ␮ ⬍ ␲ ,⫺ ␲
⬍z⬍ ␲ )
共37兲
where ␤ ⫽B/L. This Green’s function has a pole at ␤ ⫽z
⫺z ⬘ ⫽0, that is when the points X and X⬘ coincide. At far
distance, ␤ ⰇL, this Green’s function has asymptotic
(4)
GM
共 X,X⬘ 兲 ⬃
共41兲
共38兲
124025-4
冕
␮
dz ⬘
⫺␮
a⫺cos共 z ⬘ ⫺z 兲
⫽
2
冑a
2
⫺1
再 冋 冉 冊册
arctan p tan
␮ ⫹z
2
⫹ ␲␽ 共 ␮ ⫹z⫺ ␲ 兲
冋 冉 冊册
冎
⫹arctan p tan
␮ ⫺z
2
⫹ ␲␽ 共 ␮ ⫺z⫺ ␲ 兲 ,
共43兲
PHYSICAL REVIEW D 67, 124025 共2003兲
BLACK HOLES IN A COMPACTIFIED SPACETIME
where p⫽ 冑(a⫹1)/(a⫺1). We understand arctan to be the
principal value and include ␽ functions to get the correct
value over the entire interval ⫺ ␲ ⬍z⬍ ␲ . We also change
the parameter of integration W to w⫽W/L and take into
account that the integrand is an even function of w. After
these manipulations we obtain
U 共 ␳ ,z 兲 ⫽⫺
1
␲
冕
⬁
dw
0
冉
U共 ␤ ,z 兲
␮
⫺
2
␤
冑␤ ⫹b 2
冊
Note that a function ⌰(z/ ␮ ) which enters the source term
关see Eqs. 共7兲, 共8兲兴 allows the following Fourier decomposition on the circle:
⬁
⌰ 共 z/ ␮ 兲 ⫽a 0 ⫹
兺
k⫽1
a k cos共 kz 兲 ,
共51兲
where
␮
⫹
ln共 ␳ 2 ⫹b 2 兲 ,
2␲
␮
a 0⫽ ,
␲
共44兲
a k⫽
2
sin共 k ␮ 兲 .
␲k
共52兲
Using the Fourier decomposition for U
where
⬁
U共 ␤ ,z 兲 ⫽V共 ␤ ,z 兲 ⫹V共 ␤ ,⫺z 兲 ,
共45兲
冉 冊册
冋
cosh ␤ ⫹1
␮ ⫹z
tan
sinh ␤
2
V共 ␤ ,z 兲 ⫽arctan
U 共 ␳ ,z 兲 ⫽U 0 共 ␳ 兲 ⫹
␦共 ␳ 兲
d 2 U k 1 dU k
⫺k 2 U k ⫽a k
.
2 ⫹
d␳
␳ d␳
␳
Note that now ␤ which enters Eqs. 共44兲 and 共45兲 is
␤ ⫽ 冑w 2 ⫹ ␳ 2 .
共46兲
A representation similar to Eq. 共44兲 can be written for the
Schwarzschild potential U S
1
␲
冕
⬁
dw
0
US 共 ␤ ,z 兲
,
␤
共47兲
U k 共 ␳ 兲 ⫽⫺a k K 0 共 k ␳ 兲 ,
US 共 ␤ ,z 兲 ⫽VS 共 ␤ ,z 兲 ⫹VS 共 ␤ ,⫺z 兲 ,
共48兲
冉 冊
␮ ⫹z
.
␤
where K ␯ (z) is MacDonald function. For k⫽0 the solution
is
冕
␲
⬁
0
dw
冉
U共 w,z 兲 ⫺US 共 w,z 兲
w
共56兲
Thus the gravitational potential U allows the following series
representation:
⬁
One can check that this integral really gives expression 共27兲.
Using these representations we obtain the following expression for the quantity Û(z)⫽U(0,z)⫺U S (0,z) which determines the properties of the event horizon
1
共54兲
共55兲
U 0 共 ␳ 兲 ⫽a 0 ln共 ␳ 兲 .
VS 共 ␤ ,z 兲 ⫽arctan
共53兲
For k⬎0 the solutions of these equations which are decreasing at infinity are
where
Û 共 z 兲 ⫽⫺
U k 共 ␳ 兲 cos共 kz 兲 ,
we obtain the following equations for the radial functions
U k ( ␳ ):
⫹ ␲␽ 共 ␮ ⫹z⫺ ␲ 兲 .
U S 共 ␳ ,z 兲 ⫽⫺
兺
k⫽1
⫺
␮
冑w 2 ⫹1
冊
␮
sin共 k ␮ 兲
cos共 kz 兲 K 0 共 k ␳ 兲 . 共57兲
U 共 ␳ ,z 兲 ⫽ ln ␳ ⫺2 兺
␲
␲k
k⫽1
This representation is very convenient for studying the
asymptotics of the gravitational potential near the horizon.
For small ␳ one has
.
⫺K 0 共 k ␳ 兲 ⬃ln
共49兲
To obtain the redshift factor u it is sufficient to calculate
Û(z) for z⫽ ␮
u⫽Û 共 ␮ 兲 .
k␳
⫹␥,
2
共58兲
where ␥ ⬇0.57721 is Euler’s constant. Substituting these asymptotics into Eq. 共57兲 and combining the terms one obtains
冋
册
冉
␳
␮
1
ln ⫹ ␥
U 共 ␳ ,z 兲 ⬃ ln ⫹ ␥ ⌰ 共 z/ ␮ 兲 ⫺
2
␲
2
共50兲
1
⫹
␲
D. Series representation for the gravitational potential
For numerical calculations of the gravitational potential U
and study of its asymptotics near the black hole horizon it is
convenient to use another representation for U, namely, its
Fourier decomposition with respect to the periodic variable z.
冉 冊
⬁
兺
k⫽1
冊
ln k
关 sin关 k 共 ␮ ⫹z 兲兴 ⫹sin关 k 共 ␮ ⫺z 兲兴兴 .
k
共59兲
Using the relation 共see Eq. 共5.5.1.24兲 in Ref. 关20兴兲
124025-5
PHYSICAL REVIEW D 67, 124025 共2003兲
A. V. FROLOV AND V. P. FROLOV
⬁
兺
k⫽1
冉 冊冏
冏
ln k
␲
1 x
x
x⫺ ␲
sin共 kx 兲 ⫽
共 ␥ ⫹ln 2 ␲ 兲 ⫹ ln sin ⌫ 2
k
2
2
␲ 2
2␲
共60兲
valid for 0⭐x⬍2 ␲ , one gets for 兩 z 兩 ⭐ ␮
U 共 ␳ ,z 兲 ⬃ln
⫹
cient. The function V( ␳ ,z) was recovered by direct integration of differential equation 共3兲 by finite differencing in Z
direction. The gravitational potential U( ␳ ,z), function
V( ␳ ,z), and their equipotential surfaces for two different
values of ␮ are shown in Fig. 1.
冉 冊冏
冏
␳ ␮
1
␮ ⫹z 2 ␮ ⫹z
1
⌫
⫹ ln共 4 ␲ 兲 ⫹ ln sin
␲ ␲
2
␲
2
2␲
冉 冊冏
冏
1
␮ ⫺z 2 ␮ ⫺z
1
ln sin
⌫
.
2
␲
2
2␲
共61兲
Using asymptotic 共29兲 of the Schwarzschild potential U S
near horizon, one can present Û(z)⫽lim␳ →0 关 U( ␳ ,z)
⫺U S ( ␳ ,z) 兴 in the region 兩 z 兩 ⭐ ␮ in the form
Û 共 z 兲 ⫽
冋 冉 冊 冉 冊册
␮
␮ ⫹z
␮ ⫺z
1
ln共 4 ␲ 兲 ⫹ ln f
f
␲
2
2
2
A. Large distance asymptotics
Let us first analyze the asymptotic behavior of the CS
metric at large distance ␳ . For this purpose we use the integral representation 共44兲 for U. It is easy to check that the
integrand expression at large ␳ is of order of O( ␤ ⫺2 ) and
hence the integral is of order of ␳ ⫺1 . Thus the ln-term in the
square brackets in Eq. 共44兲 is leading at infinity so that
␮
ln ␳ .
␲
共68兲
␮2
ln ␳ .
␲2
共69兲
U 共 ␳ ,z 兲 兩 ␳ →⬁ ⬃
共62兲
,
Using Eq. 共3兲 we also get
where the function f (x) is defined by
冉冊
x
1
.
f 共 x 兲 ⫽ 2 x sin x⌫ 2
␲
␲
V 共 ␳ ,z 兲 兩 ␳ →⬁ ⬃
共63兲
The metric 共25兲 in the asymptotic region ␳ →⬁ is of the form
It has the following properties:
f 共 0 兲 ⫽1,
IV. PROPERTIES OF CS BLACK HOLES
冉冊
␲
1
f
⫽ ,
2
2
f 共 ␲ 兲 ⫽0.
共64兲
In fact, in the interval 0⭐x⭐ ␲ it can be approximated by a
linear function
f 共 x 兲 ⬇1⫺
x
␲
冋 册
冉
冉
冊
冊
共70兲
The proper size of a closed Killing trajectory for the vector
⳵ z is
C z ⫽2 ␲ L ␳ ⫺( ␮ / ␲ )(1⫺ ␮ / ␲ ) .
共71兲
The metric 共70兲 coincides with the special case (a 1 ⫽a 2 ) of
the Kasner solution 关19兴
ds 2 ⫽⫺ ␳ 2a 0 dt 2 ⫹ ␳ 2a 1 d ␳ 2 ⫹ ␳ 2a 2 dz 2 ⫹ ␳ 2a 3 d ␾ 2 ,
共72兲
a 1 ⫹1⫽a 2 ⫹a 3 ⫹a 0 ,
兩z兩⫺␮
1
⫹ ln
.
2
兩z兩⫹␮
共 a 1 ⫹1 兲 2 ⫽a 22 ⫹a 23 ⫹a 20 .
共66兲
An approximate value of U(0,z) in this region is
冋
⫹ ␳ ⫺2( ␮ / ␲ ) ␳ 2 d ␾ 2 .
共65兲
with an accuracy of order of 1%.
Making similar calculations for 兩 z 兩 ⭓ ␮ one obtains
兩z兩⫹␮
f
␮
2
1
U 共 0,z 兲 ⫽ ln共 4 ␲ 兲 ⫹ ln
␲
2
兩z兩⫺␮
f
2
ds 2 ⫽⫺ ␳ 2( ␮ / ␲ ) dt 2 ⫹ ␳ ⫺2( ␮ / ␲ )(1⫺ ␮ / ␲ ) 共 d ␳ 2 ⫹dz 2 兲
One can rewrite the metric 共69兲 by using the properdistance coordinate l. For small ␮
册
␮
1 关 2 ␲ ⫺ 共 兩 z 兩 ⫹ ␮ 兲兴共 兩 z 兩 ⫺ ␮ 兲
.
U 共 0,z 兲 ⬇ ln共 4 ␲ 兲 ⫹ ln
␲
2 关 2 ␲ ⫺ 共 兩 z 兩 ⫺ ␮ 兲兴共 兩 z 兩 ⫹ ␮ 兲
共67兲
l⫽
␳ 1⫺ ␮ / ␲
,
␮
1⫺
␲
共73兲
and the metric in the ( ␳ , ␾ ) sector takes the form
E. Solutions
To find the gravitational potential U( ␳ ,z) one can use
either its integral representation 共44兲 or the series 共57兲. We
used both methods. Integrals 共44兲 were evaluated using
MAPLE, while the series 共57兲 were implemented in C code
using fast Fourier transform 共FFT兲 techniques. Both methods
give results which agree with high accuracy, but of course
the C implementation is much more computationally effi-
冉 冊
dl 2 ⫹ 1⫺
␮
␲
2
l 2d ␾ 2.
共74兲
Thus the metric of the CS black hole has an angle deficit 2 ␮
at infinity.
The asymptotic form of the metric can be used to deter␮
be a timelike Killing
mine the mass of the system. Let ␰ (t)
124025-6
PHYSICAL REVIEW D 67, 124025 共2003兲
BLACK HOLES IN A COMPACTIFIED SPACETIME
FIG. 1. Compactified Schwarzschild black hole solutions for ␮ ⫽0.5 共left兲 and ␮ ⫽2.0 共right兲. The surface plots show the gravitational
potential U( ␳ ,z) 共top兲 and the function V( ␳ ,z) 共bottom兲; contours represent equipotential surfaces of U 共top/red兲 and V 共bottom/blue兲,
correspondingly.
vector and ⌺ be a 2D surface lying inside t⫽const hypersurface, then the Komar mass m is defined as
m⫽
1
4␲
冕␰
⌺
␮;␯
(t) d ␴ ␮ ␯ .
B. Redshift factor, surface gravity, and proper distance between
black hole poles
Using Eq. 共62兲, we obtain for the redshift factor u the
expression
共75兲
u⫽
For simplicity we choose ⌺ so that t⫽const and ␳ ⫽ ␳ 0
⫽const. For this choice
2a 0 ⫺1 0 1
␦ [␮␦ ␯] ,
␰
␮;␯
u⬇
共76兲
共77兲
Figure 2 共left兲 shows dependence of the redshift factor u on
parameter ␮ . Using the approximation 共65兲 we can write
1
1⫹2a 1
d ␴ ␮ ␯ ⫽ ␦ [0␮ ␦ ␯1 ] ␳ 0
dzd ␾ ,
2
␰ ␮ ; ␯ ⫽⫺2a 0 ␳ 0
␮
1
ln共 4 ␲ 兲 ⫹ ln f 共 ␮ 兲 .
␲
2
冉 冊
␮
␮
1
ln共 4 ␲ 兲 ⫹ ln 1⫺
.
␲
2
␲
The redshift factor u has maximum u
*
1
u ⫽ln共 4 ␲ 兲 ⫺ 兵 1⫹ln 2⫹ln关 ln共 4 ␲ 兲兴 其 ⬇1.22
*
2
⫺2a ⫺1
⫽2a 0 ␳ 0 1 ␦ [0␮ ␦ ␯1 ] .
共78兲
共79兲
at
Substituting these expressions into Eq. 共75兲 and taking the
integral we get m⫽ ␮ . Since all our quantities are normalized by the radius of compactification L, we obtain that the
Komar mass of our system is M ⫽L ␮ .
␮ ⫽ ␲ 兵 1⫺1/关 2 ln共 4 ␲ 兲兴 其 ⬇2.52.
共80兲
*
For ␮ ⬎ ␮ the function u rapidly falls down, becoming
*
negative and logarithmically divergent at ␮ ⫽ ␲ .
124025-7
PHYSICAL REVIEW D 67, 124025 共2003兲
A. V. FROLOV AND V. P. FROLOV
FIG. 2. Redshift factor u 共left兲 and the irreducible mass ␮ 0 ⫽ ␮ exp(⫺u) as functions of ␮ .
In the same approximation we get the following expressions for the irreducible mass ␮ 0 and the surface gravity ␬ :
冉 冊
冉 冊
␮ 0 ⫽ ␮ exp共 ⫺u 兲 ⬇ ␮ 共 4 ␲ 兲 ⫺ ␮ / ␲ 1⫺
␬⫽
␮
␲
⫺1/2
,
e ⫺2u 1
␮
⬇
.
共 2 ␲ 兲 2( ␮ / ␲ ) 1⫺
4␮ 4␮
␲
冕
␲
␮
The surface area of the distorted horizon 共23兲 written in
units L 2 is
共81兲
共82兲
For ␮ → ␲ , they behave as ␮ 0 →⬁ and ␬ →0. Figure 2
共right兲 shows the irreducible mass ␮ 0 as a function of ␮ .
Another invariant characteristic of the solution is the proper
distance between the ‘‘north pole,’’ z⫽ ␮ , and ‘‘south pole,’’
z⫽⫺ ␮ , along a geodesic connecting these poles and lying
outside the black hole. This distance l( ␮ ) is
l 共 ␮ 兲 ⫽2
C. Size and shape of the event horizon
A⫽16␲ ␮ 20 ,
where ␮ 0 is the irreducible mass 共81兲. The shape of the horizon is determined by the shape function
F共 z 兲 ⫽Û 共 z 兲 ⫺u.
⬇2 共 4 ␲ 兲 ⫺ ␮ / ␲
d ␴ 2 ⫽e 2F
冕 冑
␮
dz
共 z⫹ ␮ 兲共 2 ␲ ⫺z⫹ ␮ 兲
共 z⫺ ␮ 兲共 2 ␲ ⫺z⫺ ␮ 兲
⫽2 冑␲ 2 ⫺ ␮ 2 E 共 ␸ ,k 兲 ⫹2 ␮
冑
␲⫹␮
F 共 ␸ ,k 兲 ⫺ 共 ␲ ⫺ ␮ 兲 ,
␲⫺␮
dz 2
d␾2
⫺2F
2
2
⫹e
␮
⫺z
.
兲
共
␮ 2 ⫺z 2
␮2
K⫽e ⫺2F(z) 兵 1⫹ 共 ␮ 2 ⫺z 2 兲关 F⬙ ⫺2 共 F⬘ 兲 2 兴 ⫺4zF⬘ 其 . 共89兲
where
1
k⫽
冑1⫺ 共 ␮ / ␲ 兲 2
.
共84兲
Here F( ␸ ,k) and E( ␸ ,k) are the elliptic integrals of the first
and second kind, respectively. In particular one has
l 共 0 兲 ⫽2 ␲ ,
l 共 ␲ 兲 ⫽ ␲ /2.
共88兲
The Gaussian curvature of the metric d ␴ 2 is K⫽ 21 R, where
R is the Ricci scalar curvature. It is given by the expression
共83兲
␸ ⫽ 冑1⫺ ␮ / ␲ ,
共87兲
Figure 4 共left兲 shows a plot of exp关F(z) 兴 for several values
of ␮ . By multiplying the 2-metric on the horizon d ␥ 2 by
(2 ␮ 0 ) ⫺2 one obtains the metric of the 2-surface which has
the topology of a sphere S 2 and the surface area 4 ␲ . The
metric describing this distorted sphere is
dze ⫺U(0,z)
␲
共86兲
共85兲
Figure 3 shows l/(2 ␲ ) as a function of ␮ . It might be surprising that in the limit ␮ → ␲ , when the coordinate distance
⌬z between the poles tends to 0, the proper distance between
them remains finite. This happens because in the same limit
the surface gravity tends to 0.
124025-8
FIG. 3. l/(2 ␲ ) as a function of ␮ .
PHYSICAL REVIEW D 67, 124025 共2003兲
BLACK HOLES IN A COMPACTIFIED SPACETIME
FIG. 4. The shape function exp关F(z) 兴 共left兲 and the Gaussian curvature of the horizon K(z) 共right兲 for different values of ␮ .
冋
The Gauss-Bonnet formula gives
冕
2
dl 2 ⫽ 共 r ⫹
⫹a 2 兲 F̃ 共 x 兲 dx 2 ⫹
d 2 x 冑␴ K⫽4 ␲ .
共90兲
冋 冉 冊 冉 冊册
1
F⫽ ln
2
␮ ⫹z
␮ ⫺z
f
2
2
f 共␮兲
冋
册
␮ 2 ⫺z 2
1
⬇ ln 1⫹
.
2
4␲共 ␲⫺␮ 兲
共91兲
F̃ 共 x 兲 ⫽
1
1⫺x 2
⫺ ␤ 2,
册
共95兲
,
l eq共 ␮ 兲 ⬇2 ␲
Let us write the metric d ␴ in the form
d␾2
,
d ␴ ⫽F 共 z 兲 dz ⫹ 2
␮ F共 z 兲
2
2
1
1
2⫹
2
␮ ⫺z
4 ␲ 共 1⫺ ␮ / ␲ 兲
2
l pole共 ␮ 兲 ⬇4E
共92兲
then in this approximation one has
共93兲
␤⫽
a
冑r ⫹2 ⫹a 2
.
共96兲
Here r ⫹ ⫽M ⫹ 冑M 2 ⫺a 2 gives the position of the event horizon, and M and a are the mass and the rotation parameter of
the Kerr black hole. The line element 共92兲, 共93兲 is obtained
from the above by coordinate redefinition z⫽ ␮ x and analytic continuation ␣ ⫽i ␤ , with ␣ ⫽( ␮ /2␲ )(1⫺ ␮ / ␲ ) ⫺1/2.
Denote by l eq the proper length of the equatorial circumference, and by l pole the proper length of a closed geodesic
passing through both poles 兩 z 兩 ⫽ ␮ of the black hole horizon.
Then one has
2
F共 z 兲⬇
F̃ 共 x 兲
where
For the unperturbed black hole K⫽1. As a result of deformation, the CS black hole has K⬎1 at the poles, z⫽⫾ ␮ ,
and K⬍1 at the ‘‘equatorial plane’’ z⫽0. Figure 4 共right兲,
which shows K(z) for different values of ␮ , illustrates this
feature. This kind of behavior can be easily understood as a
result of self-attraction of the black hole because of the compactification of the coordinate z.
Using approximation 共65兲 allows one to obtain simple
analytical expressions for the shape function and the Gaussian curvature. Equations 共62兲 and 共77兲 give
f
d␾2
冑1⫺ ␮ / ␲
1⫺ ␮ / 共 2 ␲ 兲
冉
共97兲
,
i␮
2 ␲ 冑1⫺ ␮ / ␲
冊
,
where E(k) is the complete elliptic integral of the second
kind. One has l eq(0)⫽l pole(0)⫽2 ␲ and the surface is a
round sphere. For ␮ → ␲ the lengths l eq→0 and l pole→⬁.
D. Embedding diagrams for a distorted horizon
while the Gaussian curvature is
K⬇
16␲ 2 共 ␲ ⫺ ␮ 兲 2 关共 2 ␲ ⫺ ␮ 兲 2 ⫹3z 2 兴
.
关共 2 ␲ ⫺ ␮ 兲 2 ⫺z 2 兴 3
共94兲
The Gaussian curvature is positive in the interval 兩 z 兩 ⬍ ␮ .
It is interesting to note that the horizon geometry of the
CS black hole coincides 共up to a constant factor兲 with the
geometry on the 2D surface of the horizon of the Euclidean
4D Kerr black hole. This fact can be easily checked since the
induced 2D geometry of the horizon of the Kerr black hole is
共see, e.g., Eq. 共3.5.4兲 in Ref. 关18兴兲
The metric 共92兲 can be obtained as an induced geometry
on a surface of rotation ⌺ embedded in a 3-dimensional Euclidean space. Let
dl 2 ⫽dh 2 ⫹dr 2 ⫹r 2 d ␾ 2
共98兲
be the metric of the Euclidean space and the surface ⌺ be
determined by an equation h⫽h(r), then the induced metric
on ⌺ is
124025-9
冋 冉 冊册
d ␴ 2 ⫽ 1⫹
dh
dr
2
dr 2 ⫹r 2 d ␾ 2 .
共99兲
PHYSICAL REVIEW D 67, 124025 共2003兲
A. V. FROLOV AND V. P. FROLOV
FIG. 5. Embedding diagrams for the surface of the black hole horizon. By rotating a curve from a family shown at the plot around a
horizontal axis one obtains surface isometric to the surface of a black hole described by the metric d ␴ 2 . Different curves correspond to
different values of ␮ . The larger ␮ the more oblate is the form of the curve.
By comparing this metric with Eq. 共92兲 we get
1
r⫽
冑F 共 z 兲
冉 冊 冉 冊
dh
dz
2
⫹
dr
dz
共100兲
,
2
⫽F 共 z 兲 .
共101兲
These equations imply the following differential equation for
h(z):
dh
⫽
dz
冑
F⫺
F ⬘2
4F 3
.
共102兲
Figure 5 shows the embedding diagrams for the distorted
horizon surfaces of a compactified black hole for different
values of ␮ . The larger is the value ␮ the more oblate is the
surface of the horizon. For large ␮ close to ␲ it has a cigarlike form.
E. µ\ ␲ limit
Let us now discuss the properties of the spacetime in the
limiting case ␮ → ␲ . This limit can be easily taken in the
series representation 共57兲 for the gravitational potential U.
Since sin(␲k)⫽0 for k⬎0, only the logarithmic term survives in this limit. Thus U( ␳ ,z)⫽ln ␳. Since the limiting
metric is invariant under translations in the z direction, it has
the form of the Kasner solution 共70兲 with ␮ ⫽ ␲ and reads
ds 2 ⫽⫺ ␳ 2 dt 2 ⫹d ␳ 2 ⫹dz 2 ⫹d ␾ 2 .
共103兲
This is a Rindler metric with two dimensions orthogonal to
the acceleration direction being compactified
z苸 共 ⫺ ␲ , ␲ 兲 ,
␾苸共 ⫺␲,␲ 兲.
共104兲
Restoring the dimensionality we can write this metric as
dS 2 ⫽⫺
R2 2
dT ⫹dR 2 ⫹dZ 2 ⫹L 2 d ␾ 2 .
L2
共105兲
makes the horizon prolated grows with the black hole mass.
For large mass ␮ ⭓ ␲ /2 the black hole deformation becomes
profound. The pole parts of the horizon, that is parts close to
z⫽⫺ ␮ and z⫽ ␮ , attract one another. As a result of this
attraction the Gaussian curvature of regions close to black
hole poles grows, while the Gaussian curvature in the ‘‘equatorial’’ region falls down and the surface of the horizon is
‘‘flattened down’’ in this region. For large value of the mass
␮ , the ‘‘flattening’’ effects occur for a wide range of the
parameter z. Such a black hole is reminiscent of a cigar or a
part of the cylinder with two sharpened ends.
We did not include any branes in our consideration. However, we should note that the surface Z⫽0 is a solution of
the Nambu-Goto action for a test brane. This can be easily
seen, as the solution we discussed is symmetric around the
surface Z⫽0, which implies that its extrinsic curvature vanishes there. At far distances the induced gravitational field on
the Z⫽0 submanifold is asymptotically a solution of vacuum
(2⫹1)-dimensional Einstein equations. It is not so for regions close to the black hole. This ‘‘violation’’ of the vacuum
(2⫹1)-dimensional Einstein equations for the induced metric makes the existence of the (2⫹1)-dimensional black
hole on the brane possible.
In our work we did not find any indications of instability
of a black hole which might be interpreted as connected with
the Gregory-Laflamme instability 关22,23兴. It may not be surprising since these kinds of instabilities are expected in
spacetimes with higher number of dimensions 共see, e.g.,
Refs. 关21,24 –26兴兲.
On the other hand, a solution describing a black hole in a
compactified spacetime may be unstable for a different reason. The nature of this instability is the following. In our
setup we fix a radius of compactification L. In a flat spacetime we can choose parameter L arbitrarily and the energy of
the system, being equal to zero, does not depend on this
choice. The situation is different in the presence of a black
hole. Consider a black hole of a given area, that is with a
fixed parameter M 0 . Since the black hole entropy, which is
proportional to the area, remains unchanged for quasistationary adiabatic processes, one may consider different states of
a black hole with a given M 0 . L plays a role of an independent parameter, specifying a solution. In particular one has
V. DISCUSSION
The obtained results can be summarized as follows. If the
size of a black hole is much smaller that the size of compactification, its distortion is small. The deformation which
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M 0⫽
M 共 4 ␲ 兲 ⫺M /( ␲ L)
冑1⫺M / 共 ␲ L 兲
.
共106兲
PHYSICAL REVIEW D 67, 124025 共2003兲
BLACK HOLES IN A COMPACTIFIED SPACETIME
the system. In this case the lowest energy state corresponds
to L→⬁, so that a stable solution will be an isolated
Schwarzschild black hole in an empty spacetime without any
compactifications. In the opposite case L⬍L the energy
*
decreases when L→0. In this limit M ⬇ ␲ L and hence it
corresponds to a limiting solution ␮ → ␲ . The limiting metric is given by Eq. 共105兲. The corresponding spacetime is a
2D torus compactification of the Rindler metric.
This argument, based on the energy consideration, indicates a possible instability of a compactified spacetime with
a black hole with respect to compactified dimension either
‘‘unwrapping’’ completely or being ‘‘swallowed’’ by a black
hole. While ‘‘unwrapping’’ of the extra dimension may be
prevented by the usual stabilization mechanisms, the other
instability regime might not be so benign. It is interesting to
check whether this conjecture is correct by standard perturbation analysis.
FIG. 6. M as a function of L for fixed M 0 .
This relation shows that for fixed M 0 the energy of the system M depends on compactification radius L. The plot of the
function M (L) is shown in Fig. 6. For L⫽L ⫽1.345M 0 the
*
mass M has maximum M ⫽M ⫽3.3877M 0 . At the corre*
sponding value ␮ ⫽2.52 the function u( ␮ ) has its maxi*
mum. Thus if one starts with a system with L⬎L then a
*
positive variation of parameter L will decrease the energy of
This work was partly supported by the Natural Sciences
and Engineering Research Council of Canada. One of the
authors 共V.F.兲 is grateful to the Killam Trust for its financial
support.
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ACKNOWLEDGMENTS
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