793
Progress of Theoretical Physics, Vol. 114, No. 4, October 2005
Infinite Number of Soliton Solutions to 5-Dimensional Vacuum
Einstein Equation
Takao Koikawa∗)
The Niels Bohr Institute, Blegdamsvej 17, 2100 Copenhagen, Denmark
(Received January 31, 2005)
We obtain an infinite number of exact solutions to the 5-dimensional static Einstein
equation with axial symmetry using the inverse scattering method. The solutions are characterized by two integers representing the soliton numbers. The first non-trivial example of
these solutions is the static black ring solution found recently.
In recent years, interest in the higher-dimensional Einstein equation has increased. In particular, the 5-dimensional black ring solution discovered by Emparaton and Reall1)–3) has attracted considerable attention. In contrast to the 4dimensional black hole case, the 5-dimensional Einstein equations allow for more
topologies than just S 3 . The black ring solution, which has a regular event horizon characterized by S 2 × S 1 , is one interesting example. The stationary black
ring solution supported by its angular momentum is derived under stationary and
axi-symmetric conditions, while the static solution has a conical singularity.
In the 4-dimensional case, most of the well-known exact solutions to the Einstein
equation can be constructed as soliton solutions. The Einstein equations with some
symmetries are completely integrable systems, and therefore they have an infinite
number of exact solutions. Most of the well-known solutions are derived as 2-soliton
solutions in a series of multi-soliton solutions. The Kerr solution was studied using
the inverse scattering technique first in Ref. 4). We studied the multi-Weyl solutions,
including the Schwartzschild solution in Ref. 5), the solutions to the Einstein equation
with a scalar field in Ref. 6), a multi-charged Weyl solution including the multiReissner Nordström solution in Ref. 7), and the dilaton black hole solutions with
electric charge in Ref. 8). These examples show that the inverse scattering is a
powerful tool to study the Einstein equations in 4 dimensions, and it would be
interesting to determine if we can also use it to study the 5-dimensional Einstein
equations. The 2 soliton solutions represent the soliton-like objects corresponding to
the well-known solutions, and some of them are known as black hole solutions. The
2n soliton solutions can be interpreted as n soliton-like objects aligned along the z
axis. The inverse scattering method is the only systematic method of finding such
n-body solutions of the Einstein equations.
We have also studied the Kaluza-Klein-type solution,9)–11) where the extradimensions are assumed to be compactified as tori. However, in this paper we do not
distinguish the inner and outer spaces, and we do not impose such compactification
∗)
E-mail: [email protected]
On leave of absence from Otsuma Women’s University.
794
T. Koikawa
conditions on the extra dimensions. We study the 5-dimensional Einstein equations
by use of the inverse scattering method. In order to solve the static vacuum solutions to the 5-dimensional Einstein equation, we impose axial symmetry and use
the canonical variables ρ and z. Though there are several possible topologies in the
5-dimensional case, we would like to find interesting solutions, including the static
ring solution as the first non-trivial solution.
We start with the Einstein equations one of which has a matrix form. By
assuming static and axially symmetric conditions, we obtain general soliton solutions
that are characterized by two soliton numbers, and therefore call these (m, n)-soliton
solutions using two integers m and n. We then analyse the first few examples for
smaller m and n and show that the (0, 2)-soliton solution is identical to the static ring
solution expressed in terms of the axial symmetric variables ρ and z. In the following,
we discuss the characteristics of the 5-dimensional black hole solutions, which might
have more than two event horizons and may violate the No Hair Theorem.
The metric is assumed to be given by
ds2 = f (dρ2 + dz 2 ) + gab dxa dxb ,
(1)
where f and gab are functions of ρ and z, and the suffices a and b of gab run from 0
to 2. Here, we also assume the following supplementary condition for g = (gab )
detg = −ρ2 .
(2)
In the following, we represent the 3 × 3 matrix by g. The Einstein equations in the
above metric read
(3)
ρg,ρ g −1 ,ρ + ρg,z g −1 ,z = 0,
1
1
(4)
(ln f ),ρ = − + Tr(U 2 − V 2 ),
ρ 4ρ
1
Tr(U V ),
(5)
(ln f ),z =
2ρ
where g,ρ represents the partial derivative of g with respect to the variable ρ, and U
and V are 3 × 3 matrices defined by
U = ρg,ρ g −1 ,
V = ρg,z g −1 .
(6)
In the 4-dimensional case, the matrix equation (3) has a 2 × 2 form. In the higherdimensional Einstein equations, it takes a larger matrix form. In the study of KaluzaKlein general relativity,10) the matrix has a block-diagonal form composed of 2 × 2
and N × N matrices, where N represents the dimension of the extra space. The
matrix form equation is also derived in a recent paper.12) For the 5-dimensional
case, it has a 3 × 3 form. We assume the static condition in the present paper. This
condition implies that the matrix g is diagonal. Noting that g should satisfy the
supplementary condition (2), we define it as
−1
2 + z2 − z h ,
2 + z2 + z h
h
,
ρ
ρ
(7)
g = diag −h−1
1
2 ,
1
2
Infinite Number of Soliton Solutions to 5-D Vacuum Einstein Equation
795
where h1 and h2 are functions of ρ and z. The metric is explicitly written as
−1 2
ρ2 + z 2 − z h1 dx21 +
ρ2 + z 2 + z h2 dx22
ds2 = −h−1
1 h2 dt +
+ f (dρ2 + dz 2 ).
(8)
The matrix equation (3) can be rewritten in the form of equations for hi (i = 1, 2)
as
(ρ(ln hi ),ρ ) ,ρ + (ρ(ln hi ),z ) ,z = 0.
(i = 1, 2)
(9)
In order to solve these equations, we first solve (9) to obtain g, and then construct
U and V . By substituting the results into the RHS of the above Einstein equations
for f , we integrate them to obtain f . It is known that the equation for g is a zerocurvature equation,4) just like that for the soliton equations. This is the reason that
we refer to these solutions as “soliton solutions”. In contrast to the 4-dimensional
case, there are two soliton equations, as given in (9). Since each of them has soliton
solutions, we can specify a solution by its soliton integer numbers m and n. We call
these (m, n)-soliton solutions. They are given explicitly by
h1 = ρ−m
m
(iµk ),
(10)
k=1
n
−n
(iµ̃l ).
h2 = ρ
(11)
l=1
Here, µk and µ̃l are called the poles, and are given by
µk = wk − z ± ρ2 + (wk − z)2 ,
µ̃k = w̃k − z ± ρ2 + (w̃k − z)2 ,
(12)
(13)
where wk and w̃l are arbitrary constants. They can be either real or complex.
Here we restrict ourselves to real poles. We omit the deformation parameter here,
because we are now interested in a series related to the static black ring solutions.
By substituting the above forms of the (m, n)-soliton solutions into U and V , we
integrate the above Einstein equations for f to obtain
n−m
ρ2 + z 2 − z 4
f = Cmn 2
2
2
2
ρ + z + z
ρ +z
m
−m− n +2 n
−n− m +2
2
2
×
µk
µ̃k
ρ
m2 +n2 −m−n+mn
2
k=1
×
m l=1
ρ2 + z 2 − z − µk
k=1
×
2
k>l (µk − µl )
m k=1
n ρ2 + z 2 + z + µ̃l
l=1
(µ̃
−
µ̃l )2 k l (µk − µ̃l )
k
k>l
n 2
,
2
µ2k + ρ2
l=1 µ̃l + ρ
(14)
796
T. Koikawa
where Cmn is a constant to be determined by the asymptotic flat condition.
We first consider the (0, 0)-soliton solution, which is the flat solution. In this
case, the metric components h1 , h2 and f are given by
h1 = h2 = 1,
1
,
f= 2 ρ2 + z 2
(15)
(16)
where we have set C00 = 1/2 in (14). We next introduce the coordinates r and θ
defined through the relations
1
ρ = r2 sin 2θ,
2
1
z = r2 cos 2θ.
2
(17)
(18)
We then notice that
1
,
r2
dρ2 + dz 2 = r2 (dr2 + r2 dθ2 ).
f=
(19)
(20)
Then, by writing the coordinates as x1 = φ and x2 = ψ, we obtain the flat metric,
ds2 = −dt2 + r2 sin2 θdφ2 + r2 cos2 θdψ 2 + dr2 + r2 dθ2 .
(21)
Next we study the (0, 2)-soliton solution, which is given by
h1 = 1,
(22)
µ̃1 µ̃2
,
ρ2
ρ2 + z 2 − z
(µ̃2 − µ̃1 )2
f = C02 2
2
ρ +z
ρ2 + z 2 + z + µ̃1
ρ2 + z 2 + z + µ̃2
2
×
.
µ̃1 + ρ2 µ̃22 + ρ2
h2 = −
(23)
(24)
We choose the minus sign for µ̃1 and the plus sign for µ̃2 and set w̃1 = z0 − σ and
w̃2 = z0 + σ, without loss of generality. Then we have
(25)
µ̃1 = z0 − σ − z − ρ2 + (z + z0 + σ)2 ,
(26)
µ̃2 = z0 + σ − z + ρ2 + (z − z0 − σ)2 .
Because we can always shift the zero of the z axis, we are allowed to transform
z → z + z0 in the metric. Next, we define the transformed µ̃i (i = 1, 2), and µ̃3 along
with their conjugates, as
µ̃1 = −σ − z − R1 ,
(27)
Infinite Number of Soliton Solutions to 5-D Vacuum Einstein Equation
µ̃∗1
µ̃2
µ̃∗2
µ̃3
µ̃∗3
= −σ − z + R1 ,
= σ − z + R2 ,
= σ − z − R2 ,
= −z0 − z − R3 ,
= −z0 − z + R3 ,
797
(28)
(29)
(30)
(31)
(32)
where the quantities Ri are given by
R1 = ρ2 + (z + σ)2 ,
R2 = ρ2 + (z − σ)2 ,
R3 = ρ2 + (z + z0 )2 .
(33)
(34)
(35)
They satisfy µ̃i µ̃∗i = −ρ2 for i =1–3. The nonzero metric components gab and f are
given by
µ̃∗1
−σ − z + R1
=−
,
µ̃2
σ − z + R2
= µ̃∗3 = −z0 − z + R3 ,
µ̃3 µ̃2
(−z0 − z − R3 )(σ − z + R2 )
=
=
,
∗
µ̃1
−σ − z + R1
g00 = −
(36)
g11
(37)
g22
f = C02
(38)
µ̃∗3 (µ̃2 − µ̃1 )2 (−µ̃3 + µ̃1 ) (−µ̃3 + µ̃2 )
.
R3 µ̃21 + ρ2 µ̃22 + ρ2
(39)
We rewrite f in order to compare the above solutions to the known solution. Using
the relations
µ̃21 + ρ2 = −2µ̃1 R1 ,
µ̃22 + ρ2 = 2µ̃2 R2 ,
(40)
(41)
and the quadratic relations satisfied by the quantities µ̃i and µ̃∗i
µ2i − 2(wi − z)µi − ρ2 = 0,
(42)
we can rewrite f as
(R
+
R
+
2σ)
(1 + zσ0 )R1 + (1 −
1
2
C02 z0
f =−
2
R1 R2 R3
σ
z0 )R2
− 2 zσ0 R3
.
(43)
This is identical to the result given in Ref. 12).
We next derive the 5-dimensional Schwarzshild solution as a special case of the
static black ring solutions. In the above solution, the parameters z0 and σ are
independent. Now we assume the relation z0 = −σ between them. In this case,
R3 = R2 , and so f is simplified as
f = 2C02 σ
R1 + R2 + 2σ
.
R1 R2
(44)
798
T. Koikawa
Then, introducing r and θ through the relations
1
ρ=
1−
2
1
1−
z=
2
1
4σ 2 2
r sin 2θ,
r2
2σ
r2 cos 2θ,
r2
(45)
(46)
Ri can be written as
r2
− 2σ sin2 θ,
2
r2
R2 = R3 =
− 2σ cos2 θ,
2
R1 =
(47)
(48)
and therefore µ̃i (i =1–3) and their conjugates are expressed as
µ̃1
µ̃∗1
µ̃2
µ̃∗2
= −r2 cos2 θ,
= (r2 − 4σ) sin2 θ,
= µ̃∗3 = r2 sin2 θ,
= µ̃3 = −(r2 − 4σ) cos2 θ.
(49)
(50)
(51)
(52)
Then, gab and f can be rewritten as
4σ
g00 = − 1 − 2 ,
r
(53)
g11 = r2 sin2 θ,
g00 = r2 cos2 θ,
f=
r2 1 −
(54)
(55)
8C02 σ
4σ
r2
+
We also note that
2
2
dρ + dz = r
2
4σ 4σ 2 sin2 2θ
1− 2 +
r
r2
4σ 2 sin2 2θ
r2
.
4σ
1− 2
r
(56)
−1
2
2
dr + r dθ
2
.
(57)
1
, we obtain the 5-dimensional Schwartzshild
Setting the constant C02 as C02 = 8σ
solution as a special case of the (0, 2)-soliton soliton solution:
4σ
4σ −1 2
2
2
dr + r2 dθ2 + r2 sin2 θdφ2 + r2 cos2 θdψ 2 .
ds = − 1 − 2 dt + 1 − 2
r
r
(58)
We next consider the (2, 0)-soliton solution. The metric components h1 , h2 and
f are given by
h1 = −
µ1 µ2
,
ρ2
(59)
Infinite Number of Soliton Solutions to 5-D Vacuum Einstein Equation
h2 = 1,
799
(60)
ρ2 + z 2 + z
(µ2 − µ1 )2
2
2
ρ +z
ρ2 + z 2 − z − µ1
ρ2 + z 2 − z − µ2
2
.
×
µ1 + ρ2 µ22 + ρ2
f = C20
(61)
(62)
Here we choose the minus sign for µ1 and the plus sign for µ2 and set w1 = z0 − σ
and w2 = z0 + σ, as done previously for µ̃1 and µ̃2 . Then we have
µ1 = z0 − σ − z − ρ2 + (z − z0 + σ)2 ,
(63)
2
2
(64)
µ2 = z0 + σ − z + ρ + (z − z0 − σ) .
We can translate in the z direction, i.e. taking z → z +z0 , as before and we eliminate
z0 from the above expressions. Because of the slight difference from the (0, 2)-soliton
case with regard to g11 and g22 , as well as f , this solution seems to be different from
the (0, 2)-soliton solution. However, in the special case that z0 = σ, we obtain the
same 5-dimensional Schwarzschild metric by expressing the metric in terms of r and
θ defined in (45) and (46).
We have used the canonical coordinates ρ and z in solving the 5-dimensional Einstein equations, which have been used extensively in the study of the 4-dimensional
solutions in previous works. These can be found for the Kerr solution in Ref. 4), the
multi-Weyl solutions including the Schwartzschild solution in Ref. 5), the solutions
to the Einstein equation with a scalar field in Ref. 6), the multi-charged Weyl solution, including the multi-Reissner Nordström solution in Ref. 7), and the dilaton
black hole solutions with electric charge in Ref. 8). In these papers, the zeros of ρ
are either the positions of the horizons or the singularities of the black hole solution.
This is related to the fact that detg = −ρ2 . We note that the converse is not always
true, since detg is composed of the products of the metric components, and therefore
the zeros might be cancelled in the products. We list them below.
In the case of the Kerr solution, the transformation of the canonical variables ρ
and z to the spherical coordinates r and θ is given by4)
ρ = (r − m)2 − σ 2 sin θ.
(65)
Here, σ is given by
σ 2 = m2 − a2 ,
(66)
where m is the mass of the√black hole and a is the angular momentum per unit mass.
The zeros of ρ at r = m± m2 − a2 correspond to the inner and outer horizons of the
Kerr black hole solution. In the a = 0 case, we obtain the Schwartzschild solution,
which is also obtained by setting the distortion parameter to unity in the Weyl
solution. In this case, the above relation reduces to
(67)
ρ = (r − 2m)r sin θ.
800
T. Koikawa
The zeros of ρ in this case correspond to the horizon at r = 2m and the singularity
at r = 0.
In the Reisner-Nordström solution, the metric component g00 is given by
g00 = 1 −
2m e2
+ 2,
r
r
(68)
where m is the mass and e is the electric charge. This solution can also be regarded
as the 2-soliton solution of an infinite series of soliton solutions obtained with the
inverse scattering method by adopting the axially symmetric coordinates ρ and z.7)
We change the coordinates to the spherical coordinates r and θ, in terms of which
the variable ρ is expressed as
ρ = (r − m)2 − d2 sin θ,
(69)
with
d 2 = m 2 − e2 .
(70)
√
The zeros of ρ are at r = m ± m2 − e2 , which are the positions of the inner and
outer horizons, as also seen from (68).
In the dilaton black hole solution,13) the metric is given by
2M
2M −1 2
2
2
dt + 1 −
dr + r(r + 2Σ )dΩ 2 ,
(71)
ds = − 1 −
r
r
where M is the mass of the black hole and Σ is the dilaton charge. In order to
prevent the area measured at some r from vanishing, and therefore becoming singular
as r → −2Σ , the mass and the dilaton charge should satisfy the condition
M > −Σ .
(72)
We can derive an infinite number of solutions, including the above solution, using
the inverse scattering method. We use the spherical coordinates r and θ instead of
z and ρ to derive the above metric. The variable ρ is related to them by
(73)
ρ = (r − 2M )(r + 2Σ ) sin θ.
This shows that the zeros of ρ are the horizon at r = 2M and the singularity at
r = −2Σ .
These examples in 4 dimensions show that the zeros of ρ2 obtained by solving the quadratic equation of the spherical coordinates r are either the horizons or
singularities. Consider the general quadratic equation given by
r2 + c1 r + c0 = 0,
(74)
where the coefficients c0 and c1 are some constants. In order for the quadratic
equation to have two positive roots, the necessary condition is that both c0 and c1
be nonzero. These two constants turn out to express two charges of black holes.
Infinite Number of Soliton Solutions to 5-D Vacuum Einstein Equation
801
When c0 = 0 and c1 = 0, the roots are r = 0 and r = −c1 . Here, the nonzero root
corresponds to only hair, or the mass of the black hole. This shows that we see the
number of horizons and charges from the number of roots of the quadratic equation.
Also in the 5-dimensional case, we use the canonical coordinates ρ and z, but
they are different from those in the 4-dimensional case. While ρ has the dimension
of length in the 4-dimensional case, it has the dimensions of (length)2 in the 5dimensional case. The variable ρ is related to the spherical coordinates r and θ as
in Eqs. (17) and (45). The zeros of ρ2 are obtained by solving a quartic equation, in
contrast to the quadratic equations in the 4-dimensional case. This quartic equation
is
r4 + c3 r3 + c2 r2 + c1 r + c0 = 0.
(75)
Following arguments analogous to that given in the 4-dimensional case, this quartic
equation seems to suggest that the 5-dimensional black holes might have maximally
four event horizons and four charges. In the vacuum case, the black hole can have
only a mass parameter, which is due to the fact that c2 = 0, while c3 = c1 = c0 =
0. More constants would become nonzero and give rise more roots to the quartic
equation if the stationary black hole solutions of nonvacuum Einstein equations with
some fields using the canonical variables are obtained. We may incorporate such
fields as electro-magnetic fields, three-form field strengths and dilaton fields to solve
the Einstein equations, and then we would have nonzero coefficients in the quartic
equation. A study along this direction is in progress. For example, the charged
dilatonic black rings have recently been discussed in Ref. 14). Black holes with these
charges would have features that differ from those in the 4-dimensional case.
In this paper, we obtained an infinite number of soliton solutions to the 5dimensional vacuum Einstein equation by assuming static and axially symmetric
conditions, which are expressed compactly in terms of poles. A solution is characterized by the soliton numbers, and the first non-trivial solution is the static
black ring solution found recently. We have obtained general n soliton solutions
to the 5-dimensional Einstein equation, and have shown that the (0,2)- soliton solution coincides with the known static solution. From the structure of the solution,
(0, 3k + 2)-soliton (k = 1, 2, · · · ) solutions seem to form k + 1 soliton-like objects.
We need to work more in order to clarify this point. We can also apply the inverse
scattering method to the stationary case, where we may find singularity-free black
hole solutions, and hence in this case, the study of multi-soliton solutions would
be more interesting. It would also be interesting to study the non-vacuum case by
using canonical variables, where we might find more than two event horizons and
more than three charges, in contrast to the case of 4-dimensional black holes.
Appendix A
Higher-Dimensional Einstein Equations
Here we compute the Ricci tensor. We consider the D-dimensional metric
ds2 = f (dρ2 + dz 2 ) + gij dxi dxj ,
(A.1)
802
T. Koikawa
where f = f (ρ, z) and gij = gij (ρ, z) (i, j = 0, 1, · · · , D − 3). We assume that the
(D − 2) × (D − 2) matrix g = (gij ) satisfies the relation
det g = −ρ2 .
(A.2)
The nonzero Christoffel symbols are computed as
∂gij
∂gij
1
1
1 ∂gkj
i
, Γijz = − f −1
, Γρj
,
= g ik
Γijρ = − f −1
2
∂ρ
2
∂z
2
∂ρ
1 ∂gij
1
1
1
i
ρ
z
ρ
, Γρρ
= g ik
= (ln f ),ρ , Γzz
= (ln f ),z , Γρz
= (ln f ),z ,
Γzj
2
∂z
2
2
2
1
1
1
z
ρ
z
(A.3)
Γzρ = (ln f ),ρ , Γzz = − (ln f ),ρ , Γρρ = − (ln f ),z ,
2
2
2
where (ln f ),ρ = ∂ ln f /∂ρ. Then, because det g = −ρ2 , we have
g ij
∂gji
2
= ,
∂ρ
ρ
g ij
∂gji
= 0.
∂z
(A.4)
Using these, we obtain
i
=
Γρi
1
,
ρ
i
Γzi
= 0.
These relations enable us to obtain the Ricci tensors in the following forms:
2
∂
∂2
1
1 ∂
kl ∂gki ∂glj
kl ∂gki ∂glj
gij − g
Rij =
+
−g
,
+
2f
∂ρ2 ρ ∂ρ ∂z 2
∂ρ ∂ρ
∂z ∂z
∂gkj ∂gli
∂2
1
1 ∂2
1 ∂
1
+ 2 ln f + g ik g jl
,
−
Rρρ = − 2 +
2
ρ
2 ∂ρ
ρ ∂ρ ∂z
4
∂ρ ∂ρ
∂gkj ∂gli
∂2
1 ∂2
1 ∂
1
+
,
+
Rzz =
ln f + g ik g jl
2 ∂ρ2 ρ ∂ρ ∂z 2
4
∂z ∂z
∂gkj ∂gli
1
1 ∂ ln f
Rρz = −
+ g ik g jl
.
2ρ ∂z
4
∂ρ ∂z
(A.5)
(A.6)
(A.7)
(A.8)
(A.9)
We can write the vacuum Einstein equations in matrix form. They can be divided
into two groups. The first includes equations only for g, and the second includes
equations for determining f by use of the obtained solution of g:
(ρg,ρ g −1 ),ρ + (ρg,z g −1 ),z = 0,
1
1
(ln f ),ρ = − + T r(U 2 − V 2 ),
ρ 4ρ
1
T r(U V ),
(ln f ),z =
2ρ
(A.10)
(A.11)
(A.12)
where the matrices U and V are defined by
U = ρg,ρ g −1 ,
V = ρg,z g −1 .
(A.13)
Infinite Number of Soliton Solutions to 5-D Vacuum Einstein Equation
803
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