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Axially symmetric solution of the Einstein equation with solitons in extra dimensions
Volume 136. number 6
10 April 1989
PHYSICS LETTERS A
AXIALLY SYMMETRIC SOLUTION OF THE EINSTEIN WITH SOLITONS IN EXTRA DIMENSIONS
EQUATION
T. AZUMA”, M. ENDO” and T. KOIKAWA a,b a Faculty of Liberal Arts, Dokkyo University, So/w, Saitama 340, Japan b Department ofphysics, Tokyo Metropolitan University, Setagaya, Tokyo 158, Japan
Received 26 September 1988; accepted for publication 7 February 1989 Communicated by J.P. Vigier
We give the solution to the higher dimensional Einstein equation with axial symmetry by using the inverse scattering method. The solution consists of two solitons in the external dimensions and any number of solitons in the internal dimensions.
In recent years higher dimensional theories such as the unification of gravitation and gauge theories have been principal concerns of many physicists [ I]. If the realm of nature actually lies in more than four dimensions, this has great implications in the study of the early universe and black holes. Taking account of the compactification of internal space many studies have been done in the search for a black hole solution with event horizon in four dimensions. In these studies the no-hair theorem [ 2 ] that black holes with a mass, an electric charge and an angular momentum cannot be endowed with scalar charge is important, because the action of higher dimensional gravity is known [3] to be equivalent to that of the four dimensional gravity with a massless scalar field after the Weyl resealing. Lately one of the present authors and Shiraishi studied the stationery and axisymmetric higher dimensional Einstein equation [ 41 of which the metric is -d.r2=f(p,
z) (dp2+dz2)+g,,(p,
+h(p, z) i
consider either a nonstationary case or abandon the massless free scalar field. In this short note we look for a solution of the higher dimensional Einstein equation with axial symmetry, which might be a candidate for a black hole solution. For this purpose we consider the case that not all the metric coefficients of the extra dimensions are necessarily the same as above. This enables us to discuss the construction of the soliton solution in the extra dimensions. In finding the explicit solution we use the inverse scattering method [ 5 ] which has been applied to the black hole solution and gravitational solitons. We stress here that the method to obtain the soliton solutions is useful in determining the metric of the extra dimensions as well as the four dimensional part, which was not noted in the previous work. The line interval is given by #’ -dr2=f(dp2+dz2)+g,bdx”dxb+h,Bdx*dxa =f(dp2+dz2)+GijdXidXi,
z) dx” dxb
(dxQ2.
Here h(p, z) is the metric coefficient of the iV-dimensional internal space and In h plays the role of scalar field. In ref. [ 41 the solution has a naked singularity and this might be understandable by the nohair theorem. In order to evade the theorem we must
(1)
wherefl grrb has and G, are functions of p and z. (G,) *’ In the five dimensional Raluza-Klein type theory, in which
03759601/89/$ 03.50 0 Elsevier Science Publishers B.V. ( North-Holland Physics Publishing Division )
non-vanishing gauge potentials exist, some work has already been published [ 61. Since we have the field theory limit of the superstring theory in mind, we do not have those potentials corresponding to the internal-external dimensional part of the metric.
269
Volume 136, number
6
PHYSICS
is the (2 +N) x (2 +N) block diagonal matrix given by
(gib)(ho) > ,
G= (Gij)=
(2)
0
where g& is a 2 X 2 matrix and haB an NX N matrix, Then the vacuum Einstein equations read (pG,G-‘).+(pG,,G-‘),,=O, (Infig=-
(3)
i + iTr(U2-V’),
(da)
(ln_O,z= &WW,
(@I
where U and I/ are (2 + N) x (2 +N) matrices defined by U=pG,G-’
,
V=pG,,G-’
.
(5)
LETTERS
A
10 April 1989
seed metric, we can construct the soliton solution satisfying (3a) and (3b). However they do not necessarily have the same determinants as the seed metric and so renormalization is necessary. In the present case we should require that the renormalized metric satisfies the constraint ( 6’ ). This means that the renormalized metric gCph’or hcph’ and the seed metric go or ho do not necessarily have the same determinant. Of the various possibilities we discuss the case that the renormalized or physical metric gCph) and h(ph’ satisfy det g (ph)= _p2 2
(gal
det h(ph)= 1 .
(8b)
The above procedure of obtaining a new solution by the soliton technique is summarized as follows: ‘ph’=N,g’=N,B,gO,
go+g’=B,go+g
(gal
We have fixed the determinant of the matrix G without loss of generality as
ho+h’=Bzho+h(ph)=N2h’=N2B2h0,
det G= -p2,
where B1 and B2 are the 2x2 matrices given by
(6)
(9b)
which can be assured to be satisfied by ( 3 ) . By noting that G is given by (2)) the first Einstein equation (3) can be decomposed into the same type of equation: (P&C’).+
(P&g-‘),,=0
(ph,h-‘),-(ph,,h-‘),,=O,
9
(3a) (3b)
where g and h are 2 x 2 and NX N matrices of which the entries are g,b and h,, respectively. The constraint eq. (6 ) also reads det gxdet h= -p2.
(6’ )
Now we are going to find the solutions to the Ein-
stein equations (3) and (4) under some assumptions on h in ( 1). We assume that h is diagonal coiresponding to the torus compactification: h,, = h,&, .
(7)
The case that all h,‘s are the same and depend on dpw has been discussed in ref. [ 41. As we have mentioned we are interested in the case that, at least, two of the ha’s are different. Though the following procedure is quite general in discussing the arbitrary diagonal h with distinctive entries, we assume that h = diag( h,, h2) for definiteness. Starting with the 270
a, j?, c=5,6.
(lob)
For the detailed definitions of the notations in ( 1Oa) the readers are referred to ref. [ 51. Because of the parallelism in (3a) and (3b) we can construct the new solution h’ by ( 1Ob) which has definitions similar to ( lOa). Since we have assumed that h is diagonal as in (7), we set fi $” = 0 to preserve diagonality. The second step in (9) is the normalization procedure to recover the constraints given by (8 ). In the diagonal case we do not need to follow the prescription of ref. [ 5 1. Since this point is different from the standard procedure we give an explanation by returning to the general case. We start with the flat seed metric ho=diag(l,
l,..., 1))
(11)
with N times 1. The n-soliton solution hi is constructed from the (n - 1 )-soliton solution h,_ 1 by
h;=B$“‘h,_,
,
(12)
where the B&klund transformation B$“’ =diag(l,
B$“) is given by
1, .... 1, -I/&),
(13)
with N- 1 times 1 and (14)
6 =P,lP * In ( 14) ,i& is the pole trajectory given by Pn = w, -rf
[ ( w, -Z)Z+p*]
l’* .
N2 =diag(pq’, bq2, .... j?qN-‘,p-l-Q)
,
(16)
where p=det hk,
(17)
N-l
(18)
Qziz,qi*
Then it is clear that the determinant of h iph) is 1 and hiph) also satisfies eq. (3b). Now getting back to our case we obtain the n-soliton solution by setting N= 2, hiph) =diag{ [ ( - 1)“6, a2 ... C,,]” , [ (- 1 )“a, C*... &] -“} .
(19)
As for the solution of (3a), i.e., the four dimensional part, we follow ref. [ 51 which leads to the Kerr-NUT solution. In order to express this by the customary variables we introduce I and 8 by I’* sin 19,
(20a)
z-2, = (r-m)cose,
(20b)
where o and zI are real constants and express the position of the pole trajectory of the four dimensional part as pr =zr +a-z+
[ (z1 +a-zy+p*]
l’* 9
(21)
p*=z,
[ (z1 -a-z)*+p*]
I’*.
(22)
-o-z+
Now we shall determine the physically interesting part h of the metric by referring to the previously obtained formula ( 19 ) with ( 14) and ( 15 ) . We are left with the choice of the plus and minus sign in ( 15 ), the choice of the parameter W,, in ( 15) and the number of solitons. We limit ourselves to the real pole trajectory. First we consider the one-soliton solution given by hCph)=diag[ ( -81)q,
(15)
Since h; does not satisfy the constraint det h= 1, we transform h; into hiph) by N2 defined by #2
p= [ (r-m)*-a*]
II*In the cosmological model this type of renormalization has been
( -t3,)-q]
,
(23)
where d , =pI/p and ji, should be a real pole trajectory, p, = w, -z+
w, -z)*+p*]
[(
“*.
(24)
This shows that b, depends not only on r but also on 8 when p and z are replaced by (20), and that the one-soliton solution does not satisfy the asymptotic flatness. Next we consider the two-soliton solution h(ph)=diag[ (6,t?2)q, (a,C2)-q] , where Cj=fiJp
(25)
(i= 1,2) and fii is to be written as
p1 =z, +z-zf
[(Z, +z--z)*+p*]
I’*,
(26)
p* =z, -z-z*
[(Z, -z-z)*+p*]
l’*.
(27)
Here Z1 and z are different from z, and o in (2 1) and (22). However we first analyse (25 ) by assuming that they are the same and then consider the case that they differ. When we choose the plus sign of both fi, and ,C2,8, z2 in (25) becomes tY,~2=tan2(0/2),
(28)
which is incompatible with the asymptotic flatness. When we choose the minus sign of both fi, and ,i12,6, cT2is given by a,c2
=c0t2(e/2)
,
(29)
which is also inappropriate. When we choose the plus sign of fi, and minus sign of ,C2or the alternative assignment, d, C2 is expressed as CT1 a* = s
The choice of plus signs in front of the square root in (2 1) and (22) leads to the Kerr-NUT solution in the standard expression when the extra dimensions do not exist. used
10 April 1989
PHYSICS LETTERS A
Volume 136, number 6
w
r-m-lo r-m-a’
(30)
Then, by choosing the constant q in ( 25 ) to be even, h cph) reads
h(Ph)=diag[ (ST,
(so)-“]
,
(31)
in ref. [ 7 1.
271
Volume
136, number
6
PHYSICS
which goes to the flat metric as r--tco. Since (31) seems to be the physically interesting candidate for the metric we need to study f in ( 1) which is determined by integrating (4a) and (4b). For this purpose we factorize f as
f(P,~)=wT~M(P,~),
(32)
and require that F satisfy (lnF),P=-
d
+ ~Tr[(pg,g-l)‘-(pg,g-l)‘l,
(33a)
(lnF),,= 6 Tr[ @g,g-‘1 Mw-‘)I
,
(33b)
LETTERS
A
10 April 1989
A=A+A_ = (r-m+o) = (r-m)2-02
r=
(r-m)‘-a2
(r-m-a)
)
(38) (39)
c0s28,
with the constraint 02=m2-a2+b2.
(40)
As we have mentioned, q in (36) is an even number. The minimum nontrivial even number 2 corresponds to the degenerate four-soliton solution according to the renormalization by Belinskii and Sakharov [ 5 1. Setting b= 0, we obtain at the spatial infinity -d.r2=-(l-2m/r)dr2+(l-2m/r)-‘dr2
and Q satisfy
+r2(d62+sin28d@Z) +2
(ln QJp= 6 Tr[(ph,h-‘)2-(ph,=h-‘)21,
+ (1+2qa/r)
2am sin 28 dr d@ r
(dx5)2+ (1-2qalr)
(dx6)2. (41)
(34a) UnQ>,,= $Tr[(ph,&‘)
W,A-*)I
.
(34b)
Substituting (3 1) into the r.h.s. of (34) this is integrated to give
Q=L
P2(h
-P2)2
z+P')
q2
(P:+P2)
(r-m+a)
> q2
(r-m-a)
= ( (r-m+acos8)
(r-m-acosf3)
*
(35)
>
Together with the already known results in the four dimensional two-soliton solution we obtain the solution to the Einstein equations (3) and (4), -d.r2=dq2Pq20(d-’ -o-‘{
(d-a2
dr2+d6J2)
sin28) dr2
-[4dbcos6-4asin26(mr+b2)]
drdo
+[d(asin26+2bco~e)2 -sin26 (r2+b2+a2)2] + (d+/d_ )“(d?)
do2}
2+ (A-/A+ )“(cW) 2,
(36)
where ~=r2+ 272
(b-a
cos e)2,
(37)
The four dimensional part is the same as that of the Kerr solution. The effect of the extra dimensions is of the order of l/r’. We have obtained the solution consisting of the two-soliton solution in the external dimensions and the two-soliton (or degenerate four-soliton) solution in the internal dimensions. We point out some special cases of the solution (36) when the NUT parameter b = 0. (i) 8= 0 and a# 0; It takes infinite time for the light emitted outside of the surface r= m+ J_ a to arrive there from the direction &O. (ii) 8=x/2 and a=O; This is the case with zero angular momentum. It takes infinite time for the light emitted outside of not r= 2m but [= m to arrive there from the direction &n/2. This might show that the hypersurface defined by A=0 is not the event horizon, at least when a=O. (iii) m=a; This condition leads to A=T and A+ =A_. Then the metric (36) is the special case of the Kerr solution that the inner and outer event horizons coincide and the solitons do not exist in the external dimensions. In order to see the detailed structure of the metric we need to investigate the curvature invariants. We leave it together with the interesting cases such as those without (8 ) to the forthcoming paper [ 8 3. Note
Volume 136, number 6
PHYSICS LETTERS A
that the solution (36) was derived by assuming that the integration constants appearing in the pole trajectories ,E, in the internal dimensions are the same as those of ,u~in the external dimensions, which is not usually the case. By changing E and 2, from o and z, the positions of the poles or the zeros become different from those in the metric (36). This probably enables us to confine singularities, even if they exist in addition to the r=O singularity, to the event horizon. In obtaining the soliton solutions in the internal dimensions we have restricted ourselves to real pole trajectories. But even if we allow for complex trajectories we have not found new solutions up to foursoliton solutions. Note added. After finishing this Letter we got to know a Ph.D. thesis by Karasu [ 91, where the higher dimensional Kaluza-Klein type theory with axial symmetry is considered as the generalization of the paper by Dereli et al. [ 6 1. The inverse scattering technique is also applied there, which overlaps some of the observations here. But the explicit solutions are not shown there.
lOApril 1989
T.K. acknowledges Iwanami Ftijukai for financial support.
References [ 1 ] M.B. Green, J.H. Schwarz and E. Witten, Superstring theory (Cambridge Univ. Press, Cambridge, 1986), and references therein. [2] J.D. Bekenstein, Phys. Rev. D 5 (1972) 1239; C. Teitelboim, Phys. Rev. D 5 (1972) 2941; J.B. Hartle, Phys. Rev. D 3 (1971) 2938; S.W. Hawking, Commun. Math. Phys. 25 (1972) 167. [3]A.SalamandJ.Strathdee,Amt.Phys. (NY) 141 (1987) 316. [4] T. Koikawa and K. Shiraishi, Prog. Theor. Phys. 80 (1988) 108. [5] V.A. Belinskii and V.E. Sakharov, Zh. Eksp. Teor. Fiz 75 (1978) 1955; 77 (1979) 3. [6] V.A. Belinskii and R. Riftini, Phys. Lett. B 89 (1980) 195; D.J. Gross and M.J. Perry, Nucl. Phys. B 226 ( 1983) 29; R.D. So&in, Phys. Rev. Lett. 51 ( 1983) 87; T. Dereli, A. Et-isand A. Karasu, Nuovo Cimento B 93 ( 1986) 102. [7] M.C. Diaz, R.J. Gleiser and J.A. Pullin, J. Math. Phys. 29 (1988) 169. [ 81 T. Azuma, M. Endo and T. Koikawa, in preparation. [ 9 ] A. Karasu, Stationary, axi-symmetric Einstein field equations in higher dimension: harmonic mappings, uniqueness of solutions and inverse scattering transform technique, Ph.D. thesis, Middle East Technical University, Ankara (1988).
273
10 April 1989
PHYSICS LETTERS A
AXIALLY SYMMETRIC SOLUTION OF THE EINSTEIN WITH SOLITONS IN EXTRA DIMENSIONS
EQUATION
T. AZUMA”, M. ENDO” and T. KOIKAWA a,b a Faculty of Liberal Arts, Dokkyo University, So/w, Saitama 340, Japan b Department ofphysics, Tokyo Metropolitan University, Setagaya, Tokyo 158, Japan
Received 26 September 1988; accepted for publication 7 February 1989 Communicated by J.P. Vigier
We give the solution to the higher dimensional Einstein equation with axial symmetry by using the inverse scattering method. The solution consists of two solitons in the external dimensions and any number of solitons in the internal dimensions.
In recent years higher dimensional theories such as the unification of gravitation and gauge theories have been principal concerns of many physicists [ I]. If the realm of nature actually lies in more than four dimensions, this has great implications in the study of the early universe and black holes. Taking account of the compactification of internal space many studies have been done in the search for a black hole solution with event horizon in four dimensions. In these studies the no-hair theorem [ 2 ] that black holes with a mass, an electric charge and an angular momentum cannot be endowed with scalar charge is important, because the action of higher dimensional gravity is known [3] to be equivalent to that of the four dimensional gravity with a massless scalar field after the Weyl resealing. Lately one of the present authors and Shiraishi studied the stationery and axisymmetric higher dimensional Einstein equation [ 41 of which the metric is -d.r2=f(p,
z) (dp2+dz2)+g,,(p,
+h(p, z) i
consider either a nonstationary case or abandon the massless free scalar field. In this short note we look for a solution of the higher dimensional Einstein equation with axial symmetry, which might be a candidate for a black hole solution. For this purpose we consider the case that not all the metric coefficients of the extra dimensions are necessarily the same as above. This enables us to discuss the construction of the soliton solution in the extra dimensions. In finding the explicit solution we use the inverse scattering method [ 5 ] which has been applied to the black hole solution and gravitational solitons. We stress here that the method to obtain the soliton solutions is useful in determining the metric of the extra dimensions as well as the four dimensional part, which was not noted in the previous work. The line interval is given by #’ -dr2=f(dp2+dz2)+g,bdx”dxb+h,Bdx*dxa =f(dp2+dz2)+GijdXidXi,
z) dx” dxb
(dxQ2.
Here h(p, z) is the metric coefficient of the iV-dimensional internal space and In h plays the role of scalar field. In ref. [ 41 the solution has a naked singularity and this might be understandable by the nohair theorem. In order to evade the theorem we must
(1)
wherefl grrb has and G, are functions of p and z. (G,) *’ In the five dimensional Raluza-Klein type theory, in which
03759601/89/$ 03.50 0 Elsevier Science Publishers B.V. ( North-Holland Physics Publishing Division )
non-vanishing gauge potentials exist, some work has already been published [ 61. Since we have the field theory limit of the superstring theory in mind, we do not have those potentials corresponding to the internal-external dimensional part of the metric.
269
Volume 136, number
6
PHYSICS
is the (2 +N) x (2 +N) block diagonal matrix given by
(gib)(ho) > ,
G= (Gij)=
(2)
0
where g& is a 2 X 2 matrix and haB an NX N matrix, Then the vacuum Einstein equations read (pG,G-‘).+(pG,,G-‘),,=O, (Infig=-
(3)
i + iTr(U2-V’),
(da)
(ln_O,z= &WW,
(@I
where U and I/ are (2 + N) x (2 +N) matrices defined by U=pG,G-’
,
V=pG,,G-’
.
(5)
LETTERS
A
10 April 1989
seed metric, we can construct the soliton solution satisfying (3a) and (3b). However they do not necessarily have the same determinants as the seed metric and so renormalization is necessary. In the present case we should require that the renormalized metric satisfies the constraint ( 6’ ). This means that the renormalized metric gCph’or hcph’ and the seed metric go or ho do not necessarily have the same determinant. Of the various possibilities we discuss the case that the renormalized or physical metric gCph) and h(ph’ satisfy det g (ph)= _p2 2
(gal
det h(ph)= 1 .
(8b)
The above procedure of obtaining a new solution by the soliton technique is summarized as follows: ‘ph’=N,g’=N,B,gO,
go+g’=B,go+g
(gal
We have fixed the determinant of the matrix G without loss of generality as
ho+h’=Bzho+h(ph)=N2h’=N2B2h0,
det G= -p2,
where B1 and B2 are the 2x2 matrices given by
(6)
(9b)
which can be assured to be satisfied by ( 3 ) . By noting that G is given by (2)) the first Einstein equation (3) can be decomposed into the same type of equation: (P&C’).+
(P&g-‘),,=0
(ph,h-‘),-(ph,,h-‘),,=O,
9
(3a) (3b)
where g and h are 2 x 2 and NX N matrices of which the entries are g,b and h,, respectively. The constraint eq. (6 ) also reads det gxdet h= -p2.
(6’ )
Now we are going to find the solutions to the Ein-
stein equations (3) and (4) under some assumptions on h in ( 1). We assume that h is diagonal coiresponding to the torus compactification: h,, = h,&, .
(7)
The case that all h,‘s are the same and depend on dpw has been discussed in ref. [ 41. As we have mentioned we are interested in the case that, at least, two of the ha’s are different. Though the following procedure is quite general in discussing the arbitrary diagonal h with distinctive entries, we assume that h = diag( h,, h2) for definiteness. Starting with the 270
a, j?, c=5,6.
(lob)
For the detailed definitions of the notations in ( 1Oa) the readers are referred to ref. [ 51. Because of the parallelism in (3a) and (3b) we can construct the new solution h’ by ( 1Ob) which has definitions similar to ( lOa). Since we have assumed that h is diagonal as in (7), we set fi $” = 0 to preserve diagonality. The second step in (9) is the normalization procedure to recover the constraints given by (8 ). In the diagonal case we do not need to follow the prescription of ref. [ 5 1. Since this point is different from the standard procedure we give an explanation by returning to the general case. We start with the flat seed metric ho=diag(l,
l,..., 1))
(11)
with N times 1. The n-soliton solution hi is constructed from the (n - 1 )-soliton solution h,_ 1 by
h;=B$“‘h,_,
,
(12)
where the B&klund transformation B$“’ =diag(l,
B$“) is given by
1, .... 1, -I/&),
(13)
with N- 1 times 1 and (14)
6 =P,lP * In ( 14) ,i& is the pole trajectory given by Pn = w, -rf
[ ( w, -Z)Z+p*]
l’* .
N2 =diag(pq’, bq2, .... j?qN-‘,p-l-Q)
,
(16)
where p=det hk,
(17)
N-l
(18)
Qziz,qi*
Then it is clear that the determinant of h iph) is 1 and hiph) also satisfies eq. (3b). Now getting back to our case we obtain the n-soliton solution by setting N= 2, hiph) =diag{ [ ( - 1)“6, a2 ... C,,]” , [ (- 1 )“a, C*... &] -“} .
(19)
As for the solution of (3a), i.e., the four dimensional part, we follow ref. [ 51 which leads to the Kerr-NUT solution. In order to express this by the customary variables we introduce I and 8 by I’* sin 19,
(20a)
z-2, = (r-m)cose,
(20b)
where o and zI are real constants and express the position of the pole trajectory of the four dimensional part as pr =zr +a-z+
[ (z1 +a-zy+p*]
l’* 9
(21)
p*=z,
[ (z1 -a-z)*+p*]
I’*.
(22)
-o-z+
Now we shall determine the physically interesting part h of the metric by referring to the previously obtained formula ( 19 ) with ( 14) and ( 15 ) . We are left with the choice of the plus and minus sign in ( 15 ), the choice of the parameter W,, in ( 15) and the number of solitons. We limit ourselves to the real pole trajectory. First we consider the one-soliton solution given by hCph)=diag[ ( -81)q,
(15)
Since h; does not satisfy the constraint det h= 1, we transform h; into hiph) by N2 defined by #2
p= [ (r-m)*-a*]
II*In the cosmological model this type of renormalization has been
( -t3,)-q]
,
(23)
where d , =pI/p and ji, should be a real pole trajectory, p, = w, -z+
w, -z)*+p*]
[(
“*.
(24)
This shows that b, depends not only on r but also on 8 when p and z are replaced by (20), and that the one-soliton solution does not satisfy the asymptotic flatness. Next we consider the two-soliton solution h(ph)=diag[ (6,t?2)q, (a,C2)-q] , where Cj=fiJp
(25)
(i= 1,2) and fii is to be written as
p1 =z, +z-zf
[(Z, +z--z)*+p*]
I’*,
(26)
p* =z, -z-z*
[(Z, -z-z)*+p*]
l’*.
(27)
Here Z1 and z are different from z, and o in (2 1) and (22). However we first analyse (25 ) by assuming that they are the same and then consider the case that they differ. When we choose the plus sign of both fi, and ,C2,8, z2 in (25) becomes tY,~2=tan2(0/2),
(28)
which is incompatible with the asymptotic flatness. When we choose the minus sign of both fi, and ,i12,6, cT2is given by a,c2
=c0t2(e/2)
,
(29)
which is also inappropriate. When we choose the plus sign of fi, and minus sign of ,C2or the alternative assignment, d, C2 is expressed as CT1 a* = s
The choice of plus signs in front of the square root in (2 1) and (22) leads to the Kerr-NUT solution in the standard expression when the extra dimensions do not exist. used
10 April 1989
PHYSICS LETTERS A
Volume 136, number 6
w
r-m-lo r-m-a’
(30)
Then, by choosing the constant q in ( 25 ) to be even, h cph) reads
h(Ph)=diag[ (ST,
(so)-“]
,
(31)
in ref. [ 7 1.
271
Volume
136, number
6
PHYSICS
which goes to the flat metric as r--tco. Since (31) seems to be the physically interesting candidate for the metric we need to study f in ( 1) which is determined by integrating (4a) and (4b). For this purpose we factorize f as
f(P,~)=wT~M(P,~),
(32)
and require that F satisfy (lnF),P=-
d
+ ~Tr[(pg,g-l)‘-(pg,g-l)‘l,
(33a)
(lnF),,= 6 Tr[ @g,g-‘1 Mw-‘)I
,
(33b)
LETTERS
A
10 April 1989
A=A+A_ = (r-m+o) = (r-m)2-02
r=
(r-m)‘-a2
(r-m-a)
)
(38) (39)
c0s28,
with the constraint 02=m2-a2+b2.
(40)
As we have mentioned, q in (36) is an even number. The minimum nontrivial even number 2 corresponds to the degenerate four-soliton solution according to the renormalization by Belinskii and Sakharov [ 5 1. Setting b= 0, we obtain at the spatial infinity -d.r2=-(l-2m/r)dr2+(l-2m/r)-‘dr2
and Q satisfy
+r2(d62+sin28d@Z) +2
(ln QJp= 6 Tr[(ph,h-‘)2-(ph,=h-‘)21,
+ (1+2qa/r)
2am sin 28 dr d@ r
(dx5)2+ (1-2qalr)
(dx6)2. (41)
(34a) UnQ>,,= $Tr[(ph,&‘)
W,A-*)I
.
(34b)
Substituting (3 1) into the r.h.s. of (34) this is integrated to give
Q=L
P2(h
-P2)2
z+P')
q2
(P:+P2)
(r-m+a)
> q2
(r-m-a)
= ( (r-m+acos8)
(r-m-acosf3)
*
(35)
>
Together with the already known results in the four dimensional two-soliton solution we obtain the solution to the Einstein equations (3) and (4), -d.r2=dq2Pq20(d-’ -o-‘{
(d-a2
dr2+d6J2)
sin28) dr2
-[4dbcos6-4asin26(mr+b2)]
drdo
+[d(asin26+2bco~e)2 -sin26 (r2+b2+a2)2] + (d+/d_ )“(d?)
do2}
2+ (A-/A+ )“(cW) 2,
(36)
where ~=r2+ 272
(b-a
cos e)2,
(37)
The four dimensional part is the same as that of the Kerr solution. The effect of the extra dimensions is of the order of l/r’. We have obtained the solution consisting of the two-soliton solution in the external dimensions and the two-soliton (or degenerate four-soliton) solution in the internal dimensions. We point out some special cases of the solution (36) when the NUT parameter b = 0. (i) 8= 0 and a# 0; It takes infinite time for the light emitted outside of the surface r= m+ J_ a to arrive there from the direction &O. (ii) 8=x/2 and a=O; This is the case with zero angular momentum. It takes infinite time for the light emitted outside of not r= 2m but [= m to arrive there from the direction &n/2. This might show that the hypersurface defined by A=0 is not the event horizon, at least when a=O. (iii) m=a; This condition leads to A=T and A+ =A_. Then the metric (36) is the special case of the Kerr solution that the inner and outer event horizons coincide and the solitons do not exist in the external dimensions. In order to see the detailed structure of the metric we need to investigate the curvature invariants. We leave it together with the interesting cases such as those without (8 ) to the forthcoming paper [ 8 3. Note
Volume 136, number 6
PHYSICS LETTERS A
that the solution (36) was derived by assuming that the integration constants appearing in the pole trajectories ,E, in the internal dimensions are the same as those of ,u~in the external dimensions, which is not usually the case. By changing E and 2, from o and z, the positions of the poles or the zeros become different from those in the metric (36). This probably enables us to confine singularities, even if they exist in addition to the r=O singularity, to the event horizon. In obtaining the soliton solutions in the internal dimensions we have restricted ourselves to real pole trajectories. But even if we allow for complex trajectories we have not found new solutions up to foursoliton solutions. Note added. After finishing this Letter we got to know a Ph.D. thesis by Karasu [ 91, where the higher dimensional Kaluza-Klein type theory with axial symmetry is considered as the generalization of the paper by Dereli et al. [ 6 1. The inverse scattering technique is also applied there, which overlaps some of the observations here. But the explicit solutions are not shown there.
lOApril 1989
T.K. acknowledges Iwanami Ftijukai for financial support.
References [ 1 ] M.B. Green, J.H. Schwarz and E. Witten, Superstring theory (Cambridge Univ. Press, Cambridge, 1986), and references therein. [2] J.D. Bekenstein, Phys. Rev. D 5 (1972) 1239; C. Teitelboim, Phys. Rev. D 5 (1972) 2941; J.B. Hartle, Phys. Rev. D 3 (1971) 2938; S.W. Hawking, Commun. Math. Phys. 25 (1972) 167. [3]A.SalamandJ.Strathdee,Amt.Phys. (NY) 141 (1987) 316. [4] T. Koikawa and K. Shiraishi, Prog. Theor. Phys. 80 (1988) 108. [5] V.A. Belinskii and V.E. Sakharov, Zh. Eksp. Teor. Fiz 75 (1978) 1955; 77 (1979) 3. [6] V.A. Belinskii and R. Riftini, Phys. Lett. B 89 (1980) 195; D.J. Gross and M.J. Perry, Nucl. Phys. B 226 ( 1983) 29; R.D. So&in, Phys. Rev. Lett. 51 ( 1983) 87; T. Dereli, A. Et-isand A. Karasu, Nuovo Cimento B 93 ( 1986) 102. [7] M.C. Diaz, R.J. Gleiser and J.A. Pullin, J. Math. Phys. 29 (1988) 169. [ 81 T. Azuma, M. Endo and T. Koikawa, in preparation. [ 9 ] A. Karasu, Stationary, axi-symmetric Einstein field equations in higher dimension: harmonic mappings, uniqueness of solutions and inverse scattering transform technique, Ph.D. thesis, Middle East Technical University, Ankara (1988).
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