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Bäcklund transformations of the Einstein equations
Physica 114A (1982) 10&104 North-Holland
BACKLUND
Publishing Co.
TRANSFORMATIONS
OF THE EINSTEIN
EQUATIONS
Minoru OMOTE* Theoretical Physics Institute, University of Alberta, Edmonton, Alberta, Canada T6G 211 and Miki WADATI Institute of Physics,
Collego of General Education, Japan
University of Tokyo, Komaba,
Tokyo 153,
From the viewpoint of the Bkklund transformation a systematic approach to the transformation theories for the Einstein equation and for the Einstein-Maxwell equations is presented.
1. Introduction
B&zklund transformations are known to play an important r&e in obtaining the exact solutions of nonlinear evolution equations. These Bgcklund transformations have pseudopotentials which satisfy equations with Riccati-type nonlinearity. Fiom these equations we can obtain a pair of linear first-order equations which are eigenvalue equations of the inverse scattering method. In this paper we shall derive Blcklund transformations of both the Einstein and the Einstein-Maxwell equations using a systematic method.
2. B&klund
transformations
of the Einstein
equations
The axially-symmetric stationary Einstein equation (The Ernst equation) is written in terms of the Ernst potential E ‘) as ,3,&E= -+,E+a,E)++,E&E, 4P
where 2p = x1 + x2, xl = xl and 2T = E + E*. The B5cklund transformation is a pair of first-order differential equations which relates one solution E to another solution E’ and has the following form? L; = CY,L,+ (Yz, Li = p,L2+ p2
(2)
where Li = aiE, LI = aiE’. In (2) ai and pi (i = 1,2) are functions of E, E*, E’, * Permanent
address: Institute of Physics, University of Tsukuba, Sakura, Ibaraki 305, Japan.
0378-4371/82/0ooMH>00/$02.75
@ 1982 North-Holland
BACKLUND TRANSFORMATIONS
OF THE EINSTEIN EQUATIONS
101
E’*, x1 and x2, and their functional forms can be determined from the requirements: (i) E’ satisfies the integrability condition al&E = 82d1E’, (ii) E’ is a solution of the Ernst equation. These requirements give coupled partial-differential equations of the ai and pi (i = 1,2). By solving such equations we found four kinds of transformations. i) cxl=/3r=~, Integration E’=
T’
cx2=/32=0.
(3)
of (2) with (3) yields cE+id,
(4)
where c and d are real constants. T’
(5)
where 4, (which is such that $r+T = 1) satisfies (6) From this transformation E,=
we may have
E+ic
(7)
iyE + d’ where c, d and y are real constants. formation3). iii) cxl= J!LG C#Q, p, = $ $;I,
This is equivalent
a2=$(1-42),
to the Ehlers trans-
Bz=$(l-&I),
(8)
where (Pz (= 41) satisfies
(Q#)*=-~(m,L,+Lf)+++
I),
(q242=-42-1 ~(L2+42LT)++#+
1).
This transformation iv)
generates
~,=_T’+3(1+t+3), T 43+t
(9)
the rotation of the Killing vectors4). p,=_’
l+t+3 T 43c+3+
x(1+~43), p2 = &‘@3
I)’
(10)
102
MINORU
OMOTE
AND MIKI WADATI
where c2 = (il - x2)/(il + x,) and I is an arbitrary real constant. The pseudopotential (b3 satisfies a pair of Riccati-type equations
(11) from which we can derive the Lax pair of the inverse scattering method. This transformation is called the Harrison transformation5) and plays an important role because it includes three other kinds of transformation as special case@).
3. Bicklund
transformations
of the Einstein-Maxwell
The axially-symmetric stationary Einstein-Maxwell terms of two complex potentials E and @ as ~?,a,@ = -$(a,@+ a,a2E = --+,E
a*~)+~{(a,E+2~*a,~)a*~
equations
equations
are given in
+(a,E+2@*&@)&@},
+ a2E)+~{(a,E+2@*~,@)a2E+(&E+2@*&@)a,E}, (12)
where Q = E + E* + 2@@*. Backlund transformations the following form’) M; = a,M, + a2N, +
a3,
of (12) are expressed
M;= b,Mz+ b,Nz+ b,
in
(13)
and N;=c,M,+c2N,+c3,
N;=d,M2+d2N2+d3,
(14)
where Mi, Ni, MI and Ni (i = 1,2) are defined in terms of two sets of solutions (@, E) and (a’, E’) by MI = aim’,
N{ = aiE’+ 2~‘*ai~‘,
Mi = ai@, Ni = 8iE + 2@*a,@.
By using the same approach as in the previous functional forms of ai, bi, ci and d, (i = 1,2,3).
a2= a3= b2=b3=c,=d,=cJ=ds=0.
section
(15)
we can find the
(16)
BACKLUND
TRANSFORMATIONS
103
OF THE EINSTEIN EQUATIONS
This gives the transformation @‘=c’%+d,
(17)
E’=cE-2c”2d*@-dd*+iy,
where c and y are real constants
and d is an arbitrary * e:-‘},
ii) al= br=${81+2(1-02)~ cl= d,=2$1-e2)@*W-‘,
c2 =
constant.
a2= b2=82-
d2 = -Q’ e2,
Q
Q ’
a3 = b3 = c3 = d3 = 0,
(18) where 8, and 02 satisfy alet = el(l - 0,) M1, @
a,e2 =
e2- 1 Q
@*
I
e2N2+NT+2(1-e2)*Ml
I @*
e2N2+NT+2(1-e2)yjp42
a,e,=y(
This transformation transformations
I
can be integrated
(19)
,
.
and is found to generate the following
cc*
CC@ - Y)
@‘=
a2e1 = el(l - 0,) M2 @ ’
(20)
E’=E+2y*@_d+16y
E+2y*@-d’
where d = yy* + ie. In (20) c, y, 6 and E are constants. iii)
al
=
(-EL)“’($1’2e;/2,
a2= b2= a3= b3=0,
b,
=
(s)“’ (SJ1’2 e:12, d2 = 5
~~=~e,,
Q
d3=$(ei1-I),
c3=-$(e3-1),
e;‘,
(21)
cl=dl=O,
where e3 satisfies ale3 = y(e,N,+Nr)+-$ej-
l),
a2e3 = y(N2+
1).
e3Nt)+$-(e:-
(22)
This transformation generates a rotation of the Killing vector and an electromagnetic duality rotation (K/I<*).
104
MINORU
OMOTE
AND MIKI WADATI
4. Conclusion
By using the systematic approach we found four kinds of Backlund transformations of the Einstein equation and found three kinds of transformations of the Einstein-Maxwell equation. It can be easily seen that the transformations i), ii) and iii) of the Einstein-Maxwell equations are generalizations of those of the Einstein equation, respectively. The generalization of iv) of the Einstein equation, however, cannot yet be found. As we have noted at the end of section 2 the fourth transformation (The Harrison transformation) has a special significance. It is therefore an important problem to find the generalization of the Harrison transformation.
Acknowledgements
One of the authors (M.O.) wishes to thank Prof. Y. Takahashi and the members of the Theoretical Physics Institute of the University of Alberta for their hospitality. This work was supported by the Natural Sciences and Engineering Research Council of Canada.
References 1) 2) 3) 4) 5) 6) 7)
F.J. Ernst, Phys. Rev. 167 (1968) 1175. M. Omote and M. Wadati, J. Math. Phys. 22 (1981) 961. W. Kinnersley, J. Math. Phys. 18 (1970) 1529. C.M. Cosgrove, private communication. B.K. Harrison, Phys. Rev. L&t. 41 (1978) 1197. M. Omote and M. wadati, Progr. Theor. Phys. 65 (1981) 1621. M. Omote and M. Wadati, Phys. Lett. 83A (1981) 411.
BACKLUND
Publishing Co.
TRANSFORMATIONS
OF THE EINSTEIN
EQUATIONS
Minoru OMOTE* Theoretical Physics Institute, University of Alberta, Edmonton, Alberta, Canada T6G 211 and Miki WADATI Institute of Physics,
Collego of General Education, Japan
University of Tokyo, Komaba,
Tokyo 153,
From the viewpoint of the Bkklund transformation a systematic approach to the transformation theories for the Einstein equation and for the Einstein-Maxwell equations is presented.
1. Introduction
B&zklund transformations are known to play an important r&e in obtaining the exact solutions of nonlinear evolution equations. These Bgcklund transformations have pseudopotentials which satisfy equations with Riccati-type nonlinearity. Fiom these equations we can obtain a pair of linear first-order equations which are eigenvalue equations of the inverse scattering method. In this paper we shall derive Blcklund transformations of both the Einstein and the Einstein-Maxwell equations using a systematic method.
2. B&klund
transformations
of the Einstein
equations
The axially-symmetric stationary Einstein equation (The Ernst equation) is written in terms of the Ernst potential E ‘) as ,3,&E= -+,E+a,E)++,E&E, 4P
where 2p = x1 + x2, xl = xl and 2T = E + E*. The B5cklund transformation is a pair of first-order differential equations which relates one solution E to another solution E’ and has the following form? L; = CY,L,+ (Yz, Li = p,L2+ p2
(2)
where Li = aiE, LI = aiE’. In (2) ai and pi (i = 1,2) are functions of E, E*, E’, * Permanent
address: Institute of Physics, University of Tsukuba, Sakura, Ibaraki 305, Japan.
0378-4371/82/0ooMH>00/$02.75
@ 1982 North-Holland
BACKLUND TRANSFORMATIONS
OF THE EINSTEIN EQUATIONS
101
E’*, x1 and x2, and their functional forms can be determined from the requirements: (i) E’ satisfies the integrability condition al&E = 82d1E’, (ii) E’ is a solution of the Ernst equation. These requirements give coupled partial-differential equations of the ai and pi (i = 1,2). By solving such equations we found four kinds of transformations. i) cxl=/3r=~, Integration E’=
T’
cx2=/32=0.
(3)
of (2) with (3) yields cE+id,
(4)
where c and d are real constants. T’
(5)
where 4, (which is such that $r+T = 1) satisfies (6) From this transformation E,=
we may have
E+ic
(7)
iyE + d’ where c, d and y are real constants. formation3). iii) cxl= J!LG C#Q, p, = $ $;I,
This is equivalent
a2=$(1-42),
to the Ehlers trans-
Bz=$(l-&I),
(8)
where (Pz (= 41) satisfies
(Q#)*=-~(m,L,+Lf)+++
I),
(q242=-42-1 ~(L2+42LT)++#+
1).
This transformation iv)
generates
~,=_T’+3(1+t+3), T 43+t
(9)
the rotation of the Killing vectors4). p,=_’
l+t+3 T 43c+3+
x(1+~43), p2 = &‘@3
I)’
(10)
102
MINORU
OMOTE
AND MIKI WADATI
where c2 = (il - x2)/(il + x,) and I is an arbitrary real constant. The pseudopotential (b3 satisfies a pair of Riccati-type equations
(11) from which we can derive the Lax pair of the inverse scattering method. This transformation is called the Harrison transformation5) and plays an important role because it includes three other kinds of transformation as special case@).
3. Bicklund
transformations
of the Einstein-Maxwell
The axially-symmetric stationary Einstein-Maxwell terms of two complex potentials E and @ as ~?,a,@ = -$(a,@+ a,a2E = --+,E
a*~)+~{(a,E+2~*a,~)a*~
equations
equations
are given in
+(a,E+2@*&@)&@},
+ a2E)+~{(a,E+2@*~,@)a2E+(&E+2@*&@)a,E}, (12)
where Q = E + E* + 2@@*. Backlund transformations the following form’) M; = a,M, + a2N, +
a3,
of (12) are expressed
M;= b,Mz+ b,Nz+ b,
in
(13)
and N;=c,M,+c2N,+c3,
N;=d,M2+d2N2+d3,
(14)
where Mi, Ni, MI and Ni (i = 1,2) are defined in terms of two sets of solutions (@, E) and (a’, E’) by MI = aim’,
N{ = aiE’+ 2~‘*ai~‘,
Mi = ai@, Ni = 8iE + 2@*a,@.
By using the same approach as in the previous functional forms of ai, bi, ci and d, (i = 1,2,3).
a2= a3= b2=b3=c,=d,=cJ=ds=0.
section
(15)
we can find the
(16)
BACKLUND
TRANSFORMATIONS
103
OF THE EINSTEIN EQUATIONS
This gives the transformation @‘=c’%+d,
(17)
E’=cE-2c”2d*@-dd*+iy,
where c and y are real constants
and d is an arbitrary * e:-‘},
ii) al= br=${81+2(1-02)~ cl= d,=2$1-e2)@*W-‘,
c2 =
constant.
a2= b2=82-
d2 = -Q’ e2,
Q
Q ’
a3 = b3 = c3 = d3 = 0,
(18) where 8, and 02 satisfy alet = el(l - 0,) M1, @
a,e2 =
e2- 1 Q
@*
I
e2N2+NT+2(1-e2)*Ml
I @*
e2N2+NT+2(1-e2)yjp42
a,e,=y(
This transformation transformations
I
can be integrated
(19)
,
.
and is found to generate the following
cc*
CC@ - Y)
@‘=
a2e1 = el(l - 0,) M2 @ ’
(20)
E’=E+2y*@_d+16y
E+2y*@-d’
where d = yy* + ie. In (20) c, y, 6 and E are constants. iii)
al
=
(-EL)“’($1’2e;/2,
a2= b2= a3= b3=0,
b,
=
(s)“’ (SJ1’2 e:12, d2 = 5
~~=~e,,
Q
d3=$(ei1-I),
c3=-$(e3-1),
e;‘,
(21)
cl=dl=O,
where e3 satisfies ale3 = y(e,N,+Nr)+-$ej-
l),
a2e3 = y(N2+
1).
e3Nt)+$-(e:-
(22)
This transformation generates a rotation of the Killing vector and an electromagnetic duality rotation (K/I<*).
104
MINORU
OMOTE
AND MIKI WADATI
4. Conclusion
By using the systematic approach we found four kinds of Backlund transformations of the Einstein equation and found three kinds of transformations of the Einstein-Maxwell equation. It can be easily seen that the transformations i), ii) and iii) of the Einstein-Maxwell equations are generalizations of those of the Einstein equation, respectively. The generalization of iv) of the Einstein equation, however, cannot yet be found. As we have noted at the end of section 2 the fourth transformation (The Harrison transformation) has a special significance. It is therefore an important problem to find the generalization of the Harrison transformation.
Acknowledgements
One of the authors (M.O.) wishes to thank Prof. Y. Takahashi and the members of the Theoretical Physics Institute of the University of Alberta for their hospitality. This work was supported by the Natural Sciences and Engineering Research Council of Canada.
References 1) 2) 3) 4) 5) 6) 7)
F.J. Ernst, Phys. Rev. 167 (1968) 1175. M. Omote and M. Wadati, J. Math. Phys. 22 (1981) 961. W. Kinnersley, J. Math. Phys. 18 (1970) 1529. C.M. Cosgrove, private communication. B.K. Harrison, Phys. Rev. L&t. 41 (1978) 1197. M. Omote and M. wadati, Progr. Theor. Phys. 65 (1981) 1621. M. Omote and M. Wadati, Phys. Lett. 83A (1981) 411.