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The complete Virasoro algebra for the stationary and axially symmetric Einstein field equations
Volume 129, number 5,6
PHYSICS LETTERS A
30 May 1988
THE COMPLETE VIRASORO ALGEBRA FOR THE STATIONARY AND AXIALLY SYMMETRIC EINSTEIN FIELD EQUATIONS Wei LI Institute of Modern Physics, Northwest University, Xian, PR China Received 26 May 1987; revised manuscript received 25 February 1988; accepted for publication 25 March 1988 Communicated by J.P. Vigier
In this Letter it is described how to extend the Virasoro symmetry for the stationary, axially symmetric Einstein field equations given by Hou and the present author to the full Virasoro symmetry.
In our previous papers [1,2] we found the existence of the Virasoro symmetry in the solution space of the stationary and axially symmetric Einstein field equations and constructed the infinitesimal transformation of this symmetry to establish the representation of the Virasoro algebra. We also pointed out that our symmetry is different from the Geroch symmetry [3] which corresponds to the Kac—Moody algebra [4], and these two symmetries, therefore, can be used to form a larger infinite-dimensional Lie algebra. With the aid of these symmetries we developed a new systematic generating-solution method for the stationary and axially symmetric Einstein field equations to enlarge the known family of exact solutions. The further investigation showed that the set of transformations are available for positive integers after being expanded in powers ofa parameter of our transformation. We failed to get another set of the transformations for negative integers. The situation is not similar to the case of the Geroch symmetry transformation because for negative integers the latter is concerned with the gauge transformation though it is trivial. So the Virasoro algebra under our construction is only the half one. The purpose of this Letter is to find a way to extend this incomplete Virasoro algebra to the full one. As the similar work [5] has been accomplished for the Geroch symme-
try we are motivated to use the same procedure to settle our problems without difficulty. In this way we are able to construct the full Virasoro algebra. For the convenience of discussion, we prefer to adapt the Hauser—Ernst formalism to the present Letter. We start with the metric under the form (1) th2=g dx’dx’+f((dx3)2+ (dx4)2) where g,~(i, 1= 1, 2) and fare functions of x3 and x4 only. Then one can define a 2 x 2 real and symmetric matrix as g= (g,~)and 2 2 det g= a ( ) For the stationary and axially symmetric Einstein fields, the vacuum field equations are reduced to the Ernst equation —
.
2(fl+ a*) dE= (E+E + )Q dE, / 0 i\
~
—
o)’
(3)
where the matrix Ernst potential E is defined as E=g+ isv, tr (EQ) = 2fl, (4) with the twist potential d a ‘gQ dg
yi
given by (5)
— —
The notations * and + represent the two-dimensional duality operation with properties *~3 = dx4 and *dx4 = dx3, and the hermitian conjugation, respectively. —
Present address: Department of Mathematics and Computer Science, Clarkson University, Potsdam, NY 13676, USA.
0375-9601/88/s 03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
301
Volume 129, number 5.6
PHYSICS LETTERS A
The Ernst equation implies the existence of the Hauser—Ernst linearization equation dF(t) =1(t) QF(t),
(6)
where
30 May 1988
and P(t) is a 2x2 matrix function of x3, x4 and t. For this formulation, the auxiliary conditions of P(t) should be given by detP(t)= [(t—2fl)2+(2a)2]~~2
(7)
F(t)=t[l_2t(fl+a*)]~dE
and tis aparameterandF(t) a 2x2 matrixfunction of x3, x4 and t. F(t) is subject to auxiliary conditions given in refs. [3], if F( t) can be used to determine E. In the previous papers [1,2] we gave the expressions for the infinitesimal transformations of the Ernst potential
(14)
and P(t)~Q(t—(E+E~)Q) P(t)=Q.
(15)
Using a solution of the new linearization equation (12) we can propose a new infinitesimal transformation ~E = s[ sF(s) P ‘(s) + ~]. (16) —
According to eq. (16), it follows that 8E=_P(s)F_l(s)Q,
where
F(t)=~F(t)
__________
(8) and for a solution F(t) of eq. (6)
—sF(s) F ‘(s)] F(t), —
~a=s2
(9)
where F(s) also satisfies the Hauser—Ernst linearization equation with another parameter s. Note that we neglect writing out the infinitesimal constants for the above transformations. We demonstrated that both the Ernst equation and its linearization equation are invariant under these transformations, i.e., they are the symmetry transformations. By setting ~
~ (s_2$)2+(2a)2 /3(s—2J3)—2a~
(17)
and
[tF(t) F-1(t)
—
—
(10)
a
(18)
(s_213)2+ (2a)2’ It is easy to check that the new transformation is also
the symmetry one to the Ernst equation, by making use of tI’e same treatment as we did before. That is, if E is a solution for eq. (3), then so is E+JE, 2(fl+a*) d(~E)+2(~/3+~a*) dE = (E+E~)Q d(~E)+ (~E+~E~ )Q dE. (19) In order to explore the Lie algebra of the new
transformation we need to give of P(t) in eq. (12). We find thatthe transformation
k~O
commutations of a infinite set of operators {ö”~}are evaluated to be [~(k) ~1)] E= (k—I) ~(k+/)E, k, 1?~0. (11)
~P(t)
It is apparent that our transformation indeed forms the half Virasoro algebra. Following Hauser’s work [6], we learn that there is another linearization equation for the Ernst equation. If t in eq. (6) is replaced by 1/1, we obtain
keeps the linearization equation (12) invariant, i.e., d(~P(t)) =~f(t)QP(t)+f(t)Q~P(t). (21)
dP(t) =P(t) QP(t),
(12)
S
t—s [tP(t)P’(l) —sP(s)P ‘(s)] P(t)
(20)
By setting ~
k= I
which is also the linearization equation to the Ernst equation, where
we can express the transformations (16) and (20) as the infinitesimal Riemann—Hilbert transforma-
P(t)= [t_2(/J+a*)]_I
tions, respectively,
302
dE
(13)
Volume 129, number 5,6
PHYSICS LETTERS A
we write a very neat relation
1 (22)
~JY_k+IP(Y)P_1(Y)dyQ
~(k)E_
d an
f
1
~(k)
P(t)=— ~
yk+I
.
y—t P(y)P
d ~ (y) y (t), (23)
where we denote C0, as a circle surrounding y = 0 and y=t in the y-complex plane. It should be emphasized that the subscript k is not permitted to be zero or negative integers because it violates the invanance of eqs. (3) and (12). We then use the method described in ref. [2] to obtain the following commutations [~(k) ~ k 1~l. (24) It means that the present-proposed transformation can constitute the structure of the half Virasoro algebra as the previous one. It is natural for us to consider the relationships between these two different types of the transformations 8E and ~E.First, we need to compute the commutator of ~ and ~1) on E. For this purpose, it is not difficult to set up st [tF(t) F~(t) —st
[Lk,L,] E=(k—I) Lk+/E,
+sP(s)P’(s)+fl F(t),
(25)
6F(t)—~P(t), t—+l/t,s—+l/s
(31)
and ~ oF(t)—~.SF(t), t—1/t,s--*l/s
(32)
respectively. The same correspondence exists between eq. (8) and eq. (16) except an additional constant. Toourknowledgethericherstructureofoursymmetry, like the Geroch symmetry, must be connected with the Riemann—Hilbert problem. Through the way it is possible to exponentiate our infinitesimal symmetry transformations into the finite forms and to establish the representations of the Virasoro group rather than the Virasoro algebra, it will become easier to apprehend the relationships between two different types of our Virasoro symmetries. More important, we are able to create a new generating-
[7] we observe that L~, ~
d(~F(t))=~T’(t)QF(t)+f’(t) Q~F(t).
(26)
In a similar way, [tP( t) P ‘(t) (27)
Therefore, the commutators are evaluated to be E=(k+I)
~(k_!)E,
=
L0,
~t= L_1,
(33)
where we have used the same notations as Cosgrov. Thus we can understand the fact that the algebra sl(2,R) of the Cosgrove transformations is one of subalgebras of the complete Virasoro algebra.
—
+sF(s)F~(s)+flP(t).
[ö(k)~(I)]
(30)
by summarizing all the listed commutations. Apparently, it is nothing but the full Virasoro algebra. Finally, we notice that the transformations (20) and (27) canbederivedfromeqs. (9) and (25) by substituting
which is proved to satisfy
-~—~—-~
~
solution method to get more solutions of the Ernst equation. By comparison with the Cosgrove transformation
.
6P( t) =
30 May 1988
The author is grateful to Professor B.Y. Hou for discussion and his encouragement. The author also thanks the referee for his helpful suggestion.
.
k>~I>~l,
(k+I)~”~E, I~>k+l>~l. (28)
References
If we define Lk=ö~, (—k) =~
if k~0, 29
‘ ,
if
k<0,
Ll]B.Y. Hou and W. Li, Lett. Math. Phys. 13 (1987) 1; 14 (1987)372(E). [2] B.Y. Houand W. Li, J. Phys. A20 (1987) L897. [3] R.Geroch,J.Math.Phys. 12(1971)918; 13(1972)394.
303
Volume 129, number 5,6
PHYSICS LETTERS A
[4] W. Kinnersley and D.M. Chitre, J. Math. Phys. 18 (1977) 1538; 19 (1978) 1926; I. Hauser and F. Ernst, J. Math. Phys. 21(1980) 1126; Y.S. Wu and M.L. Ge, J. Math. Phys. 24 (1983)1187. [5] Y.S. Wu, Commun. Math. Phys. 90 (1983) 461; NucI. Phys. B2ll (1983) 160;
304
30 May 1988
Y.S. Wu and ML. Ge, Vertex operators in mathematics and physics, MSRI publication #3 (Springer, Berlin, 1984) p. 329. [6] 1. Hauser, Lecture notes in physics, Vol. 205, eds. C. Hoenselaers and W. Dietz (Springer, Berlin, 1984) p. 113. [71C.M.Cosgrove,J.Math.Phys.2l (1980)2417.
PHYSICS LETTERS A
30 May 1988
THE COMPLETE VIRASORO ALGEBRA FOR THE STATIONARY AND AXIALLY SYMMETRIC EINSTEIN FIELD EQUATIONS Wei LI Institute of Modern Physics, Northwest University, Xian, PR China Received 26 May 1987; revised manuscript received 25 February 1988; accepted for publication 25 March 1988 Communicated by J.P. Vigier
In this Letter it is described how to extend the Virasoro symmetry for the stationary, axially symmetric Einstein field equations given by Hou and the present author to the full Virasoro symmetry.
In our previous papers [1,2] we found the existence of the Virasoro symmetry in the solution space of the stationary and axially symmetric Einstein field equations and constructed the infinitesimal transformation of this symmetry to establish the representation of the Virasoro algebra. We also pointed out that our symmetry is different from the Geroch symmetry [3] which corresponds to the Kac—Moody algebra [4], and these two symmetries, therefore, can be used to form a larger infinite-dimensional Lie algebra. With the aid of these symmetries we developed a new systematic generating-solution method for the stationary and axially symmetric Einstein field equations to enlarge the known family of exact solutions. The further investigation showed that the set of transformations are available for positive integers after being expanded in powers ofa parameter of our transformation. We failed to get another set of the transformations for negative integers. The situation is not similar to the case of the Geroch symmetry transformation because for negative integers the latter is concerned with the gauge transformation though it is trivial. So the Virasoro algebra under our construction is only the half one. The purpose of this Letter is to find a way to extend this incomplete Virasoro algebra to the full one. As the similar work [5] has been accomplished for the Geroch symme-
try we are motivated to use the same procedure to settle our problems without difficulty. In this way we are able to construct the full Virasoro algebra. For the convenience of discussion, we prefer to adapt the Hauser—Ernst formalism to the present Letter. We start with the metric under the form (1) th2=g dx’dx’+f((dx3)2+ (dx4)2) where g,~(i, 1= 1, 2) and fare functions of x3 and x4 only. Then one can define a 2 x 2 real and symmetric matrix as g= (g,~)and 2 2 det g= a ( ) For the stationary and axially symmetric Einstein fields, the vacuum field equations are reduced to the Ernst equation —
.
2(fl+ a*) dE= (E+E + )Q dE, / 0 i\
~
—
o)’
(3)
where the matrix Ernst potential E is defined as E=g+ isv, tr (EQ) = 2fl, (4) with the twist potential d a ‘gQ dg
yi
given by (5)
— —
The notations * and + represent the two-dimensional duality operation with properties *~3 = dx4 and *dx4 = dx3, and the hermitian conjugation, respectively. —
Present address: Department of Mathematics and Computer Science, Clarkson University, Potsdam, NY 13676, USA.
0375-9601/88/s 03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
301
Volume 129, number 5.6
PHYSICS LETTERS A
The Ernst equation implies the existence of the Hauser—Ernst linearization equation dF(t) =1(t) QF(t),
(6)
where
30 May 1988
and P(t) is a 2x2 matrix function of x3, x4 and t. For this formulation, the auxiliary conditions of P(t) should be given by detP(t)= [(t—2fl)2+(2a)2]~~2
(7)
F(t)=t[l_2t(fl+a*)]~dE
and tis aparameterandF(t) a 2x2 matrixfunction of x3, x4 and t. F(t) is subject to auxiliary conditions given in refs. [3], if F( t) can be used to determine E. In the previous papers [1,2] we gave the expressions for the infinitesimal transformations of the Ernst potential
(14)
and P(t)~Q(t—(E+E~)Q) P(t)=Q.
(15)
Using a solution of the new linearization equation (12) we can propose a new infinitesimal transformation ~E = s[ sF(s) P ‘(s) + ~]. (16) —
According to eq. (16), it follows that 8E=_P(s)F_l(s)Q,
where
F(t)=~F(t)
__________
(8) and for a solution F(t) of eq. (6)
—sF(s) F ‘(s)] F(t), —
~a=s2
(9)
where F(s) also satisfies the Hauser—Ernst linearization equation with another parameter s. Note that we neglect writing out the infinitesimal constants for the above transformations. We demonstrated that both the Ernst equation and its linearization equation are invariant under these transformations, i.e., they are the symmetry transformations. By setting ~
~ (s_2$)2+(2a)2 /3(s—2J3)—2a~
(17)
and
[tF(t) F-1(t)
—
—
(10)
a
(18)
(s_213)2+ (2a)2’ It is easy to check that the new transformation is also
the symmetry one to the Ernst equation, by making use of tI’e same treatment as we did before. That is, if E is a solution for eq. (3), then so is E+JE, 2(fl+a*) d(~E)+2(~/3+~a*) dE = (E+E~)Q d(~E)+ (~E+~E~ )Q dE. (19) In order to explore the Lie algebra of the new
transformation we need to give of P(t) in eq. (12). We find thatthe transformation
k~O
commutations of a infinite set of operators {ö”~}are evaluated to be [~(k) ~1)] E= (k—I) ~(k+/)E, k, 1?~0. (11)
~P(t)
It is apparent that our transformation indeed forms the half Virasoro algebra. Following Hauser’s work [6], we learn that there is another linearization equation for the Ernst equation. If t in eq. (6) is replaced by 1/1, we obtain
keeps the linearization equation (12) invariant, i.e., d(~P(t)) =~f(t)QP(t)+f(t)Q~P(t). (21)
dP(t) =P(t) QP(t),
(12)
S
t—s [tP(t)P’(l) —sP(s)P ‘(s)] P(t)
(20)
By setting ~
k= I
which is also the linearization equation to the Ernst equation, where
we can express the transformations (16) and (20) as the infinitesimal Riemann—Hilbert transforma-
P(t)= [t_2(/J+a*)]_I
tions, respectively,
302
dE
(13)
Volume 129, number 5,6
PHYSICS LETTERS A
we write a very neat relation
1 (22)
~JY_k+IP(Y)P_1(Y)dyQ
~(k)E_
d an
f
1
~(k)
P(t)=— ~
yk+I
.
y—t P(y)P
d ~ (y) y (t), (23)
where we denote C0, as a circle surrounding y = 0 and y=t in the y-complex plane. It should be emphasized that the subscript k is not permitted to be zero or negative integers because it violates the invanance of eqs. (3) and (12). We then use the method described in ref. [2] to obtain the following commutations [~(k) ~ k 1~l. (24) It means that the present-proposed transformation can constitute the structure of the half Virasoro algebra as the previous one. It is natural for us to consider the relationships between these two different types of the transformations 8E and ~E.First, we need to compute the commutator of ~ and ~1) on E. For this purpose, it is not difficult to set up st [tF(t) F~(t) —st
[Lk,L,] E=(k—I) Lk+/E,
+sP(s)P’(s)+fl F(t),
(25)
6F(t)—~P(t), t—+l/t,s—+l/s
(31)
and ~ oF(t)—~.SF(t), t—1/t,s--*l/s
(32)
respectively. The same correspondence exists between eq. (8) and eq. (16) except an additional constant. Toourknowledgethericherstructureofoursymmetry, like the Geroch symmetry, must be connected with the Riemann—Hilbert problem. Through the way it is possible to exponentiate our infinitesimal symmetry transformations into the finite forms and to establish the representations of the Virasoro group rather than the Virasoro algebra, it will become easier to apprehend the relationships between two different types of our Virasoro symmetries. More important, we are able to create a new generating-
[7] we observe that L~, ~
d(~F(t))=~T’(t)QF(t)+f’(t) Q~F(t).
(26)
In a similar way, [tP( t) P ‘(t) (27)
Therefore, the commutators are evaluated to be E=(k+I)
~(k_!)E,
=
L0,
~t= L_1,
(33)
where we have used the same notations as Cosgrov. Thus we can understand the fact that the algebra sl(2,R) of the Cosgrove transformations is one of subalgebras of the complete Virasoro algebra.
—
+sF(s)F~(s)+flP(t).
[ö(k)~(I)]
(30)
by summarizing all the listed commutations. Apparently, it is nothing but the full Virasoro algebra. Finally, we notice that the transformations (20) and (27) canbederivedfromeqs. (9) and (25) by substituting
which is proved to satisfy
-~—~—-~
~
solution method to get more solutions of the Ernst equation. By comparison with the Cosgrove transformation
.
6P( t) =
30 May 1988
The author is grateful to Professor B.Y. Hou for discussion and his encouragement. The author also thanks the referee for his helpful suggestion.
.
k>~I>~l,
(k+I)~”~E, I~>k+l>~l. (28)
References
If we define Lk=ö~, (—k) =~
if k~0, 29
‘ ,
if
k<0,
Ll]B.Y. Hou and W. Li, Lett. Math. Phys. 13 (1987) 1; 14 (1987)372(E). [2] B.Y. Houand W. Li, J. Phys. A20 (1987) L897. [3] R.Geroch,J.Math.Phys. 12(1971)918; 13(1972)394.
303
Volume 129, number 5,6
PHYSICS LETTERS A
[4] W. Kinnersley and D.M. Chitre, J. Math. Phys. 18 (1977) 1538; 19 (1978) 1926; I. Hauser and F. Ernst, J. Math. Phys. 21(1980) 1126; Y.S. Wu and M.L. Ge, J. Math. Phys. 24 (1983)1187. [5] Y.S. Wu, Commun. Math. Phys. 90 (1983) 461; NucI. Phys. B2ll (1983) 160;
304
30 May 1988
Y.S. Wu and ML. Ge, Vertex operators in mathematics and physics, MSRI publication #3 (Springer, Berlin, 1984) p. 329. [6] 1. Hauser, Lecture notes in physics, Vol. 205, eds. C. Hoenselaers and W. Dietz (Springer, Berlin, 1984) p. 113. [71C.M.Cosgrove,J.Math.Phys.2l (1980)2417.