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Axisymmetric Stationary Solutions as Harmonic Maps

We present a method for generating exact solutions of Einstein equations in vacuum using harmonic maps, when the spacetime possesses two commutating Killing vectors. This method consists in writing the axisymmetric stationry Einstein equations in vacuum as a harmonic map which belongs to the group SL(2,R), and decomposing it in its harmonic"submaps". This method provides a natural classification of the solutions in classes (Weil's class, Lewis' class etc).


Introduction
Stationary Einstein equations in terms of harmonic maps (HM) [1] are well-known. Sánchez [2] found that the stationary Einstein equations in electrovacuum are in fact HM, but that the corresponding differential equations are non-linear. Neugebauer and Kramer [3] showed that this HM has a Lax pairsrepresentation (equivalent linear problem) in terms of a HM which is an element of the group SU(2,1), i.e. axisymmetric stationary Einstein-Maxwell equations in potential space (Ernst potential space) are HM G : M 4 → V p ⊗ SU (2, 1), being V p a p-dimensional Riemannian space. On the other hand, Belinsky and Zakharov [4] have also given the empty Einstein equations in terms of HM, but in a SL(2, R) representation of spacetime. Recently, Whitman and Stoeger gave a HM representation of these equations [5]. It is also possible to give the empty Einstein equation in potential space in terms of HM if we reduce the group SU (2,1) to SU (1,1). The isomorphism SU (1, 1) ≃ SL(2, R) allows us to give a representation of these equations in terms of the group SL(2, R) [6]. Thus we can write the Einstein equations in vacuum as a SL(2, R) -HM in the spacetime, and in the potential space.
In this work, we will start from the HM representation of the axisymmetric stationary Einstein equations in vacuum, and develop a method for generating exact solutions. It consists in separating the group SL(2, R) into its one-parametric subgroups and itself giving a representation of the HM, G : M 4 → V p ⊗ SL(2, R) in its tangent space (see section 3 or ref. [7]). The result is that each subgroup corresponds to a well-known class of solutions, explaining why do these classes exist.
One could also extend this analysis to the Einstein-Maxwell equations. It may be done in potential space [8] where the HM belongs to the group SU (2, 1), but there is no HM representation in spacetime. Nevertheless, five-dimensional gravity admits a HM re presentation for the stationary equations [9]. Both the spacetime and the potential space have a SL(3, R) HM representation of the field equations [10] (see also ref. [9]). Thus an analysis with harmonic " submaps ′′ can be done [11], [12]. This work pretends to be self-contained. In sections 1 and 2 we give the Einstein's spacetime field equations in vacuum when the metric depends only on two coordinates. In section 3 we briefly outline the stationary Einstein equations in potential space (Ernst potential). Section 4 is devoted to explain the method of subgroups (harmonic " submaps ′′ ) which will be applied in section 5.

Generalities
We consider a V 4 manifold with two commuting Killing-vectors and local coordinates {X µ } = (X r , y A ), with X r = (X 1 , X 2 ) being the " active ′′ (and arbitrary) coordinates and y A = (y 1 , y 2 ) playing the role of the " passive ′′ coordinates (i.e., with ∂/∂y A being the two commuting Killing vectors). The metric can be given in the quasi-diagonalizable form The essential "Killigian ′′ part of the metric is thus described by the three real objects We use for this part of the metric the spinorial notation, because evidently the constant transformations of the passive variables y A do not affect the structure considered in any manner. We assume the rules of manipulating the spinorial indices as follows: We assume the " Killingian sector ′′ to be of signature (+,−). To ensure this, it is necessary and sufficent to assume that The 2-dimensional metric V 2 : g 2 := g rs dX r ⊗ dX s (1.6) with g rs = g rs (X 1 , X 2 ) = g (rs) has then the signature (+,+). The condition for this is assuming, of course, that the coordinates X r are real. The V 2 structure is thus just a proper 2-dimensional Riemannian space. The covariant derivative with respect to g 2 shall be denoted by "|r ′′ . The indices r, s, ... are to be manipulated by g rs and its inverse, the contravariant metric g rs (g rs g st = δ r t ). Our problem consists now in studying the Einstein empty spacetime equations working with the chart which we can evidently also take in the form with R µν = R σ µνσ being the Ricci tensor. The Ricci tensor of metric (1.1) can be easily evaluated in the covariant form with respect to the chart used in the description of the metric g 2 ,i.e., the coordinates {X r } = {X 1 , X 2 }.
Indeed, one finds that all " mixed ′′ components of R µν identically vanish For the " Killingian ′′ components of R µν one finds the V 2 -covariant basic expression R being the scalar curvature of the non-Killingian sector (g 2 ) of the metric. Knowing this, one works out the " non-Killingian ′′ components of R µν ,i.e. R rs , in the form The next thing to do is to work out the components of R AB and R rs in an apropiate form for our further purposes.
For R AB we notice first that we have the spinorial identity Employing it, we find that (1.14) Equality (1.14) says then that We arrive in this way at the identity Using this in (1.10) we have We bring this last expression to the slightly simpler form The expression above is already satisfactory for many purposes. We can obtain a more satisfactory equivalent expression by executing the contraction K AP R AB , remembering that because of det(K AB ) < 0, the matrix K AB is invertible. Indeed, employing (1.13) in the form K AP K AB = Kδ P B , we have The last term in this expression, with the index P lowered, |r AP is antysymmetric in P B and hence equal to so that, rising again the index P Consequently, the terms in the last line of (1.19) simply cancel out, and we arrive at This is perhaps the most condensed form concerning the analytic form of the " Killingian part ′′ of the Ricci tensor. We would like to observe at this point that the expression E AB defined above has the trace so that the objects E AB defined by (1.22) are linearly dependent.
In the next part of this section, we would like to study the structure of the " non-Killingian ′′ part of the Ricci tensor, (1.2).
Because of (1.13), we have and therefor the last term in (1.12) transforms to With this identity, we have This can be still slightly simplified, observing that Hence, we have for R rs We can now conveniently evaluate the scalar curvature of the g 4 metric. Indeed, it is obvious from (1.1) that (1.30) and consequently remembering (1.9). From (1.29) we evaluate now and from (1.23) we have Knowing R, we can now evaluate the components of the Einstein tensor (1.35) In particular, the (rs) components (i.e., the non-Killingian part) of this object are The cancellation of the term which involves 0 R in this expression is of basic importance, the second derivatives of the g 2 metric (g rs,tu ) do not enter in the structure of G rs .
Summarizing, we can now state the basic equations of the problem R µν = 0 written covariantly with respect to the chart {X r } = {X 1 , X 2 } in terms of which the metric g 2 is described as the following hierarchy of differential equations. First we have the " K-sector ′′ Einstein equations These equations imply so they are not independent because of the identity Then we have the equations and finally where 0 R is the scalar curvature of g rs defined by (1.11).
2. The Equations R µν = 0 in Weyl's Coordinates. The metric g 2 can be of course always expressed in the conformally flat form or, introducing the complex coordinates in the simple form We shall call ξ andξ the Weyl coordinates. They are a sort of " null variables ′′ and they are arbitrary up to the transformations with f (z) being an arbitrary analytic function such that f ′ (z) = 0. Under the transformation ξ → ξ ′ the real structural function φ transforms according to Experience with the theory of the exact solutions, e.g., the cases of the D-type solutions or the Tomimatsu-Sato solutions, indicates that the Weyl variables are certainly not the best variables in practice, i.e., in the description of the physically pertinent solutions. On the other hand, these variables are theoretically important in decoding the essential structure of the differential problem stated at the end of the previous section.
When we employ the Weyl coordinates, a simple manner of proceeding (and taking advantage of the covariant formulation of the problem outlined at the end of the previous section) is this: we simply take the metric g 2 as given in the form understanding X 1 ≡ ξ and X 2 ≡ξ. With these complex coordinates all the apparatus of the classical differential geometry in two dimensions, formally implicates the condition on g 2 to be positive definite. This implys that the determinant of the metric must be negative, For the inverse metric we have then so that the tensor density is just a numeric matrix G rs = 0 1 1 0 . (2.12) The Weyl coordinates are thus harmonic in the sense that G rs ,s = 0. (2.13) The Christoffel symbols Γ r st computed from the metric (2.7) and its inverse (2.10) vanish, excepting the two components This permits us to determine easily the scalar curvature of g 2 , the object 0 R. We arrive at Now, equations of the type (AB |r ) |r = 0 in the present coordinates, being equivalent (when A and B are g 2 scalars) to (G rs AB ,r ) ,s = 0, simply transform to (AB ,ξ ) ,ξ + (AB ,ξ ) ,ξ = 0 . Therefor, the fundamental equations (1.37) take the form The essential fact emerges here since these equations do not involve in any manner the structural function φ, being them the autonomous " K-sector ′′ equations. The necessary implication of these equations (1.38), can be now simply stated in the form Consider now equations (1.40), their (11) component transforms to 18) or explicitly The complex conjugate of this equation transforms of course to The (12) But because of Γ r 12 ≡ 0 and ( √ −K) |12 = ( √ −K) ,ξξ , cancelling the terms with first derivatives, we have Including that R AB = 0 implies ( √ −K) ,ξξ = 0, it follows that so that the conditions G 12 = 0 are automatically fulfilled on the " K-sector ′′ equations. The last of the field equations which must be described in terms of the Weyl coordiantes is (1.41).
Assuming that as a consequence of R AB = 0 we know already that √ −K |r |r = 0 ⇔ ( √ −K) ,ξξ = 0 , and using (2.15), this equation transforms to or simply to demand that Summarizing, we conclude that working in the Weyl coordinates (ξ,ξ) , in order to fulfill the field equations R µν = 0 , we must demand that the real structural functions K AB = K AB (ξ,ξ) = K (AB) , φ = φ(ξ,ξ) should be subject to the sequence of differential conditions: (a) The complete set of these conditions, complemented by the basic requirement −K = −det(K AB ) > 0, is sufficient to assure R µν = 0, but we do not claim that all of these conditions are (independently) necessary.
To end this section we rewrite equations (2.26) in matrix notation. Let (γ) AB = K AB be the 2x2 symmetric matrix corresponding to the Killingian part of the metric (1.1). Observe that K A R = γǫ and K AB = −ρ 2 γ −1 with ǫ = ǫ AB , then equations (2.26) can be cast into the form where we have set K = −ρ 2 . Equation (2,27c) is a consistency equation of (2,27b)(see also ref. [9]).

3.The Potential Space.
The Ernst potential [13] E has proved to be a very useful tool for finding exact solutions [14]. To define the potential E, we need the existence of a time-like Killing vector Z µ . Then one finds that the Einstein equations in vacuum where Q µν =Z µ;ν = Z µ;ν + 1 2 ǫ µναβ Z α;β . Furthermore one finds that the Lie derivative of Q µν with respect to Z vanishes. This allows us to define the Ernst potential E µ = Z α Q αµ , whose integrability conditions are just (3.2) and the vanishing of the Lie derivative of Q µν , i.e.

E →Ē
These equations can be derived from the Lagrangian Generation techniques consist in finding invariant transformations of the Lagrangian (3.5). This is equivalent to finding the isometry group of the metric This isometry group is SU (1, 1) [14] which is isomorphic to SL(2, R). A straightforward calculation shows that the metric (3.6) of the potential space (defined by this metric) can be cast into the form [3] where the 2x2 matrix G is given by (3.8) The matrix (3.8) is an element of the group SU (1, 1) restricted to G 2 = 1. An other parametrization of (3.7) belongs to the group SL(2, R), it reads [6], [10] G where E = f + iǫ. Observe that detG = 1 in both cases, but G in (3.9) is symmetric. Now we return to the field equations in the potential space. It is clear that the field equation (3.4) can be derived from the Lagrangian We are interested on fields depending on two coordinates X 1 and X 2 . Let z = X 1 + iX 2 be the complex variable which G depends on. Equation (3.4) transforms into [6], [10] (ρG ,z G −1 ) ,z + (ρG ,z G −1 ) ,z = 0 (3.11) derived from Lagrangian (3.10). They are the chiral equations with GǫSL(2, R) and G = G T .

4.The SL(2, R) − Chiral Equations
It is now clear why we are interested in developing a technique for solving chiral equations with the group SL(2, R). Solitonic methods for doing so are given in ref. [15]. We want to give an another method using an ansatz due to Neugebauer and Kramer [8]. Let us outline it [7].
Suppose G is an element of a Lie group H which depends on a set of parameters λ a ; a = 1, ..., p with λ a = λ a (z,z) such that these parameters form minimal surfaces on a Riemannian space V p (HM , λ a : M 4 → V p ): where γ a bc are the Christoffel symbols of V p . In terms of these parameters the chiral equation (3.11) reads a) That means that A 1 is a constant matrix of G = sl(2, R). We need then the matrices G to be symmetric. Hence it is convenient to choose the left action in order to have L c (G) also symmetric. With this left action the equivalence relation reads The representatives of the classes can be the 2x2, traceless and real Jordan matrices. There is only one representative, i.e. (4.8) Now we map A 1 into the group H, using its definition A 1 = (∂ λ G)G −1 , i.e., solving this matrix differential equation. We have and get G 11,λ = G 12 G 12,λ = G 22 G 12,λ = αG 11 G 22,λ = αG 12 which imply the differential equations G 11,λλ − αG 11 = 0 , G 12,λλ − αG 12 = 0.
We suppose now p = 2 and V 2 a 2-dimensional Riemannian space. But all V 2 Riemannian spaces are conformally flat and therefore have a metric which may be written in the form (4.10) In reference [2] and [8] it was shown that the V p spaces must be symmetric (i.e. all covariant derivatives of the Riemannian tensor vanish), this implies that k must be a constant. Now we choose a base of the Killing vector space on V 2 . Let this base be ξ 1 = 1 2V 2 (kτ 2 + 1, kλ 2 + 1) (4.11) With this set of Killing vectors the commutation relations (4.2b) read We have to put k = −1 in order to have the commutation relations of sl (2, R). A representation of sl (2, R) is which is a base of the vector space G = sl(2, R). Now we map the algebra into the group. From the definitions where λ and τ must fulfill the geodesic equation (4.1). With the metric (4.10) we get (ρλ ,z ) ,z + (ρλ ,z ) ,z + 4τ 1 − λτ ρλ ,z λ ,z = 0, (4.14) and the other one by changing λ for τ and viceversa. The other possibility is to put k = 0 in (4.12), but this case corresponds to the one-dimensional subalgebras of sl (2, R) studied before with λ → λ + τ . The corresponding equation (4.1) will be the Laplace equation for λ and τ separately. Observe that det G = 1 if we put d = −1, c = 1 in (4.13) and det G = −1 if we put d = 1 = c. Let us introduce a real potential α by , and a similar equation for α ,z , withz in plase of z. Since the chiral equations imply α ,zz = α ,zz , the integrability condition of α follows from the chiral equation (3.11) [8]. This potential can be calculated separately for each case. For the one dimensional subspaces we have and for the two-dimensional subspaces, a straightforward calculation gives These potentials will be important in the following section.

5.The Unified Point of View
In this section we apply the technique developed in section 4 to the results of sections 2 and 3. Equations (2.27a) and (3.11) are chiral equations, but there is a difference between them. In the first case detγ = −ρ 2 , in the second one det G = 1. We can transform equation (2.27a) in order to have a (3.11)-like equation. First observe that √ −K = ρ = 1 2 (ξ +ξ). is a solution of equation (2.17). Then we can write ξ = X 1 + iX 2 = z = ρ + iζ. Now define a matrix G such that γ = ρG with det G = −1. It is easy to see that G fulfills just the chiral equation (3.11). Let us parametrize the matrix G in Papapetrou form where F = −ρ/f. Compare (5.1) with (3.9). That means the following: if we have a solution of the chiral equations in the potential space, it is also a solution of the chiral equation in the spacetime. This transformation was given first in ref. [16] (see also ref. [17]). We study the subspaces on the spacetime. There are the three cases of where we have used the Papapetrou parametrization (5.1). This means that (5.2) belongs to the Weyl's class (see ref. [14] eq. (18.22)) or to the van Stockum class [18] by a → ∞. We can write the solution (5.2) in a better-known form. Take a (4.6) transformation with we find that γ = bρ(C 1 + aC 2 ) 2 e aλ + cρ(C 1 − aC 2 ) 2 e −aλ 2ρba(C 1 + aC 2 )C 4 e aλ 2ρba(C 1 + aC 2 )C 4 e aλ 4ba 2 C 2 4 ρe aλ , (5.6) then which in the Papapetrou form reads 2u = aλ + lnρ, 4ba 2 C 2 4 = −1, ω = constant. An other example on this subspace is the solution of the Laplace equation λ = n a lnρ. If we substitute this λ in (5.2) then we find a cylindrically symmetric solution of the Einstein's equations with one restriction (equation (20.7) in ref. [14] with B 2 a 1 a 2 = −1/4. See also ref. [9]). Now we apply the transformation (4.6) to get which is the general cylindrically symmetric solution of the Einstein's equations in vacuum, (equation (20.7) in ref. [14] with c = C − /d − , a 1 = bd 2 + , a 2 = − 1 4ba 2 d 2 − ). The second case in table 1 corresponds to the solution γ = − ρ a cos(aλ + ψ 0 ) ρsin(aλ + ψ 0 ) ρsin(aλ + ψ 0 ) aρcos(aλ + ψ 0 ) where b 2 a 2 = 1, in order to have detG = −1. The function φ is then a solution of the differential equation It is easy to see that ρ 2 e −4u = ω 2 + 1, (5.10) which corresponds to the Lewis' class (see eq. (18.22) in ref. [14]). The third case of table 1 corresponds to the degenerated class f = 0 [8].

Conclusions
We have shown that the solutions of the Einstein's equations with two commuting Killing vectors can be separated naturally in equivalent classes given by the subgroups of SL(2, R). Those classes coincide with the classes of solutions, the one parametric subgroups of SL(2, R) give the Weyl's class, van Stockum class, the general cylindrically symmetric solutions, the Lewis' and the degenerated classes in the spacetime, and the Papapetrou's class in the potential space. The group SL(2, R) itself gives the Tomimatsu-Sato class in the potential space and a corresponding class in the space time. The left action of the group SL(2, R) C over SL(2, R) is just the invariance transformations of Neugebauer and Kramer [16], i.e. they are the Ehlers and the Buchdahl transformations to obtain new solutions from a seed one. Then we have shown that a classification of these solutions can be done by the classification of the subgroups of the invariance group.