The Diffiety Institute Preprint Series

Preprint DIPS–1/2004 January 8, 2004

Available via INTERNET: http://diffiety.ac.ru/ http://diffiety.org/

The Diffiety Institute

Polevaya 6-45, Pereslavl-Zalessky, 152140 Russia.

Recursion operators for vacuum Einstein equations with symmetries by

Michal Marvan

Recursion operators for vacuum Einstein equations with symmetries

M. Marvan

Mathematical Institute, Silesian University in Opava,

Na Rybníčku 1, 746 01 Opava, Czech Republic

Abstract

Direct and inverse recursion operator is derived for the vacuum Einstein equations for metrics with two commuting Killing vectors that are orthogonal to a foliation by 2-dimensional leaves.

1 Introduction

In the past decade, inverse recursion operators became a subject of a number of papers [7, 8, 14, 15]. In particular, the work of Guthrie [7] opened a new perspective on recursion operators by essentially identifying them with auto-Bäcklund transformations for linearized equation [18].

It is known for a long time that recursion operators of integrable systems are obtainable from their Lax pairs (see [6] and references therein) and ZCR’s (see [10, 24]). However, recently it became clear that zero curvature representations are related much closer to inverse recursion operators than to their ‘direct’ counterparts [19, 20]. Examples that have been already published elsewhere include the Korteweg–de Vries and Tzitzéica equation [19] and the stationary Nizhnik–Novikov–Veselov equation [20]. In the present paper, the methods of [19, 20] are applied to equations of General Relativity.

2 Recursion operators

Let E = {F^{l} = 0} be a system of PDE’s in unknown functions u^{k} of two independent variables x, y. We assume that F^{l} are functions of x, y, u^{k} and a finite number of the derivatives u_{ij}^{k} = ∂^{i+j}u^{k}/∂x^{i}∂y^{j} (u_{00}^{k} = u^{k}). Consider the infinite-dimensional jet space J^{∞} with local coordinates x, y, u_{ij}^{k} along with the commuting vector fields D_{x} = ∂/∂x + ∑_{ij} u_{i+1,j}^{k}(∂/∂u_{ij}^{k}), D_{y} = ∂/∂y + ∑_{ij} u_{i,j+1}^{k}(∂/∂u_{ij}^{k}), called total derivatives. The submanifold E determined by equations F^{l} = 0 and their differential consequences D_{x}F^{l} = 0, D_{y}F^{l} = 0, D_{x}^{2}F^{l} = 0, D_{x}D_{y}F^{l} = 0, D_{y}^{2}F^{l} = 0, …, is called the equation manifold (and is an underlying space of the diffiety structure [12] employed in [19]). In this context, infinitesimal symmetries (more precisely, their generating functions) are functions U^{k} defined on E such that ∑_{k,i,j}(∂F^{l}/∂u_{ij}^{k})D_{x}^{i}D_{y}^{j}U^{k} = 0.

It is then natural to consider the jet space with coordinates x, y, u_{ij}^{k}, U_{ij}^{k}, and denote

LF^{l} = ∑_{k,i,j} \frac{∂F^{l}}{∂u_{ij}^{k}} U_{ij}^{k}.  (1)

The system LE := {F^{l} = 0, LF^{l} = 0} on unknowns u^{k}, U^{k} will be called the linearized equation. Now, Guthrie’s recursion operators [7] may be interpreted as auto-Bäcklund transformations of the linearized equation LE that keep variables u^{k} unchanged [19].

In now standard formalism [21], recursion operators are pseudodifferential operators, characterized by the occurrence of inverses of total derivatives D_{x}^{−1}. Under Guthrie’s approach, p = D_{x}^{−1}f is introduced as an auxiliary nonlocal variable satisfying

p_{x} = f,  p_{y} = g,  (2)

provided such a g exists, and it actually does without known exception; see Sergyeyev [23] for a proof in case of evolution systems. Thus, p is a potential of a conservation law f dx + g dy of the linearized equation LE.

For the inverse recursion operators, the nonlocalities tend to be genuinely nonabelian pseudopotentials related to a zero curvature representation of the system in question. Let 𝔤 be a matrix Lie algebra. Let α = A dx + B dy be a 𝔤-valued zero curvature representation (ZCR) for the system E. This means that A, B are 𝔤-valued functions on the equation submanifold E and D_{y}A − D_{x}B + [A, B] = 0 holds on E. Let us introduce the associated pseudopotential P as a 𝔤-valued solution of the compatible system

P_{x} = [A, P] + LA,  P_{y} = [B, P] + LB.  (3)

A recursion operator R is then a linear operator in U^{k} and P such that U′ = R(U, P) solves the linearized system LE whenever U does and P satisfies (3) (see [19, 20]). In this way, the inverse recursion operator can be found without previous knowledge of the direct recursion operator. A remarkable aspect of this approach is that R(U, P) tends to be a very simple expression.

For the above scheme to work, it is not necessary that the ZCR α a priori depends on the “spectral parameter.” However, if R is a recursion operator related to the ZCR α as above, then (R^{−1} + μ Id)^{−1} is another recursion operator, associated with a ZCR α_{μ} which depends on μ.

3 The results

We consider vacuum Einstein equations for a space-time with two commuting Killing vectors that are orthogonal to a foliation by 2-dimensional surfaces [4, 5]. Our presentation will be restricted to the case when both Killing vectors are space-like. The case when one of the Killing vectors is time-like is equivalent to ours via an appropriate complex transformation of coordinates.

As is well known, there exist coordinates x, y, z^{1}, z^{2} such that the metric in question can be written in the form ds^{2} = 2f(x, y) dx dy + g_{ij}(x, y) dz^{i} dz^{j} (the Lewis [13] metric). The vacuum Einstein equations essentially reduce to

(√det g g_{x}g^{−1})_{y} + (√det g g_{y}g^{−1})_{x} = 0,  (4)

while f can be obtained by quadrature. Using the standard normalization det g = (x + y)^{2} compatible with Eq. (4), we parametrize g as follows: g_{11} = (x + y)/u, g_{12} = (x + y)v/u, g_{22} = (x + y)(u^{2} + v^{2})/u. Equation (4) then becomes

\begin{matrix}u_{xy} = \frac{u_{x}u_{y} − v_{x}v_{y}}{u} − \frac{1}{2} \frac{u_{x} + u_{y}}{x + y} , \\ v_{xy} = \frac{v_{x}u_{y} + u_{x}v_{y}}{u} − \frac{1}{2} \frac{v_{x} + v_{y}}{x + y} . \end{matrix}  (5)

As is well known, Eq. (5) has a ZCR and a Bäcklund transformation [1, 2, 3, 9, 16, 17, 22]. The ZCR reads

\begin{matrix}A = \frac{1}{2} (\begin{matrix}−(θ + 1)u_{x}/u & (θ + 1)v_{x}/u^{2} \\ (θ − 1)v_{x} & (θ + 1)u_{x}/u \end{matrix}), \\ B = \frac{1}{2θ} (\begin{matrix}−(θ + 1)u_{y}/u & (θ + 1)v_{y}/u^{2} \\ (−θ + 1)v_{y} & (θ + 1)u_{y}/u \end{matrix}), \end{matrix}

where θ = \begin{matrix}√ \\ (μ + y)/(μ − x) \end{matrix}, μ being the spectral parameter.

The main result of this paper, obtained by the methods of [19, 20], is as follows: If nonlocal variables p_{11}, p_{12}, p_{21} satisfy

\begin{matrix}p_{11,x} = −\frac{θ − 1}{2}v_{x}p_{12} + \frac{θ + 1}{2} \frac{v_{x}}{u^{2}} p_{21} − \frac{θ + 1}{2} \frac{1}{u}U_{x} + \frac{θ + 1}{2} \frac{u_{x}}{u^{2}} U, \\ p_{12,x} = −(θ + 1)\frac{v_{x}}{u^{2}} p_{11} − (θ + 1)\frac{u_{x}}{u} p_{12} \\ − (θ + 1)\frac{v_{x}}{u^{3}} U + \frac{θ + 1}{2} \frac{1}{u^{2}} V_{x}, \\ p_{21,x} = (θ − 1)v_{x}p_{11} + (θ + 1)\frac{u_{x}}{u} p_{21} + \frac{θ − 1}{2}V_{x}, \\ p_{11,y} = \frac{θ − 1}{2θ}v_{y}p_{12} + \frac{θ + 1}{2θ} \frac{v_{y}}{u^{2}} p_{21} + \frac{θ + 1}{2θ} \frac{u_{y}}{u^{2}} U − \frac{θ + 1}{2θ} \frac{1}{u}U_{y}, \\ p_{12,y} = −\frac{θ + 1}{θ} \frac{v_{y}}{u^{2}} p_{11} − \frac{θ + 1}{θ} \frac{u_{y}}{u} p_{12} \\ − \frac{θ + 1}{θ} \frac{v_{y}}{u^{3}} U + \frac{θ + 1}{2θ} \frac{1}{u^{2}} V_{y}, \\ p_{21,y} = −\frac{θ − 1}{θ}v_{y}p_{11} + \frac{θ + 1}{θ} \frac{u_{y}}{u} p_{21} − \frac{θ − 1}{2θ}V_{y}, \end{matrix}  (6)

then

\begin{matrix}U′ = 2 \frac{u}{_{√(μ − x)(μ + y)}} p_{11} + \frac{1}{_{√(μ − x)(μ + y)}} U, \\ V′ = − \frac{u^{2}}{_{√(μ − x)(μ + y)}} p_{12} − \frac{1}{_{√(μ − x)(μ + y)}} p_{21} \end{matrix}  (7)

is a recursion operator for Eq. (5), namely, it sends symmetries to symmetries if the latter are viewed as solutions of the linearized system

U_{xy} = (\frac{u_{y}}{u} − \frac{1}{2(x + y)})U_{x} + (\frac{u_{x}}{u} − \frac{1}{2(x + y)})U_{y}

− \frac{u_{x}u_{y} − v_{x}v_{y}}{u^{2}} U − \frac{v_{y}}{u} V_{x} − \frac{v_{x}}{u} V_{y},

V_{xy} = \frac{v_{y}}{u} U_{x} + \frac{v_{x}}{u} U_{y} − \frac{v_{x}u_{y} + u_{x}v_{y}}{u^{2}} U

+ (\frac{u_{y}}{u} − \frac{1}{2(x + y)})V_{x} + (\frac{u_{x}}{u} − \frac{1}{2(x + y)})V_{y}.

The ‘direct’ recursion operator for this equation seems to be missing in the literature; we can obtain it by inverting the operator (7), the result being

\begin{matrix}U′ = uvp_{1} − up_{2} + (y − x)U, \\ V′ = − \frac{1}{2} (u^{2} − v^{2})p_{1} − vp_{2} − \frac{1}{2} p_{3} + (y − x)V, \end{matrix}

where p_{1}, p_{2}, p_{3} satisfy

p_{1,x} = (x + y)(−2\frac{v_{x}}{u^{3}} U + \frac{1}{u^{2}} V_{x}),

p_{2,x} = (x + y)(−\frac{uu_{x} + 2vv_{x}}{u^{3}} U + \frac{1}{u}U_{x} + \frac{v_{x}}{u^{2}} V + \frac{v}{u^{2}} V_{x}),

p_{3,x} = (x + y)(\begin{matrix}2\frac{(uu_{x} + vv_{x})v}{u^{3}} U − 2\frac{v}{u}U_{x} \\ − 2\frac{uu_{x} + vv_{x}}{u^{2}} V + \frac{u^{2} − v^{2}}{u^{2}} V_{x}), \end{matrix}

p_{1,y} = (x + y)(2\frac{v_{y}}{u^{3}} U − \frac{1}{u^{2}} V_{y}),

p_{2,y} = (x + y)(\frac{uu_{y} + 2vv_{y}}{u^{3}} U − \frac{1}{u}U_{y} − \frac{v_{y}}{u^{2}} V − \frac{v}{u^{2}} V_{y}),

p_{3,y} = (x + y)(\begin{matrix}−2\frac{(uu_{y} + vv_{y})v}{u^{3}} U + 2\frac{v}{u}U_{y} \\ + 2\frac{uu_{y} + vv_{y}}{u^{2}} V − \frac{u^{2} − v^{2}}{u^{2}} V_{y}). \end{matrix}

It is readily seen that p_{i} are potentials of the linearizations [18] of the three obvious conservation laws of Eq. (4).

Quite unusually, neither of the recursion operators found generates an infinite series of local symmetries (and no such series is known). The action of our operators on the infinite-dimensional Geroch group of nonlocal symmetries [5, 11] remains to be investigated.

It is convenient to rewrite system (6) in triangular form. To achieve this, we introduce the Riccati pseudopotential q by

\begin{matrix}q_{x} = \frac{θ − 1}{2}v_{x}q^{2} − (θ + 1)\frac{u_{x}}{u} q − \frac{θ + 1}{2} \frac{v_{x}}{u^{2}} , \\ q_{y} = −\frac{θ − 1}{2θ}v_{y}q^{2} − \frac{θ + 1}{2θ} \frac{u_{y}}{u} q − \frac{θ + 1}{2θ} \frac{v_{y}}{u^{2}} \end{matrix}

and a nonlocal potential r by

\begin{matrix}r_{x} = (θ − 1)v_{x}q − (θ + 1)\frac{u_{x}}{u} , \\ r_{y} = −\frac{θ − 1}{θ}v_{y}q − \frac{θ + 1}{θ} \frac{u_{y}}{u} . \end{matrix}

Then the inverse recursion operator assumes the form

\begin{matrix}U′ = \frac{1}{_{√(μ − x)(μ + y)}} (2uQ − 2\frac{uq}{e^{r}} R + U), \\ V′ = \frac{1}{_{√(μ − x)(μ + y)}} (−u^{2}e^{r}P − 2u^{2}qQ + \frac{u^{2}q^{2} − 1}{e^{r}}R), \end{matrix}

where P, Q, R are supposed to satisfy

P_{x} = (θ + 1)\frac{q}{u}e^{−r}U_{x} + (\frac{θ + 1}{2} \frac{1}{u^{2}} − \frac{θ − 1}{2}q^{2})e^{−r}V_{x}

− (θ + 1)(\frac{qu_{x}}{u^{2}} + \frac{v_{x}}{u^{3}} )e^{−r}U,

Q_{x} = −\frac{θ + 1}{2} \frac{1}{u}U_{x} + \frac{θ − 1}{2}qV_{x} + \frac{θ + 1}{2} \frac{u_{x}}{u^{2}} U − \frac{θ − 1}{2}v_{x}e^{r}P,

R_{x} = \frac{θ − 1}{2}e^{r}V_{x} + (θ − 1)v_{x}e^{r}Q,

P_{y} = \frac{θ + 1}{θ} \frac{q}{u}e^{−r}U_{y} + (\frac{θ + 1}{2θ} \frac{1}{u^{2}} + \frac{θ − 1}{2θ}q^{2})e^{−r}V_{y}

− \frac{θ + 1}{θ}(\frac{qu_{y}}{u^{2}} + \frac{v_{y}}{u^{3}} )e^{−r}U,

Q_{y} = −\frac{θ + 1}{2θ} \frac{1}{u}U_{y} − \frac{θ − 1}{2θ}qV_{y} + \frac{θ + 1}{2θ} \frac{u_{y}}{u^{2}} U + \frac{θ − 1}{2θ}v_{y}e^{r}P,

R_{y} = −\frac{θ − 1}{2θ}e^{r}V_{y} − \frac{θ − 1}{θ}v_{y}e^{r}Q.

This form of the inverse recursion operator is better adapted to generation of symmetries, which is, however, beyond the scope of this paper.

Acknowledgements

I would like to thank G. Vilasi for drawing my attention to the problem. The support from the grant MSM:J10/98:192400002 is gratefully acknowledged.

References

[1] V.A. Belinskiĭ and V.E. Zakharov, Integration of the Einstein equations by means of the inverse scattering problem, Soviet Phys. JETP 75 (1978) (6) 1955–1971.

[2] F.J. Chinea, Bäcklund transformations in general relativity, in: Nonlinear Phenomena, Proc., Oaxtepec, México, 1982, Lecture Notes in Phys. 189 (Springer, Berlin, 1983) 342–353.

[3] R.K. Dodd, J. Kinoulty and H.C. Morris, Bäcklund transformation for the Ernst equation of general relativity, in: Advanced Nonlinear Waves I. (Boston, 1984) 254–272.

[4] R. Geroch, A method for generating solutions of Einstein’s equations, J. Math. Phys. 12 (1971) 918–924.

[5] R. Geroch, A method for generating new solutions of Einstein’s equations. II, J. Math. Phys. 13 (1972) 394–404.

[6] M. Gürses, A. Karasu and V.V. Sokolov, On construction of recursion operators from Lax representation, J. Math. Phys. 40 (1999) 6473–6490.

[7] G.A. Guthrie, Recursion operators and non-local symmetries, Proc. R. Soc. London A 446 (1994) 107–114.

[8] G.A. Guthrie and M.S. Hickman, Nonlocal symmetries of the KdV equation, J. Math. Phys. 34 (1993) 193–205.

[9] B.K. Harrison, Bäcklund transformation for the Ernst equation of general relativity, Phys. Rev. Lett. 41 (1978) (18) 1197–1200.

[10] A. Karasu (Kalkanli), A. Karasu and S.Yu. Sakovich, A strange recursion operator for a new integrable system of coupled Korteweg–de Vries equations, nlin.SI/0203036; Acta Appl. Math., to appear.

[11] W. Kinnersley, Symmetries of the stationary Einstein–Maxwell field equations, J. Math. Phys. 18 (1977) 1529–1537.

[12] I.S. Krasil’shchik and A.M. Vinogradov, eds., Symmetries and Conservation Laws for Differential Equations of Mathematical Physics, Translations of Mathematical Monographs 182 (American Mathematical Society, Providence, 1999).

[13] T. Lewis, Some special solutions of the equations of axially symmetric gravitational fields, Proc. Roy. Soc. London A 136 (1932) 176–192.

[14] S. Lou, Recursion operator and symmetry structure of the Kawamoto-type equation, Phys. Lett. A 181 (1993) 13–16.

[15] S. Lou and W. Chen, Inverse recursion operator of the AKNS hierarchy, Phys. Lett. A 179 (1993) 271–274.

[16] D. Maison, Are the stationary, axially symmetric Einstein equations completely integrable?, Phys. Rev. Lett. 41 (1978) (8) 521–522.

[17] D. Maison, On the complete integrability of the stationary, axially symmetric Einstein equations, J. Math. Phys. 20 (1979) (5) 871–877.

[18] M. Marvan, Another look on recursion operators, in: Differential Geometry and Applications, Proc. Conf. Brno 1995 (Masaryk University, Brno, 1997) 393–402; electronic edition in ELibEMS, www.emis.de/proceedings/6ICDGA/IV.

[19] M. Marvan, Reducibility of zero curvature representations with application to recursion operators, nlin.SI/0306006; Acta Appl. Math., to appear.

[20] M. Marvan and A. Sergyeyev, Recursion operator for the Nizhnik– Veselov–Novikov equation, J. Phys. A: Math. Gen. 36 (2003) L87–L92; nlin.SI/0210028.

[21] P.J. Olver, Evolution equations possessing infinitely many symmetries, J. Math. Phys. 18 (1977) 1212–1215.

[22] M. Omote and M. Wadati, Bäcklund transformations for the Einstein equation, in: Advanced Nonlinear Waves I. 242–253.

[23] A. Sergyeyev, On recursion operators and nonlocal symmetries of evolution equations, in: Proceedings of the Seminar on Differential Geometry (Opava, 2000), Edited by D. Krupka (Opava, Silesian University in Opava, 2000) 159–173, nlin.SI/0012011.

[24] S.Yu. Sakovich, Cyclic bases of zero-curvature representations: five illustrations to one concept, nlin.SI/0212019; Acta Appl. Math., to appear.