Binary Darboux Transformations in Bidifferential Calculus and Integrable Reductions of Vacuum Einstein Equations

We present a general solution-generating result within the bidifferential calculus approach to integrable partial differential and difference equations, based on a binary Darboux-type transformation. This is then applied to the non-autonomous chiral model, a certain reduction of which is known to ap...

Повний опис

Збережено в:

Бібліографічні деталі
Дата:	2013
Автори:	Dimakis, A., Müller-Hoissen, F.
Формат:	Стаття
Мова:	English
Опубліковано:	Інститут математики НАН України 2013
Назва видання:	Symmetry, Integrability and Geometry: Methods and Applications
Онлайн доступ:	http://dspace.nbuv.gov.ua/handle/123456789/149219
Теги:	Додати тег Немає тегів, Будьте першим, хто поставить тег для цього запису!
Назва журналу:	Digital Library of Periodicals of National Academy of Sciences of Ukraine
Цитувати:	Binary Darboux Transformations in Bidifferential Calculus and Integrable Reductions of Vacuum Einstein Equations / A. Dimakis, F. Müller-Hoissen // Symmetry, Integrability and Geometry: Methods and Applications. — 2013. — Т. 9. — Бібліогр.: 80 назв. — англ.

Репозитарії

Digital Library of Periodicals of National Academy of Sciences of Ukraine

id	irk-123456789-149219
record_format	dspace
spelling	irk-123456789-1492192019-02-22T01:24:13Z Binary Darboux Transformations in Bidifferential Calculus and Integrable Reductions of Vacuum Einstein Equations Dimakis, A. Müller-Hoissen, F. We present a general solution-generating result within the bidifferential calculus approach to integrable partial differential and difference equations, based on a binary Darboux-type transformation. This is then applied to the non-autonomous chiral model, a certain reduction of which is known to appear in the case of the D-dimensional vacuum Einstein equations with D−2 commuting Killing vector fields. A large class of exact solutions is obtained, and the aforementioned reduction is implemented. This results in an alternative to the well-known Belinski-Zakharov formalism. We recover relevant examples of space-times in dimensions four (Kerr-NUT, Tomimatsu-Sato) and five (single and double Myers-Perry black holes, black saturn, bicycling black rings). 2013 Article Binary Darboux Transformations in Bidifferential Calculus and Integrable Reductions of Vacuum Einstein Equations / A. Dimakis, F. Müller-Hoissen // Symmetry, Integrability and Geometry: Methods and Applications. — 2013. — Т. 9. — Бібліогр.: 80 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 37K10; 16E45 DOI: http://dx.doi.org/10.3842/SIGMA.2013.009 http://dspace.nbuv.gov.ua/handle/123456789/149219 en Symmetry, Integrability and Geometry: Methods and Applications Інститут математики НАН України
institution	Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection	DSpace DC
language	English
description	We present a general solution-generating result within the bidifferential calculus approach to integrable partial differential and difference equations, based on a binary Darboux-type transformation. This is then applied to the non-autonomous chiral model, a certain reduction of which is known to appear in the case of the D-dimensional vacuum Einstein equations with D−2 commuting Killing vector fields. A large class of exact solutions is obtained, and the aforementioned reduction is implemented. This results in an alternative to the well-known Belinski-Zakharov formalism. We recover relevant examples of space-times in dimensions four (Kerr-NUT, Tomimatsu-Sato) and five (single and double Myers-Perry black holes, black saturn, bicycling black rings).
format	Article
author	Dimakis, A. Müller-Hoissen, F.
spellingShingle	Dimakis, A. Müller-Hoissen, F. Binary Darboux Transformations in Bidifferential Calculus and Integrable Reductions of Vacuum Einstein Equations Symmetry, Integrability and Geometry: Methods and Applications
author_facet	Dimakis, A. Müller-Hoissen, F.
author_sort	Dimakis, A.
title	Binary Darboux Transformations in Bidifferential Calculus and Integrable Reductions of Vacuum Einstein Equations
title_short	Binary Darboux Transformations in Bidifferential Calculus and Integrable Reductions of Vacuum Einstein Equations
title_full	Binary Darboux Transformations in Bidifferential Calculus and Integrable Reductions of Vacuum Einstein Equations
title_fullStr	Binary Darboux Transformations in Bidifferential Calculus and Integrable Reductions of Vacuum Einstein Equations
title_full_unstemmed	Binary Darboux Transformations in Bidifferential Calculus and Integrable Reductions of Vacuum Einstein Equations
title_sort	binary darboux transformations in bidifferential calculus and integrable reductions of vacuum einstein equations
publisher	Інститут математики НАН України
publishDate	2013
url	http://dspace.nbuv.gov.ua/handle/123456789/149219
citation_txt	Binary Darboux Transformations in Bidifferential Calculus and Integrable Reductions of Vacuum Einstein Equations / A. Dimakis, F. Müller-Hoissen // Symmetry, Integrability and Geometry: Methods and Applications. — 2013. — Т. 9. — Бібліогр.: 80 назв. — англ.
series	Symmetry, Integrability and Geometry: Methods and Applications
work_keys_str_mv	AT dimakisa binarydarbouxtransformationsinbidifferentialcalculusandintegrablereductionsofvacuumeinsteinequations AT mullerhoissenf binarydarbouxtransformationsinbidifferentialcalculusandintegrablereductionsofvacuumeinsteinequations
first_indexed	2025-07-12T21:06:53Z
last_indexed	2025-07-12T21:06:53Z
_version_	1837476790583427072
fulltext	Symmetry, Integrability and Geometry: Methods and Applications SIGMA 9 (2013), 009, 31 pages Binary Darboux Transformations in Bidifferential Calculus and Integrable Reductions of Vacuum Einstein Equations Aristophanes DIMAKIS † and Folkert MÜLLER-HOISSEN ‡ † Department of Financial and Management Engineering, University of the Aegean, 82100 Chios, Greece E-mail: dimakis@aegean.gr ‡ Max-Planck-Institute for Dynamics and Self-Organization, 37077 Göttingen, Germany E-mail: folkert.mueller-hoissen@ds.mpg.de Received November 12, 2012, in final form January 29, 2013; Published online February 02, 2013 http://dx.doi.org/10.3842/SIGMA.2013.009 Abstract. We present a general solution-generating result within the bidifferential calcu- lus approach to integrable partial differential and difference equations, based on a binary Darboux-type transformation. This is then applied to the non-autonomous chiral model, a certain reduction of which is known to appear in the case of the D-dimensional vacuum Einstein equations with D − 2 commuting Killing vector fields. A large class of exact so- lutions is obtained, and the aforementioned reduction is implemented. This results in an alternative to the well-known Belinski–Zakharov formalism. We recover relevant examples of space-times in dimensions four (Kerr-NUT, Tomimatsu–Sato) and five (single and double Myers–Perry black holes, black saturn, bicycling black rings). Key words: bidifferential calculus; binary Darboux transformation; chiral model; Einstein equations; black ring 2010 Mathematics Subject Classification: 37K10; 16E45 1 Introduction The bidifferential calculus formalism aims to understand integrability features and solution- generating methods for (at least a large class of) integrable partial differential or difference equations (PDDEs) resolved from the particularities of examples, i.e., on an as far as possible universal level [16, 17, 19]. The most basic ingredient is a graded associative algebra, supplied with two anti-commuting graded derivations of degree one. It can and should be regarded as a generalization (in the spirit of noncommutative geometry) of the algebra of differential forms on a manifold, but supplied with two analogs of the exterior derivative. Once a PDDE is translated to this framework, it is simple to elaborate its integrability conditions. In fact, it could not be simpler. In this framework, Darboux transformations (see [35,54,62] and the references therein) have first been addressed in [19]. In the latter work, we had also obtained a very simple solution gene- rating result that evolved into improved versions in recent applications [15, 18, 20]. Its relation with Darboux transformations has been further clarified in [15] (see Appendix A therein), an essential step toward the much more general result that we present in this work. The resulting class of solutions is expressed in a universal way, in the sense that the corresponding formula holds simultaneously for all integrable PDDEs possessing a bidifferential calculus formulation. Choosing a bidifferential calculus associated with a specific PDDE, we can generate infinite families of (soliton-like) solutions. mailto:dimakis@aegean.gr mailto:folkert.mueller-hoissen@ds.mpg.de http://dx.doi.org/10.3842/SIGMA.2013.009 2 A. Dimakis and F. Müller-Hoissen In Section 2 we recall some basics of bidifferential calculus and set up the stage in a concise way to formulate our general result about Darboux transformations. This hardly requires pre- vious knowledge and can be taken as an independent and self-contained step into the world of integrable PDDEs. That this is indeed a very powerful tool, is demonstrated in the rest of this work, where we concentrate on one of the more tricky examples of integrable PDEs. In Section 3 we elaborate in detail the example of the non-autonomous chiral model equation( ρgzg −1) z + ε ( ρgρg −1) ρ = 0 (1.1) for an m ×m matrix g, where ρ > 0 and z are independent real variables and ε = ±1.1 With ε = 1, this governs the case of the stationary, axially symmetric vacuum Einstein (m = 2) and Einstein–Maxwell (m = 3) equations in four dimensions, where we have two commuting Killing vector fields, one spacelike and the other one asymptotically timelike (see, e.g., [2,3,48,68]). In an analogous way, (1.1) with m > 3 appears in the dimensional reduction of (the bosonic part of) higher-dimensional supergravity theories to two dimensions (see, e.g., [5, 25, 30, 36, 45, 73]). With ε = −1, the above equation appears in the case of vacuum solutions of Einstein’s equations in four dimensions with two commuting spacelike Killing vector fields [2,4,34,77], describing in particular cylindrical gravitational waves. In Section 3, we also present a reduction condition that, imposed on the obtained family of solutions, achieves that g is symmetric. Any such real and symmetric g determines a solution of the vacuum Einstein equations in m + 2 dimensions, as recalled in Section 4. The resulting recipe to construct solutions of the vacuum Einstein equations is close to the familiar Belinski– Zakharov method [2–4], which has been applied in numerous publications. In a sense, what we obtained is a kind of matrix version of the latter. A well-known reformulation of the equations obtained from the integrable reduction of the four-dimensional vacuum Einstein (and also the Einstein–Maxwell) equations by introduction of the so-called twist potential (a crucial step toward the Ernst equation), connects space-time metrics in a different way with the non-autonomous chiral model (see, e.g., [58]). Solutions of the latter are then required to have a constant determinant. This allows a constant seed and thus an application of the restricted solution-generating result in [15] (also see Remark 4.3 below). It does not work in the more direct approach we take in the present work, but here we resolve the restriction in [15]. In Section 4 we elaborate several examples in four and five space-time dimensions, and show that the resulting metrics include important solutions of Einstein’s equations, in particular some of the more recently found black objects in five dimensions, which have no counterpart in four dimensions (see, e.g., [25, 42,45,73]). Here we used Mathematica2 in a substantial way. Section 5 contains some concluding remarks. 2 Binary Darboux transformations in bidifferential calculus A graded associative algebra is an associative algebra Ω over C with a direct sum decomposi- tion Ω = ⊕ r≥0 Ωr into a subalgebra A := Ω0 and A-bimodules Ωr, such that ΩrΩs ⊆ Ωr+s. A bidifferential calculus (or bidifferential graded algebra) is a unital graded associative algebra Ω, equipped with two (C-linear) graded derivations3 d, d̄ : Ω→ Ω of degree one (hence dΩr ⊆ Ωr+1, d̄Ωr ⊆ Ωr+1), with the properties d2 = d̄2 = dd̄ + d̄d = 0. (2.1) 1A subscript indicates a partial derivative with respect to the corresponding variable. Formally, ρ 7→ iρ relates the two values of ε in (1.1). 2Mathematica Edition: Version 8, Wolfram Research, Inc., Champaign, Illinois, 2010. 3Hence d and d̄ both satisfy the graded Leibniz rule d(χχ′) = (dχ)χ′ + (−1)rχdχ′, for all χ ∈ Ωr and χ′ ∈ Ω. Binary Darboux Transformations and Vacuum Einstein Equations 3 In the bidifferential calculus approach to integrable PDDEs we are looking for a choice (Ω, d, d̄) and some φ ∈ A, or some invertible g ∈ A, such that either d̄dφ = dφdφ (2.2) or d [ (d̄g)g−1 ] = 0 (2.3) is equivalent to a certain PDDE. The two equations are related by the Miura equation (d̄g)g−1 = dφ, (2.4) which establishes a kind of Miura transformation4 between the two equations (2.2) and (2.3). This equation has both, (2.2) and (2.3), as integrability conditions. As a consequence, if we find a solution pair (φ, g) of (2.4), then φ solves (2.2) and g solves (2.3). In the following theorem, we formulate a solution-generating result which amounts to a trans- formation that takes a given solution (φ0, g0) of the Miura equation (2.4) to a new solution of (2.4), (φ0, g0) 7→ (φ, g). This induces corresponding transformations of solutions of (2.2), respectively (2.3). Let nowA be the algebra of all finite-dimensional5 matrices, with entries in a unital algebra B. The product of two matrices is defined to be zero if the sizes of the two matrices do not match. In the following we assume that there is a graded algebra Ω with Ω0 = A, and a bidifferential calculus (Ω,d, d̄), and such that d and d̄ preserve the size of matrices. I = Im and I = In denote the m×m, respectively n×n, identity matrix, and we assume that they are annihilated by d and d̄. Furthermore, Mat(m,n,B) denotes the set of m× n matrices over B. Theorem 2.1. Let φ0, g0 ∈ Mat(m,m,B) solve the Miura equation (2.4). Let P ,Q ∈ Mat(n, n,B) be invertible solutions of d̄P = (dP )P , d̄Q = QdQ, (2.5) which are independent in the sense that QY = Y P implies Y = 0. Let U ∈ Mat(m,n,B) and V ∈ Mat(n,m,B) be solutions of the linear equations6 d̄U = (dU)P + (dφ0)U , d̄V = QdV − V dφ0. (2.6) Furthermore, let X ∈ Mat(n, n,B) be an invertible solution of the Sylvester-type equation XP −QX = V U . (2.7) Then φ = φ0 + UX−1V , g = ( I + U(QX)−1V ) g0 (2.8) solve the Miura equation (2.4), and thus also (2.2), respectively (2.3). 4The original Miura transformation maps solutions of the modified Korteweg–de Vries equation to solutions of the Korteweg–de Vries equation. 5An extension to a suitable class of ∞-dimensional matrices, respectively operators, is certainly possible. 6If we are primarily interested in solving (2.3), it is more convenient to replace dφ0 by (d̄g0)g−1 0 in these equations, by use of the Miura equation for (φ0, g0). 4 A. Dimakis and F. Müller-Hoissen Proof. Acting on (2.7) with d̄, using the rules of bidifferential calculus, (2.5), (2.6), and (2.7) again, we find Q[d̄X − (dX)P + (dQ)X + (dV )U ] = [d̄X − (dX)P + (dQ)X + (dV )U ]P . Since P and Q are assumed to be independent, this implies7 d̄X − (dX)P + (dQ)X + (dV )U = 0. (2.9) Using also (2.5) and (2.7), we easily deduce that d̄(QX)−1 = −X−1 [ (dX)X−1(XP )− (dV )U ] (QX)−1 = (dX−1) [ I + V U(QX)−1 ] + X−1(dV )U(QX)−1. This is then used in the elaboration of d̄g = d̄ [ (I + U(QX)−1V )g0 ] via the graded derivation (Leibniz) rule for d̄. We eliminate all further terms involving a d̄ with the help of (2.6), and apply (2.7) to eliminate a P in favor of a Q. Finally we obtain d̄g = (dφ)g with φ given by (2.8). Our assumptions ensure that the inverse of g exists. It is given by g−1 = g−10 ( I −U(XP )−1V ) . The integrability conditions of (2.6) and (2.9) are satisfied as a consequence of (2.5) and d̄dφ0 = dφ0dφ0 (which follows from (d̄g0)g −1 0 = dφ0). � For a reader acquainted with corresponding results about binary Darboux transformations in the literature on integrable systems, the similarity will be evident (see, e.g., [12, 35, 54, 59, 62]). It is close to results in [63–65]8. A new feature, however, is the dependence of the linear equations (2.6) on solutions P and Q of the nonlinear equations (2.5). This generalization is crucial for our application to the Einstein equations. As already mentioned in the introduction, Theorem 2.1 considerably generalizes previous, more restricted results, see in particular [15]. Remark 2.1. As a consequence of (2.5), P and −Q solve the n× n version of (2.2). Since we assume that they are invertible, P and Q−1 solve the n× n version of (2.3). If U is a solution of the first of (2.6), then also UP , by use of the first of (2.5). If V is a solution of the second of (2.6), then also QV , by use of the second of (2.5). Remark 2.2. By direct computations one verifies the following “scaling property”. Let U and V solve (2.6) and let ĝ0 = wg0, Û = UW 1, V̂ = W 2V , with a solution (ϕ,w) of the m = 1 version of (2.4), and invertible W i ∈ Mat(n, n,B), subject to [W 1,P ] = 0, [W 2,Q] = 0. We further assume that ϕ and w are in the center of Ω.9 If W 1 and W 2 satisfy d̄W 1 = (dW 1)P + (d̄w)w−1W 1, d̄W 2 = QdW 2 −W 2(d̄w)w−1, 7It is a key feature of Theorem 2.1 that this equation is automatically solved if P and Q are independent. Relaxing the latter assumption, the theorem remains valid if this equation is added to the assumptions. 8The precise relation has still to be clarified. 9This holds, e.g., in the case of the non-autonomous chiral model treated in Section 3. But this assumption will be a (perhaps too) severe restriction if B involves differential or difference operators. Binary Darboux Transformations and Vacuum Einstein Equations 5 then Û and V̂ solve (2.6), with φ0 replaced by φ̂0 = φ0 + ϕI. Furthermore, X̂ = W 2XW 1 solves (2.7) with U , V replaced by Û , V̂ . For the new solutions of (2.4) (and thus (2.2), respectively (2.3)), determined by the theorem, we find φ̂ = φ+ ϕI, ĝ = wg, where φ and g are given by (2.8). Hence, under the stated conditions, multiplication of g0 by a “scalar” solution w of (2.3) simplify amounts to multiplication of g by w. In the special case where w is the identity element and g thus remains unchanged, the above equations for W 1 and W 2 still allow non-trivial transformations of U and V . These results will be used in Section 4. Remark 2.3. Let (φi, gi), i = 1, 2, be two solutions obtained via Theorem 2.1 from the same seed solution (φ0, g0). Let (P i,Qi,U i,V i) be corresponding solutions of (2.5) and (2.6). Then P = ( P 1 0 0 P 2 ) , Q = ( Q1 0 0 Q2 ) , U = (U1,U2), V = ( V 1 V 2 ) also solve (2.5) and (2.6). If P and Q are independent and if (2.7) has an invertible solution X, then (2.8) determines a new solution of (2.4), and thus new solutions of (2.2) and (2.3). This expresses a nonlinear superposition principle. By iteration, one can build superpositions of an arbitrary number of “elementary” solutions, hence (analogs of) “multi-solitons”. Remark 2.4. We note that the first of equations (2.6) is a special case (and n × n matrix version) of the general linear equation d̄ψ − Aψ = dψP, where A = dφ. (2.10) Here P is a solution of the nonlinear equation d̄P = (dP )P. An n× n matrix version of it appears in (2.5). The integrability condition of (2.10) is (2.2). If we set A = (d̄g)g−1, the integrability condition is (2.3) instead. A similar statement holds for the second of equations (2.6), the “adjoint linear system”. The Darboux transformation (2.8) is “binary” since it involves solutions of the linear system as well as of the adjoint. Remark 2.5. In the special case where Ω is the algebra of matrix-valued differential forms on a manifold, and d the exterior derivative, any tensor field N of type (1, 1) with vanishing Nijenhuis torsion determines a map d̄N satisfying the above conditions [7, 31]. Moreover, ac- cording to Frölicher–Nijenhuis theory [31], any d̄ such that (2.1) holds has to be of this form. Finite-dimensional integrable systems have been considered in this framework in [13]. The ge- neralization to Lie algebroid structures is nicely described in [6]. See also [1, 51, 52] for related aspects. We should stress, however, that in our central examples we depart from differential geometry and consider a differential calculus in a weaker sense (e.g., of noncommutative geo- metry). The setting of the above theorem allows in principle structures far away from classical calculus and in particular differential geometry, then dealing with equations beyond differential and difference equations. A somewhat more restricted framework is given by setting Ω = A⊗ ∧( CN ) , where ∧ (CN ) denotes the exterior (Grassmann) algebra of the vector space CN . In this case it is sufficient to define suitable operators d and d̄ on A, since they extend to Ω in an evident way. Many integrable PDDEs have been treated in this framework and in the next section we turn to an important example. 6 A. Dimakis and F. Müller-Hoissen 3 Solutions of the non-autonomous chiral model The non-autonomous chiral model is well-known to be a reduction of the (anti-) self-dual Yang– Mills equations, for which a very simple bidifferential calculus exists [19] (that may actually be considered as a prototype). From the latter one can then derive a bidifferential calculus for (1.1). It is determined by df = −fzξ1 + eθ ( fρ − ρ−1fθ ) ξ2, d̄f = e−θ ( fρ + ρ−1fθ ) ξ1 + εfzξ2 for f ∈ C∞(R3) (cf. [15]). ξ1, ξ2 is a basis of ∧1(C2). Choosing B = C∞(R3), d and d̄ extend to A⊗ ∧ (C2) via d(f1ξ1 + f2ξ2) = (df1) ∧ ξ1 + (df2) ∧ ξ2 and d(fξ1 ∧ ξ2) = (df) ∧ ξ1 ∧ ξ2 = 0, and correspondingly for d̄. For an m×m matrix-valued function g, (2.3) now takes the form( ρgzg −1) z + ε ( ρgρg −1) ρ − [( gρ + ρ−1gθ ) g−1 ] θ + ( gθg −1) ρ = 0, which reduces to (1.1) if g does not depend on θ. The coordinate θ is needed to have the prop- erties of a bidifferential calculus, but addressing (1.1) we are primarily interested in equations for objects that do not depend on it. Using this bidifferential calculus, the Miura equation (2.4) decomposes into( gρ + ρ−1gθ ) g−1 = −eθφz, gzg −1 = εeθ ( φρ − ρ−1φθ ) . If g is θ-independent, this requires φ = e−θφ̃, (3.1) with θ-independent φ̃, and then reduces to gρg −1 = −φ̃z, εgzg −1 = φ̃ρ + ρ−1φ̃. (3.2) In the following, we elaborate Theorem 2.1 using the above bidifferential calculus. Section 3.1 provides the complete solution of the two nonlinear equations (2.5) under the condition that P (respectively Q) has geometrically simple spectrum (i.e., for each eigenvalue there is a unique corresponding Jordan block in the Jordan normal form), also see [15]. In this case the two equations actually coincide. Then it only remains to solve linear equations, see Section 3.2 and Section 3.3 below. In Section 3.4 we present a condition to be imposed on the data in order to achieve that the solution g is symmetric (or Hermitian). This reduction is crucial in the context of Einstein’s equations, see Section 4. Most of the following is, however, independent of additional assumptions and provides a general procedure to construct solutions of the non- autonomous chiral model. The corresponding equations cannot be solved explicitly without some restrictions, in particular on the form of the seed solution. We content ourselves with providing illustrative and important examples relevant in the context of gravity in Section 4. 3.1 The equations for P and Q In terms of P̃ := eθP , the first of the two equations (2.5) decomposes into the following pair of equations, P̃ ρ + ρ−1 ( P̃ θ − P̃ ) = −P̃ zP̃ , εP̃ z = [ P̃ ρ − ρ−1 ( P̃ θ − P̃ )] P̃ , (3.3) Binary Darboux Transformations and Vacuum Einstein Equations 7 which are autonomous in the variable θ. Assuming that P̃ and I + εP̃ 2 are invertible, and that P̃ is θ-independent, (3.3) implies P̃ 2 − 2ρ−1(zI −A)P̃ − εI = 0, (3.4) with an arbitrary constant n × n matrix A (also see [15]). This is a matrix version of the pole trajectories in the Belinski–Zakharov approach [2,4]. A well-known symmetry of the latter extends to the matrix case: (3.4) is invariant under P̃ 7→ −εP̃−1. For the following result, see Lemma 4.1 in [15]. Lemma 3.1. Any θ-independent, invertible solution of (3.4), which commutes with its deriva- tives with respect to ρ and z, solves (3.3). A can be taken in Jordan normal form A = block-diag(An1 , . . . ,Ans), without restriction of generality, and a solution of (3.4) is then given by P̃ = block-diag(P̃ n1 , . . . , P̃ ns), where the block P̃ ni is a solution of (3.4) with A replaced by the Jordan block Ani , see the next examples. Under the assumption that P̃ has geometrically simple spectrum, this is the most general solution of d̄P = (dP )P with θ-independent P̃ = eθP [15]. Example 3.1. If A is diagonal, i.e., A = diag(a1, . . . , an) with constants ai, corresponding solutions of (3.4) are given by P̃ = diag(p̃1, . . . , p̃n), where p̃i is any of pi = ρ−1(z − ai + Ri), p̄i = ρ−1(z − ai −Ri), (3.5) with Ri = √ (z − ai)2 + ερ2. Note that p̄i = −ε/pi. Remark 3.1. For better comparison with the relevant literature, we will sometimes write pi = ρ µi , p̄i = ρ µ̄i , where µi = √ (z − ai)2 + ερ2 − (z − ai), µ̄i = − √ (z − ai)2 + ερ2 − (z − ai). (3.6) µi and µ̄i are often referred to as (indicating the presence of) a soliton and an anti-soliton, respectively. Note that µ̄i = −ερ2/µi. If ε = 1, then µi is non-negative and only vanishes on a subset of {ρ = 0}. Example 3.2. For an r × r Jordan block Ar = aIr + N r, N r =  0 1 0 · · · 0 0 0 1 . . . ... ... . . . . . . ... ... . . . 1 0 · · · · · · · · · 0  , (3.4) has the solutions (see [15]) P̃ r = ρ−1 ( zIr −Ar + r−1∑ k=0 ( 1/2 k ) (±R)1−2k [ 2(a− z)N r + N2 r ]k) , 8 A. Dimakis and F. Müller-Hoissen where R = √ (z − a)2 + ερ2. This is an upper-triangular Toeplitz matrix and thus commutes with its derivatives. In particular, we have P̃ 1 = p̃ = ρ−1[z − a±R] and P̃ 2 = p̃ ( 1 ∓R−1 0 1 ) , P̃ 3 = p̃ 1 ∓R−1 ± ε 2ρR −3p̃−1 0 1 ∓R−1 0 0 1  . For the solutions P̃ obtained in this way, and thus P , we have d̄P = (dP )P = P (dP ), so that they also provide us with solutions Q = e−θQ̃, with θ-independent Q̃, of the second of (2.5), also see [15]. Recall that on the way to the above results we assumed that P̃ and I + εP̃ 2 are invertible, and then also Q̃ and I + εQ̃ 2 . These assumption will also be made throughout in the following. Remark 3.2. The source matrix A for P̃ and the corresponding source matrix A′ for Q̃ can be assumed to be simultaneously in Jordan normal form, without restriction of generality. Since this is achieved by similarity transformations with constant transformation matrices, the latter can be absorbed by redefinitions that restore all the equations obtained from Theorem 2.1. But in general this will no longer be so if we impose a reduction condition that relates P̃ and Q̃, as in Section 3.4 below. 3.2 The equations for U and V Setting φ0 = e−θφ̃0 with a θ-independent φ̃0 (cf. (3.1)), (2.6) is autonomous in the variable θ. Assuming that U , V are θ-independent, and using the fact that φ̃0 has to solve the Miura equations (3.2) together with some g0, we obtain the following systems of linear differential equations for U and V , Uρ = ( g0,ρg −1 0 U − g0,zg−10 UP̃ )( I + εP̃ 2)−1 , U z = ( g0,zg −1 0 U + εg0,ρg −1 0 UP̃ )( I + εP̃ 2)−1 , (3.7) and V ρ = ( I + εQ̃ 2)−1( Q̃V g0,zg −1 0 − V g0,ρg −1 0 ) , V z = − ( I + εQ̃ 2)−1( V g0,zg −1 0 + εQ̃V g0,ρg −1 0 ) . (3.8) These equations have to be solved for the given seed solution g0 of (2.3). For diagonal g0, this is done in the next example for the V -equations. Similar results are easily obtained for the U -equations. Example 3.3. Let Q̃ = diag(p̃1, . . . , p̃n), g0 = diag(w1, . . . , wm), where p̃i is either pi or p̄i in (3.5), and wα is a non-vanishing solution of the scalar (i.e., m = 1) version [ρ(lnw)ρ]ρ = −ε[ρ(lnw)z]z (3.9) Binary Darboux Transformations and Vacuum Einstein Equations 9 of the non-autonomous chiral model (1.1). Writing V = (Viα), where i = 1, . . . , n and α = 1, . . . ,m, (3.8) reads (lnViα)ρ = 1 1 + εp̃2i (p̃i(lnwα)z − (lnwα)ρ) , (lnViα)z = − 1 1 + εp̃2i ((lnwα)z + εp̃i(lnwα)ρ) . Let us list some simple solutions. If wα is constant, then also Viα. If p̃i = pi, then Viα = kiα { (ρpi) −1/2 if wα = ρ, (ρpi) 1/2 ( 1 + εp−1i p−1α ) if wα = pα, where pα shall be given by the same expression as some pi, but with in general different constant, say a′α. kiα is an arbitrary constant. If p̃i = p̄i, then Viα = kiα { (pi/ρ)1/2 if wα = ρ, (pi/ρ)1/2(1 + εpipα)−1 if wα = pα. More complicated solutions are now obtained by noting the following. • If Viα is a solution for wα, then V −1iα is a solution for w−1α . • If wα is the product of two solutions of (3.9), then Viα is the product of the respective solutions for the factors. 3.3 The Sylvester equation and the solution formula Recalling that U , V and Q̃ are θ-independent, the formula for (θ-independent) g in (2.8) requires X = eθX̃ with θ-independent X̃. (2.7) becomes the θ-independent Sylvester equation X̃P̃ − Q̃X̃ = V U . (3.10) If spec(P̃ ) ∩ spec(Q̃) = ∅, then (3.10) has a unique solution, for any choice of the matrices on the right hand side. The two matrices P and Q are then independent, hence Theorem 2.1 implies that10 g = ( I + U ( Q̃X̃ )−1 V ) g0 (3.11) solves the non-autonomous chiral model equation (1.1). Obviously, scaling U or V with an arbitrary non-zero constant leaves g invariant. We recall from [15] (see Remark 4.4 therein) that det g = det P̃ det Q̃ det g0. (3.12) Example 3.4. Let P̃ be diagonal, as in Example 3.1, and also Q̃, with eigenvalues q̃i, given by an expression of the same form as p̃i. If they have no eigenvalue in common, i.e., {p̃i}∩{q̃i} = ∅, then the unique solution of (3.10) is given by the Cauchy-like matrix X̃ij = (V U)ij p̃j − q̃i . 10We have to choose Q̃ invertible and make sure that the solution X̃ of (3.10) is invertible. See [14, 38] for conditions that guarantee the latter. 10 A. Dimakis and F. Müller-Hoissen A vast literature exists on solutions of the Sylvester equation (3.10), more generally with non-diagonal matrices P̃ and Q̃ (and not necessarily satisfying the spectrum condition that guarantees a unique solution). Proposition 3.1 ( [14,37,44]). Let spec(P̃ ) ∩ spec(Q̃) = ∅ and P(λ) = n∑ k=0 Pkλ k be the characteristic polynomial of P̃ . Then the unique solution of the Sylvester equation (3.10) is given by X̃ = −P(Q̃)−1 n∑ k=1 Pk k−1∑ i=0 Q̃ k−1−i V UP̃ i . (3.13) Remark 3.3. (2.9) takes the form X̃ρ + ρ−1X̃ + X̃zP̃ − Q̃zX̃ − V zU = 0, εX̃z − ( X̃ρ − ρ−1X̃ ) P̃ + ( Q̃ρ + ρ−1Q̃ ) X̃ + V ρU = 0. (3.14) If we drop the spectrum condition for P̃ and Q̃, these equations also have to be solved. Otherwise they are a consequence of our assumptions (see the proof of Theorem 2.1). (3.14) will only be used in the proof of Proposition 4.1 in Appendix B. Remark 3.4. Using the above results, we also obtain solutions φ of (2.2), given by the expression in (2.8), which in the case under consideration and via (3.1) takes the form εφ̃zz + ( φ̃ρ + ρ−1φ̃ ) ρ = [ φ̃ρ + ρ−1φ̃, φ̃z ] (which corrects a typo in (4.3) of [15]). 3.4 A reduction condition It is of particular interest (see Section 4) to find a convenient condition which guarantees that the solution matrix g given by (3.11) is symmetric, i.e., gᵀ = g, where ᵀ means matrix transpose. The following result is easily verified by a direct computation. Lemma 3.2. Let P̃ ᵀ = −εQ̃−1 and gᵀ0 = g0. If V solves (3.8), then U = (V g0) ᵀ solves (3.7). Proposition 3.2. Let P̃ ᵀ = −εQ̃−1, spec(P̃ )∩ spec(Q̃) = ∅, gᵀ0 = g0, and U = (V g0) ᵀ. Then g given by (3.11) is symmetric. Proof. Using gᵀ0 = g0 and U = (V g0) ᵀ, the Sylvester equation (3.10) and its transpose lead to Q̃ [ Q̃X̃ − (Q̃X̃)ᵀ ] = Q̃ [ X̃P̃ − P̃ ᵀ X̃ ᵀ] = [ Q̃X̃ + εX̃ ᵀ P̃ −1] P̃ = [ Q̃X̃ − (Q̃X̃)ᵀ ] P̃ . In the last two steps we used P̃ ᵀ = −εQ̃−1. Now the spectrum condition implies Q̃X̃ = (Q̃X̃)ᵀ. Together with gᵀ0 = g0 and U = (V g0) ᵀ, this shows that g given by (3.11) is symmetric. � Remark 3.5. Lemma 3.2 and Proposition 3.2 also hold with transposition replaced by any involutory anti-automorphism of the matrix algebra, hence in particular for Hermitian conju- gation. The Hermitian reduction of the non-autonomous chiral model appears in particular in the context of the (electro-vacuum) Einstein–Maxwell equations in four dimensions with two commuting Killing vector fields (also see [15]). Binary Darboux Transformations and Vacuum Einstein Equations 11 In terms of the matrix Γ := −QX̃ = ε ( P̃ ᵀ)−1 X̃, (3.15) which is symmetric under the assumptions of Proposition 3.2 (see the proof of Proposition 3.2), the Sylvester equation (3.10) takes the form of a Stein equation, Γ + εP̃ ᵀ ΓP̃ = V g0V ᵀ. (3.16) Implementing the assumptions of Proposition 3.2 in the solution formula (3.11), we have g = ( I − g0V ᵀΓ−1V ) g0. (3.17) If P̃ is diagonal, i.e., P̃ = diag(p̃1, . . . , p̃n), and if p̃ip̃j 6= −ε for all i, j, then the solution of (3.16) is given (via Example 3.4) by Γij = ε (V g0V ᵀ)ij p̃ip̃j + ε . (3.18) This is essentially the corresponding matrix (usually denoted by Γ) in the Belinski–Zakharov method [2, 3]. Remark 3.6. From Remark 2.3 we deduce the following superposition result. Let (P̃ i,V i), i = 1, . . . , N , be solutions of (3.3) and (3.8) with Q̃i = −ε(P̃ ᵀi )−1, for the same seed solution g0. Then P̃ = block-diag(P̃ 1, . . . , P̃N ) and V = (V ᵀ1, . . . ,V ᵀ N )ᵀ solve (3.3) and (3.8) with Q̃ = −ε(P̃ ᵀ)−1. If spec(P̃ )∩ spec(−ε(P̃ ᵀ)−1) = ∅, and if (3.16) has an invertible solution Γ, then (3.17) is again a symmetric solution of (1.1). 4 Solutions of the vacuum Einstein equations In D dimensions, let us consider a space-time metric of the form ds2 = gαβdx αdxβ + f ( εdρ2 + dz2 ) , where the (real) components gαβ, α, β = 1, . . . ,m, and the function f only depend on the coordinates ρ and z (and thus not on x1, . . . , xm). The metric then obviously admits m = D−2 commuting Killing vector fields11. For the next result, see, e.g., [2, 5, 25,73]12. If (1) the matrix g = (gαβ) satisfies det g = −ερ2, (2) g solves the m×m non-autonomous chiral model equation (1.1), (3) f is a solution of the compatible system of linear equations (ln f)ρ = −1 ρ + 1 4ρ tr ( U2 − εV2 ) , (ln f)z = 1 2ρ tr(UV), (4.1) where U := ρgρg −1 and V := ρgzg −1, 11It is not the most general metric admitting m = D − 2 commuting Killing vector fields. See, e.g., [2]. 12There are also reductions of the Einstein vacuum equations to the non-autonomous chiral model equation using a non-Abelian Lie algebra of Killing vector fields. See the case with a null (i.e., lightlike, or isotropic) Killing vector field in four dimensions treated in [67]. 12 A. Dimakis and F. Müller-Hoissen then the metric is Ricci-flat, hence a solution of the vacuum Einstein equations (with vanishing cosmological constant). Since g has to be real and symmetric, corresponding conditions have to be imposed on the matrix data of the class of solutions obtained in Section 3, so that these solutions determine Ricci- flat metrics (also see [15]). Such conditions have been found in Section 3.4, and, accordingly, in all examples of this section we shall set Q̃ = − ( P̃ −1)ᵀ , U = (V g0) ᵀ. (4.2) Typically we will regard these equations as defining Q̃ and U in terms of P̃ , V and g0. In most of the examples below, P̃ is diagonal. The (symmetric) solution of the non-autonomous chiral model is then given by (3.17) with Γ in (3.18). If P̃ is not diagonal, we have to proceed via the solution (3.13) of the Sylvester equation (still assuming that the spectrum condition holds) and (3.15). According to the following remark, the determinant condition (1) can always be achieved if m is odd. There is a slight restriction if m is even. Remark 4.1. By taking the trace of (1.1), one finds that det g and any power of it is a solution of (1.1) in the scalar case (also see [2,4]). We further note that ĝ = wg, with any non-vanishing solution w of the scalar equation, is again a solution of (1.1). For a solution g given by (3.17), using (3.12) and (4.2) we find that w = ( ρ2 (det P̃ )2(−1)n+1εdet g0 )1/m achieves that det ĝ = −ερ2. Since w has to be real, for even m this requires (−1)n(−εdet g0) > 0. In the next subsection, we address (4.1). Then we sketch a useful procedure to construct non-diagonal metrics from diagonal ones in such a way that the diagonal metric is recovered by setting some parameters to zero. A collection of relevant examples in four and five space-time dimensions follows. The method is indeed of most interest for D = 4 and D = 5. The higher the number of dimensions, the more restrictive is the assumption of D − 2 commuting Killing vector fields for the set of solutions of the vacuum Einstein equations. 4.1 Solutions of the equations for the metric function f Example 4.1. For a diagonal solution g0 of (1.1), so that (g0)αα solves (3.9), (4.1) leads to f0 = κρ−1 m∏ α=1 fα, where κ is an arbitrary constant and fα has to be a solution of (ln f)ρ = ρ 4 ( (lnw)2ρ − ε(lnw)2z ) , (ln f)z = ρ 2 (lnw)ρ(lnw)z, (4.3) with w replaced by (g0)αα. If w is a constant, then also f. Let us write f[w] for the solution of the above equations for a given solution w of (3.9), and let ∝ denote equality up to a non-zero constant factor. In particular, we have f[µ̃i] ∝ µ̃i√ µ̃2i + ερ2 , where µ̃i = ± √ (z − ai)2 + ερ2 − (z − ai) Binary Darboux Transformations and Vacuum Einstein Equations 13 (cf. (3.6)). More generally, we find f [ ρk µ̃1 · · · µ̃r µ̃′1 · · · µ̃′s ] ∝ ρk2/4 ( µ̃1 · · · µ̃r µ̃′1 · · · µ̃′s )k/2( r∏ i=1 f[µ̃i] ) s∏ j=1 f[µ̃′j ] (∏ i<k F[µ̃i, µ̃k] ) × ∏ j<l F[µ̃′j , µ̃ ′ l]  r∏ i s∏ j F[µ̃i, µ̃ ′ j ] −1, where µ̃′i is µ̃i with the constant ai replaced by some constant a′i, and we introduced the abbre- viation F[µ̃i, µ̃j ] = µ̃iµ̃j µ̃iµ̃j + ερ2 = µ̃i − µ̃j 2(ai − aj) , where the last equality holds if ai 6= aj . The µ̃i (and also the µ̃′i) need not be distinct in the above formula. Using these results in our formula for f0, we have the solution of (4.1) for a large class of diagonal solutions of (1.1). See Appendix A for some details, and also [3]. For the proofs of the following propositions, see Appendix B. Proposition 4.1. Let f0 be a solution of (4.1) for a seed g0 (solution of the Miura equation). Let g be a corresponding solution of the non-autonomous chiral model from the family obtained in Section 3 (i.e., given by (3.11), where the ingredients are subject to the respective equations). Then f = κf0ρ −n det P̃ det Q̃det X̃[ det ( I + εP̃ 2) det ( I + εQ̃ 2)]1/2 , with an arbitrary constant κ, solves (4.1) with the right hand sides evaluated with g. Proposition 4.1 does neither assume that g0 and g are symmetric, nor the reduction condi- tions (4.2). The following corollary now specializes to the case of this reduction. Corollary 4.1. Let g0 be symmetric, Q̃ ᵀ = −εP̃−1, spec(P̃ ) ∩ spec(Q̃) = ∅, and U = (V g0) ᵀ. Then the solution of (4.1) corresponding to g given by (3.17) is f = κf0ρ −n (det P̃ )2 det ( I + εP̃ 2) det Γ, with an arbitrary constant κ. A convenient formula for the determinant of the solution Γ of the Stein equation seems not to be available in the literature. If we have to modify a solution g in order to achieve the determinant condition (1), the following result is of great help. Proposition 4.2. Let f be a solution of (4.1) for a given solution g from the class obtained in Section 3, and let f(w) be a solution of (4.1) for a scalar solution w of the non-autonomous chiral model (1.1). Then f̂ = (ρf(w)) m−1 f(w det g) f(det g) f (4.4) solves (4.1) with ĝ = wg (which also solves (1.1) according to Remark 4.1). If det ĝ = −ερ2, this reduces to f̂ = wf (ρf(w))m . (4.5) 14 A. Dimakis and F. Müller-Hoissen Due to these results, the problem of explicitly computing solutions of the vacuum Einstein equations, based on the results in Section 3 and with a diagonal seed, essentially boils down to that of solving the linear equations (3.8) for V . For diagonal Q̃ = −εP̃−1, solutions of the latter have already been obtained in Example 3.3. Although we do not yet have general results in case of a non-diagonal Q̃, corresponding examples are treated below in Examples 4.3 and 4.4. 4.2 From diagonal to non-diagonal solutions In the following, we impose the reduction conditions (4.2), which determine Q̃ in terms of P̃ , and U in terms of V and g0, and restrict our considerations to the case ε = 1. We translate a procedure due to Pomeransky [60], which has been formulated and applied in the Belinski–Zakharov (BZ) approach, to our framework. We start with a diagonal (hence in particular static) solution of the above dimensionally reduced vacuum Einstein equations. The corresponding solution g̃ of (1.1) is thus diagonal. According to Remark 2.2 we may still simplify it by multiplication with a suitable scalar solution of (1.1). Now some (anti-) solitons are removed from it in the following way. • Removal of a soliton pi = ρµ−1i (i.e., a soliton at z = ai) from g̃αα. This means multipli- cation of g̃αα by −p−2i . In the BZ language, the corresponding trivial BZ vector has a 1 at the αth position and otherwise zeros. • Removal of an anti-soliton p̄j = −p−1j = −ρ−1µj (i.e., an anti-soliton at z = aj) from g̃ββ . This means multiplication of g̃ββ by p̄−2j = −p2j . Afterwards these solitons are reintroduced, with more freedom, in the following way. • In order to reintroduce the corresponding soliton, set P̃ kk = pi, for some k. The kth row of the matrix V that solves (3.8) (with Q̃ ᵀ = −P̃−1), or rather the vector formed by its constant coefficients, generalizes the aforementioned trivial BZ vector. It’s αth component has to be non-zero. • To reintroduce the corresponding anti-soliton, set P̃ ll = p̄j , for some l. The lth row of the matrix V generalizes the corresponding trivial BZ vector if its βth component is non-zero. After multiplication with a suitable scalar solution of (1.1), in order to achieve the determinant condition, this leads to a non-diagonal (and thus typically stationary) generalization of the original diagonal metric. We recover the latter by suitably fixing the parameters introduced in the second step. In particular, this offers the possibility to reconstruct a known non-diagonal solution from its diagonal specialization (provided the latter can be achieved) via the solution-generating tech- nique. Some well-known examples of four-dimensional space-times are recovered in Section 4.3 in this way. The above procedure will be applied in the five-dimensional case in Section 4.4. Remark 4.2. According to an observation at the end of Remark 2.2, any transformation V 7→ WV with a constant n × n matrix W that commutes with P̃ ᵀ (i.e., [W , P̃ ᵀ ] = 0) leaves g given by (3.17) invariant. If P̃ is diagonal, any diagonal W commutes with it. We can then use such a transformation of V to scale in each row of V one of the non-zero constant coefficients to a fixed value like 1. In the above step of reintroduction of solitons, without restriction of generality we can thus fix the constants appearing in the respective positions of the rows of V . If P̃ = P̃ r (see Example 3.2), then W is an arbitrary constant, lower-triangular Toeplitz matrix and we can again rescale one non-zero coefficient in each row of the corresponding matrix V to a chosen value (different from zero), without changing the corresponding solution g. This feature generalizes to a P̃ that is block-diagonally composed of matrices P̃ r. Binary Darboux Transformations and Vacuum Einstein Equations 15 4.3 Solutions of the vacuum Einstein equations in four dimensions Let D = 4, hence m = 2, and ε = 1. We thus consider stationary and axially symmetric solutions of the vacuum Einstein equations. The metric components gαβ, α, β = 1, 2, should then refer to coordinates x1 = t (time) and x2 = ϕ (angle around the symmetry z-axis), ρ is the coordinate distance from the axis. This interpretation in general imposes additional conditions on the metric. The simple solution g0 = ( −1 0 0 ρ2 ) (4.6) of (1.1) corresponds to the four-dimensional Minkowski metric ds2 = −dt2 + ρ2dϕ2 + dρ2 + dz2 in cylindrical coordinates. The solution f0 of (4.1) with g given by (4.6) is simply a constant, which can be set to 1 (since this can be achieved by a coordinate transformation). In order to facilitate comparison with the relevant literature, in the following we will mainly use the µi introduced in (3.6), instead of the pi, see Remark 3.1. Furthermore, we will frequently use the abbreviation (also see [73]) Rij = ρ2 + µiµj = 2(ai − aj) µiµj µi − µj , where the last equality holds if ai 6= aj . Example 4.2 (Kerr-NUT). In order to recover the Kerr-NUT metric, we start with its static specialization, the Schwarzschild metric. The latter corresponds to the following solution of (1.1), g̃ = −µ2µ1 0 0 ρ2 µ1 µ2  . Next we remove a soliton at z = a1 from g̃11 and a soliton at z = a2 from g̃22. The resulting matrix is simplified by rescaling it with the inverse of w = ρ−2µ1µ2. We obtain the seed solution g0 = w−1g̃ diag ( −(ρ/µ1) −2,−(ρ/µ2) −2) , which is nothing but (4.6). Next we reintroduce the two solitons (n = 2) via P̃ = diag ( ρ µ1 , ρ µ2 ) . The respective solution of (3.8) (with Q̃ = −P̃−1) is V = ( v11 v12µ −1 1 v21 v22µ −1 2 ) , with constants viα (see Example 3.3). According to Remark 4.2, we can set v11 = v22 = 1 without restriction of generality. Let us rename the remaining parameters in V , c1 = v12, c2 = v21. 16 A. Dimakis and F. Müller-Hoissen Now we obtain the matrix Γ from (3.18). Then ĝ = wg, with g given by (3.17), satisfies the determinant condition, i.e., det ĝ = −ρ2, and has the components ĝ11 = 1 h (( c21 + c22 ) ρ2µ1µ2(µ1 − µ2)2 − µ1µ2 (( 1 + c21c 2 2 ) R2 12 − 2c1c2R11R22 )) , ĝ12 = (µ1 − µ2)R12 h ( c1 ( c22µ 2 2 − ρ2 ) R11 + c2 ( c21ρ 2 − µ21 ) R22 ) , ĝ22 = 1 h ( −(µ1 − µ2)2 µ1µ2 ( c21ρ 8 + c22µ 4 1µ 4 2 ) + ρ2 µ1µ2 (( µ41 + c21c 2 2µ 4 2 ) R2 12 − 2c1c2µ 2 1µ 2 2R11R22 )) , where h = (µ1 − µ2)2 ( c21ρ 4 + c22µ 2 1µ 2 2 ) + ( µ21 + c21c 2 2µ 2 2 ) R2 12 − 2c1c2µ1µ2R11R22. Furthermore, according to Example 4.1, we have f(w) ∝ µ1µ2√ R11R22R12 . Corollary 4.1 and Proposition 4.2 then yield f̂ = − κ µ1µ2R11R22 h, with a constant κ. Setting a2 = −a1 = σ, c1 = m + σ a + b , c2 = a + b m− σ , κ = C (m− σ)2 4σ2 , σ = √ m2 − a2 + b2, with new constants a, b, m and C, and passing over from the coordinates ρ and z to new coordinates r and θ via ρ = √ (r −m)2 − σ2 sin θ, z = (r −m) cos θ, we obtain the Kerr-NUT metric in Boyer–Lindquist coordinates, as given by (8.48) and (8.49) in [2]. We can compose N Kerr-NUT data blockwise into larger matrices (the new P̃ and V are then 2N × 2N , respectively 2N × 2 matrices), and obtain again a symmetric solution of the non-autonomous chiral model equation, see Remark 3.6. The determinant condition is again achieved by multiplication with a suitable scalar. In this way we can recover the multi-Kerr-NUT solutions [50]. Choosing for P̃ the non-diagonal solution of (3.4) resulting from an n × n Jordan block matrix An (see Example 3.2), we should expect that the resulting family of solutions of the non-autonomous chiral model can be obtained alternatively via a “soliton coincidence” limit (“pole fusion” in the Belinski–Zakharov formalism [2]) of the family of solutions obtained with a diagonal P̃ (with distinct eigenvalues). This is confirmed by the following example. Example 4.3. Again, let n = 2 and g0 as in (4.6). Now we choose P̃ = ρ µ1 ( 1 −R−1 0 1 ) , where µ1 = R − z with R = √ z2 + ρ2. This is P̃ 2 in Example 3.2 (with a = 0). We find the solution V = ( v11 v12µ −1 1 v21 v22µ −1 1 − 2v12R−111 ) Binary Darboux Transformations and Vacuum Einstein Equations 17 of (3.8), with constants viα. Computing the solution of the Sylvester equation via (3.13), the symmetric matrix Γ is obtained from (3.15). We have to rescale the resulting g, given by (3.17), to ĝ = wg with w = ρ−2µ21, in order to arrange the determinant condition. It turns out that ĝ only depends on the parameters viα via α = v12/v11 and β = det(viα)/v211. Furthermore, we find that ĝ also results from the corresponding (Kerr-NUT) solution in Example 4.2 in the limit σ → 0, after setting c1 = α and c2 = α−2(α− 2βσ). The Kerr-NUT metric consists of two solitons of multiplicity one. The Tomimatsu–Sato metrics [72] are known to generalize it to two solitons of multiplicity δ ≥ 1 (see, e.g., [2])13. We should then expect that, for fixed δ, this solution can be obtained alternatively with P̃ consisting of two blocks P̃ δ as given in Example 3.2 (corresponding to the δ × δ Jordan block Aδ). For δ = 2 this is confirmed in the next example. Example 4.4 (Tomimatsu–Sato). In the static limit, the δ = 2 Tomimatsu–Sato metric (also see [49]) reduces to the Zipoy–Voorhees metric. The latter corresponds to the diagonal solution g̃ = diag ( −µ−21 µ22, ρ 2µ21µ −2 2 ) of (1.1). Now we remove twice a soliton at z = a1 = σ from g̃11, and also twice a soliton at z = a2 = −σ from g̃22. With a simplifying scaling we obtain the seed g0 = w−1g̃ diag ( ρ−4µ41, ρ −4µ42 ) , w = ρ−4µ21µ 2 2. The resulting g0 is again (4.6). Next we reintroduce the two double-solitons via14 P̃ =  ρµ−11 −2ρR−111 0 0 0 ρµ−11 0 0 0 0 ρµ−12 −2ρR−122 0 0 0 ρµ−12  . The solution of (3.8) is then given by V =  v11 v12µ −1 1 v21 v22µ −1 1 − 2v12R−111 v31 v32µ −1 2 v41 v42µ −1 2 − 2v32R−122  . Again, the matrix Γ is obtained via (3.13) and (3.15). Next we have to compute ĝ = wg, with g given by (3.17), and f̂ = κρ−6wf−2(w) (det P̃ )2 det Γ det(I + P̃ 2 ) , where f(w) = ρ3 ( µ21µ 2 2 R11R12R22 )2 . According to Remark 4.2, without restriction of generality we can set v11 = v21 = v32 = v42 = 1. Choosing v12 = v22 = v31 = v41 = q 1 + p , κ = σ−2 ( 1 + p 4p )4 , 13The Tomimatsu–Sato metrics are in fact limiting cases of multiple Kerr-NUT solutions [21,50]. 14Noting that p1/R1 = 2ρ/R11, this consists of two blocks of the form P̃ 2 in Example 3.2. 18 A. Dimakis and F. Müller-Hoissen in terms of prolate spheroidal coordinates x, y, given by ρ = σ √ (x2 − 1)(1− y2), z = σxy, (4.7) and after a coordinate transformation t 7→ t − 4σp−1qϕ, we obtain the δ = 2 Tomimatsu–Sato metric ds2 = −A B dt2 + 8σ ( 1− y2 )qC pB dtdϕ+ σ2 ( 1− y2 ) D p2B dϕ2 + B p4(x2 − y2)3 ( dx2 x2 − 1 + dy2 1− y2 ) , with A = ( 1− y2 )4 g(Z)g(−Z), B = ( g(x) + q2y4 )2 + 4q2y2 ( px3 + 1− (px+ 1)y2 )2 , C = q2(px+ 1)y4 ( −y2 + 3 ) + ( −2q2 ( px3 + 1 ) + (px+ 1)g(x) ) y2 − ( 2px3 − px+ 1 ) g(x), D = p2q4 ( x2 − 1 ) y8 + 4q2(px+ 1) ( p3x3 + 3p2x2 − p3x+ 4px− 3p2 + 4 ) y6 − 2q2 ( 3p4x6 + 4p3x5 − 3p4x4 + 8p3x3 + 37p2x2 − 12p3x+ 48px− 13p2 + 24 ) y4 + 4q2 ( p4x8 − p4x6 + 4p3x5 − 4p3x3 + 15p2x2 + 24px− 3p2 + 12 ) y2 + g(x) ( p4x6+ 6p3x5− p4x4+ 16p2x4− 12p3x3+ 32px3+ 15p2x2+ 6p3x− 15p2+ 16 ) , where p2 + q2 = 1 and g(x) = p2x4 + 2px3 − 2px− 1, Z2 = x2 − y2 1− y2 . These are the expressions (32)–(35) in [49]15. Remark 4.3. The m = 2 non-autonomous chiral model is also related to the D = 4 vacuum Einstein equations with two commuting Killing vector fields in another way, via the Ernst equation (see, e.g., [68]). In this case, one has to impose a different determinant condition: det g = 1. This allows a constant seed g0, and thus constant matrices U , V , in which case the application of Theorem 2.1 is considerably simplified. The solutions obtained in this way include the multi-Kerr-NUT metrics (see [15] for details)16. A similar construction with m = 3 leads to the (electrically and magnetically) charged generalizations, the multi-Demianski–Newman metrics [15]. 4.4 Solutions of the vacuum Einstein equations in five dimensions Let m = 3 and ε = 1. In this case it seems that no relevant solutions can be obtained by choosing as the seed a diagonal solution of the non-autonomous chiral model corresponding to five-dimensional Minkowski space-time. It then becomes a subtle problem to choose a suitable seed solution. Here insights about the “rod structure” of the putative axis ρ = 0 are of great help [2,9,27,36,60]. Some important solutions that have been obtained or recovered previously in the Belinski–Zakharov approach will now be presented in our framework. We use the procedure outlined in Section 4.2. 15We obtained a factor q4 in the expression for D, whereas there is a factor q2 in [49]. We believe that the latter is a typo. 16In [78, 80], solutions of the five-dimensional Einstein–Maxwell equations are constructed from a pair of solu- tions of the m = 2 non-autonomous chiral model with symmetric g and det g = 1. Binary Darboux Transformations and Vacuum Einstein Equations 19 4.4.1 Myers–Perry black holes The higher-dimensional generalization of a static black hole is given by the Schwarzschild– Tangherlini solution [71]. The corresponding diagonal solution of (1.1) is g̃ = diag ( −µ1 µ2 , µ2, ρ2 µ1 ) . The seed g0 is obtained from g̃ as follows: • Remove a soliton at z = a1 from g̃33. • Remove an anti-soliton at z = a2 from g̃22. • Multiply the resulting matrix by w−1, with w = −µ1/µ2, to achieve a simpler form. This results in g0 = diag ( 1, ρ2 µ1 , µ2 ) . According to Example 4.1, the associated solution of (4.1) is (up to a constant factor) f0 = µ2√ R11R22 . Next we reintroduce the soliton and the anti-soliton via P̃ = diag ( ρ µ1 ,−µ2 ρ ) . The first row of the 2×3 matrix V should have a non-zero entry in the third component, which means we recreate the soliton at z = a1 in the respective diagonal component of the seed. The second row of V should have a non-zero entry in the second component, which means we recreate the anti-soliton at z = a2. According to Example 3.3, the solution of (3.8) (with Q̃ = −P̃−1) is given by V = v11 v12 µ1 R11 v13 R12 µ1µ2 v21 v22 R12 ρ2 v23 µ2 R22  , with constants viα. The corresponding solution of the Stein equation (3.16) is obtained from (3.18). With g given by (3.17), the new solution ĝ = wg satisfies the determinant condition: det ĝ = −ρ2. We find f(w) = R12 ρ √ R11R22 (up to a constant factor). Without restriction of generality, we can set v13 = v22 = 1 (see Remark 4.2). With the restrictions and renamings v12 = v23 = 0, c1 = v11, c2 = v21, 20 A. Dimakis and F. Müller-Hoissen use of Corollary 4.1 and Proposition 4.2 yields f̂ = κµ2 c21c 2 2µ 3 1µ2 − (µ1 − µ2)2 ( µ1 ( c21µ1µ2 + c22ρ 2 ) +R2 12 ) (µ1 − µ2)2R11R12R22 , with a constant κ. The corresponding metric is given by ds2 = ĝ11dt 2 + ĝ22dϕ 2 + ĝ33dψ 2 + 2ĝ12dtdϕ+ 2ĝ13dtdψ + 2ĝ23dϕdψ + f̂ ( dρ2 + dz2 ) . Setting a1 = −σ, a2 = σ, with a constant σ, in terms of prolate spheroidal coordinates x, y, given by (4.7), we have ĝ11 = − ( 4σx+ ( a21 − a22 ) y − ρ20 ) ω−1, ĝ12 = −a1ρ20(1− y)ω−1, ĝ13 = −a2ρ20(1 + y)ω−1, ĝ22 = 1− y 4 ( 4σx+ a21 − a22 + ρ20 + 2a21ρ 2 0(1− y)ω−1 ) , ĝ23 = 1 2 a1a2ρ 2 0 ( 1− y2 ) ω−1, ĝ33 = 1 + y 4 ( 4σx− a21 + a22 + ρ20 + 2a22ρ 2 0(1 + y)ω−1 ) , f̂ = ω 8 , ω = 4σx+ (a21 − a22)y + ρ20. Performing a linear transformation of the coordinates t, ϕ, ψ, (t, ϕ, ψ) 7→ ( t− a1ϕ− a2ψ,− a1 c2 ϕ+ a2c2 4σ ψ, a1c1 4σ ϕ− a2 c1 ψ ) , and setting σ = 1 4 √ (ρ20 − a21 − a22) 2 − 4a21a 2 2, κ = (a21 − a22) 2 − (4σ + ρ20) 2 16σρ20 , c21 = ∣∣∣∣4σ(ρ20 − a21 + a22 − 4σ) ρ20 + a21 − a22 + 4σ) ∣∣∣∣ , c22 = ∣∣∣∣4σ(ρ20 + a21 − a22 − 4σ) ρ20 − a21 + a22 + 4σ ∣∣∣∣ , we recover the analogue of the Kerr metric, i.e., the Myers–Perry solution [36,56,57] of the five- dimensional vacuum Einstein equations, precisely in the form of equation (19) in [60]. Switching on the constants v12 and v23 (which we set to zero above), leads to a more general class of space-times. 4.4.2 Black saturn A black saturn [22] (also see [11, 69, 79]) is a black hole surrounded by a black ring17. In the following, we show how the black saturn solution, originally obtained in [22], can be recovered in our approach. Let us start with the same static solution of the non-autonomous chiral model as in [22], g̃ = diag ( −µ1µ3 µ2µ4 , ρ2µ4 µ3µ5 , µ2µ5 µ1 ) . This is motivated by a rod structure analysis (also see [9, 27, 36]). A suitable seed solution is then constructed as follows. 17Black rings are similar to black holes, but with horizon topology S1 × SD−3. They only appear in D > 4 dimensions. See [8, 10,23,24,26,28,29,32,33,40,41,43,46,47,55,61,74–76]. Binary Darboux Transformations and Vacuum Einstein Equations 21 • Remove anti-solitons at z = a1 and at z = a3 from g̃11. • Remove a soliton at z = a2 from g̃11. • Multiply the resulting matrix by w−1 with w = ρ2µ2 µ1µ3 to simplify its form. This results in g0 = w−1g̃ diag ( −ρ 2µ22 µ21µ 2 3 , 1, 1 ) = diag ( 1 µ4 , µ1µ4 µ2µ5 , µ3µ5 ρ2 ) . The corresponding solution of (4.1) is obtained via Example 4.1, f0 = k2µ3µ4µ5 R12R15R24R45√ R11R22R33R14R25R35R44R55 , with a constant k. We have n = 3 and choose P̃ diagonal with P̃ 11 = −µ1/ρ, P̃ 22 = ρ/µ2, P̃ 33 = −µ3/ρ. The solution of the equations for V is then given by V =  v11 R14 µ1 v12 R12R15 R11R14 v13 ρ2µ1 R13R15 v21 µ2µ4 R24 v22 µ2µ5R12R24 µ1µ4R22R25 v23 µ1R23R25 µ22µ3µ5 v31 R34 µ3 v32 R23R35 R13R34 v33 ρ2µ3 R33R35  , with constants viα. Without restriction of generality, we can set v11 = v21 = v31 = 1. The case considered in Section 2.2 of [22] should then correspond to the subclass of solutions given by v12 = v22 = v33 = 0. The authors of [22] then only elaborate the special case v32 = 0 further. In this case, however, the removal of the anti-soliton at z = a3 from the original static metric and the subsequent reintroduction via P̃ 33 is actually redundant. This means that the black saturn space-time can already be obtained from n = 2 data. The solution Γ of the Stein equation (3.16) is given by (3.18). With g given by (3.17), we set ĝ = wg to satisfy the determinant condition. Then ĝ reduces to g̃ if v13 = v23 = 0, as expected. We find f(w) = R12R23√ R11R22R33R13 (up to a constant factor) and, using Corollary 4.1 and Proposition 4.2, f̂ = −κρ−6wf0f−3(w) (det P̃ )2 det Γ det ( I + P̃ 2) , 22 A. Dimakis and F. Müller-Hoissen which turns out to be a lengthy expression. Setting v13 = 2c1(a1 − a5), v23 = c2[4(a2 − a3)(a2 − a4)]−1, κ = 4 (a1 − a2)2(a2 − a3)2 (a1 − a3)2 , with constants ci, we obtain the metric ds2 = −H2 H1 ( dt+ ω H2 dψ )2 +H1 ( k2P(dρ2 + dz2) + G1 H1 dϕ2 + G2 H2 dψ2 ) , where G1 = ρ2µ4 µ3µ5 , G2 = µ3µ5 µ4 , P = R15R2 34R45, H1 = F−1 ( M0 + c21M1 + c22M2 + c1c2M3 + c21c 2 2M4 ) , H2 = µ3 µ4F ( µ1 µ2 M0 − c21 ρ2 µ1µ2 M1 − c22 µ1µ2 ρ2 M2 + c1c2M3 + c21c 2 2 µ2 µ1 M4 ) , F = µ1µ5(µ1 − µ3)2(µ2 − µ4)2 ( 5∏ i=1 Rii ) R13R14R23R24R25R35, with M0 = µ2µ 2 5(µ1 − µ3)2(µ2 − µ4)2R2 12R2 14R2 23, M1 = ρ2µ21µ2µ3µ4µ5(µ1 − µ2)2(µ1 − µ5)2(µ2 − µ4)2R2 23, M2 = ρ2µ21µ −1 2 µ3µ4µ5(µ1 − µ2)2(µ1 − µ3)2R2 14R2 25, M3 = 2µ21µ3µ4µ5(µ1 − µ3)(µ1 − µ5)(µ2 − µ4)R11R22R14R23R25, M4 = µ41µ −1 2 µ23µ 2 4(µ1 − µ5)2R2 12R2 25, ω = 2 F √ G1 ( c1R1 √ M0M1 − c2R2 √ M0M2 + c21c2R2 √ M1M4 − c1c22R1 √ M2M4 ) , and Ri = √ (z − ai)2 + ρ2. This is the black saturn metric18 as given in [22, 73], with some obvious changes in notation, but some deviations in the factors µi in the expressions for M2, M3 and M4. 4.4.3 Double Myers–Perry black hole solution In order to recover the double Myers–Perry black hole solution obtained in [39], we start with the matrix that determines a static two-black hole solution (cf. (3.1) in [70]), g̃ = diag ( −µ1µ4 µ2µ5 , ρ2µ3 µ1µ4 , µ2µ5 µ3 ) . Removal of two solitons µ2, µ5 and two anti-solitons µ1, µ4 from g̃11, and simplification with a suitable factor, leads to g0 = w−1g̃ diag ( − ( µ21µ 2 4 µ22µ 2 5 )−1 , 1, 1 ) = diag ( −1, ρ2µ3 µ2µ5 , µ1µ4 µ3 ) , w = µ2µ5 µ1µ4 . 18Additional conditions have to be imposed on the remaining parameters in order to achieve asymptotic flatness and absence of naked and conical singularities, see [11,22]. Binary Darboux Transformations and Vacuum Einstein Equations 23 According to Example 4.1, the corresponding solution of (4.1) is given by f0 = µ1µ4 µ3  5∏ i=1 i 6=3 √ Rii  −1 R13R23R34R35 R14R25R33 . We have n = 4 and reintroduce the removed solitons and anti-solitons via P̃ = diag ( −ρ−1µ1,−ρ−1µ4, ρµ−12 , ρµ−15 ) . The solution of (3.8) is given by V =  v11 v12 R12R15 ρ2R13 v13 µ1R13 R11R14 v21 v22 R24R45 ρ2R34 v23 µ4R34 R14R44 v31 v32 µ2µ5R23 µ3R22R25 v33 µ3R12R24 µ1µ2µ4R23 v41 v42 µ2µ5R35 µ3R25R55 v43 µ3R15R45 µ1µ4µ5R35  . In order to recover the double Myers–Perry black hole solution presented in [39], we reduce the set of solutions obtained in this way by restricting V to V =  1 v12 R12R15 ρ2R13 0 1 v22 R24R45 ρ2R34 0 1 0 0 1 0 0  . We immediately notice that this means in particular “trivializing” the solitons at z = a2 and z = a5. Hence, the solution obtained with this special choice of V can already be obtained from n = 2 data. We should expect, however, a 4-soliton transformation to be necessary in order to generate a (sufficiently general) double black hole solution, which suggests to explore the above more general solution. This will not be done here, and we return to the special case with the above restricted V . Again, we obtain Γ from (3.18). Let ĝ be the resulting solution (3.17), multiplied by w to achieve the determinant condition. From Example 4.1, we obtain f(w) = ρ−1  5∏ i=1 i 6=3 √ Rii  −1 R12R15R24R45 R14R25 , and then f̂ via Corollary 4.1 and Proposition 4.2. Setting v12 = b(a1 − a3) 2(a1 − a2)(a1 − a5) , v22 = c(a3 − a4) 2(a2 − a4)(a4 − a5) , κ = ( 4 (a1 − a2)(a1 − a5)(a2 − a4)(a4 − a5) (a1 − a4)(a2 − a5) )2 , with constants b, c, this results in the metric ds2 = −H2 H1 ( dt+ ω H2 dϕ )2 + ρ2µ3H1 µ2µ5H2 dϕ2 + µ2µ5 µ3 dψ2 + k H1 F ( dρ2 + dz2 ) , 24 A. Dimakis and F. Müller-Hoissen where H1 = M0 + b2M1 + c2M2 + bcM3 + b2c2M4, H2 = ρ2 µ2µ5 ( µ1µ4 ρ2 M0 − b2 µ4 µ1 M1 − c2 µ1 µ4 M2 − bcM3 + b2c2 ρ2 µ1µ4 M4 ) , F = µ23(µ1 − µ4)2 ( 5∏ i=1 Rii ) R12R2 14R15R24R2 25R45 R13R23R34R35 , ω = −2 ( µ3 µ2µ5 )1/2 ( bR1 √ M0M1 + cR4 √ M0M2 − b2cR4 √ M1M4 − bc2R1 √ M2M4 ) , with M0 = µ2µ 2 3µ5(µ1 − µ4)2R2 12R2 15R2 24R2 45, M1 = µ21µ 2 2µ3µ 2 5(µ1 − µ3)2R2 14R2 24R2 45, M2 = µ22µ3µ 2 4µ 2 5(µ3 − µ4)2R2 12R2 14R2 15, M3 = 2µ1µ 2 2µ3µ4µ 2 5(µ1 − µ3)(µ3 − µ4)R11R12R15R24R44R45, M4 = ρ4µ21µ 3 2µ 2 4µ 3 5(µ1 − µ3)2(µ1 − µ4)2(µ3 − µ4)2. With obvious changes in notation, this is the metric obtained in [39]. Remark 4.4. It is plausible that one can start with the diagonal solution of the non-autonomous chiral model corresponding to a static triple black hole space-time (see (4.1) in [70]) and construct a space-time with three Myers–Perry black holes. This procedure should continue to produce solutions with an arbitrary number of rotating black holes. 4.4.4 Bicycling black rings Let us start with the solution g̃ = diag ( −µ1µ5 µ3µ7 , ρ2µ3µ7 µ2µ4µ6 , µ2µ4µ6 µ1µ5 ) of (1.1), which corresponds to a static metric. Removal of a soliton at z = a7 and an anti-soliton at z = a1 from g̃11, and a rescaling, leads to the seed metric g0 = w−1g̃ diag ( µ27 µ21 , 1, 1 ) = diag ( µ5 µ3 ,−ρ 2µ1µ3 µ2µ4µ6 ,−µ2µ4µ6 µ5µ7 ) , w = −µ7 µ1 , with the following solution of (4.1) (up to a constant factor), f0 = µ2µ4µ6 µ5µ7 R12R14R16R23R25R27R34R35R36R45R47R56R67√ R11R13R22R2 24R2 26R33R44R2 46R55R57R66 √ R77 . We have n = 2 and shall set P̃ = diag ( −ρ−1µ1, ρµ−17 ) . The solution of (3.8) is then given by (see Example 3.3) V =  v11 R13 R15 v12 R12R14R16 µ1R11R13 v13 µ1R15R17 R12R14R16 v21 µ3R57 µ5R37 v22 µ2µ4µ6R17R37 µ1µ3R27R47R67 v23 µ5R27R47R67 µ2µ4µ6R57R77  . Binary Darboux Transformations and Vacuum Einstein Equations 25 Without restriction of generality, we can set v11 = v21 = 1. But we do restrict the class of solutions by setting v12 = v23 = 0. Again, the solution Γ of the Stein equation is obtained from (3.18). The resulting solution (3.17) of the non-autonomous chiral model has to be modified to ĝ = wg, with w as given above, in order to achieve the determinant condition. We have (disregarding a constant factor) f(w) = ρ−1 R17√ R11R77 , and, from Corollary 4.1 and Proposition 4.2, f̂ = κρ−5wf0f −3 (w) (det P̃ )2 det Γ det ( I + P̃ 2) , which results in a lengthy expression. Setting v13 = c1, v22 = b2 (a7 − a4)(a7 − a5)(a7 − a6) (a7 − a1)(a7 − a3)2 , κ = 4(a7 − a1)2, we obtain the metric19 ds2 = −H2 H1 ( dt− ω1 H2 dϕ− ω2 H2 dψ )2 + 1 H2 ( G1dϕ 2 +G2dψ 2 − 2Jdϕdψ ) + PH1 ( dρ2 + dz2 ) , where H1 = M0 + c21M1 + b22M2 − b22c21M3, H2 = µ5 µ3 ( µ1 µ7 M0 − c21 ρ2 µ1µ7 M1 − b22 µ1µ7 ρ2 M2 − b22c21 µ7 µ1 M3 ) , G1 = ρ2µ1µ5 µ2µ4µ6 ( M0 − c21 ρ2 µ21 M1 + b22M2 + b22c 2 1 ρ2 µ21 M3 ) , G2 = µ2µ4µ6 µ3µ7 ( M0 + c21M1 − b22 µ27 ρ2 M2 + b22c 2 1 µ27 ρ2 M3 ) , ω1 = b2 R77 µ7 ( µ1µ5 µ2µ4µ6 )1/2(√ M0M2 − c21 ρ µ1 √ M1M3 ) , ω2 = c1 R11 µ1 ( µ2µ4µ6 ρ2µ3µ7 )1/2(√ M0M1 − b22 µ7 ρ √ M2M3 ) , J = b2c1ρ 2µ1µ2µ3µ4µ 2 5µ6(µ3 − µ7)2(µ4 − µ7)(µ5 − µ7)(µ6 − µ7) ×R11R12R13R14R2 15R16R17R27R77, and M0 = µ4µ 3 5µ6µ7(µ3 − µ7)4R2 12R2 13R2 14R2 16R2 17R2 27, M1 = ρ2µ21µ2µ3µ 2 4µ5µ 2 6(µ1 − µ7)2(µ3 − µ7)4R4 15R2 17R2 27, 19Here we used ai − aj = (µi − µj)(ρ 2 + µiµj)/(2µiµj) to eliminate ai − aj . 26 A. Dimakis and F. Müller-Hoissen M2 = ρ4µ1µ2µ 2 3µ 2 5µ7(µ4 − µ7)2(µ5 − µ7)2(µ6 − µ7)2R2 12R2 13R2 14R2 16, M3 = ρ4µ31µ 2 2µ 3 3µ4µ6(µ4 − µ7)2(µ5 − µ7)2(µ6 − µ7)2R4 15R2 17, P = µ2 µ1µ45µ7(µ3 − µ7)4 R23R25R34R35R36R45R47R56R57R67 R12R13R14R2 15R16R17R2 24R2 26R27R2 37R2 46 . With obvious changes in notation, this is the “bicycling” black bi-ring solution obtained in [23] (also see [47]), except for the fact that we have a minus sign instead of a plus in the expressions for ω1 and ω2. 5 Final remarks We presented a general formulation of binary Darboux-type transformations in the bidifferential calculus framework. Whenever a PDDE can be cast into the form (2.2) or (2.3), Theorem 2.1 can be applied and it will typically generate a large class of exact solutions. Meanwhile a bidifferential calculus formulation is available for quite a number of integrable PDDEs. We elaborated this general result for the case of the non-autonomous chiral model, consid- erably extending previous results in [15]. We also presented conditions that, imposed on the matrix data that determine the general class of solutions, guarantee that the resulting solution of the non-autonomous chiral model is symmetric (or Hermitian). If the solution is also real, then it is known to determine a Ricci-flat metric, i.e., a solution of the vacuum Einstein equa- tions, dimensionally reduced to two dimensions. We essentially solved the equations resulting from the assumptions in Theorem 2.1 in the case of a diagonal seed metric, though not yet the V -equations in sufficient generality if P̃ is non-diagonal (but see Examples 4.3 and 4.4). All this provides a working recipe to compute quite easily solutions of the vacuum Einstein equations. In particular, in the four-dimensional case we recovered (multi-) Kerr-NUT (in a different way than in [15]) and the δ = 2 Tomimatsu–Sato solution. In the five-dimensional case we recovered single and double Myers–Perry black holes, the “black saturn” and the “bicycling black ring” solutions. The more general solutions still have to be explored. In view of the complexity of the latter solutions, it is certainly an advantage to have now an independent method at our disposal to derive, verify or generalize them. Surely further important solutions of Einstein’s equations in D ≥ 4 space-time dimensions can be recovered using this method and there is a chance to discover interesting new solutions. We concentrated on examples in the stationary case ε = 1, but developed the formalism as well for the wave case ε = −1 [2,4]. It is not difficult to recover relevant examples in this case too. The recipe to construct solutions of the non-autonomous chiral model and the dimensionally reduced vacuum Einstein equations, obtained from Theorem 2.1, is – not surprisingly – a variant, a sort of matrix version, of the well-known method of Belinski and Zakharov [2–4]. One should look for suitable ways to spot physically relevant solutions within the plethora of solutions. How are desired properties of solutions, like asymptotic flatness, absence of naked singularities and proper axis conditions encoded in the (matrix) data that determine a solution? Here the rod structure analysis [2, 9, 27, 36], developed for the Belinski–Zakharov approach and frequently used, is of great help. Section 3 also paved the way toward a treatment of other reductions of the non-autonomous chiral model, which, e.g., are relevant in the Einstein–Maxwell case and supergravity theories. In this work we only elaborated Theorem 2.1 for a particular example of an integrable equation in the bidifferential calculus framework. Although we already applied a more restricted version of it previously to several other integrable equations, it will be worth to reconsider them and to also explore further equations, using the much more general solution-generating tool we now have at our disposal. Furthermore, it should be clarified whether, e.g., the examples in [63–65] fit into the framework of Theorem 2.1. We should also mention that Sylvester equations, like Binary Darboux Transformations and Vacuum Einstein Equations 27 those that arise from (2.7), and more generally operator versions of them, are ubiquitous in the theory of integrable systems. In particular, they are related to a Riemann–Hilbert factorization problem [66] and they are at the roots of Marchenko’s operator approach [53]. Appendix A. Addendum to Example 4.1 From (4.3) we find that f[ρ] ∝ ρ1/4, f[wk] ∝ f[w]k 2 , f[w1w2] ∝ f[w1]f[w2]F[w1, w2], where F[w1, w2] has to solve (lnF[w1, w2])ρ = ρ 2 ( (lnw1)ρ(lnw2)ρ − ε(lnw1)z(lnw2)z ) , (lnF[w1, w2])z = ρ 2 ( (lnw1)ρ(lnw2)z + (lnw1)z(lnw2)ρ ) (also see [47]). It is easy to verify that F[w1 · · ·wr, w′1 · · ·w′s] ∝ r∏ i=1 s∏ j=1 F[wi, w ′ j ]. In particular, F[wk1 , w l 2] ∝ F[w1, w2] kl. Furthermore, we have F [ ρkw1, ρ lw2 ] ∝ ρkl/2wl/21 w k/2 2 F[w1, w2]. It follows that f[ρkw] ∝ ρk2/4wk/2f[w], f[µ̃1 · · · µ̃r] ∝ ( r∏ k=1 f[µ̃k] )∏ i<j F[µ̃i, µ̃j ]  , and f [ µ̃1 · · · µ̃r µ̃′1 · · · µ̃′s ] ∝ f[µ̃1 · · · µ̃r]f[µ̃′1 · · · µ̃′r] r∏ i s∏ j F[µ̃i, µ̃′j ] , from which the main result in Example 4.1 is easily deduced. Appendix B. Some proofs Proof of Proposition 4.1. Using (ln detY )ρ = tr(YρY −1) for an invertible (and differentiable) matrix function Y , we obtain (ln f)ρ = (ln f0)ρ + tr ( −ρ−1I + X̃ρX̃ −1 + P̃ ρP̃ −1 + Q̃ρQ̃ −1 − εP̃ ρP̃ ( 1 + εP̃ 2)−1 − εQ̃ρQ̃ ( 1 + εQ̃ 2)−1) , (ln f)z = (ln f0)z + tr ( X̃zX̃ −1 + P̃ zP̃ −1 + Q̃zQ̃ −1 − εP̃ zP̃ ( 1 + εP̃ 2)−1 − εQ̃zQ̃ ( 1 + εQ̃ 2)−1) , 28 A. Dimakis and F. Müller-Hoissen for the expression of f in Proposition 4.1. In order to verify that (4.1) holds, we have to show that these expressions equal the corresponding right hand sides of (4.1), evaluated with U = −ρφ̃z = −ρ [ φ̃0,z + ( UX̃ −1 V ) z ] = U0 − ρ ( U zX̃ −1 V −UX̃ −1 X̃zX̃ −1 V + UX̃ −1 V z ) , V = V0 + ε [ ρ ( UρX̃ −1 V −UX̃ −1 X̃ρX̃ −1 V + UX̃ −1 V ρ ) + UX̃ −1 V ] , where U0 = −ρφ̃0,z and V0 = ε(ρφ̃0,ρ + φ̃0). 20 We work on the right hand sides of (4.1) and, as intermediate steps, we consider tr(U2 − U2 0 ), tr(V2 − V20 ) and tr(UV − U0V0) separately. First we eliminate derivatives of φ0 with the help of (3.7) and (3.8). Then we eliminate V U using the Sylvester equation, and V zU , V ρU via (3.14). Rewriting (3.14) with the help of (3.10), we obtain a version that allows us to also replace all occurencies of V U z and V Uρ. Several times one has to exploit the cyclicity of the trace. Finally we use P̃ ρ = ρ−1P̃ ( I − εP̃ 2)( I + εP̃ 2)−1 , P̃ z = 2ερ−1P̃ 2( I + εP̃ 2)−1 (which follows from (3.3)), and the corresponding equations for Q̃, to show that (4.1) holds. Proof of Proposition 4.2. Using Û = U + ρ(lnw)ρI and V̂ = V + ρ(lnw)zI, we have tr ( Û2 − U2 ) = 2ρ(lnw)ρ trU +mρ2(lnw)2ρ = 2ρ2(lnw)ρ(ln det g)ρ +mρ2(lnw)2ρ, and a corresponding expression for tr(V̂2 − V2). Furthermore, tr ( Û V̂ − UV ) = ρ2 ( (lnw)ρ(ln det g)z + (lnw)z(ln det g)ρ +m(lnw)ρ(lnw)z ) . With the help of (4.1) we obtain( ln f̂ f ) ρ = 1 4ρ tr ( Û2 − U2 − ε ( V̂2 − V2 )) = ρ 2 ( (lnw)ρ(ln det g)ρ − ε(lnw)z(ln det g)z ) +m ρ 4 ( (lnw)2ρ − ε(lnw)2z ) ,( ln f̂ f ) z = 1 2ρ tr ( Û V̂ − UV ) = ρ 2 ( (lnw)ρ(ln det g)z + (lnw)z(ln det g)ρ ) +m ρ 2 (lnw)ρ(lnw)z. Next we use (ln(ρf(w)))ρ = ρ 4 ( (lnw)2ρ − ε(lnw)2z ) , (ln(ρf(w)))z = ρ 2 (lnw)ρ(lnw)z, and the corresponding equations for f(det g) and f(w det g), to deduce that( ln f(w det g) ρf(w)f(det g) ) ρ = ρ 2 ( (lnw)ρ(ln det g)ρ − ε(lnw)z(ln det g)z ) ,( ln f(w det g) ρf(w)f(det g) ) z = ρ 2 ( (lnw)ρ(ln det g)z + (lnw)z(ln det g)ρ ) . Inserting the last expressions in our previous results, we obtain (4.4) by integration. Let us now assume that det ĝ = −ερ2, and thus det g = −ερ2w−m. With f(ρ2wk) ∝ wk(ρf(w))k 2 (see Example 4.1), (4.5) results from (4.4). 20This is quite a tour de force and a more elegant proof would be desirable. Binary Darboux Transformations and Vacuum Einstein Equations 29 References [1] Arsie A., Lorenzoni P., F -manifolds with eventual identities, bidifferential calculus and twisted Lenard– Magri chains, Int. Math. Res. Not., to appear, arXiv:1110.2461. [2] Belinski V.A., Verdaguer E., Gravitational solitons, Cambridge Monographs on Mathematical Physics, Cam- bridge University Press, Cambridge, 2001. [3] Belinskĭı V.A., Sakharov V.E., Stationary gravitational solitons with axial symmetry, Sov. Phys. JETP 50 (1979), 1–9. [4] Belinskĭı V.A., Zakharov V.E., Integration of the Einstein equations by means of the inverse scattering problem technique and construction of exact soliton solutions, Sov. Phys. JETP 48 (1978), 985–994. [5] Breitenlohner P., Maison D., Gibbons G., 4-dimensional black holes from Kaluza–Klein theories, Comm. Math. Phys. 120 (1988), 295–333. [6] Camacaro J., Cariñena J., Alternative Lie algebroid structures and bi-differential calculi, in Applied Differential Geometry and Mechanics, Editors W. Sarlet, F. Cantrijn, University of Gent, 2003, 1–20. [7] Chavchanidze G., Non-Noether symmetries in Hamiltonian dynamical systems, Mem. Differential Equations Math. Phys. 36 (2005), 81–134, math-ph/0405003. [8] Chen Y., Hong K., Teo E., Unbalanced Pomeransky–Sen’kov black ring, Phys. Rev. D 84 (2011), 084030, 11 pages, arXiv:1108.1849. [9] Chen Y., Teo E., Rod-structure classification of gravitational instantons with U(1)×U(1) isometry, Nuclear Phys. B 838 (2010), 207–237, arXiv:1004.2750. [10] Chruściel P., Cortier J., Maximal analytic extensions of the Emparan–Reall black ring, J. Phys. Conf. Ser. 229 (2010), 012030, 4 pages, arXiv:0807.2309. [11] Chruściel P.T., Eckstein M., Szybka S.J., On smoothness of black saturns, J. High Energy Phys. 2010 (2010), no. 11, 048, 39 pages, arXiv:1007.3668. [12] Cieśliński J.L., Algebraic construction of the Darboux matrix revisited, J. Phys. A: Math. Theor. 42 (2009), 404003, 40 pages, arXiv:0904.3987. [13] Crampin M., Sarlet W., Thompson G., Bi-differential calculi, bi-Hamiltonian systems and conformal Killing tensors, J. Phys. A: Math. Gen. 33 (2000), 8755–8770. [14] de Souza E., Bhattacharyya S.P., Controllability, observability and the solution of AX −XB = C, Linear Algebra Appl. 39 (1981), 167–188. [15] Dimakis A., Kanning N., Müller-Hoissen F., The non-autonomous chiral model and the Ernst equation of general relativity in the bidifferential calculus framework, SIGMA 7 (2011), 118, 27 pages, arXiv:1106.4122. [16] Dimakis A., Müller-Hoissen F., Bi-differential calculi and integrable models, J. Phys. A: Math. Gen. 33 (2000), 957–974, math-ph/9908015. [17] Dimakis A., Müller-Hoissen F., Bicomplexes and integrable models, J. Phys. A: Math. Gen. 33 (2000), 6579–6591, nlin.SI/0006029. [18] Dimakis A., Müller-Hoissen F., Bidifferential calculus approach to AKNS hierarchies and their solutions, SIGMA 6 (2010), 055, 27 pages, arXiv:1004.1627. [19] Dimakis A., Müller-Hoissen F., Bidifferential graded algebras and integrable systems, Discrete Contin. Dyn. Syst. (2009), suppl., 208–219, arXiv:0805.4553. [20] Dimakis A., Müller-Hoissen F., Solutions of matrix NLS systems and their discretizations: a unified treat- ment, Inverse Problems 26 (2010), 095007, 55 pages, arXiv:1001.0133. [21] Economou A., Tsoubelis D., Multiple-soliton solutions of Einstein’s equations, J. Math. Phys. 30 (1989), 1562–1569. [22] Elvang H., Figueras P., Black saturn, J. High Energy Phys. 2007 (2007), no. 5, 050, 48 pages, hep-th/0701035. [23] Elvang H., Rodriguez M.J., Bicycling black rings, J. High Energy Phys. 2008 (2008), no. 4, 045, 30 pages, arXiv:0712.2425. [24] Emparan R., Reall H.S., A rotating black ring solution in five dimensions, Phys. Rev. Lett. 88 (2002), 101101, 4 pages, hep-th/0110260. [25] Emparan R., Reall H.S., Black holes in higher dimensions, Living Rev. Relativ. 11 (2008), 6, 87 pages, arXiv:0801.3471. http://dx.doi.org/10.1093/imrn/rns172 http://arxiv.org/abs/1110.2461 http://dx.doi.org/10.1017/CBO9780511535253 http://dx.doi.org/10.1007/BF01217967 http://dx.doi.org/10.1007/BF01217967 http://arxiv.org/abs/math-ph/0405003 http://dx.doi.org/10.1103/PhysRevD.84.084030 http://arxiv.org/abs/1108.1849 http://dx.doi.org/10.1016/j.nuclphysb.2010.05.017 http://dx.doi.org/10.1016/j.nuclphysb.2010.05.017 http://arxiv.org/abs/1004.2750 http://dx.doi.org/10.1088/1742-6596/229/1/012030 http://arxiv.org/abs/0807.2309 http://dx.doi.org/10.1007/JHEP11(2010)048 http://arxiv.org/abs/1007.3668 http://dx.doi.org/10.1088/1751-8113/42/40/404003 http://arxiv.org/abs/0904.3987 http://dx.doi.org/10.1088/0305-4470/33/48/313 http://dx.doi.org/10.1016/0024-3795(81)90301-3 http://dx.doi.org/10.1016/0024-3795(81)90301-3 http://dx.doi.org/10.3842/SIGMA.2011.118 http://arxiv.org/abs/1106.4122 http://dx.doi.org/10.1088/0305-4470/33/5/311 http://arxiv.org/abs/math-ph/9908015 http://dx.doi.org/10.1088/0305-4470/33/37/310 http://arxiv.org/abs/nlin.SI/0006029 http://dx.doi.org/10.3842/SIGMA.2010.055 http://arxiv.org/abs/1004.1627 http://arxiv.org/abs/0805.4553 http://dx.doi.org/10.1088/0266-5611/26/9/095007 http://arxiv.org/abs/1001.0133 http://dx.doi.org/10.1063/1.528597 http://dx.doi.org/10.1088/1126-6708/2007/05/050 http://arxiv.org/abs/hep-th/0701035 http://dx.doi.org/10.1088/1126-6708/2008/04/045 http://arxiv.org/abs/0712.2425 http://dx.doi.org/10.1103/PhysRevLett.88.101101 http://arxiv.org/abs/hep-th/0110260 http://arxiv.org/abs/0801.3471 30 A. Dimakis and F. Müller-Hoissen [26] Emparan R., Reall H.S., Black rings, Classical Quantum Gravity 23 (2006), R169–R197, hep-th/0608012. [27] Emparan R., Reall H.S., Generalized Weyl solutions, Phys. Rev. D 65 (2002), 084025, 26 pages, hep-th/0110258. [28] Evslin J., Krishnan C., The black di-ring: an inverse scattering construction, Classical Quantum Gravity 26 (2009), 125018, 13 pages, arXiv:0706.1231. [29] Figueras P., A black ring with a rotating 2-sphere, J. High Energy Phys. 2005 (2005), no. 7, 039, 9 pages, hep-th/0505244. [30] Figueras P., Jamsin E., Rocha J.V., Virmani A., Integrability of five-dimensional minimal supergravity and charged rotating black holes, Classical Quantum Gravity 27 (2010), 135011, 37 pages, arXiv:0912.3199. [31] Frölicher A., Nijenhuis A., Theory of vector-valued differential forms. I. Derivations in the graded ring of differential forms, Proc. Koninkl. Ned. Acad. Wetensch. Ser. A 59 (1956), 338–359. [32] Frolov V.P., Goswami R., Surface geometry of 5D black holes and black rings, Phys. Rev. D 75 (2007), 124001, 11 pages, gr-qc/0612033. [33] Gauntlett J.P., Gutowski J.B., Concentric black rings, Phys. Rev. D 71 (2005), 025013, 7 pages, hep-th/0408010. [34] Griffiths J.B., Colliding plane waves in general relativity, Oxford Mathematical Monographs, Oxford Science Publications, The Clarendon Press, Oxford University Press, New York, 1991. [35] Gu C., Hu H., Zhou Z., Darboux transformations in integrable systems. Theory and their applications to geometry, Mathematical Physics Studies, Vol. 26, Springer, Dordrecht, 2005. [36] Harmark T., Stationary and axisymmetric solutions of higher-dimensional general relativity, Phys. Rev. D 70 (2004), 124002, 25 pages, hep-th/0408141. [37] Hartwig R.E., Resultants and the solution of AX −XB = −C, SIAM J. Appl. Math. 23 (1972), 104–117. [38] Hearon J.Z., Nonsingular solutions of TA−BT = C, Linear Algebra Appl. 16 (1977), 57–63. [39] Herdeiro C., Rebelo C., Zilhão M., Costa M., A double Myers–Perry black hole in five dimensions, J. High Energy Phys. 2008 (2008), no. 7, 009, 24 pages, arXiv:0805.1206. [40] Hollands S., Yazadjiev S., Uniqueness theorem for 5-dimensional black holes with two axial Killing fields, Comm. Math. Phys. 283 (2008), 749–768, arXiv:0707.2775. [41] Hong K., Teo E., A new form of the C-metric, Classical Quantum Gravity 20 (2003), 3269–3277, gr-qc/0305089. [42] Horowitz G. (Editor), Black holes in higher dimensions, Cambridge University Press, Cambridge, 2012. [43] Hoskisson J., Explorations of four and five dimensional black hole spacetimes, Ph.D. thesis, Durham Uni- versity, 2009, available at http://etheses.dur.ac.uk/2115/. [44] Hu Q., Cheng D., The polynomial solution to the Sylvester matrix equation, Appl. Math. Lett. 19 (2006), 859–864. [45] Iguchi H., Izumi K., Mishima T., Systematic solution-generation of five-dimensional black holes, Progr. Theoret. Phys. Suppl. 189 (2011), 93–125, arXiv:1106.0387. [46] Iguchi H., Mishima T., Solitonic generation of vacuum solutions in five-dimensional general relativity, Phys. Rev. D 74 (2006), 024029, 17 pages, hep-th/0605090. [47] Izumi K., Orthogonal black di-ring solution, Progr. Theoret. Phys. 119 (2008), 757–774, arXiv:0712.0902. [48] Klein C., Richter O., Ernst equation and Riemann surfaces. Analytical and numerical methods, Lecture Notes in Physics, Vol. 685, Springer-Verlag, Berlin, 2005. [49] Kodama H., Hikida W., Global structure of the Zipoy–Voorhees–Weyl spacetime and the δ = 2 Tomimatsu– Sato spacetime, Classical Quantum Gravity 20 (2003), 5121–5140, gr-qc/0304064. [50] Kramer D., Neugebauer G., The superposition of two Kerr solutions, Phys. Lett. A 75 (1980), 259–261. [51] Lorenzoni P., Flat bidifferential ideals and semi-Hamiltonian PDEs, J. Phys. A: Math. Gen. 39 (2006), 13701–13715, nlin.SI/0604053. [52] Lorenzoni P., Magri F., A cohomological construction of integrable hierarchies of hydrodynamic type, Int. Math. Res. Not. 2005 (2005), no. 34, 2087–2100, nlin.SI/0504064. [53] Marchenko V.A., Nonlinear equations and operator algebras, Mathematics and its Applications (Soviet Series), Vol. 17, D. Reidel Publishing Co., Dordrecht, 1988. http://dx.doi.org/10.1088/0264-9381/23/20/R01 http://arxiv.org/abs/hep-th/0608012 http://dx.doi.org/10.1103/PhysRevD.65.084025 http://arxiv.org/abs/hep-th/0110258 http://dx.doi.org/10.1088/0264-9381/26/12/125018 http://arxiv.org/abs/0706.1231 http://dx.doi.org/10.1088/1126-6708/2005/07/039 http://arxiv.org/abs/hep-th/0505244 http://dx.doi.org/10.1088/0264-9381/27/13/135011 http://arxiv.org/abs/0912.3199 http://dx.doi.org/10.1103/PhysRevD.75.124001 http://arxiv.org/abs/gr-qc/0612033 http://dx.doi.org/10.1103/PhysRevD.71.025013 http://arxiv.org/abs/hep-th/0408010 http://dx.doi.org/10.1103/PhysRevD.70.124002 http://arxiv.org/abs/hep-th/0408141 http://dx.doi.org/10.1137/0123012 http://dx.doi.org/10.1016/0024-3795(77)90019-2 http://dx.doi.org/10.1088/1126-6708/2008/07/009 http://dx.doi.org/10.1088/1126-6708/2008/07/009 http://arxiv.org/abs/0805.1206 http://dx.doi.org/10.1007/s00220-008-0516-3 http://arxiv.org/abs/0707.2775 http://dx.doi.org/10.1088/0264-9381/20/14/321 http://arxiv.org/abs/gr-qc/0305089 http://etheses.dur.ac.uk/2115/ http://dx.doi.org/10.1016/j.aml.2005.09.005 http://dx.doi.org/10.1143/PTPS.189.93 http://dx.doi.org/10.1143/PTPS.189.93 http://arxiv.org/abs/1106.0387 http://dx.doi.org/10.1103/PhysRevD.74.024029 http://dx.doi.org/10.1103/PhysRevD.74.024029 http://arxiv.org/abs/hep-th/0605090 http://dx.doi.org/10.1143/PTP.119.757 http://arxiv.org/abs/0712.0902 http://dx.doi.org/10.1088/0264-9381/20/23/011 http://arxiv.org/abs/gr-qc/0304064 http://dx.doi.org/10.1016/0375-9601(80)90556-3 http://dx.doi.org/10.1088/0305-4470/39/44/006 http://arxiv.org/abs/nlin.SI/0604053 http://dx.doi.org/10.1155/IMRN.2005.2087 http://dx.doi.org/10.1155/IMRN.2005.2087 http://arxiv.org/abs/nlin.SI/0504064 http://dx.doi.org/10.1007/978-94-009-2887-9 http://dx.doi.org/10.1007/978-94-009-2887-9 Binary Darboux Transformations and Vacuum Einstein Equations 31 [54] Matveev V.B., Salle M.A., Darboux transformations and solitons, Springer Series in Nonlinear Dynamics, Springer-Verlag, Berlin, 1991. [55] Mishima T., Iguchi H., New axisymmetric stationary solutions of five-dimensional vacuum Einstein equations with asymptotic flatness, Phys. Rev. D 73 (2006), 044030, 6 pages, hep-th/0504018. [56] Myers R.C., Myers–Perry black holes, in Black Holes in Higher Dimensions, Editor G. Horowitz, Cambridge University Press, Cambridge, 2012, 101–133, arXiv:1111.1903. [57] Myers R.C., Perry M.J., Black holes in higher-dimensional space-times, Ann. Physics 172 (1986), 304–347. [58] Nakamura Y., Symmetries of stationary axially symmetric vacuum Einstein equations and the new family of exact solutions, J. Math. Phys. 24 (1983), 606–609. [59] Nimmo J.J.C., Gilson C., Ohta Y., Applications of Darboux transformations to the self-dual Yang–Mills equations, Theoret. and Math. Phys. 122 (2000), 239–246. [60] Pomeransky A.A., Complete integrability of higher-dimensional Einstein equations with additional symme- try and rotating black holes, Phys. Rev. D 73 (2006), 044004, 5 pages, hep-th/0507250. [61] Pomeransky A.A., Sen’kov R.A., Black ring with two angular momenta, hep-th/0612005. [62] Rogers C., Schief W.K., Bäcklund and Darboux transformations. Geometry and modern applications in soliton theory, Cambridge Texts in Applied Mathematics, Cambridge University Press, Cambridge, 2002. [63] Sakhnovich A.L., Dressing procedure for solutions of nonlinear equations and the method of operator iden- tities, Inverse Problems 10 (1994), 699–710. [64] Sakhnovich A.L., Generalized Bäcklund–Darboux transformation: spectral properties and nonlinear equa- tions, J. Math. Anal. Appl. 262 (2001), 274–306. [65] Sakhnovich A.L., On the GBDT version of the Bäcklund–Darboux transformation and its applications to linear and nonlinear equations and Weyl theory, Math. Model. Nat. Phenom. 5 (2010), 340–389, arXiv:0909.1537. [66] Sakhnovich L.A., Problems of factorization and operator identities, Russ. Math. Surv. 41 (1986), no. 1, 1–64. [67] Sparano G., Vilasi G., Vinogradov A.M., Vacuum Einstein metrics with bidimensional Killing leaves. I. Local aspects, Differential Geom. Appl. 16 (2002), 95–120, gr-qc/0301020. [68] Stephani H., Kramer D., MacCallum M., Hoenselaers C., Herlt E., Exact solutions of Einstein’s field equa- tions, 2nd ed., Cambridge Monographs on Mathematical Physics, Cambridge University Press, Cambridge, 2003. [69] Szybka S.J., Stable causality of Black Saturns, J. High Energy Phys. 2011 (2011), no. 5, 052, 9 pages, arXiv:1102.3942. [70] Tan H.S., Teo E., Multi-black-hole solutions in five dimensions, Phys. Rev. D 68 (2003), 044021, 11 pages, hep-th/0306044. [71] Tangherlini F.R., Schwarzschild field in n dimensions and the dimensionality of space problem, Nuovo Cimento 27 (1963), 636–651. [72] Tomimatsu A., Sato H., New exact solution for the gravitational field of a spinning mass, Phys. Rev. Lett. 29 (1972), 1344–1345. [73] Tomizawa S., Ishihara H., Exact solutions of higher dimensional black holes, arXiv:1104.1468. [74] Tomizawa S., Morisawa Y., Yasui Y., Vacuum solutions of five dimensional Einstein equations generated by inverse scattering method, Phys. Rev. D 73 (2006), 064009, 8 pages, hep-th/0512252. [75] Tomizawa S., Nozawa M., Vacuum solutions of five dimensional Einstein equations generated by inverse scattering method. II. Production of the black ring solution, Phys. Rev. D 73 (2006), 124034, 10 pages, hep-th/0604067. [76] Tomizawa S., Uchida Y., Shiromizu M., Twist of stationary black hole or ring in five dimensions, Phys. Rev. D 70 (2004), 064020, 5 pages, gr-qc/0405134. [77] Verdaguer E., Soliton solutions in spacetime with two spacelike Killing fields, Phys. Rep. 229 (1993), 1–80. [78] Yazadjiev S.S., 5D Einstein–Maxwell solitons and concentric rotating dipole black rings, Phys. Rev. D 78 (2008), 064032, 11 pages, arXiv:0805.1600. [79] Yazadjiev S.S., Black saturn with a dipole ring, Phys. Rev. D 76 (2007), 064011, 8 pages, arXiv:0705.1840. [80] Yazadjiev S.S., Completely integrable sector in 5D Einstein–Maxwell gravity and derivation of the dipole black ring solutions, Phys. Rev. D 73 (2006), 104007, 7 pages, hep-th/0602116. http://dx.doi.org/10.1103/PhysRevD.73.044030 http://arxiv.org/abs/hep-th/0504018 http://arxiv.org/abs/1111.1903 http://dx.doi.org/10.1016/0003-4916(86)90186-7 http://dx.doi.org/10.1063/1.525735 http://dx.doi.org/10.1007/BF02551200 http://dx.doi.org/10.1103/PhysRevD.73.044004 http://arxiv.org/abs/hep-th/0507250 http://arxiv.org/abs/hep-th/0612005 http://dx.doi.org/10.1017/CBO9780511606359 http://dx.doi.org/10.1088/0266-5611/10/3/013 http://dx.doi.org/10.1006/jmaa.2001.7577 http://dx.doi.org/10.1051/mmnp/20105415 http://arxiv.org/abs/0909.1537 http://dx.doi.org/10.1070/RM1986v041n01ABEH003200 http://dx.doi.org/10.1016/S0926-2245(01)00062-6 http://arxiv.org/abs/gr-qc/0301020 http://dx.doi.org/10.1017/CBO9780511535185 http://dx.doi.org/10.1007/JHEP05(2011)052 http://arxiv.org/abs/1102.3942 http://dx.doi.org/10.1103/PhysRevD.68.044021 http://arxiv.org/abs/hep-th/0306044 http://dx.doi.org/10.1007/BF02784569 http://dx.doi.org/10.1007/BF02784569 http://dx.doi.org/10.1103/PhysRevLett.29.1344 http://arxiv.org/abs/1104.1468 http://dx.doi.org/10.1103/PhysRevD.73.064009 http://arxiv.org/abs/hep-th/0512252 http://dx.doi.org/10.1103/PhysRevD.73.124034 http://arxiv.org/abs/hep-th/0604067 http://dx.doi.org/10.1103/PhysRevD.70.064020 http://dx.doi.org/10.1103/PhysRevD.70.064020 http://arxiv.org/abs/gr-qc/0405134 http://dx.doi.org/10.1016/0370-1573(93)90139-5 http://dx.doi.org/10.1103/PhysRevD.78.064032 http://arxiv.org/abs/0805.1600 http://dx.doi.org/10.1103/PhysRevD.76.064011 http://arxiv.org/abs/0705.1840 http://dx.doi.org/10.1103/PhysRevD.73.104007 http://arxiv.org/abs/hep-th/0602116 1 Introduction 2 Binary Darboux transformations in bidifferential calculus 3 Solutions of the non-autonomous chiral model 3.1 The equations for P and Q 3.2 The equations for U and V 3.3 The Sylvester equation and the solution formula 3.4 A reduction condition 4 Solutions of the vacuum Einstein equations 4.1 Solutions of the equations for the metric function f 4.2 From diagonal to non-diagonal solutions 4.3 Solutions of the vacuum Einstein equations in four dimensions 4.4 Solutions of the vacuum Einstein equations in five dimensions 4.4.1 Myers-Perry black holes 4.4.2 Black saturn 4.4.3 Double Myers-Perry black hole solution 4.4.4 Bicycling black rings 5 Final remarks Appendix A. Addendum to Example 4.1 Appendix B. Some proofs References

Binary Darboux Transformations in Bidifferential Calculus and Integrable Reductions of Vacuum Einstein Equations

Репозитарії

Схожі ресурси