Self-Dual Solutions of Einstein's Equation  

In General > s.a. actions for general relativity; self-dual gauge fields; solutions of Einstein's equation.
$ Riemannian metrics: (Anti-)self-dual Riemannian metrics are those whose Riemann tensor is (anti-)self-dual,

*Rabcd:= \(1\over2\)εabmn Rmncd = ± Rabcd ;

This implies the same condition on the Weyl tensor and that Rab = 0, so all such metrics are solutions of the (Euclideanized) Einstein equation.
$ Lorentzian metrics: (Anti-)self-dual Lorentzian metrics are those whose Riemann tensor is (anti-)self-dual up to multiplication by i,

*Rabcd:= \(1\over2\)εabmn Rmncd = ± i Rabcd ;

Notice that the metrics must be complex; As in the Riemannian case, they are solutions of the (complex) Einstein equation.
$ Conformally (anti-)self-dual solutions: Those whose Weyl tensor is (anti-)self-dual,

*Cabcd:= \(1\over2\)εabmn Cmncd = ± Cabcd (Euclidean case), or ± i Cabcd (complex Lorentzian case).

* History: The most general were found independently by Penrose's, Newman's and Plebański's groups.
* Properties: If one gives data on \(\cal I\) for (anti-)self-dual solutions, and one evolves them, one finds at \(\cal I\)+ the same data, i.e., the classical S-matrix is trivial.
* Complex Lorentzian: We cannot distinguish self-dual from anti-self-dual solutions, because we cannot resolve the sign ambiguity in εabcd by comparing it with a reference one; So we call all these solutions half-flat.
@ Solutions: Devchand & Ogievetsky CQG(96)ht/94; García-Compeán & Matos PRD(95)ht/94; Koshti & Dadhich gq/94 [general solution]; Rosly & Selivanov ht/97 [perturbiner]; Calderbank & Pedersen JDG(02)m.DG/01 [2 Killing vector fields]; > s.a. black holes in modified theories; wormhole solutions.

Alternative Characterizations, Relationships
* Relationships: Self-dual gravity can be expressed in terms of the Moyal bracket [@ Strachan].
* Triad: An equivalent formulation is obtained by giving, on a 3-slice Σ with π2(Σ) = 0, three linearly independent vectors eai, divergenceless with respect to some reference metric 0qab, and evolving them by e·ai:= 2–1/2 εijk [ej, ek]a; Then the metric gab:= ∑i eai ebi + ta tb is a solution of Einstein's equation (all self-dual metrics are locally like this); If the es become linearly dependent after a while, g becomes degenerate, and we could have a model for topology change.
* Connection variables: In Ashtekar variables, a self-dual metric gab is equivalent to the vanishing, FabAB = 0, of the curvature defined by

2 \(\cal D\)[a \(\cal D\)b] λA = FabAB λB, where the covariant derivative \(\cal D\) is defined by \(\cal D\)a εAB = 0 .

@ Connection variables: Bengtsson CQG(90) [and Yang-Mills Hamiltonian].
@ And integrable theories: Strachan JPA(96)ht [deformation and Toda lattice]; Ueno MPLA(96)ht/95.

References > s.a. 3D gravity; bianchi IX models; born-infeld theory; conformal gravity; perturbations in general relativity.
@ General: Ashtekar JMP(86), in(86), in(88), et al CMP(88); Mason & Newman CMP(89) [self-dual Einstein and Yang-Mills theory]; Koshti & Dadhich CQG(90); Kalitzin & Sokatchev PLB(91); Grant PRD(93)gq; Husain CQG(93), PRL(94)gq; Strachan CQG(93); Abe MPLA(95) [moduli spaces]; Devchand & Ogievetsky CQG(96)ht/94; García-Compeán et al RMF(96)ht/94 [Hopf algebra structure]; Tafel gq/06 [description]; Jakimowicz & Tafel CQG(06)gq [Husain and Plebański equations]; Mansi et al CQG(09)-a0808 [and 3+1 split]; Malykh & Sheftel JPA(11)-a1011 [and the general heavenly equation].
@ Symmetries and conservation laws: Boyer & Plebański JMP(85); Boyer & Winternitz JMP(89); Husain JMP(95); Strachan JMP(95)ht/94; Popov et al PLB(96)ht.
@ Lagrangian: Plebański & Przanowski PLA(96)ht/95, García-Compeán et al PLA(96) [chiral approach, WZW-like action].
@ Integrability of self-dual Einstein equation: in Penrose GRG(76); Nutku Sigma(07)n.SI [completely integrable]; Nutku et al JPA(08)-a0802 [multi-Hamiltonian structure].
@ Deformations: Takasaki PLB(92), JGP(94); García-Compeán et al APPB(98)ht/97, ht/97-MG8.
@ Of Einstein-Yang-Mills theory: Selivanov PLB(98)ht/97 [perturbiner]; > s.a. solitons.
@ Riemannian metrics: Torre JMP(90) [linearization stability]; Malykh et al CQG(03)gq [on K3, anti-self-dual].
@ Generalizations: Dunajski PRS(02)m.DG/01 [(+,+,–,–) metrics, with extra structure]; Fino & Nurowski a1109 [in 9D, based on an irreducible representation of SO(3) × SO(3)].

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