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,

*R_abcd:=  _ab^mn R_mncd = R_abcd ;

This implies the same condition on the Weyl tensor and that R_ab = 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,

*R_abcd:=  _ab^mn R_mncd = i R_abcd ;

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,

*C_abcd:=  _ab^mn C_mncd = C_abcd (Euclidean case), or i C_abcd (complex Lorentzian case).

* History: The most general were found independently by Penrose's, Newman's and Plebanski's groups.
* Properties: If one gives data on ^– for (anti-)self-dual solutions, and one evolves them, one finds at ⁺ 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 _abcd 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, wormholes.

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 with ₂() = 0, three linearly independent vectors e^a_i, divergenceless wrt some reference metric ⁰q_ab, and evolving them by e^·a_i:= 2^–1/2 _ijk [e_j, e_k]^a; Then the metric g^ab:= _i e^a_i e^b_i + t^a t^b is a solution of Einstein's equation (all self-dual metrics are locally like this); If the e's 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 g_ab is equivalent to the vanishing of the curvature F_abA^B, 2 _{[a b] A} = F_abA^B _B, with defined by _{a AB} = 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; 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 Plebanski equations]; Mansi et al a0808 [and 3+1 split].
@ Symmetries and conservation laws: Boyer & Plebanski JMP(85); Boyer & Winternitz JMP(89); Husain JMP(95); Strachan JMP(95)ht/94; Popov et al PLB(96)ht.
@ Lagrangian: Plebanski & 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-in.
@ Of Einstein-Yang-Mills: Selivanov PLB(98)ht/97 [perturbiner]; > s.a. solitons.
@ Riemannian metrics: Torre JMP(90) [linearization stability]; Malykh et al gq/03/CQG [on K3, anti-self-dual].
@ (+,+,–,–) metrics: Dunajski PRS(02)m.DG/01 [with extra structure].
{& A A, seminar 21.01.1987; D Robinson, seminar 8.04.1987}.

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