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}:= \(1\over2\)*ε*_{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}:= \(1\over2\)*ε*_{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}:= \(1\over2\)*ε*_{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 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 *ε*_{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; 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 π_{2}(Σ)
= 0, three linearly independent vectors *e*^{a}_{i},
divergenceless with respect to some reference metric ^{0}*q*_{ab},
and evolving them by *e*^{·a}_{i}:=
2^{–1/2} *ε*_{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, *F*_{abA}^{B} = 0, of the curvature defined by

2 \(\cal D\)_{[a} \(\cal D\)_{b]} *λ*_{A}
= *F*_{abA}^{B} *λ*_{B},
where the covariant derivative \(\cal D\) is defined
by \(\cal D\)_{a} *ε*_{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; 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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