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}
±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];
Chacón & García-Compeán a1812 [via Hitchin's equations];
> 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 \(^0q\)_{ab}, and
evolving them by \(\dot e\)^{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;
differential forms; 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];
Krasil'shchik & Sergyeyev a1901 [with non-zero cosmological constant].

@ __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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send feedback and suggestions to bombelli at olemiss.edu – modified 25 jun 2019