Self-Dual Solutions of Einstein's Equation| 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 ±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];
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 π2(Σ) = 0, three linearly
independent vectors eai,
divergenceless with respect to some reference metric \(^0q\)ab, and
evolving them by \(\dot e\)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;
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)].
main page
– abbreviations
– journals – comments
– other sites – acknowledgements
send feedback and suggestions to bombelli at olemiss.edu – modified 25 jun 2019