On self-dual gravity

Abstract

We study the Ashtekar-Jacobson-Smolin equations that characterise four dimensional complex metrics with self-dual Riemann tensor.We find that we can characterise any self-dual metric by a function that satisfies a non-linear evolution equation, to which the general solution can be found iteratively. This formal solution depends on two arbitrary functions of three coordinates. We construct explicitly some families of solutions that depend on two free functions of two coordinates, included in which are the multi-centre metrics of Gibbons and Hawking.

Symmetries, currents and conservation laws of self-dual gravity

“…The self-duality equations on the curvature of a metric in four dimensions are an important example of a multidimensional integrable system, which can be solved by a twistor geometric construction [8][9][10][11]. The discussion of hidden symmetries of this model was started in the papers [12] on the basis of Plebański's equations [13], and has been continued by many authors (see, e.g., [14][15][16][17][18]). For the study of hidden symmetries the reformulation of the self-dual gravity equations as (reduced) self-dual Yang-Mills equations with infinite-dimensional gauge group was very useful [19,16,20] (see also the clear exposition in [21]).…”

“…For the study of hidden symmetries the reformulation of the self-dual gravity equations as (reduced) self-dual Yang-Mills equations with infinite-dimensional gauge group was very useful [19,16,20] (see also the clear exposition in [21]). It was shown that the self-dual gravity equations are invariant with respect to a group, whose generators form the affine Lie algebra w ∞ ⊗ C[λ, λ −1 ], λ ∈ C, associated with the Lie algebra w ∞ of area-preserving diffeomorphisms of a certain (null) surface [12,[14][15][16][17][18].…”

Evanescent ergosurfaces and ambipolar hyperkähler metrics

“…Biquard's construction is a modification of the Ashtekar-Jacobson-Smolin (AJS) initial value construction of hyperkähler manifolds [14][15][16], which we review briefly here and it more detail in appendix A. The AJS construction consists of choosing three linearly independent vector fields V i on S which preserve a fixed volume form v on S. One then extends these vector fields off S using the Nahm evolution equations 11) and, defining a fourth vector V 0 = ∂/∂x, one obtains the hyperkähler 2-forms and metric via…”

Solutions in self-dual gravity constructed via chiral equations

“…Several authors [3] [4], beginning with the AJS formulation made contact with the Plebański approach to self-dual gravity [5]. In [3] Grant has shown that Eqs. (1) are related in a very close way with the first heavenly equation of ref.…”