A new characterization of half-flat solutions to Einstein's equation

Abstract

A 3+1 formulation of complex Einstein's equation is first obtained on a real 4-manifoldM, topologically σ×R, where σ is an arbitrary 3-manifold. The resulting constraint and evolution equations are then simplified by using variables that capture the (anti-) self dual part of the 4-dimensional Weyl curvature. As a result, to obtain a vacuum self-dual solution, one has just to solve one constraint and one “evolution” equation on a field of triads on σ:

$$Div V_i^a = 0 and \dot V_i^a = \varepsilon _{ijk} \left[ {V_j ,V_k } \right]^a , with i \equiv 1,2,3,$$

where Div denotes divergence with respect to a fixed, non-dynamical volume element. If the triad is real, the resulting self-dual metric is real and positive definite. This characterization of self-dual solutions in terms of triads appears to be particularly well suited for analysing the issues of exact integrability of the (anti-) self-dual Einstein system. Finally, although the use of a 3+1 decomposition seems artificial from a strict mathematical viewpoint, as David C. Robinson has recently shown, the resulting triad description is closely related to the hyperkähler geometry that (anti-) self-dual vacuum solutions naturally admit.