On the integrability of spherical gravitational waves in vacuum

The general class of Robinson-Trautman metrics that describe gravitational radiation in the exterior of bounded sources in four space-time dimensions is shown to admit zero curvature formulation in terms of appropriately chosen two-dimensional gauge connections. The result, which is valid for either type II or III metrics, implies that the gravitational analogue of the Lienard-Wiechert fields of Maxwell equations form a new integrable sector of Einstein equations for any value of the cosmological constant. The method of investigation is similar to that used for integrating the Ricci flow in two dimensions. The zero modes of the gauge symmetry (factored by the center) generate Kac’s K2 simple Lie algebra with infinite growth. ∗On sabbatical leave from Department of Physics, University of Patras, GR-26500 Patras, Greece; e-mail: [email protected] The theory of gravitational waves is an old subject. The existence of general asymptotically flat radiative space-times and the explicit construction of various classes of solutions have received considerable attention over the years. They have several applications that serve as testing bed for the physical properties of gravitation. Spherical gravitational waves are quite interesting in this respect, as they are thought to represent an isolated gravitationally radiating system. Yet, there has been no systematic way to understand the algebraic structure of the non-linear equations that govern their propagation in vacuum, apart from global considerations and the behavior of solutions in the asymptotic future. A few explicit solutions are also known to this day. It is the purpose of the present work to provide a new formulation of the problem by casting the corresponding sector of Einstein equations into zero curvature form, thus making a decisive step toward their integration. The mathematical framework that achieves this purpose points to the relevance of a novel class of infinite dimensional Lie algebras that might be of more general value in dynamical gravitational problems. Recall that the most general gravitational vacuum solution in four space-time dimensions which admits a geodesic, shear-free, twist-free (but diverging) null congruence is described by the class of Robinson-Trautman metrics with line element, [1, 2], ds = 2redzdz̄ − 2dtdr −Hdt , (1) where H(z, z̄; t) has the special form H = r∂tΦ−∆Φ− 2m(t) r − Λr 2 3 . (2) The affine parameter r varies along the rays of the repeated null eigenvector and t is a retarded time coordinate. Closed surfaces of constant r and t represent distorted two-dimensional spheres Σ with metric coefficient given by Φ(z, z̄; t) in the system of conformally flat coordinates (z, z̄). The parameter m(t), which in some cases represents the physical mass of the system, can be set equal to a constant when differs from zero; in the sequel m is taken positive and set equal to 1/3 by appropriate relabeling of the null hyper-surfaces in the Robinson-Trautman class of metrics. The special case m = 0 will be considered separately at the end. Finally, Λ is the value of the cosmological constant in space-time, which can be positive, negative or zero depending on the physical circumstances. There is a close analogy between the Lienard-Wiechert fields of Maxwell equations and the Robinson-Trautman metrics of Einstein equations for they both admit a principal null vector field which is geodesic, shear and twist free with non-vanishing divergence, thus giving a more intuitive meaning to the general ansatz for the metrics above, [3]. In the gravitational case, the field equations imply that all vacuum solutions in this class with cosmological constant Λ satisfy the following parabolic fourth order differential equation for the unknown function Φ, ∂tΦ = −∆∆Φ , (3) where ∆ = exp(−Φ)∂∂̄ denotes the Laplace-Beltrami operator on the distorted twodimensional spheres Σ. Note that this non-linear equation, which is named after Robinson