Separation of variables and Hamiltonian formulation for the Ernst equation.

It is shown that the vacuum Einstein equations for an arbitrary stationary axisymmetric space-time can be completely separated by re-formulating the Ernst equation and its associated linear system in terms of a non-autonomous Schlesinger-type dynamical system. The conformal factor of the metric coincides (up to some explicitly computable factor) with the τ -function of the Ernst equation in the presence of finitely many regular singularities. We also present a canonical formulation of these results, which is based on a “two-time” Hamiltonian approach, and which opens new avenues for the quantization of such systems. Introduction. In this letter we demonstrate that the vacuum Einstein equations for space-times with two commuting Killing vectors can be re-formulated in terms of a pair of compatible ordinary matrix differential equations. Similar results can be shown to hold for the more general equations obtained by dimensional reduction from higher-dimensional theories of gravity and supergravity with Supported by Alexander von Humboldt Foundation 1 matter couplings to two dimensions. As a by-product, we establish a previously unknown relation between the conformal factor of the associated metric and the so-called τ -function, which plays a pivotal role in the modern formulation of integrable systems [1, 2]. Thirdly, we present a canonical formulation of these results, which avoids certain technical difficulties encountered in previous treatments. Our results suggest that an exact quantization of axisymmetric stationary (matter-coupled) gravity by exploiting techniques developed for flat space (quantum) integrable systems [3, 4] is now within reach. The Ernst equation and related linear system. We start from the following metric on stationary axisymmetric space-time [5] ds = f[e(dx + dρ) + ρdφ]− f(dt+ Fdφ) (1) where (x, ρ) are Weyl canonical coordinates; t and φ are the time and angular coordinates, respectively. The functions f(x, ρ), F (x, ρ) and k(x, ρ) are related to the complex Ernst potential E(x, ρ) by f = ReE , Fξ = 2ρ (E − Ē)ξ (E + Ē)2 , kξ = 2iρ EξĒξ (E + Ē)2 , (2) where ξ = x+ iρ , ξ̄ = x− iρ; hereafter subscripts ξ, ξ̄ denote partial derivatives with respect to these variables. In terms of the potential E(ξ, ξ̄), Einstein’s equations for the metric (1) in particular imply the Ernst equation [5] ((ξ − ξ̄)gξg )ξ̄ + ((ξ − ξ̄)gξ̄g )ξ = 0 (3) with the symmetric matrix g = 1 E + Ē   2 i(E − Ē) i(E − Ē) 2EĒ   . (4)