Bianchi-IX, Darboux–Halphen and Chazy–Ramanujan

Bianchi-IX four metrics are SU(2) invariant solutions of vacuum Einstein equation, for which the connection-wise self-dual case describes the Euler top, while the curvature-wise self-dual case yields the Ricci flat classical Darboux–Halphen system. It is possible to see such a solution exhibiting Ricci flow. The classical Darboux–Halphen system is a special case of the generalized one that arises from a reduction of the self-dual Yang–Mills equation and the solutions to the related homogeneous quadratic differential equations provide the desired metric. A few integrable and near-integrable dynamical systems related to the Darboux–Halphen system and occurring in the study of Bianchi-IX gravitational instanton have been listed as well. We explore in details whether self-duality implies integrability.

SCHWARZSCHILD–DE SITTER METRIC AND INERTIAL BELTRAMI COORDINATES

Under consideration of coordinate conditions, we get the Schwarzschild–Beltrami–de Sitter (S-BdS) metric solution of the Einstein field equations with a cosmological constant Λ. A brief review to the de Sitter invariant special relativity (dS-SR), and de Sitter general relativity (dS-GR, or GR with a Λ) is presented. The Beltrami metric Bμν provides inertial reference frame for the dS-spacetime. By examining the Schwarzschild–de Sitter (S-dS) metric [Formula: see text] existed in literatures since 1918, we find that the existed S-dS metric [Formula: see text] describes some mixing effects of gravity and inertial-force, instead of a pure gravity effect arisen from "solar mass" M in dS-GR. In this paper, we solve the vacuum Einstein equation of dS-GR, with the requirement of gravity-free metric [Formula: see text]. In this way we find S-BdS solution of dS-GR, written in inertial Beltrami coordinates. This is a new form of S-dS metric. Its physical meaning and possible applications are discussed.