Gravity of a static massless scalar field and a limiting   Schwarzschild-like geometry
  We study a set of static solutions of the Einstein equations in presence of a massless scalar field and establish their connection to the Kantowski-Sachs cosmological solutions based on some kind of duality transformations. The physical properties of the limiting case of an empty hyperbolic spacetime (pseudo-Schwarzschild geometry) are analyzed in some detail. 
M. Gaudin, V. Gorini, A. Kamenshchik, U. Moschella, V. Pasquier