Hamiltonian reduction and exact solutions to Einstein's equations

As an application of the recently proposed Hamiltonian reduction in the (2+2) formalism of general relativity under no symmetry assumptions, I present three exact solutions to Einstein's equations, namely, the Minkowski spacetime, the plane symmetric solution of Taub, and the Kasner spacetime, by solving Hamilton's equations of motion governed by the non-vanishing true gravitational Hamiltonian in the privileged spacetime coordinates. The true Hamiltonian is a local function of the free data only, namely, the conformal two metric and its conjugate momentum, and depends explicitly on the physical time which is identified as the area element of the cross section of a null hypersurface. Possibilities of constructing the general solution to Einstein's equations in terms of the freely specifiable true gravitational degrees of freedom are discussed.