Paper detail

Dynamical determination of the gravitational degrees of freedom

$[n+1]$-dimensional ($n\geq 3$) smooth Einsteinian spaces of Euclidean and Lorentzian signature are considered. The base manifold $M$ is supposed to be smoothly foliated by a two-parameter family of codimension-two-surfaces which are orientable and compact without boundary in $M$. By applying a pair of nested $1+n$ and $1+[n-1]$ decompositions, the canonical form of the metric and the conformal structure of the foliating codimension-two-surfaces a gauge fixing, analogous to the one applied in arXiv:1409.4914, is introduced. In verifying that the true degrees of freedom of gravity may conveniently be represented by the conformal structure it is shown first that regardless whether the primary space is Riemannian or Lorentzian, in terms of the chosen geometrically distinguished new variables, the $1+n$ momentum constraint can be written as a first order symmetric hyperbolic system. It is also argued that in the generic case the Hamiltonian constraint can be solved as an algebraic equation. By combining the $1+n$ constraints with the part of the reduced system of the secondary $1+[n-1]$ decomposition that governs the evolution of the conformal structure---in the Riemannian case with an additional introduction of an imaginary `time'---a well-posed mixed hyperbolic-algebraic-hyperbolic system is formed. It is shown that regardless whether the primary space is of Riemannian or Lorentzian if a regular origin exists solutions to this system are also solutions to the full set of Einstein's equations. This, in particular, offers the possibility of developing a new method for solving Einstein's equations in the Riemannian case. The true degrees of freedom of gravity are also found to be subject of a nonlinear wave equation.