The quantization of gravity in globally hyperbolic spacetimes

Understand

We apply the ADM approach to obtain a Hamiltonian description of the Einstein-Hilbert action.

• In doing so we add four new ingredients: (i) We eliminate the diffeomorphism constraints.
• (ii) We replace the densities $\sqrt g$ by a function $\f(x,g_{ij})$ with the help of a fixed metric $\chi$ such that the Lagrangian and hence the Hamiltonian are functions.
• (iii) We consider the Lagrangian to be defined in a fiber bundle with base space $\so$ and fibers F(x) which can be treated as Lorentzian manifolds equipped with the Wheeler-DeWitt metric.