On Schrödinger equations in Einstein's theory of gravity (Q1576217)

This paper deals with the Hamiltonian formalism and consecutive canonical quantization of a system composed by incoherent matter coupled with its own gravitational field. The system is studied within the framework of general relativity. Incoherent matter is described by giving a line congruence and assuming that the energy of the matter satisfies a continuity equation. Then, a formalism similar to the ADM approach is built up. As an example, quantization of Friedmann-Robertson-Walker models where gravity is generated by incoherent matter are discussed. It is shown that, in that case, a strict application of Dirac's constraint formalism to the complete Lagrangian leads to a Schrödinger-like equation for a wavefunction of the matter and gravity. It is remarked that the WKB approximation is not applicable whenever the scale factor of FRW model under consideration approaches the null value, no matter what the value of the spatial curvature is. Moreover, for positive curvature and closed spacelike section of the spacetime, WKB approximation is applicable to \(A\) near its maximum.