Ultralocal limit of the gravitational field coupled to a scalar field

The ultralocal limit of the gravitational field and the gravitational field

The ultralocal limit of the gravitational field and the gravitational field coupled to a scalar field are studied from both the classical and quantum standpoints. For the classical cases, use will be made of the minisuperspace formalism. In the quantum case, techniques for quantizing ultralocal self-interacting scalar fields are extended to deal with the gravitational field. It will be shown that although the singularity behavior of the gravitational field by itself is characterized by Kasner solutions at each spatial point classically, the quantum case requires the addition of a repulsive scattering potential in order to have well-defined self-adjoint field operators. For the gravitational field coupled to a massive scalar field, both the singularity behavior and when the mass terms become important in the ultralocal limit are studied classically and from the quantum standpoint. Classically, the singularity behavior of the gravitational field is modified by the scalar field so that the solutions are Kasner-like in a four-dimensional hyperspace minisuperspace which allows for expansions in all three axes in space-time. The quantum case for the singularity behavior has solutions which are confined by repulsive potential barriers to regions where only the states with the lowest ‘‘angular momentum’’ can evolve out of. Adding matter terms introduces solutions that show the periodic motion of a harmonic oscillator. For the quantum case, a discrete eigenvalue spectrum occurs for the scalar degree of freedom. It is therefore seen that in the ultralocal limit, the quantum and classical cases are significantly different.