A new paradigm for dealing with the dynamics of totally constrained theories like general relativity will continue to be developed both at a classical and quantum level. The paradigm consists in constructing discrete theories such that the continuum theory of interest arises as a well defined limit. The specific proposal for constructing the discrete theories, called "uniform discretizations" has several attractive features. In particular the discrete theories are free of constraints and therefore avoid many of the hard conceptual problems that complicate traditional quantum gravity, and nevertheless manage to provide a well defined limit in which one recovers the continuum theory of interest. A paradigm emerges that is analogous to that of lattice gauge theory, where discrete theories are used in a limiting procedure to define a continuum theory of interest. This approach has already been applied successfully in some situations where the space-time has some spatial dependence (i.e. more complicated than homogeneous cosmologies) but without genuine degrees of freedom, most notably providing a characterization of the complete space-time of a non-singular loop quantum gravity black hole. This NSF award will focus on applying the paradigm in situations of increasing complexity. This includes the collapse of scalar fields to form black holes and related models. It is also proposed to further other topics related to the dynamics of quantum gravity like the problem of time and the issue of measurement in quantum mechanics. This research program also aids in the creation of human resources in US physics through the training of a postdoctoral researcher and provides the possibility that the discretization techniques introduced here may have applicability in other areas of science and engineering as well.
We have applied the rules of quantum mechanics to Einstein's general theory of relativity for situations that are spherical in nature and where space-time is empty of matter. The approach we use is known as loop quantum gravity. We have been able to solve the equations of the theory. The resulting solution can be viewed as a quantum version of a black hole. Black holes in the general theory of relativity exhibit what is known a singularity in their interior where the equations of the theory break down under the intense graviational fields present there. In the quantum solution we found, the equations continue to work in the region where general relativity broke down, although that region cannot be understood as an ordinary space time. Continuing the equations of the theory beyond that region leads us to another region where space time looks like that in the exterior of the black hole. As has been known since the pioneering work of Hawking in 1975, black holes emit radiation an should eventually evaporate. The process through which this happens is not yet understood. We have taken the first steps in loop quantum gravity towards understanding it by rederiving Hawking's result in this context. Loop quantum gravity predicts the same result as Hawking with small corrections due to the quantum nature of space-time. Next steps in this line of research will be to understand how the emitted radiation shrinks the black hole, leading ot its ultimate disappearance. This is considered by many as the most vexing problem of fundamental physics today.
