Topology and volume effects in quantum gravity: Wheeler–DeWitt theory

Abstract

We consider the quantization of space-times which can possess different topologies within a symmetry reduced version of Wheeler-DeWitt theory. The quantum states are defined from a natural decomposition as an outer-product of a topological state, dictating the topology of the two-surfaces of the space-time, and a geometric state, which controls the geometry and is comprised of solutions to the Wheeler-DeWitt constraints. Within this symmetry reduced theory an eigenvalue equation is derived for the two-volume of spacetime, which for spherical topology is fixed to a value of 4π. However, for the other topologies it is found that the spectrum can be discrete and hence the universe, if in one of these other topological states, may only possess certain possible values for the two-volume, whereas classically all values are allowed. We analyze this result in the context of pure gravity (black holes).

Non-commutative black holes of various genera in the connection formalism

“…Such black hole solutions have been studied in detail in [47] - [50], and within quantum gravity theories in [55] - [58].…”

Quantum relativistic cosmology: dynamical interpretation and tunneling universe

“…The WdW equation, since that time when it was proposed until now, has inspired several researches in accordance with the original proposals of Wheeler and DeWitt. Among these researches, we can mention investigations on quantum cosmology in the 1980's [5][6][7] and recently [8][9][10][11][12][13][14][15][16][17][18][19]. Along this line of research which uses the WdW equation, we can also mention very recent works with studies concerning the tunneling wave function of the universe [20], the interaction of multiverse and effects on the cosmic microwave background [21] and about the consistence of the WdW equation in a modified theory of gravity [22].…”

Class of solutions of the Wheeler-DeWitt equation in the Friedmann-Robertson-Walker universe

“…From that time up to now, this equation has inspired a lot of investigations in quantum cosmology [4][5][6], in which context the evolution of the universe is determined by the quantum states which obey the WDW equation [7][8][9][10][11], in such a way that in the appropriate classical limit the Friedmann solutions are recovered. Along this line of research, some results using different approaches have been obtained [12][13][14][15][16][17][18][19]. Especially, we emphasize the line of research which uses the WDW equation as a fundamental tool to formulate the so-called loop quantum gravity [20].…”