A Topological Extension of General Relativity

Abstract

A set of algebraic equations for the topological properties of space-time is derived, and used to extend general relativity into the Planck domain. A unique basis set of three-dimensional prime manifolds is constructed which consists of $S^3$, $S^1\times S^2$, and $T^3$. The action of a loop algebra on these prime manifolds yields topological invariants which constrain the dynamics of the four-dimensional space-time manifold. An extended formulation of Mach's principle and Einstein's equivalence of inertial and gravitational mass is proposed which leads to the correct classical limit of the theory. It is found that the vacuum possesses four topological degrees of freedom corresponding to a lattice of three-tori. This structure for the quantum foam naturally leads to gauge groups O(n) and SU(n) for the fields, a boundary condition for the universe, and an initial state characterized by local thermal equilibrium. The current observational estimate of the cosmological constant is reproduced without fine-tuning and found to be proportional to the number of macroscopic black holes. The black hole entropy follows immediately from the theory and the quantum corrections to its Schwarzschild horizon are computed.