The sum over spacetimes: gravity and inflation

[Philip Thrift]

https://arxiv.org/abs/1808.10448

Action Principle for Isotropic General Relativity

Thomas C. Bachlechner

(Submitted on 30 Aug 2018)

We study the generally covariant theory governing an isotropic spacetime region with uniform energy density. Gibbons, Hawking and York showed that fixing the induced boundary metric yields a well-posed variational problem. However, as we demonstrate, fixing the boundary metric violates general covariance and allows the mass of a back hole to vary. This observation has dramatic consequences for path integrals: A sum over spacetimes with fixed boundary metrics is a sum over classically distinct black holes. Instead, we merely demand that coordinates exist such that the metric at the boundary is the Schwarzschild-(A)dS metric of fixed mass M and two-sphere radius R. We derive the action that yields a well-posed variational problem for these physical boundary conditions. The action vanishes for all stationary and isotropic spacetimes. A vanishing action implies that both a Schwarzschild black hole and pure de Sitter space each have one unique semiclassical state. Our results provide a novel and radically conservative approach to several long-standing issues in quantum gravity, such as the wavefunction of the universe, the black hole information paradox, vacuum decay rates and the measure problem of eternal inflation.

@philipthrift

[Lawrence Crowell]

The partition is 

Z = ∫D[g]exp(-i[S_∂M + S_M])

where the sum is over not just metric configurations of the manifold M, but its boundary as well. This tends to suggest the boundary conditions are themselves dynamics. A pocket world in an internal inflating spacetime would have this feature. I though conjecture this may be a transient state of affairs. The boundary action may transition into an action over a field theory. This has some holographic content, and it would replace these boundary dynamics with a quantum field. Further, the pocket world becomes a complete manifold without boundary.

LC