Vacua of the gravitational field

TL;DR

The authors show that the Poincaré vacuum in 4D gravity is not unique because BMS supertranslations are spontaneously broken, yielding a Goldstone boson C(z,\bar{z}) that labels inequivalent vacua and defines a bulk phase space interpolating between null infinities.They construct exact vacuum metrics in BMS gauge dependent on C, revealing a bulk defect carrying superrotation charges; a bulk source described by a Liouville stress-tensor arises when including finite superrotations, but physicality requires T_{AB}=0 to ensure energy boundedness and a well-defined variational principle.Supertranslations are shown to be symplectic (tangent) symmetries with zero charges, while superrotations act as external symplectic symmetries with finite conserved charges that probe the Goldstone boson; they also formulate a consistent counterterm-based derivation yielding vanishing on-shell presymplectic form for vacua without sources.The framework extends to multi-centered vacua with multiple bulk defects, each contributing its own superrotation charge, suggesting a rich degeneracy of flat-space vacua and potential connections to holography and black hole physics in asymptotically flat spacetimes.

Abstract

The Poincaré invariant vacuum is not unique in quantum gravity. The BMS supertranslation symmetry originally defined at null infinity is spontaneously broken and results in inequivalent Poincaré vacua. In this paper we construct the unique vacua which interpolate between past and future null infinity in BMS gauge and which are entirely characterized by an arbitary Goldstone boson defined on the sphere which breaks BMS invariance. We show that these vacua contain a defect which carries no Poincaré charges but which generically carries superrotation charges. We argue that there is a huge degeneracy of vacua with multiple defects. We also present the single defect vacua with its canonically conjugated source which can be constructed from a Liouville boson on the stereographic plane. We show that positivity of the energy forces the stress-tensor of the boson to vanish as a boundary condition. Finite superrotations, which turn on the sources, are therefore physically ruled out as canonical transformations around the vacua. Yet, infinitesimal superrotations are external symplectic symmetries which are associated with conserved charges which characterize the Goldstone boson.