Constraints are not enough

TL;DR

This paper demonstrates that fixing a foliation and imposing constraints do not resolve the unboundedness of the Euclidean Einstein-Hilbert action under various boundary conditions, challenging previous assumptions.

Contribution

It shows that common methods to bound the gravitational action via foliation constraints fail across multiple boundary conditions, questioning their effectiveness.

Findings

Foliation fixing does not bound the Euclidean Einstein-Hilbert action.

Boundary conditions like compact, asymptotically flat, or AdS slices do not resolve unboundedness.

Fixing scalar curvature and Wick rotation also fail to produce a bounded action.

Abstract

The Euclidean Einstein-Hilbert action is well-known to be unbounded below and thus to raise many questions regarding the definition of the gravitational path integral. A variety of works since the late 1980's have suggested that this problem disappears when one fixes a foliation of the spacetime and imposes the corresponding gravitational constraints. However, we show here that this approach fails with various classes of boundary conditions imposed on the foliation: compact slices without boundary, asymptotically flat, or asymptotically locally anti-de Sitter slices. We also discuss the idea of fixing the scalar curvature and Wick-rotating the conformal factor, and show that it also fails to produce an action bounded from below.