A Note On Complex Spacetime Metrics

TL;DR

This work argues that including complex spacetime metrics in the gravitational path integral requires a restricting principle to avoid unphysical saddles. By adopting Kontsevich-Segal allowable metrics, the paper shows many exotic complex solutions are excluded, while identifying several physically meaningful complex saddles such as topology-changing Lorentzian cases, Hartle-Hawking contours, timefolds, the double-cone geometry, and rotating black holes as allowable. It then outlines a program to define the integration cycle via gradient flow that remains inside the allowable sector, aiming to extend semiclassical gravity beyond perturbation theory with better convergence properties. The discussion also highlights open issues regarding diffeomorphism gauge, flux sectors, and possible nonperturbative extensions of positivity theorems, signaling a path toward a more coherent quantum gravity framework grounded in QFT consistency on complex backgrounds.

Abstract

For various reasons, it seems necessary to include complex saddle points in the "Euclidean" path integral of General Relativity. But some sort of restriction on the allowed complex saddle points is needed to avoid various unphysical examples. In this article, a speculative proposal is made concerning a possible restriction on the allowed saddle points in the gravitational path integral. The proposal is motivated by recent work of Kontsevich and Segal on complex metrics in quantum field theory, and earlier work of Louko and Sorkin on topology change from a real time point of view.

Figures (6)

• Figure 1: Some choices of curve $r(u)$ in the complex $r$-plane, leading to various complex flat metrics. (a) A loop that starts and ends at $r=0$, leading to a complex flat metric on ${\sf S}^D$. (b) A homotopically inequivalent immersed loop that starts and ends at $r=0$, leading to another complex flat metric on ${\sf S}^D$. This example is equivalent to the one in (a) if equivalence is based on homology rather than homotopy. (c) A path from $r=-\infty$ to $r=+\infty$, avoiding the origin in the complex plane. It leads to a flat "wormhole" metric, as sketched in fig. \ref{['Wormhole']}(a).
• Figure 2: (a) Two copies of ${\mathbb{R}}^D$ have been connected via a "wormhole." (b) Two possibly distant regions of the same ${\mathbb{R}}^D$ are connected by a wormhole.
• Figure 3: (a) Creation of a universe from nothing. (b) Splitting of a universe in two. In each case the time coordinate $t$ runs vertically, as shown.
• Figure 4: A torus embedded in ${\mathbb{R}}^3$ in such a way that the coordinate $t$, running vertically, has a maximum, a minimum, and two saddle points. An analogous embedding of a surface of genus $g$ has a maximum, a minimum, and $2g$ saddle points.
• Figure 5: The genus 1 contribution to the one-boundary partition function in JT gravity with a negative cosmological constant, as studied in SSSJT. There is no conventional critical point; in searching for one, one finds that the length $b$ of the indicated geodesic that separates the disc from the genus 1 surface tends to shrink to 0, because of a contribution to the action proportional to $b^2$. Instead of a conventional critical point there is a critical point at infinity -- or more precisely, at $b=0$. This critical point describes a hyperbolic disc with one cusp and a hyperbolic torus with one cusp, with the cusp points identified. It controls the semiclassical behavior of the path integral.