Uses of complex metrics in cosmology

Abstract

Complex metrics are a double-edged sword: they allow one to replace singular spacetimes, such as those containing a big bang, with regular metrics, yet they can also describe unphysical solutions in which quantum transitions may be more probable than ordinary classical evolution. In the cosmological context, we investigate a criterion proposed by Witten (based on works of Kontsevich & Segal and of Louko & Sorkin) to decide whether a complex metric is allowable or not. Because of the freedom to deform complex metrics using Cauchy’s theorem, deciding whether a metric is allowable in general requires solving a complicated optimisation problem. We describe a method that allows one to quickly determine the allowability of minisuperspace metrics. This enables us to study the off-shell structure of minisuperspace path integrals, which we investigate for various boundary conditions. Classical transitions always reside on the boundary of the domain of allowable metrics, and care must be taken in defining appropriate integration contours for the corresponding gravitational path integral. Perhaps more surprisingly, we find that proposed quantum (‘tunnelling’) transitions from a contracting to an expanding universe violate the allowability criterion and may thus be unphysical. No-boundary solutions, by contrast, are found to be allowable, and moreover we demonstrate that with an initial momentum condition an integration contour over allowable metrics may be explicitly described in arbitrary spacetime dimensions.

On the initial singularity and extendibility of flat quasi-de Sitter spacetimes

“…Certainly, the issues of extendibility, geodesic incompleteness, and spacetime singularity change if the notion of spacetime continuum as we know it completely breaks down or, perhaps less drastically, simply if it is no longer real and Lorentzian. 19,20 For example, the Hartle-Hawking no-boundary proposal is a well-known example of quantum prelude to inflation, in which the 'pre-inflationary phase' is conjectured to be Riemannian [122] (or at least, the metric may have a complex transition from Riemannian to Lorentziansee, e.g., [123] and references therein for how such complex metrics might be theoretically constrained). Nevertheless, classical physics is still relevant in the context of quantum cosmology: if we imagine performing a semi-classical path integral of gravity (as a proxy for 17 Assuming the Bousso bound on entropy [90] (which is motivated by holography and quantum gravity [91]), a general (semi-classical) singularity theorem has been proved [92], from which the following corollary follows: the slightest radiation added to global de Sitter sufficiently far into the future or past results in a null geodesically incomplete spacetime.…”

Review of the No-Boundary Wave Function

“…Thus, it makes sense to investigate the consequences that this criterion would have on no-boundary path integrals. This question was studied in some detail in [84,175,176,85] and we will review the main results below.…”

“…The previous example should make it clear that it is in general difficult to assess whether a metric is allowable or not, if we permit such changes of time path. Techniques were developed in [176] to deal with this situation, and we refer to this paper for details. The results for no-boundary integrals, both with a Neumann initial condition and for a Dirichlet initial condition, are shown in Fig.…”

Truly Lorentzian quantum cosmology

“…This produces the action (4) which is quadratic in q. In many previous works, such as [24,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48], the integration range of the squared scale factor is taken to be over the whole real line. Then the path integral in q with the quadratic action can be evaluated analytically, just like the path integral of a free quantum particle [54].…”

“…More recently, Feldbrugge et al proposed [24] to define the gravitational path integrals by the Lorentzian contour, and use Picard-Lefschetz theory to study complex contour deformations only as a computational trick for the fundamentally Lorentzian theory (see Sorkin [25] for a closely related discussion). This has led to renewed interest in investigating old topics with new methods [24,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48]. In these works of "Lorentzian quantum cosmology ", it is common to adopt the minisuperspace metric…”