The Emergence of Spacetime, or, Quantum Gravity on Your Desktop

TL;DR

Causal Dynamical Triangulations (CDT) offer a nonperturbative, background-free route to quantum gravity by summing over causal geometries discretized into equilateral 4-simplices. Monte Carlo investigations reveal an emergent four-dimensional de Sitter-like spacetime at large scales, alongside a fractal, effectively lower-dimensional structure at Planckian distances, evidenced by scale-dependent $D_S(\sigma)$ and a Hausdorff dimension $D_H$. The results exhibit universality, showing independence from discretization details, and demonstrate a viable path to coupling gravity with matter within a computable framework. This approach provides concrete, testable predictions and a robust computational platform for exploring quantum gravitational dynamics and its classical limit.

Abstract

Is there an approach to quantum gravity which is conceptually simple, relies on very few fundamental physical principles and ingredients, emphasizes geometric (as opposed to algebraic) properties, comes with a definite numerical approximation scheme, and produces robust results, which go beyond showing mere internal consistency of the formalism? The answer is a resounding yes: it is the attempt to construct a nonperturbative theory of quantum gravity, valid on all scales, with the technique of so-called Causal Dynamical Triangulations. Despite its conceptual simplicity, the results obtained up to now are far from trivial. Most remarkable at this stage is perhaps the fully dynamical emergence of a classical background (and solution to the Einstein equations) from a nonperturbative sum over geometries, without putting in any preferred geometric background at the outset. In addition, there is concrete evidence for the presence of a fractal spacetime foam on Planckian distance scales. The availability of a computational framework provides built-in reality checks of the approach, whose importance can hardly be overestimated.

Figures (11)

• Figure 1: Practitioner of quantum gravity, studying the Planck-scale microstructure of spacetime through her ultra-powerful microscope. (Courtesy of E. Rijke.)
• Figure 2: Triangular building blocks, or $d$-simplices, can be used to construct piecewise flat $d$-dimensional spaces. From left to right: triangles, tetrahedra and four-simplices.
• Figure 3: Forbidden path integral configurations in causal dynamical triangulations. The spacetime topology is fixed, and the sum over histories does not contain a sum over topologies, including, for example, wormhole configurations of the kind depicted in (a). Moreover, the spatial topology is not allowed to change as a function of time, thus eliminating "baby universes" and the associated branching and merging points where the light cones and therefore the causal structure are degenerate (b).
• Figure 4: Emergence of extended geometry: a snapshot from Monte-Carlo simulations of CDT at fixed four-volume $N=91.000$, which shows the spatial three-volume as a function of cosmological proper time (vertical axis).
• Figure 5: The qualitative behaviour of the correlator $\langle VV\rangle$ defined in eq. (\ref{['correl']}), for small (a) and large (b) four-volume $V_4$.