Trying to connect a fundamentally noncommutative spacetime with the conservative perturbative approach to quantum gravity, we are led to the natural question: are noncommutative geometrical effects already present in the regime where perturbative quantum gravity provides a predictive framework? Moreover, is it necessary to introduce noncommutativity by hand, or does it arise through quantum-gravitational effects? We show that the first question can be answered in the affirmative, and the second one in the negative: perturbative quantum gravity predicts noncommutativity at the Planck scale, once one clarifies the structure of observables in the quantum theory.

We derive an entropy-area law for the future horizon of an observer in diamonds inside the static patch of de Sitter spacetime, taking into account the backreaction of quantum matter fields. We prove positivity and convexity of the relative entropy for coherent states using Tomita–Takesaki modular theory, from which the QNEC for diamonds follows. Furthermore, we show that the generalized entropy conjecture holds. Finally, we reveal that the local temperature which is measured by an observer at rest exhibits subleading quantum corrections with respect to the well-known cosmological horizon temperature H/(2π).

We compute the relative entropy between the vacuum and a coherent state for a massive scalar field in de Sitter spacetime, using Tomita-Takesaki modular theory and the Araki-Uhlmann formula for the relative entropy. Embedding de Sitter spacetime as a hyperboloid in the ambient Minkowski space, we can restrict the Minkowski wedge and the corresponding modular operator to de Sitter, and we verify that this construction gives the correct modular flow. We check that the relative entropy is positive and jointly convex, relate it to the Noether charge of translations along the trajectories of the modular flow, and determine the local temperature as seen by an observer that moves along these trajectories.

We give a topological condition for a generic sliced space to be globally hyperbolic without any hypothesis on lapse function, shift function, and spatial metric.