Causal Geometry

Information geometry has offered a way to formally study the efficacy of scientific models by quantifying the impact of model parameters on the predicted effects. However, there has been little formal investigation of causation in this framework, despite causal models being a fundamental part of science and explanation. Here, we introduce causal geometry, which formalizes not only how outcomes are impacted by parameters, but also how the parameters of a model can be intervened upon. Therefore, we introduce a geometric version of “effective information”—a known measure of the informativeness of a causal relationship. We show that it is given by the matching between the space of effects and the space of interventions, in the form of their geometric congruence. Therefore, given a fixed intervention capability, an effective causal model is one that is well matched to those interventions. This is a consequence of “causal emergence,” wherein macroscopic causal relationships may carry more information than “fundamental” microscopic ones. We thus argue that a coarse-grained model may, paradoxically, be more informative than the microscopic one, especially when it better matches the scale of accessible interventions—as we illustrate on toy examples.

New tensorial estimates in Besov spaces for time-dependent (2 + 1)-dimensional problems

In this paper, we consider various tensorial estimates in geometric Besov-type norms on a one-parameter foliation of surfaces with evolving geometries. Moreover, we wish to accomplish this with only very weak control on these geometries. Several of these estimates were proved in [S. Klainerman and I. Rodnianski, Causal geometry of Einstein-vacuum spacetimes with finite curvature flux, Invent. Math. 159 (2005) 437–529; S. Klainerman and I. Rodnianski, Sharp trace theorems for null hypersurfaces on Einstein metrics with finite curvature flux, Geom. Funct. Anal. 16(3) (2006) 164–229], but in very specific settings. A primary objective of this paper is to significantly simplify and make more robust the proofs of the estimates. Another goal is to generalize these estimates to more abstract settings. In [S. Alexakis and A. Shao, On the geometry of null cones to infinity under curvature flux bounds, Class. Quantum Grav. 31 (2014) 195012], we will apply these estimates in order to consider a variant of the problem in [S. Klainerman and I. Rodnianski, Causal geometry of Einstein-vacuum spacetimes with finite curvature flux, Invent. Math. 159 (2005) 437–529], that of a truncated null cone in an Einstein-vacuum spacetime extending to infinity. This analysis will then be used in [S. Alexakis and A. Shao, Bounds on the Bondi energy by a flux of curvature, to appear in J. Eur. Math. Soc.] to study and to control the Bondi mass and the angular momentum under minimal conditions.