  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 matches 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.
