Space-time nature of causality

Abstract

In a causal world the direction of the time arrow dictates how past causal events in a variable $X$ produce future effects in $Y$. $X$ is said to cause an effect in $Y$, if the predictability (uncertainty) about the future states of $Y$ increases (decreases) as its own past and the past of $X$ are taken into consideration. Causality is thus intrinsic dependent on the observation of the past events of both variables involved, to the prediction (or uncertainty reduction) of future event of the other variable. We will show that this {temporal} notion of causality leads to another natural spatio-temporal definition for it, and that can be exploited to detect the arrow of influence from $X$ to $Y$, either by considering shorter time-series of $X$ and longer time-series of $Y$ (an approach that explores the time nature of causality) or lower precision measured time-series in $X$ and higher precision measured time-series in $Y$ (an approach that explores the spatial nature of causality). Causality has thus space and time signatures, causing a break of symmetry in the topology of the probabilistic space, or causing a break of symmetry in the length of the measured time-series, a consequence of the fact that information flows from $X$ to $Y$.