Towards a better understanding of space-time causality: Kolmogorov complexity and causality as a matter of degree

Abstract

Abstract-Space-time causality is one of the fundamental notions of modern physics; however, it is difficult to define in observational physical terms. Intuitively, the fact that a space-time event e = (t, x) can causally influence an eventmeans that what we do in the vicinity of e changes what we observe at e ′ . If we had two copies of the Universe, we could perform some action at e in one copy but not in another copy; if we then observe the difference at e ′ , this would be an indication of causality. However, we only observe one Universe, in which we either perform the action or we do not. At first glance, it may seem that in this case, there is no meaningful way to provide an operational definition of causality. In this paper, we show that such a definition is possible if we use the notions of algorithmic randomness and Kolmogorov complexity. The resulting definition leads to a somewhat unexpected consequence: that space-time causality is a matter of degree. I. DEFINING CAUSALITY IS IMPORTANTSpace-time causality is important. Causality relation between space-time events (i.e., points in space-time) is one of the fundamental notions of physics; see, e.g., [1], [3]. Because of this, many fundamental physical theories describe, among other things, the causality relation between space-time events.According to modern physics, space-time causality relation is non-trivial. In Newton's physics, it was assumed that influences can propagate with an arbitrary speed, constituting, in effect, immediate action-at-a-distance. Under this assumption, an event e = (t, x) occurring at moment t at location x can influence an event e ′ = (t ′ , x ′ ) occurring at moment t ′ at location x ′ if and only if the second event occurs later than the first one, i.e., if and only if t < t ′ . In special relativity, the speeds of all the processes are limited by the speed of light c. In this theory, an event e = (t, x) can influence an event e ′ = (t ′ , x ′ ) if during the time t ′ − t, the faster possible process -light -can cover the distance d(x, x ′ ) between locations x and x ′ , i.e., ifIn the general relativity theory, the space-time is curved, so the corresponding causality relation is even more com-