The Dome: A Simple Violation of Determinism in Newtonian Mechanics
John D. Norton Department of History and Philosophy of Science, University of Pittsburgh Pittsburgh PA 15260. Homepage: www.pitt.edu/~jdnorton This page is available at www.pitt.edu/~jdnorton/goodies
This page is based on Section 3 "Acausality in Classical Physics" of "Causation as Folk Science," Philosophers' Imprint Vol. 3, No. 4 http://www.philosophersimprint.org/003004/; to be reprinted in H. Price and R. Corry, Causation and the Constitution of Reality. Oxford University Press.
While exotic theories like quantum mechanics and general relativity violate our common expectations of causation and determinism, one routinely assumes that ordinary Newtonian mechanics will violate these expectations only in extreme circumstances if at all. That is not so. Even quite simple Newtonian systems can harbor uncaused events and ones for which the theory cannot even supply probabilities. Because of such systems, ordinary Newtonian mechanics cannot license a principle or law of causality. Here is an example of such a system fully in accord with Newtonian mechanics. It is a mass that remains at rest in a physical environment that is completely unchanging for an arbitrary amount of time--a day, a month, an eon. Then, without any external intervention or any change in the physical environment, the mass spontaneously moves off in an arbitrary direction, with the theory supplying no probabilities for the time or direction of the motion.
The dome of Figure 1a sits in a downward directed gravitational field, with acceleration due to gravity g. The dome has a radial coordinate r inscribed on its surface and is rotationally symmetric about the origin r=0, which is also the highest point of the dome. The shape of the dome is given by specifying h, how far the dome surface lies below this highest point, as a function of the radial coordinate in the surface, r. For simplicity of the mathematics, we shall set h = (2/3g)r3/2. (Many other profiles, though not all, exhibit analogous acausality.) Figure 1a. Mass sliding on a dome
