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 mass on the dome
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
A point-like unit mass slides frictionlessly over the surface under the action of gravity. The gravitational force can only accelerate the mass along the surface. At any point, the magnitude of the gravitational force tangential to the surface is F = d(gh)/dr = r1/2 and is directed radially outward. There is no tangential force at r = 0. That is, on the surface the mass experiences a net outward directed force field of magnitude r1/2. Newton’s second law, F = ma, applied to the mass on the surface, sets the radial acceleration d2r/dt2 equal to the magnitude of the force field:
(1) d2r/dt2 = r1/2
If the mass is initially located at rest at the apex r = 0, then there is one obvious solution of Newton’s second law for all times t:
(2) r(t) = 0
The mass simply remains at rest at the apex for all time as shown:
Simplest solution: no motion
However, there is another large class of unexpected solutions. For any radial direction:
(3) r(t) = (1/144) (t-T)4 for t greater than or equal to T
= 0 for t less than or equal to T
where T is an arbitrarily chosen, positive constant. One readily confirms that the motion of (3) solves Newton’s second law (1). See Note 6
If we describe the solutions of (3) in words, we see they amount to a violation of the natural expectation that some cause must set the mass in motion. Equation (3) describes a point mass sitting at rest at the apex of the dome, whereupon at an arbitrary time t=T it spontaneously moves off in some arbitrary radial direction.
Spontaneous motion
Properties
Two distinct features of this spontaneous excitation require mention.
No cause. No cause determines when the mass will spontaneously accelerate or the direction of its motion. T
