1 The Dome: An Unexpectedly Simple Failure of Determinism John D. Norton1 Department of History and Philosophy of Science University of Pittsburgh Newton’s equations of motion tell us that a mass at rest at the apex of a dome with the shape specified here can spontaneously move. It has been suggested that this indeterminism should be discounted since it draws on an incomplete rendering of Newtonian physics; or it is “unphysical”; or it employs illicit idealizations. I analyze and reject each of these reasons. 1. Introduction It has been widely recognized for over two decades that, contrary to the long-standing lore, Newtonian mechanics is not a deterministic theory. The clarion call came in John Earman’s (1986, Ch. III), which recounted the failure of determinism, including the then recent discovery by Mather and McGehee of “space invader” systems of interacting particles that spontaneously rush into an empty space from spatial infinity. Further, simpler violations of determinism emerged. Pérez Laraudogoitia (1996) described an especially simple example of “supertask” indeterminism in which a countable infinity of masses confined to a unit interval are spontaneously energized; and Norton (1999) described a correspondingly simple example of a countable infinity of masses connected by springs that are spontaneously energized. In this developing tradition, the simplest example so far of indeterminism in Newtonian physics is what has come to be known as “the dome,” described in Norton (2003, §3). The indeterminism involves none of the complications of infinitely many systems interacting or masses appearing with unbounded speed from spatial infinity. A mass sits on a dome in a 1 My thanks to my co-symposiasts, John Earman and David Malament; to Michael Dickson, Bernie Goldstein, Stephan Hartmann, Alexandre Korolev and Dmitri Tymoczko; and to Visiting Fellows of the Center for Philosophy of Science: Boris Grozdanoff, Antigone Nounou, Hernan Pringe and Stéphanie Ruphy.2 gravitational field. After remaining motionless for an arbitrary time, it spontaneously moves in an arbitrary direction, with these indeterministic motions compatible with Newtonian mechanics. This note will consider whether the dome depends upon some improper maneuver in Newtonian theory. I will argue that it does not. Resolving this issue proves to be of unexpected philosophical interest. It requires a careful appraisal of three questions: Just what is Newtonian theory? What do we mean by the notion “unphysical”? Are some idealizations improper? 2. The Dome in Brief 2.1 Description The dome is a radially symmetric surface shown in Figure 1. Its shape is defined by h = (2/3g) r3/2 (1) where r is the radial distance coordinate in the surface of the dome, h is the vertical distance below the apex at r=0 and g is the constant acceleration of free unit mass in the vertical gravitational field surrounding the surface. A point-like, unit mass slides frictionlessly over the surface. Initially, at time t=0, it is at rest exactly at the apex. Figure 1. The Dome The net force F acting on the unit mass is directed radially outward. It is the component of the gravitational force tangent to the surface, which is g sin θ, where θ is the angle between the tangent to the surface in the radial direction and the horizontal. Since sin θ = dh/dr, we have3 F = g dh/dr = r1/2. Newton’s second law sets this force equal to the acceleration a(t) = d2r/dt2 and yields the equation of motion of the mass: d2r/dt2 = r1/2 (2) The expected solution is r(t) = 0 (3) in which the mass simply remains at rest for all times t. Another family of solutions represents spontaneous motion at an arbitrary time T in an arbitrary radial direction: r(t) = (1/144) (t – T)4 for t ≥ T (4) = 0 for t ≤ T We see that (4) satisfies (2) if we compute the radial acceleration a(t) = d2r(t)/dt2, which is a(t) = (1/12) (t – T)2 for t ≥ T (5) = 0 for t ≤ T and note that a(t) as given in (5) is the square root of r(t) as given in (4). The dome manifests indeterminism2 in the standard sense that a single past can be followed by many futures. The mass may be at rest for all times up to t=0. It then may or may not move spontaneously at any time after that.3 Elsewhere (Norton, 2003, §3) I have responded to concerns about the moment t=T. Briefly, since no net force acts on the mass at t=T, does Newton’s first law obtain? It does, since (5) tells us the mass has zero acceleration at t=T. Those who yearn for a first cause to initiate the motion should recall that a requirement of a first cause is not a part of Newton’s laws of motion. Further, there is no first moment of acceleration at which this first cause might act. The moment t=T is the last moment of unaccelerated motion. 2 Equation (2) with initial conditions r(0) = dr(0)/dt = 0 fails to satisfy a familiar condition sufficient for existence and uniqueness of a solution, a Lipshitz condition. I originally concocted the dome example by starting with a text-book example of a violation of a Lipshitz condition; and then worked backwards to a plausible physical instantiation. 3 I have used the fact that Newtonian theory assigns no probabilities to these different outcomes as a way of illustrating my claim elsewhere that inductive inference need not be probabilistic. See Norton (forthcoming).4 2.2 Individual versus Collective Indeterminism This manifestation of indeterminism differs from those already in the literature. In the case of supertask indeterminism, each component is well-behaved. If a component is set into motion, it is because is has been, struck, pushed or pulled by another component. The space invader form of indeterminism is odder in the sense that it involves components that pop into being “from spatial infinity” with unbounded speeds. However each individual component is well-behaved locally. If we look at the motion of any component in some part of spacetime of finite spatial and temporal extension, the component will only change its motion if it is struck, pushed or pulled by another component. In both supertask and space invader cases, the