The Machian Problem
What's the Machian Problem?
In the contemporary physics, physical ontology is by no means unified; in some context, we talk about particles, in others, fields, spacetime, events, and so on. In this situation, some may argue that Mach's idea is obsolete since his criticism of Newton was primarily in terms of particle physics, and he did not pay enough attention to the concept of field. However, Julian Barbour argues that Mach's idea is not outdated even in such new fields as quantum gravity (general relativity, if applicable to a tiny region, especially to an instant very close to the singular point of a black hole, must be quantized because of the smallness in question; further, there is a deeper question, how to unite relativity to quantum physics). As long as dynamics is concerned with motion and its problems, Mach's fundamental problem is alive, he claims. His point may be best illustrated in terms of the "2 snapshots problem".
The criterion for saying that a dynamical theory is Machian (Machianity) is formulated by Barbour as follows:
Given a dynamical system (say, classical, consisting of a number of particles interacting), the initial data of the masses of the particles, their separations, and the rates of change of those separations are sufficient for determining the future of the system.
This formulation owes a great deal to Poincare (his formulation of initial value problem), as well as to Mach. See Barbour & Pfister (1995), 107. However, in order to make such a Machian program of dynamics feasible, it must be kept in mind that such a theory "must be applied to the entire universe", as Poincare clearly points out (and as Mach frequently and naturally assumes). See relevant quotations from Mach and Poincare, extracted in Barbour & Pfister (1995), 109-112.
In the classical particle physics, we wish to know the change of motions of particles. Suppose we are given a relative configuration of three particles at some instant; and at another instant, we are given another configuration, as in the figure. (We do not know which comes first, nor the "perspective" from which these shots are taken; forget about your "prejudice" of the Newtonian mechanics! Incidentally, you should also discard the "Marxist" prejudice against Mach that his "idealism" or instrumentalism, or whatever, is inferior to the dialectical materialism--a sort of realism--, which underlies sound sciences; to me, the dialectical materialism does not seem to be "sound" at all.)
What we can observe is the relative distance between any two particles and their relations; absolute space and time are invisible. Then, given such information, what can we say about their motion? This question is basic, and this is what Mach was concerned with, although he may not have treated the question precisely in these terms. Notice that, with these two snapshots, we cannot say how each point moved, nor trace its trajectory through time. Then how should we construct dynamics of particles? What is the law, and how can we tell, e.g., which snapshot comes first?
The situation is exactly similar if we turn to physics in terms of fields (such as electrodynamics), although the underlying ontology has certainly changed. Suppose we have two snapshots of a field, with different patterns of intensities. Supposing such a snapshot contain all information at a given moment, how should we construct (without presupposing absolute space and time) dynamics of the field?
Again, the situation looks the same, even if we turn to the Einsteinian theory of gravity in terms of geometry of spacetime and fields. Granted, the ontology is quite different, but the question takes essentially the same form. Given two snapshots of evolving geometry (space), how should we construct dynamics, or geometrodynamics? (This is known as the "thin-sandwitch formulation". Recall a slice of 4 dimensional spacetime is a 3-geometry, and taking two snapshots means taking two slices, like sandwitch; and "thin", because an infinitesimal change is given by two, very close slices! See Evolution of 3-space.)
In order to make such dynamics thoroughly relational, we have got to do a lot of things (e.g., is distance or interval definable relationally? What about fields and geometry, should not we construct them only from relations?). But as long as the form of question is the same in each case, the Machian idea is alive. Wheeler and his students pursued a similar question.
According to Barbour's most recent research (see http://www.platonia.com/books.html, comments towards the bottom), size (or distance, so far presupposed by Barbour and Bertotti) can be eliminated from Newtonian dynamics, and likewise scale can be eliminated from general relativity (if matter and fields are disregarded); and in order to pursue this line, he is now working on dynamics in conformal superspace ("conformal" roughly means, one geometry or metric can be transformed to another by change of scale and a couple of other changes; thus conformal superspace is an arena for this, like superspace). Whether this succeeds for general relativity with matter and fields (the coupling of matter and fields with general relativity, in terms of superspace, is regarded as successful), and whether or not agrees with essential features of general relativity, are still under investigation. In any case, very interesting research is still going on. (For superspace, see Evolution of 3-Space.)
References
Barbour, Julian (1989) Absolute or Relative Motion? vol. 1, Cambridge Univ. Press, 1989, 8-12. (Vol. 2 still to appear)
Barbour, J. B. and Pfister, H., eds. (1995) Mach's Principle, Birhaeuser.
Mach, E. (1960) The Science of Mechanics, Open Court.
Misner, Thorne, and Wheeler (1973) Gravitation, Freeman, 1973, chh. 21, 43.
Poincare, H. (1905) Science and Hypothesis (English translation of 1902), Dover, 1952.
Last modified March 30, 2003. (c) Soshichi Uchii
