Mach's Principle and Machian Relationism

The relationist's boldest answer to Newton's bucket came from Ernst Mach in The Science of Mechanics (1883). Where Newton concluded that inertial effects reveal motion relative to absolute space, Mach replied that they reveal motion relative to the rest of the matter in the universe — above all the "fixed stars." On this view inertia is not a relation between a body and a substantival space but a relation between a body and all other bodies; remove the other bodies and the inertial effects would vanish. This is the seed of what later came to be called Mach's principle, a family of ideas about the cosmic origin of inertia that inspired Einstein, shaped the reception of general relativity, and remains the most developed attempt to complete the relationist programme.

This page states the principle in its several forms and traces its fortunes; it presupposes the bucket argument and connects forward to the dynamical spacetime of general relativity.

Mach against absolute space

Mach's objection to Newton is at bottom empiricist and epistemological: absolute space is unobservable, plays no role we can detect, and should be banished from physics as a metaphysical idol. When Newton says the water climbs the bucket's walls because it rotates relative to absolute space, Mach answers that this is an untestable posit. What we can actually observe is the water's rotation relative to the Earth and the fixed stars. Mach's challenge:

No one is competent to say how the experiment would turn out if the sides of the vessel [were] increased till they were ultimately several leagues thick.

The suggestion is provocative: if the bucket's walls were as massive as the heavens, perhaps rotating them around the still water would raise its surface. The inertial effect, Mach proposes, is produced by relative motion with respect to all the matter of the cosmos, and the fixed stars dominate simply because they are so massive and so numerous. Newton's two-globes argument then fails at its crucial step: in a genuinely empty universe there would be no inertia and no cord tension, because there would be nothing for the globes to be inertial with respect to.

The many "Mach's principles"

"Mach's principle" is notoriously not a single precise thesis — Hermann Bondi and others catalogued a dozen inequivalent versions. The main ones:

The strongest, most relationist version is the last: strip the universe of all matter but one particle, and there is no fact about whether it accelerates, because acceleration is motion relative to other matter and there is none. This directly contradicts Newton (and, as it turns out, unmodified general relativity, which permits empty or nearly empty solutions with well-defined inertial structure).

Mach, Einstein, and general relativity

Mach's critique of absolute space was one of Einstein's acknowledged inspirations; Einstein coined the name "Mach's principle" and hoped general relativity would vindicate it — that the metric field (which fixes the local inertial frames) would be fully determined by the matter and energy distribution, so that inertia would arise from matter as Mach demanded. The theory delivers this only partially:

• Machian successes. General relativity exhibits genuinely Machian effects. Frame-dragging (the Lense–Thirring effect): a rotating mass literally drags the local inertial frames around with it, so that "non-rotating" near a spinning body is tilted toward co-rotation — measured by Gravity Probe B. The local standard of non-rotation is thus influenced by the motion of matter, exactly as Mach envisaged.
• Machian failures. But the influence is not total. General relativity has vacuum solutions — Minkowski spacetime itself, and gravitational waves in empty space — that possess a perfectly definite inertial structure with no matter at all to ground it. Worse, Gödel's rotating universe is a solution in which the whole matter content rotates relative to the local inertial frames, which is precisely what Mach's principle should forbid. The metric field in general relativity has degrees of freedom of its own; it is not a mere bookkeeping of matter's relations. In this sense general relativity is a theory of a dynamical spacetime substance/structure, not a purely relational theory.

So general relativity is more Machian than Newtonian mechanics — inertial frames are affected by matter — but less Machian than Mach wanted, since spacetime retains an independent reality.

Barbour–Bertotti relational dynamics

The most serious modern attempt to build a genuinely Machian, fully relational physics is due to Julian Barbour and Bruno Bertotti. Their strategy ("best matching") formulates dynamics on the space of relative configurations of the universe — the shapes formed by all the matter, with absolute position, orientation, and scale quotiented out — so that only relational facts are ever used, and there is no background space or time. Time itself is not a primitive but is read off from change (Barbour's "time is change"), a view with radical consequences for the nature of time and the problem of time in quantum gravity. Barbour's programme shows that a relational reconstruction is possible and can recover much of standard physics; whether it can recover all of general relativity, and whether its elimination of primitive time is an advantage or a liability, remains contested.

Where this sits

Mach's principle is the relationist's answer to the one argument — the bucket — that the shift arguments could not dispose of, and so it is central to the ontological question and the substantivalism–relationism debate. Its partial realisation in general relativity is one of the subtlest facts in the philosophy of physics: the theory that made spacetime dynamical also made inertia partly a matter of the cosmic matter distribution, without ever quite delivering the pure relationism Mach sought. Barbour's timeless, relational physics carries the programme into the problem of time. The next page turns from the relational side to the substantival, and to the argument that most sharply threatens it in the modern setting — the hole argument.