Mach's Principle and the Structure of Dynamical Theories
| Scientific Paper | |
|---|---|
| Title | Mach's Principle and the Structure of Dynamical Theories |
| Read in full | Link to paper |
| Author(s) | Julian B Barbour, Bruno Bertotti |
| Keywords | Mach's Principle, Theory of Relativity, angular momentum |
| Published | 1982 |
| Volume | 382 |
| No. of pages | 13 |
| Pages | 295-306 |
Read the full paper here
Abstract
Proc. R. Soc. Lond. A 1982 382, 295-306, doi: 10.1098/rspa.1982.0102. A structure of dynamical theories is proposed that implements Mach's ideas by being relational in its treatment of both motion and time. The resulting general dynamics, which is called intrinsic dynamics and by construction treats the evolution of the entire Universe, is shown to admit as special cases Newtonian dynamics and Lorentz-invariant field theory provided the angular momentum of the Universe is zero in the frame in which its momentum is zero. The formal structure of Einstein's general theory of relativity also fits the pattern of intrinsic dynamics and is Machian according to the criteria of this paper provided the so-called thin-sandwich conjecture is generically correct.
Overview
This is the paper — communicated by Roger Penrose, received 6 March 1981 and revised 18 February 1982, written while both authors were at the Istituto di Fisica Teorica of the University of Pavia — in which Mach's Principle is finally given a precise formal meaning rather than a rhetorical one. Its achievement is to turn Mach's intuition that inertia is somehow determined by the whole universe into an explicit variational construction, and then to ask which existing theories satisfy it.
The strategy is to abandon the usual starting point of dynamics. Conventional theory begins with a configuration space Q whose points are the positions of N particles in a frame of reference, and with a time t supplied by a clock outside the system. Barbour and Bertotti argue that for the universe as a whole neither is legitimate: a frame of reference has no referent when there is nothing outside to refer to, and a time supplied from outside is meaningless because "it is hard to see what meaning could be attached to saying that absolutely everything is speeded up by the same amount: all the observable relations are still run through in the identical sequence." They therefore replace Q by the space of orbits of Q under the Euclidean group, which they call the intrinsic (or relative) configuration space Q0, and they let the history be a bare curve in Q0 labelled by an arbitrary monotonic parameter λ carrying no metrical properties at all. A theory formulated in Q0 implements what they call the first Mach principle; one that dispenses with an independent time implements the second Mach principle. The resulting dynamics is named intrinsic dynamics.
The argument
Two Mach principles and the Leibniz group
For N point particles only the relative distances rij are physically given, and because the underlying geometry is Euclidean only 3N − 6 of them are independent. The Cartesian descriptions q = (ri) that correspond to one set of rij form a six-parameter family related by the Euclidean symmetry group E0: r → r′ = A·r + h, with A an orthogonal matrix and h a vector. The orbits {q} of E0 are the points of Q0.
The treatment of time follows Leibniz explicitly — "time as merely the successive order of things", with instants defined by the successive relative configurations of the universe (the authors cite the Leibniz–Clarke correspondence of 1716). Combining the two modifications gives invariance under what Barbour and Bertotti call the Leibniz group:
- r → r′ = A(λ)·r + h(λ), λ → λ′ = f(λ), df/dλ ≠ 0
— seven arbitrary functions of the label, since A, h and f may all vary along the history. The programme of the paper is to show that theories of the universe invariant under this enormous group nevertheless yield theories of subsystems invariant only under the much smaller, finite-parameter Galileo or Lorentz group.
The intrinsic differential and "stacking"
The new technical tool is the intrinsic differential. Barbour and Bertotti motivate it with a vivid image: take two successive "photographs" φ1 and φ2 of a scalar field on the plane. Each photograph's relative pattern of intensities fixes a point of Q0; laying a Cartesian grid on it in all possible ways generates the orbit. To ask how much the field has changed, one must fix the grid on the second photograph relative to the first — "the problem that led Newton to introduce his concept of absolute space." Their rule dispenses with absolute space: compute dφ = φ2 − φ1 with arbitrary grids, form the L2 distance ds2 = ∫dx∫dy (dφ)2, and then slide one grid over the other by the action of E0 until ds is minimized. This procedure they call stacking; the minimal ds is a coordinate-independent, globally determined distance between the two configurations.
In general, with a positive definite metric ⟨dq|dq⟩ on Q and the operators Oα of infinitesimal translations and rotations, minimizing over the group parameters defines the intrinsic differential dIq = dq + Σε0αOαq, which is simply the part of dq orthogonal to the group orbit: ⟨dIq|Oβq⟩ = 0. The Machian action is then the geodesic principle δS = 0 with S = ∫⟨dIq|dIq⟩1/2, whose homogeneity of degree one in the λ-derivatives makes it reparametrization invariant, as the second Mach principle demands. In the equivalent Q-form the auxiliary quantities aα(λ) appear without their λ-derivatives; they are therefore not dynamical but primary first-class constraints in Dirac's sense.
The central theorem, and why the universe has no angular momentum
The paper's pivotal result is stated as a theorem: the physically distinct solutions to the Q0-problem are the geodesics of the Q'-problem that cut the orbits orthogonally. The proof runs through Noether's theorem: the Q-principle, being invariant under the six-parameter Euclidean group, conserves the quantities Pα = ⟨qλ|qλ⟩−1/2⟨qλ|Oαq⟩, and the vanishing of the Pα is precisely the stacking condition. Geometrically, if a geodesic cuts an orbit orthogonally at one instant it does so at all instants.
But the Pα are the total momentum and total Angular Momentum of the universe. The relational requirement therefore does not merely permit but selects the solutions with vanishing momentum and vanishing angular momentum. Barbour and Bertotti draw attention to "the striking fact that the Universe does not appear to have any appreciable angular momentum, in agreement with the prediction of intrinsic dynamics."
Recovering Newton, and a zero-energy universe
For N gravitating particles the flat metric ⟨dq|dq⟩ = Σmidri·dri is too simple to give non-trivial motion, so it is multiplied by the "conformal factor" V(q) = Σi<jmimj/rij. The Euler–Lagrange equations of the resulting action take an especially simple form for one distinguished choice of the arbitrary label, namely when T1/2 = V1/2; with the derivative taken with respect to that distinguished time label the equations become m'r̈i = ½∂V/∂ri — Newton's second law for gravitating point particles (the factor ½ being an inconsequential choice of units). The physically significant content is the condition T = V itself, which says that the total energy of the system is exactly zero.
The interpretive payoff is stated plainly: inertial frames and absolute time are not abolished but derived. Inertial frames "arise from the fully Machian theory when we perform the purely kinematic operation of stacking… though the inertial frames have no absolute significance and are determined through the stacking procedure by the distribution and relative motion of the matter in the Universe." Absolute time likewise is constructed by choosing the Leibnizian label to enforce a simplicity requirement.
The authors are careful, crediting Karel Kuchař, to note that zero total energy is a consequence of the particular Lagrangian, not of intrinsic dynamics as such: using Jacobi's variational principle with a fixed constant W = T − V and replacing dq by dIq gives a Machian formulation for any W. They call the possible appearance of that arbitrary constant "a weakness of the theory".
Poincaré's principle
Here the paper introduces what it names Poincaré's principle, quoting Science and Hypothesis (1905) at length: the state of bodies and their mutual distances at any moment, and the rates at which those distances are changing, should depend only on the initial mutual distances and their initial rates of change — nothing more. Poincaré found it "curious" that Newtonian evolution is nearly but not quite fixed by the observable initial data, requiring in addition arbitrary constants such as the total angular momentum. Barbour and Bertotti's answer is that the difficulty disappears exactly when those constants vanish, which is what a theory built on the intrinsic differential enforces.
Extending the same scheme to a scalar field φ(r) with T = ∫d3r φλ2 and −V = ∫d3r(∇φ)2 yields solutions of the wave equation with vanishing total angular momentum in the frame in which the momentum vanishes — that is, Lorentz-invariant field theory. Here W must be positive, since the wave field is a set of harmonic oscillators, and the authors note candidly that the condition P = 0, W ≠ 0 for the whole universe is not Lorentz invariant.
Gauge theory and geometrodynamics
The final section shows that intrinsic dynamics has the same structure as gauge theory. Adjoining the gauge group A → A + ∇Λ to E0 and applying the same recipe, the intrinsic variation with respect to Λ turns out to be identical to variation with respect to the scalar potential of Maxwell's theory once one sets Λ = −A0 (Maxwell's Equations, Electrodynamics). The stacking condition with respect to the gauge group is div At = 0. Their point is that "Newtonian or Lorentzian dynamics can be made to satisfy Poincaré's principle in exactly the same way that electrodynamics is gauge invariant."
General relativity is then treated as pure geometrodynamics. Taking the three-metric gij as the basic variable, the orbits under three-dimensional coordinate transformations are the points of Q0 — DeWitt's superspace. Slicing spacetime with lapse N and shift Ni, varying the Hilbert action with respect to N and eliminating it gives the reparametrization-invariant Baierlein–Sharp–Wheeler form of the action, in which the shift plays exactly the role of the constraint variables aα and the integrand is homogeneous of degree one in the time derivatives. Both Mach principles are thus satisfied. Two differences are flagged: the geometrodynamic action is a sum of square roots, hence a Finsler rather than a Riemannian metric, so orthogonality has no scalar-product definition (only stationarity with respect to Ni); and by the result of Hojman, Kuchař and Teitelboim (1976) the action is almost uniquely fixed by requiring that the evolving three-geometries stack into a four-dimensional space–time — so general relativity is a very special member of the family, and the more general scheme "could therefore provide a framework to study theoretically violations of general covariance, in particular Lorentz invariance."
The paper closes with two explicit reservations. For an infinite universe the principle needs boundary conditions at spatial infinity that are "quite alien to our general scheme"; indeed Q0 cannot be meaningfully defined unless the universe is finite and preferably closed, an assumption "implicit in our entire work". And even for a closed universe it is unknown whether the corresponding Cauchy problem — the thin-sandwich problem of Baierlein, Sharp and Wheeler (1962) — is generically uniquely solvable; only a conditional uniqueness proof (Belasco and Ohanian 1969) exists. Subject to that, they revise the more pessimistic verdict of their 1977 paper and conclude that "in its basic structure general relativity is Machian and gives expression to Poincaré's principle as a theory describing the evolution of closed three-geometries from intrinsically specified initial data."
Assessment
The distinctive achievement here is definitional rather than merely critical. Mach's Principle had for eighty years been a slogan that everyone invoked and nobody could state; Barbour and Bertotti give it a testable formal content in two clean parts — dynamics on the quotient space Q0, and reparametrization invariance of the action — and then apply the test. That is a much better standard of argument than the usual debate about whether Einstein's theory "is" Machian. The construction is also constructive in the strict sense: stacking is an explicit minimization, the intrinsic differential an explicit projection, and the theorem relating the Q0- and Q-problems is proved rather than asserted.
The prediction that the universe has zero total angular momentum is a genuine, falsifiable consequence rather than an accommodation, and it is one of the few places where a philosophical principle about Inertia issues in an astronomical number. The recovery of Newtonian mechanics with inertial frames as derived structures — determined by the actual matter distribution through a purely kinematic operation — is exactly what Ernst Mach asked for and what Newton's bucket argument was thought to forbid. The identification of the shift vector with the Machian constraint variables, and of the Baierlein–Sharp–Wheeler action as the geometrodynamic form of the intrinsic principle, is a real structural insight; it is one of the roots of the later "problem of time" literature and of Barbour's subsequent work on timeless physics.
The honest difficulties are largely the ones the authors themselves name. Everything depends on the universe being finite and closed: Q0 is not defined otherwise, so the scheme cannot even be posed for an open universe, and this is assumed rather than established. The general-relativistic case is conditional on the thin-sandwich conjecture, which was unproven in 1982 and has since been shown to fail in general — a uniqueness result exists only under restricted conditions — so the paper's headline claim about Einstein's theory remains hostage to a mathematical question it does not settle. The zero-energy result for the particle model is, as Kuchař pointed out to the authors and as they report, an artefact of the chosen Lagrangian rather than a consequence of Leibniz invariance; the arbitrary constant W reappears in the general case and they concede this is "a weakness of the theory". The field-theoretic extension is uncomfortable in a way the paper states but does not resolve: the condition P = 0 with W ≠ 0 for the whole universe is not Lorentz invariant, so global Machian conditions and local Lorentz symmetry sit together awkwardly. And the recovery of the restricted relativity principle is attributed frankly not to the Machian requirements but "to the particular structure of the metric defined on Q", with counterexamples said to be easy — so Galilean and Lorentz invariance are not derived from Mach's ideas here, only shown to be compatible with them.
Finally, an earlier version of the programme (Barbour & Bertotti 1977) predicted anisotropic effective masses "in contradiction with experiment"; the present paper's motivation for introducing the intrinsic differential is precisely to remove that defect. This is a case where a Machian proposal was tested against measurement — the Hughes–Drever-type limits on mass anisotropy are among the tightest in physics — and revised accordingly, which speaks well for the seriousness of the enterprise.