Quantum Field Theory and Gravity: Conceptual and Mathematical ...

272 J. Barbour
Figure 6. The distinguished representation of bestmatched
shape dynamics for the 3-body problem. For the
initial shape, one chooses an arbitrary position in Euclidean
space. Each successive shape is placed on its predecessor in
the best-matched position (‘horizontal stacking’). The ‘vertical’
separation is chosen in accordance with the distinguished
curve parameter t determined by the condition (7). In the
framework thus created, the particles behave exactly as Newtonian
particles in an inertial frame of reference with total
momentum and angular momentum zero.
Best matching is a process that determines a metric on Q N ss . For this,
three things are needed: a supermetric on Q N , best matching to find the
orthogonal inter-orbit separations determined by it, and the equivariance
property that ensures identity of them at all positions on the orbits. Nature
gives us the metric of Euclidean space, and hence the supermetric on Q N ;the
second and third requirements arise from the desire to implement Poincaré
type dynamics in Q N ss. The orbit orthogonality, leading to the constraints (12),
(13), and (14), distinguishes best-matched dynamics from Newtonian theory,
which imposes no such requirements. Moreover, the constraint propagation
needed for consistency of best matching enforces symmetries of the potential
that in Newtonian theory have to be taken as facts additional to the basic
structure of the theory.
It is important that the constraints (12), (13), and (14) apply only to the
‘island universe’ of the complete N-body system. Subsystems within it that
are isolated from each other, i.e., exert negligible forces on each other, can
perfectly well have nonvanishing values of P, L,D. It is merely necessary that
their values for all of the subsystems add up to zero. However, the consistency