Kiefer C. Quantum gravity

94 PARAMETRIZED AND RELATIONAL SYSTEMS
because the two sets of relative separations give no information about the angular
momentum or kinetic energy of the system, both of which affect the future
evolution.
This ‘Poincaré defect’ (because Poincaré pronounced this lack of predictability)
motivated Barbour and Bertotti (1982) to look for a slight generalization of
Newtonian mechanics in which the future can be predicted solely on the basis
of relative separations (and their rates of change). The key idea is to introduce
a ‘gauge freedom’ with respect to translations and rotations (because these
transformations leave the relative distances invariant) and the choice of the time
parameter τ. The theory should thus be invariant under the following gauge
transformations,
x k ↦→ x ′ k = x k + a(τ)+α(τ) × x k , (3.100)
where a parametrizes translations, α rotations, and x k is the position vector
of particle k. They depend on the ‘label time’ τ which can be arbitrarily
reparametrized,
τ ↦→ f(τ) , f>0 ˙ . (3.101)
Due to (3.100) one has instead of the original 3n only 3n − 6 parameters to
describe the relative distances. Equations (3.100) and (3.101) define the ‘Leibniz
group’ (Barbour and Bertotti 1982; Barbour 1986). One can now define a total
velocity for each particle according to
Dx k
Dτ
≡ ∂x k
∂τ + ȧ(τ)+ ˙α(τ) × x k , (3.102)
in which the first term on the right-hand side denotes the rate of change in
some chosen frame, and the second and third terms the rate of change due to a
τ-dependent change of frame. This velocity is not yet gauge invariant. A gaugeinvariant
quantity can be constructed by minimizing the ‘kinetic energy’
n∑
k=1
Dx k
Dτ
Dx k
Dτ
with respect to a and α. This procedure is also called ‘horizontal stacking’ (Barbour
1986). Intuitively it can be understood as putting two slides with the particle
positions marked on them on top of each other and moving them relative to
each other until the centres of mass coincide and there is no overall rotation. The
result of the horizontal stacking is a gauge-invariant ‘intrinsic velocity’, dx/dτ.
Having these velocities for each particle at one’s disposal one can construct the
kinetic term
T = 1 n∑
( ) 2 dxk
m k . (3.103)
2 dτ
k=1
The potential is the standard Newtonian potential
V = −G ∑ k