Kiefer C. Quantum gravity

96 PARAMETRIZED AND RELATIONAL SYSTEMS
in this connection that in 1905 Henri Poincaré argued for a definition of time
that makes the equations of motion assume their simplest form. He writes 6
Time must be defined in such a way that the equations of mechanics are as simple as
possible. In other words, there is no way to measure time that is more true than any
other; the one that is usually adopted is only more convenient.
It is, however, a fact that the choice (3.109) is not only distinguished because
then the equations of motion (3.106) take their simplest form but also because
only such a choice will ensure that the various clocks of (approximately isolated)
subsystems march in step, since ∑ k (T k +V k )= ∑ k E k = 0 (Barbour 1994). The
only truly isolated system is the universe as a whole and to determine time it is
(in principle) necessary to monitor the whole universe. In practice this is done
even when atomic clocks are employed, for example, in the determination of the
pulse arrival times from binary pulsars (Damour and Taylor 1991).
In this approach, the inertial frame and absolute time of Newtonian mechanics
are constructed from observations through the minimization of the kinetic
energy and the above choice of τ. One could call this a Leibnizian or Machian
point of view. The operational time defined by (3.109) corresponds to the notion
of ‘ephemeris time’ used in astronomy. That time must be defined such that the
equations of motion be simple was already known by Ptolemy (Barbour 1989).
His theory of eclipses only took a simple form if sidereal time (defined by the
rotation of the heavens, i.e. the rotation of the Earth) is used. This choice corresponds
to a ‘uniform flow of time’.
Time-reparametrization invariant systems have already been discussed in Section
3.1.1 in connection with the parametrized non-relativistic particle. In contrast
to there, however, no absolute time is present here and the theory relies
exclusively on observational elements.
A formal analogy of the action (3.105) is given by Jacobi’s action 7 in classical
mechanics (Barbour 1986; Lanczos 1986; Brown and York 1989),
∫
S J = ds √ E − V , (3.110)
where E is the total energy, and s parametrizes the paths in configuration space,
ds 2 ≡
n∑
m k dx k dx k .
k=1
Writing ds =ṡdτ, onegetsṡ = √ 2T ;ifE = 0, then, Jacobi’s action is proportional
to the action (3.105).
6 ‘Le temps doit être défini de telle façon que les équations de la mécanique soient aussi
simples que possible. En d’autres termes, il n’y a pas une manière de mesurer le temps qui
soit plus vraie qu’ une autre; celle qui est généralement adoptée est seulement plus commode.’
(Poincaré 1970)
7 There is a close analogy of this formulation with Fermat’s principle of least time in geometrical
optics.