Both Newton's dynamics and General Relativity (GR)
keep, to a different degree, some reference to an absolute framework that is external
to the system. In Newton this role is played by his absolute space and time, which
provide a fixed notion of inertial frames of reference and of the flow of time (fixed means not influenced by the state of the matter present in the
system). Einstein linked the flow of time and the inertial forces to a dynamical field, the metric,
which is sourced by matter. This removes Newton's absolutes locally
but, crucially, GR depends on boundary conditions that have the same character
as Newton's absolutes. This spoils the self-containedness of the theory, when
one is dealing with the whole Universe as a dynamical system (of course,
when dealing with subsystems, one expects to need some boundary conditions
summarizing the state of the rest of the Universe).
Shape Dynamics (SD) is a relational framework aiming to remove all the absolute
substructures from dynamics, that has developed over more than 30 years
Newton's absolute space and time were disposed of first [1
] finding a reformulation of the N-body problem, considered as a toy model for the
whole Universe, which generates spontaneously a preferred notion of inertial frames
of reference where Newton's laws can be applied. The determination of these frames
of reference can be performed within the system, referring only to observable relational
data. The same happens for time, which emerges as a distillation of all the changes
in the Universe. Considering only subsystems, one recovers the familiar situation where
inertial frames of reference and the flow of time are given from the exterior of the system
and cannot be deduced from within it.
All of this is a consequence of the single powerful principle that is at the basis of SD, the `Mach-Poincaré principle'
Initial data in the physical configuration space S
in the form of a point and a direction (strong form) or a point and tangent vector
(weak form) must determine the physical evolution curve in S uniquely.
The same principles cannot be applied to GR in its canonical formulation due to Arnowitt, Deser and Misner
], because the system fails to satisfy the Mach-Poincaré principle in Wheeler's
superspace, its natural configuration space. York's work on the initial value problem
] provided convincing evidence that a suitable configuration space
for GR is conformal superspace
(or shape space
), the space of conformal 3-geometries.
York proved that the solutions of Einstein's equation could be generated
by conformal data on an initial Cauchy surface with spatially constant mean extrinsic curvature
(CMC). In a neighborhood of the initial Cauchy surface, each solution is generated by one and only one
set of initial data. This consist of the conformal class of the metric, its associated momenta,
which are transverse-traceless
(TT), plus a number, called York time
is the momentum conjugate to the spatial volume.
An amount of work accumulated over the last decade
allowed to prove that the Mach-Poincaré principle is satisfied in shape space, and to assess
SD as a reformulation of GR on the configuration space of conformal three-geometries.
The last bit of this process was completed in [7
where it is shown that, expressing the dynamics of GR in a self-contained way on shape space, one ends up
with a time-dependent Hamiltonian.
I am currently investigating the consequences of this time-dependence, both in the N-body problem
and in GR. Click here