Three denials of time in the interpretation of canonical gravity

Abstract

The analysis of the temporal structure of canonical general relativity and the connected interpretational questions with regard to the role of time within the theory both rest upon the need to respect the fundamentally dual role of the Hamiltonian constraints found within the formalism. Any consistent philosophical approach towards the theory must pay dues to the role of these constrains in both generating dynamics, in the context of phase space, and generating unphysical symmetry transformations, in the context of a hypersurface embedded within a solution. A first denial of time in the context of a position of reductive temporal relationalism can be shown to be troubled by failure on the first count, and second denial in the context Machian temporal relationalism can be found to be hampered by failure on the second. A third denial of time, consistent with both the of the Hamiltonian constraints roles, is constituted by the implementation of a scheme for constructing observables in terms of correlations. The motivation for and implications of each of these three denials are investigated.

Change in Hamiltonian general relativity from the lack of a time-like Killing vector field

“…When familiar general presumptions about gauge freedom and first-class constraints disappear, less work is required to motivate taking GR as partly violating those presumptive conclusions. For example, Kuchař' (Thébault, 2012b) is clarified when one notices that H0 does neither of these jobs by itself; both are accomplished by teaming up with other constraints, whether in Hp or in G. These teaming arrangements are easy enough to see when one retains the lapse, shift vector, and associated canonical momenta p, pi and associated primary constraints, but impossible to see when one truncates the phase space in the way common since Dirac (Dirac, 1958;Salisbury, 2006;Salisbury, 2010). The idea of extending phase space by t in order to accommodate velocity-dependent gauge transformations also fits well with Thébault's project.…”

Regarding the ‘Hole Argument’ and the ‘Problem of Time’

“…What is more, in the context of this pure Hamiltonian constraint canonical hole argument, the spectre of underdetermination looms over both relationalist and substantivalist interpretations alike. In particular, as pointed out by Pooley (2001), in the context of the canonical formalism, Machian relationalism about time is also confronted by indeterminism, unless some other steps are taken -see also (Thébault 2012). Our point here is not to reopen these debates, nor to blunt Weatherall's attack on the hole argument per se.…”

“…Points within this reduced phase space are then taken to represent diffeomorphism invariant spacetimes since there is an bijection between points in the reduced phase space and points in a space of diffeomorphism invariant spacetimes, defined via the covariant formalism. However, as discussed in (Thébault 2012), the existence of such a mapping between points in two representative spaces is far from a sufficient condition for them to play equivalent roles (although it could in some cases be taken to be necessary) since we can trivially find such relationships between manifestly inequivalent structures. It is much more plausible for the representational role of a space within a theory to be fixed primarily by its relationship to the representative structures from which it is derived rather than to a space utilised in the context of a different formalism.…”

“…The problem was first noticed by Bergmann and Dirac in the late 1950s, and is still a topic of debate in contemporary physics (Isham 1992, Kuchaȓ 1991, Anderson 2012). Although there is not broad agreement about precisely what the problem is supposed to be, one plausible formulation of (at least one aspect of) the problem is in terms of a dilemma in which one must choose between an ontology without time, by treating the Hamiltonian constraints as purely gauge generating, and an underdetermination problem, by treating the Hamiltonian constraints as generating physical change (Pooley 2006, Thébault 2012. The problem derives, for the most part, from a formal deficiency in the representational language of the canonical theory: one cannot unambiguously specify refoliation symmetries as invariances of mathematical objects we define on phase space.…”

Change in Hamiltonian General Relativity with Spinors

“…Given the velocity-dependent character of foliation-changing coordinate transformations, one can agree with Thébault that it is not very obvious what it would be be to construct a reduced phase space for GR [70] and that traditional descriptions [3] merit reconsideration. 3 Gryb and Thébault find that "time remains" and propose an alternative means of quantization which gives the Hamiltonian constraint a distinctive role [72,73].…”