Classical Machian Resolution of the Spacetime Reconstruction Problem

Abstract

Following from a question of Wheeler, why does the Hamiltonian constraint $\mathcal{H}$ of GR have the particular form it does? A first answer, by Hojman, Kuchař and Teitelboim, is that using embeddability into spacetime as a principle gives the form of $\mathcal{H}$. The present paper culminates a second Machian answer - initially by Barbour, Foster and ó Murchadha - in which space but not spacetime are assumed. Thus this answer is additionally a classical-level resolution of the spacetime reconstruction problem. In this approach, mere consistency imposed by the Dirac procedure whittles down a general ansatz to one of four alternatives: Lorentzian, Galilean, or Carrollian relativity, or constant mean curvature slicing. These arise together as the different ways to kill off a 4-factor obstruction term. It is novel for such an alternative to arise from principles of dynamics considerations (in contrast with the historical form of the dichotomy between universal local Galilean or Lorentzian relativity). It is furthermore intriguing that it gives constant mean curvature slicing - familiar from York's work on the initial value problem -- as a further option on a similar footing. That is related to a number of recent alternative theories/formulations of GR known collectively as `shape dynamics'. The original work did not treat this with Poisson brackets and a proper systematic Dirac-type analysis; we rectify this in this paper. It is also the first demonstration of how this approach solves the classical spacetime reconstruction problem via `hypersurface tensor dual nationality' and what can be interpreted as embedding equations arising.

28 pages including 4 figures. Minor text and figure improvements

References in corpus (11)

• Constraints and gauge transformations: Dirac's theorem is not always valid
• Stable Homology as an Indicator of Manifoldlikeness in Causal Set Theory
• Conformal Superspace: the configuration space of general relativity
• New interpretation of variational principles for gauge theories. I. Cyclic coordinate alternative to ADM split
• Does relationalism alone control geometrodynamics with sources?
• Geometrodynamics: Spacetime or Space ?
• Machian Time Is To Be Abstracted From What Change?
• On Background Independence
• Shape dynamics and Mach's principles: Gravity from conformal geometrodynamics
• Conformal geometrodynamics regained: gravity from duality
• Quantum Cosmology will need to become a Numerical Subject

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