Cauchy problem as a two-surface based 'geometrodynamics'

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Four-dimensional spacetimes foliated by a two-parameter family of homologous two-surfaces are considered in Einstein?s theory of gravity. By combining a 1 + (1 + 2) decomposition, the canonical form of the spacetime metric and a suitable specification of the conformal structure of the foliating two-surfaces, a gauge fixing is introduced. It is shown that, in terms of the chosen geometrically distinguished variables, the 1 + 3 Hamiltonian and momentum constraints can be recast into the form of a parabolic equation and a first order symmetric hyperbolic system, respectively. Initial data to this system can be given on one of the two-surfaces foliating the three-dimensional initial data surface. The 1 + 3 reduced Einstein?s equations are also determined. By combining the 1 + 3 momentum constraint with the reduced system of the secondary 1 + 2 decomposition, a mixed hyperbolic-hyperbolic system is formed. It is shown that solutions to this mixed hyperbolic-hyperbolic system are also solutions to the full set of Einstein?s equations provided that the 1 + 3 Hamiltonian constraint is solved on the initial data surface ∑0 and the 1 + 2 Hamiltonian and momentum type expressions vanish on a world-tube yielded by the Lie transport of one of the two-surfaces foliating ∑0 along the time evolution vector field. Whenever the foliating two-surfaces are compact without boundary in the spacetime and a regular origin exists on the time-slices-this is the location where the foliating two-surfaces smoothly reduce to a point-it suffices to guarantee that the 1 + 3 Hamiltonian constraint holds on the initial data surface. A short discussion on the use of the geometrically distinguished variables in identifying the degrees of freedom of gravity are also included.

Original languageEnglish
Article number015006
JournalClassical and Quantum Gravity
Issue number1
Publication statusPublished - Jan 8 2015


  • Cauchy problem
  • degrees of freedom
  • gauge fixing

ASJC Scopus subject areas

  • Physics and Astronomy (miscellaneous)

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