### Abstract

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 language | English |
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Article number | 015006 |

Journal | Classical and Quantum Gravity |

Volume | 32 |

Issue number | 1 |

DOIs | |

Publication status | Published - Jan 8 2015 |

### Keywords

- Cauchy problem
- degrees of freedom
- gauge fixing

### ASJC Scopus subject areas

- Physics and Astronomy (miscellaneous)