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We adopt a reference-metric approach to generalize a covariant and conformalversion of the Z4 system of the Einstein equations. We refer to the resultingsystem as ``fully covariant and conformal", or fCCZ4 for short, since it iswell suited for curvilinear as well as Cartesian coordinates. We implement thisfCCZ4 formalism in spherical polar coordinates under the assumption ofspherical symmetry using a partially-implicit Runge-Kutta (PIRK) method andshow that our code can evolve both vacuum and non-vacuum spacetimes withoutencountering instabilities. Our method does not require regularization of theequations to handle coordinate singularities, nor does it depend onconstraint-preserving outer boundary conditions, nor does it need anymodifications of the equations for evolutions of black holes. We performseveral tests and compare the performance of the fCCZ4 system, for differentchoices of certain free parameters, with that of BSSN. Confirming earlierresults we find that, for an optimal choice of these parameters, and forneutron-star spacetimes, the violations of the Hamiltonian constraint can bebetween 1 and 3 orders of magnitude smaller in the fCCZ4 system than in theBSSN formulation. For black-hole spacetimes, on the other hand, any advantagesof fCCZ4 over BSSN are less evident.

physical review. d. particles, fields, gravitation, and cosmology/physical review. d, particles, fields, gravitation, and cosmology

Fully covariant and conformal formulation of the Z4 system in a  reference-metric approach: comparison with the BSSN formulation in spherical  symmetry