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

We adopt a reference-metric approach to generalize a covariant and conformal version of the Z4 system of the Einstein equations. We refer to the resulting system as “fully covariant and conformal,” or fCCZ4 for short, since it is well suited for curvilinear as well as Cartesian coordinates. We implement this fCCZ4 formalism in spherical polar coordinates under the assumption of spherical symmetry using a partially implicit Runge-Kutta method and show that our code can evolve both vacuum and nonvacuum spacetimes without encountering instabilities. Our method does not require regularization of the equations to handle coordinate singularities, nor does it depend on constraint-preserving outer boundary conditions. It also does not need any modifications of the equations for evolutions of black holes. We perform several tests and compare the performance of the fCCZ4 system, for different choices of certain free parameters, with that of BSSN. Confirming earlier results we find that, for an optimal choice of these parameters and for neutron-star spacetimes, the violations of the Hamiltonian constraint can be between 1 and 3 orders of magnitude smaller in the fCCZ4 system than in the BSSN formulation. For black hole spacetimes, on the other hand, any advantages of fCCZ4 over BSSN are less evident.

Figure 1

Hamiltonian constraint violation for a pure gauge pulse test as a function of radius at four different times for both BSSN (solid line) and fCCZ4 (dashed lines).

Figure 2

L2 norm of the Hamiltonian constraint for a pure gauge pulse test for BSSN (solid line) and fCCZ4 (dashed and dotted lines) as a function of time and for different choices of the damping parameters.

Figure 3

Pure gauge pulse: Hamiltonian constraint violations at $t=10.5\lambda$ for three different resolutions $\mathrm{\Delta }r/\lambda =\{0.1,0.05,0.025\}$, rescaled by the factors corresponding to second-order convergence.

Figure 4

Time evolution of the mass of the AH in the single puncture black hole simulation. The lower panel shows the evolution of the AH mass during the stationary phase.

Figure 5

(Upper panel) L2 norm of the Hamiltonian constraint in the single puncture black hole simulation. The inset shows a magnified view of the initial $100M$ in the evolution. (Lower panel) Same quantity but computed outside the AH.

Figure 6

(Upper panel) Time evolution of the normalized central density with fCCZ4 for different values of ${\kappa }_1$ and BSSN. (Lower panel) The L1 norm between the evolved rest-mass density and the initial density as a function of time, rescaled for three different resolutions $\mathrm{\Delta }r=\{0.2,0.1,0.05\}$ for the fCCZ4 system.

Figure 7

Comparison of the time evolution of the L2 norm of the Hamiltonian constraint for the stable spherical relativistic star for BSSN (solid line) and fCCZ4 (dashed lines) with ${\kappa }_1=\{0,0.07,0.2\}$ (in our code units).

Figure 8

Time evolution of the normalized central density (upper panel) and of the L2 norm of the Hamiltonian constraint (lower panel) for the migration test for both BSSN and fCCZ4 with ${\kappa }_1=\{0,0.07,0.2\}$ (in our code units).

Figure 9

Time evolution of the normalized central density (upper panel) and of the irreducible mass of the black hole (lower panel) for BSSN and fCCZ4 with ${\kappa }_1=\{0,0.02,0.07,0.2\}$ until $t=500$ (in our code units).

Figure 10

Time evolution of the L2 norm of the Hamiltonian constraint for BSSN and fCCZ4 computed outside of the AH. The time coordinate is relative to the time $t_{\mathrm{AH}}\sim 167$ when an apparent horizon forms for the BSSN formulation.