Z4 and Constraint-Damping Formulations

Z4-type formulations enlarge Einstein’s evolution system so that constraint violations become explicit propagating fields. Added lower-order terms can damp finite-wavelength violations, potentially carrying numerical contamination away rather than leaving zero-speed modes. The benefit is conditional: constant modes need not damp, boundaries can return outgoing violations, and large damping parameters can destabilize a discretization. A meaningful comparison with BSSN therefore matches gauge, grid, and numerical methods and tests the predicted mode behavior directly.

Helpful background. Formulations of General Relativity develops Z4 and conformal formulations, Ordinary Differential Equations supplies stability and damping analysis, Computational Relativity supplies common verification standards, and Black-Hole Dynamics and Collapse supplies the nonlinear applications.

Constraint fields as dynamical variables

In covariant Z4, the condition $Z_a=0$ recovers Einstein’s equations. In four dimensions with $\Lambda=0$, a representative damped equation adds terms of the form

$$\begin{aligned} R_{ab}+\nabla_a Z_b+\nabla_b Z_a &-\kappa_1\left[2n_{(a}Z_{b)} -(1+\kappa_2)g_{ab}n^cZ_c\right]\\ &=8\pi G\left(T_{ab}-\frac12g_{ab}T\right). \end{aligned}$$

Conformal variants such as Z4c and CCZ4 introduce variables and constraint additions that differ in detail. Record the exact equations, conformal factor, algebraic projections, covariance parameter, and signs and dimensions of $\kappa_1,\kappa_2$. Naming a family is not enough to reproduce a run.

Here $n^a$ is future directed, $\kappa_1$ has inverse-length units, and $\kappa_2$ is dimensionless. A transparent test follows from a transverse constraint perturbation on Minkowski spacetime with constant $\kappa_1\gt 0$. Its linear equation is

$$\partial_t^2Z_\perp-\partial_x^2Z_\perp +\kappa_1\partial_tZ_\perp=0.$$

For $Z_\perp\propto e^{st+ikx}$,

$$s_\pm=-\frac{\kappa_1}{2} \pm\sqrt{\frac{\kappa_1^2}{4}-k^2}.$$

Thus short waves decay with envelope $e^{-\kappa_1t/2}$, while at $k=0$ one root is zero. For $|k|\ll\kappa_1$, the slow root is $s_+\simeq-k^2/\kappa_1$: increasing damping can actually slow removal of long waves. This is a linear flat-background result; curvature and nonlinear terms require separate checks. It follows from Gundlach et al. 2005, §III, Eqs. (17)–(23).

Linear Fourier analysis predicts decay for classes of nonzero-wave-number constraint perturbations when damping parameters have suitable signs. Gundlach et al. 2005, §III and §V derives these rates and identifies undamped constant modes. A numerical test should therefore inject a known finite-$k$ violation and compare its measured decay rate with the prediction, while also injecting a constant component to ensure the code does not falsely claim universal damping.

Match the comparison with BSSN

To compare Z4c or CCZ4 with BSSN, hold initial data, lapse and shift, stencil, dissipation, time integrator, CFL factor, boundary placement, mesh, and output norms fixed. Use robust stability, linear wave, gauge wave, shifted gauge wave, and Gowdy tests before a black-hole evolution. Hilditch et al. 2013, §II, §§IV–V, Figures 1–11 and Appendix B compares Z4c and BSSN with attention to constraints and boundaries.

Scale initial random perturbations consistently with resolution. Report physical errors and constraint norms rather than ranking formulations by one norm alone. A Z4 run can have smaller constraints but larger phase error if its damping or dissipation modifies under-resolved modes. Conversely, a BSSN run may retain a stationary constraint component while preserving a selected observable. The supported claim should name both axes.

Damping, boundaries, and refinement

The primary benchmark measures decay for several wavelengths and damping values. Resolve the damping timescale with the time step and the shortest wavelength with the grid. At an outer boundary, impose conditions on incoming characteristic constraint fields; otherwise the boundary can replenish what the interior damps. Move the boundary to distinguish propagation and reflection from local growth.

For stationary Schwarzschild or Kerr, monitor Hamiltonian, momentum, Z4, and algebraic constraints together with horizon mass, spin, and curvature invariants. Use at least three resolutions, vary $\kappa_1$ and $\kappa_2$, and include the zero-damping control. The continuum assumptions are the declared gauge and formulation. Constraint damping is not physical dissipation and does not validate a strong-field solution by itself.

Compare observed order before and after constraint pulses cross refinement and outer boundaries. If convergence degrades only at an interface, the prolongation or penalty operator—not the continuum damping design—is the likely failure. Preserve spatial profiles so a decreasing global norm cannot hide a growing localized violation.

Environment and failure criteria

Pin the formulation variant, exact constraint additions, damping profiles, gauge, stencil or flux, dissipation, time integrator, CFL factor, interface and boundary conditions, mesh hierarchy, precision, test perturbation, and code revision. A clean interface could be evolve-z4 --config constraint-pulse.toml --output run/, emitting every constraint field, analytic decay target, physical errors, boundary fluxes, refinement sweeps, and environment metadata. This is a rerun specification, not evidence that a supplied code was executed.

Reject a claim of damping when only a global norm falls because the violation exits the domain, when constant modes are said to decay contrary to the analysis, or when higher resolution grows faster. Other failure modes include sign errors in $Z_a$, stiffness from excessive $\kappa_1$, incompatible algebraic projection, and boundary reflection. A formulation comparison is invalid if gauge or dissipation differs between runs.

Diagnostic

A sinusoidal constraint perturbation decays at the predicted rate, but a spatially constant offset remains. The report labels the constant offset a code failure because all constraints were expected to damp. Is that conclusion justified?

Solution

No. The linear damping analysis admits undamped constant modes under the stated conditions. The finite-$k$ decay verifies the targeted mechanism; the constant control verifies its limitation. Report both. A drifting constant mode can still become practically harmful, so test boundaries, nonlinear coupling, and invariant observables rather than misstate the analytic expectation.

References

• Gundlach, C., Martín-García, J. M., Calabrese, G., and Hinder, I. (2005). “Constraint damping in the Z4 formulation and harmonic gauge.” Classical and Quantum Gravity 22, 3767–3774. DOI. Open PDF.
• Hilditch, D., Bernuzzi, S., Thierfelder, M., Cao, Z., Tichy, W., and Brügmann, B. (2013). “Compact binary evolutions with the Z4c formulation.” Physical Review D 88, 084057. arXiv.