Z4 Systems and Constraint Damping

The Z4 formulation represents departures from the Einstein constraint surface by an additional spacetime covector $Z_\mu$ and evolves that covector together with the metric. Exact Einstein solutions are retained at $Z_\mu=0$, while suitable lower-order terms can make small nonconstant violations decay in a linearized continuum analysis. The clean result is conditional: for the standard damping choice with positive coefficient and the usual parameter value, every nonzero Fourier mode decays about Minkowski spacetime, but a spatially constant mode does not. The proof does not establish nonlinear damping on strongly curved backgrounds, suppress boundary-injected violations, or guarantee decay after discretization.

Required background. Partial differential equations and Green functions supplies subsidiary equations and Fourier-mode analysis. The Einstein equation supplies the constraint surface that Z4 extends.

Helpful background. Hypersurface normals and tangential projectors supplies the normal–spatial split of $Z_\mu$. Formulations of General Relativity explains why lower-order off-constraint changes leave exact Einstein solutions intact. Differential equations reviews damped-wave roots and stability.

The Z4 extension of Einstein’s equation

Set $\Lambda=0$ for this page. In the covariant Z4 system, the trace-reversed field equation is

$$R_{\mu\nu} +\nabla_\mu Z_\nu +\nabla_\nu Z_\mu =8\pi G\left(T_{\mu\nu}-\frac12g_{\mu\nu}T\right).$$

The algebraic condition

$$Z_\mu=0$$

reduces this equation to Einstein’s equation. Conversely, the extended system includes off-constraint solutions with nonzero $Z_\mu$; in special spacetimes a Killing covector can create a degeneracy, so $Z_\mu=0$ remains the defining Einstein constraint rather than an optional diagnostic. The four-dimensional system and its $3+1$ split into the scalar $\Theta=-n^\mu Z_\mu$ and spatial covector $Z_i$ are given in Bona, Ledvinka, Palenzuela, and Žáček 2004, §§I–II, equations (1)–(11), pp. 1–3, PDF.

Taking a divergence, using the contracted Bianchi identity and assuming stress-energy conservation gives the undamped subsidiary equation

$$\Box_g Z_\mu+R_\mu{}^\nu Z_\nu=0.$$

It is homogeneous, so data with $Z_\mu=0$ and vanishing normal derivative retain the zero solution in a domain of dependence with compatible boundary data. The equation also shows that an arbitrary nonzero violation propagates as a vector wave; propagation alone does not force its amplitude toward zero.

Lower-order constraint damping

In vacuum, introduce a smooth nonvanishing timelike vector $t^\mu$, a constant damping scale $\kappa$, and a constant dimensionless parameter $\rho$. With the global $(-,+,+,+)$ signature, a natural choice is the future unit normal $t^\mu=n^\mu$. The damped Z4 equation is

$$R_{\mu\nu} +\nabla_\mu Z_\nu +\nabla_\nu Z_\mu -\kappa\left[ t_\mu Z_\nu+t_\nu Z_\mu -(1+\rho)g_{\mu\nu}t^\lambda Z_\lambda \right]=0.$$

Every added term is proportional to $Z_\mu$, so the exact Einstein sector is unchanged. The terms contain no derivatives of the evolved fields beyond those already present; consequently they do not change the metric–Z4 principal symbol. They can change growth rates, which are lower-order properties. Hyperbolicity still depends on the gauge equations and any first-order reduction.

The associated subsidiary equation is

$$\Box_g Z_\nu+R_{\mu\nu}Z^\mu -\kappa\nabla^\mu\left( t_\mu Z_\nu+t_\nu Z_\mu +\rho g_{\mu\nu}t^\lambda Z_\lambda \right)=0.$$

This follows directly from the trace-reversed damped equation; the derivation and the equivalent $3+1$ constraint system appear in Gundlach, Calabrese, Hinder, and Martín-García 2005, §II, equations (1)–(15), pp. 1–3, PDF. The sign of $\kappa$ is fixed by requiring friction toward the future. For the $\rho=0$ flat-background test below, $\kappa\gt0$ gives decay of nonzero Fourier modes and $\kappa\lt0$ gives growth.

Linear Fourier test on Minkowski spacetime

Linearize about a vacuum Minkowski background with $Z_\mu=0$, choose inertial coordinates and $t^\mu=(1,0,0,0)$, and freeze $\kappa$ and $\rho$. For the standard choice $\rho=0$, insert

$$Z_\mu(t,x)=\widehat Z_\mu e^{st+i\omega_jx^j}.$$

Resolve the spatial field into a longitudinal component $Z_n$ along $\omega_i$ and two transverse components $Z_A$. The transverse roots are

$$s=-\frac\kappa2 \pm\sqrt{\left(\frac\kappa2\right)^2-\omega^2},$$

The temporal–longitudinal sector contains both this pair and the additional pair

$$s=-\kappa\pm\sqrt{\kappa^2-\omega^2}.$$

To count every mode, put $q=\lvert\omega\rvert$. The two component equations at $\rho=0$ are

$$\begin{aligned} \Box Z_0-2\kappa\partial_t Z_0+\kappa\partial^iZ_i&=0,\\ \Box Z_n-\kappa\partial_t Z_n&=0. \end{aligned}$$

Their Fourier matrix is triangular, so its determinant factors as

$$(s^2+2\kappa s+q^2)(s^2+\kappa s+q^2)=0.$$

There are four roots in this sector, in addition to the four roots supplied by the two transverse components. For $\kappa\gt0$ and $q\ne0$, all have negative real part. At high frequency the two damping rates approach $\kappa/2$ and $\kappa$. At low frequency the slow roots are $-q^2/\kappa+O(q^4/\kappa^3)$ and $-q^2/(2\kappa)+O(q^4/\kappa^3)$. At $q=0$ four roots are exactly zero, so constant violations can persist. The starting component equations and matrix are Gundlach, Calabrese, Hinder, and Martín-García 2005, §III, Eqs. (16)–(19), p. 3, PDF; the factorization above follows directly from that matrix and retains both pairs in the longitudinal sector. At a repeated root the general solution can also contain a polynomial prefactor such as $t e^{st}$; this does not prevent decay when $\Re s\lt0$.

The parameter restriction matters. For general $\rho$, the same matrix gives

$$\begin{aligned} p(s)={}&[s^2+\kappa(2+\rho)s+q^2] [s^2+\kappa s+q^2]\\ &-\kappa^2\rho q^2. \end{aligned}$$

For $\rho\gt0$ and $0\lt q^2\lt\kappa^2\rho$, one has $p(0)\lt0$ while $p(s)\to+\infty$ for real $s\to+\infty$. There is consequently a growing positive real root. With the displayed equations, the damping conclusion cannot be extended to every $\rho\gt-1$; the all-wavelength result established here uses $\rho=0$. This direct polynomial check also avoids relying on the incomplete root multiplicities printed after Eq. (21) of the source.

This mode calculation supplies two independent failure tests. Reversing the sign of $\kappa$ reverses the real parts and turns friction into exponential amplification. Choosing $\rho=-1$ leaves undamped modes even at nonzero wave number. Either failure can be found without evolving the nonlinear equations.

Domain of validity

On a curved Einstein background, the same conclusion applies in a frozen-coefficient, high-frequency regime where the wavelength is short compared with the curvature scale. Long-wavelength modes couple to curvature and lower-order terms, and large violations activate nonlinearities omitted by the linearization. Therefore “the constraint surface is an attractor” must always be accompanied by the background, amplitude, wavelength, and parameter assumptions.

A boundary requires an additional characteristic analysis. Incoming Z4 or subsidiary fields can continually replenish violations faster than the interior terms damp them; compatible constraint-preserving boundary data are not supplied by the damping term. Likewise, a stable continuum decay rate does not establish stability of a finite-difference, finite-volume, or spectral discretization. Variants such as Z4c or conformal covariant Z4 change variables and add further terms, so their damping and hyperbolicity claims require their own equations and parameter ranges.

Exercise

For a transverse mode at $\rho=0$, derive the two growth rates and analyze the limits of large, small, and zero wave number. What changes when $\kappa$ has the wrong sign?

Solution

The transverse subsidiary equation is

$$\left(-\partial_t^2+\partial^j\partial_j\right)Z_A -\kappa\partial_tZ_A=0.$$

Substitution of $Z_A=\widehat Z_Ae^{st+i\omega_jx^j}$ gives

$$s^2+\kappa s+\omega^2=0,$$

and hence

$$s=-\frac\kappa2 \pm\sqrt{\left(\frac\kappa2\right)^2-\omega^2}.$$

When $|\omega|$ is much larger than $\kappa$, the roots are approximately $-\kappa/2\pm i|\omega|$, so the wave oscillates with an exponentially decaying envelope. When $|\omega|$ is much smaller than $\kappa$, the roots are approximately $-\kappa$ and $-\omega^2/\kappa$. At $\omega=0$ they are $-\kappa$ and zero, confirming the undamped constant mode. If $\kappa$ is negative, at least one root has positive real part; the proposed damping term amplifies rather than suppresses the violation.

References

• Bona, Carles, Tomáš Ledvinka, Carlos Palenzuela, and Miquel Žáček. “A Symmetry-Breaking Mechanism for the Z4 General-Covariant Evolution System.” Physical Review D 69 (2004): 064036. arXiv:gr-qc/0307067.
• Gundlach, Carsten, Gioel Calabrese, Ian Hinder, and José M. Martín-García. “Constraint Damping in the Z4 Formulation and Harmonic Gauge.” Classical and Quantum Gravity 22 (2005): 3767–3774. arXiv:gr-qc/0504114.