Spacetime Evolution, Gauge, and Boundaries

Spacetime evolution is a coupled numerical design: formulation, gauge, constraint treatment, spatial and temporal discretization, outer boundary, and black-hole interior handling must work together. A run that remains finite is not necessarily accurate, and coordinate smoothness is not invariant stability. This chapter compares three major evolution families, then treats gauge, outer boundaries, excision, and punctures through standard testbeds and strong-field checks. Continuum derivations remain with Formulations of General Relativity.

Helpful background. Formulations of General Relativity supplies hyperbolic systems and gauge, Ordinary Differential Equations supplies time integration, Computational Relativity supplies shared verification practices, and Black-Hole Dynamics and Collapse supplies the nonlinear physical setting.

Enter this chapter

Each evolution is specified by exact equations and constraint additions, not a family name. Gauge source functions, damping parameters, algebraic projections, characteristic fields, stencils or fluxes, dissipation, integrator, CFL factor, mesh hierarchy, and boundary conditions all belong in the computational input. The environment and code revision accompany raw constraint and invariant time series so the finite claim can be rerun independently.

The benchmark ladder starts with robust stability, linear waves, gauge and shifted-gauge waves, and Gowdy spacetime. It then advances to stationary Schwarzschild or Kerr, where horizon mass, spin, curvature invariants, constraints, and spurious radiation can be monitored. Daverio, Dirian, and Mitsou 2018, §§4–8 illustrates formulation comparison using shared test problems; application runs still require their own convergence evidence.

Route through the six pages

BSSN Evolution Systems defines conformal fields, algebraic constraints, and the standard test sequence. Numerical Generalized-Harmonic Evolution adds gauge-source functions, reduction constraints, damping, and characteristic boundaries.

Z4 and Constraint-Damping Formulations tests explicit propagation and damping of finite-wavelength constraint errors while retaining the undamped constant-mode control. Lapse and Shift Conditions then distinguishes coordinate regularity from invariant agreement and compares singularity-avoiding, harmonic, and driver gauges.

Outer Boundaries and Cauchy–Characteristic Matching measures reflection and incoming constraint flux rather than assuming a boundary is absorbing. Excision and Moving-Puncture Black Holes closes the chapter by comparing causal interior removal with gauge compression through horizon and waveform invariants.

Evidence across formulations

Use at least three resolutions and vary one physical-numerical control independently: damping, gauge parameter, boundary radius, excision margin, or puncture near-zone resolution. Evaluate constraints with a second derivative operator and preserve spatial profiles; a decreasing volume norm can hide a localized growing mode. Track invariant observables alongside coordinate fields.

Independent checks change representation or mechanism. Examples include a predicted Fourier damping rate, an exact gauge wave, matched BSSN and Z4c discretizations, constraint-preserving versus frozen boundaries, or excision versus puncture waveforms in one asymptotic frame. Lindblom et al. 2006, §§2–5 shows how formulation, damping, and characteristic structure connect in one generalized-harmonic system.

The primary failure signatures are resolution-accelerating constraints, loss of designed convergence, boundary returns at causal times, stiffness from excessive damping, gauge shocks or zero-speed modes, incoming excision characteristics, and stable coordinates with drifting horizon quantities. No amount of waveform alignment repairs one of these failures.

Limitations and related methods

The chapter establishes numerical behavior only in tested gauges, formulations, resolutions, and boundary regimes. It does not prove continuum well-posedness, validate matter models, or interpret merger physics. Wave extraction, horizon measurement, mesh adaptivity, and GRMHD receive dedicated treatment later in the volume. The executable commands described on child pages are reproducibility interfaces, not claims that planned software ran.

Review the chapter

Diagnostic. A binary evolution has convergent waveform phase and stable coordinate tracks, but the horizon mass drifts monotonically with increasing resolution. Which evidence controls the acceptance decision?

Solution

First determine whether the mass change is physical: binary horizons can absorb gravitational radiation, so their masses need not remain constant. Compare the change with horizon fluxes, its resolution limit, and horizon-finder uncertainty. A change that agrees with a convergent absorption budget is expected; an unexplained change that grows or fails to converge is an unresolved error even when a selected waveform interval converges. Check constraints, near-horizon resolution, gauge and interior treatment before accepting or rejecting the strong-field result.

Exercises

A transverse linear constraint mode on Minkowski spacetime obeys $\ddot Z+\kappa\dot Z+k^2Z=0$. In a periodic domain, take $k=1/(2L)$ and compare the slowest decay rate for $\kappa=2/L$ and $\kappa=20/L$. Does a tenfold increase in damping remove this mode faster?

Solution

The roots are $s_\pm=-\kappa/2\pm\sqrt{\kappa^2/4-k^2}$. The slow root is $s_+=(-1+\sqrt3/2)/L\simeq-0.134/L$ in the first case and $s_+=(-10+\sqrt{99.75})/L\simeq-0.0125/L$ in the second. The larger coefficient damps this long-wave component about ten times more slowly, while its fast branch creates a shorter explicit time-step scale. The result concerns this linear constraint subsystem, not physical gravitational radiation or every nonlinear Z4 mode.

References

• Daverio, D., Dirian, Y., and Mitsou, E. (2018). “Apples with Apples comparison of 3+1 conformal numerical relativity schemes.” arXiv:1810.12346. arXiv.
• Lindblom, L., Scheel, M. A., Kidder, L. E., Owen, R., and Rinne, O. (2006). “A new generalized harmonic evolution system.” Classical and Quantum Gravity 23, S447–S462. arXiv.