Towards an understanding of the stability properties of the 3+1 evolution equations in general relativity

TL;DR

This work analyzes the stability of the standard ADM $3+1$ Einstein evolution equations under linear perturbations of flat spacetime, focusing on zero-speed modes that include pure gauge and constraint-violating families. Through a Fourier/linear analysis, they show that gauge modes can be decoupled via a conformal-traceless decomposition ($\tilde{g}_{ij}=e^{-4\phi}g_{ij}$, $\tilde{A}_{ij}=e^{-4\phi}(K_{ij}-\frac{1}{3}g_{ij}K)$) and that constraint-violating modes acquire finite speeds when momentum constraints are used to modify evolution equations for extra variables ($\tilde{\Gamma}_i$), with stability enhanced by choosing parameters ($\sigma=1$, $\xi=0$, $m\ge 1/2$ or $m=1$). Numerical Minkowski tests corroborate the theory: standard ADM is unstable; conformal-traceless formulations (e.g., BSSN) show greatly improved stability, consistent with the predicted dynamics of zero-speed modes. The results provide a physical, mathematically grounded explanation for the observed stability of CT formulations and guide design choices for robust $3+1$ evolutions in numerical relativity.

Abstract

We study the stability properties of the standard ADM formulation of the 3+1 evolution equations of general relativity through linear perturbations of flat spacetime. We focus attention on modes with zero speed of propagation and conjecture that they are responsible for instabilities encountered in numerical evolutions of the ADM formulation. These zero speed modes are of two kinds: pure gauge modes and constraint violating modes. We show how the decoupling of the gauge by a conformal rescaling can eliminate the problem with the gauge modes. The zero speed constraint violating modes can be dealt with by using the momentum constraints to give them a finite speed of propagation. This analysis sheds some light on the question of why some recent reformulations of the 3+1 evolution equations have better stability properties than the standard ADM formulation.

Figures (9)

• Figure 1: Evolution of $\phi$ described by Eq. (\ref{['eq:scalar2']}), with $\epsilon=1$ and $\delta=-0.01$ at various times (from $t=0$ to $t=30$ in equal time intervals).
• Figure 2: Evolution of $\phi$ described by Eq. (\ref{['eq:scalar']}), with $\epsilon=0$ and $\delta=-0.01$ at various times (from $t=0$ to $t=5$ in equal time intervals).
• Figure 3: Surface plot of $g_{xx}$ along the $x$ axis as a function of time for the simulation using the standard ADM formulation.
• Figure 4: Root mean square of the hamiltonian constraint as a function of time for the simulation using the standard ADM formulation.
• Figure 5: Root mean square of the hamiltonian constraint as a function of time for the simulation using the standard CT formulation with $\xi$=0 and two different values of $\sigma$.