Dynamics of Gravitational Waves in 3D: Formulations, Methods, and Tests

TL;DR

This paper explores the complex behavior of gravitational waves in 3+1 dimensional numerical relativity, focusing on non-linear effects, gauge issues, and the effectiveness of various numerical strategies and diagnostics.

Contribution

It provides a comprehensive analysis of numerical methods, gauge choices, and diagnostics for simulating gravitational waves in full 3D relativity, including a scalar model to understand non-linearities.

Findings

Different numerical schemes and gauge conditions significantly affect wave evolution.

Monitoring tools like Riemann invariants and Fourier analysis are crucial for analysis.

Non-linear scalar models help understand error coupling in simulations.

Abstract

The dynamics of gravitational waves is investigated in full 3+1 dimensional numerical relativity, emphasizing the difficulties that one might encounter in numerical evolutions, particularly those arising from non-linearities and gauge degrees of freedom. Using gravitational waves with amplitudes low enough that one has a good understanding of the physics involved, but large enough to enable non-linear effects to emerge, we study the coupling between numerical errors, coordinate effects, and the nonlinearities of the theory. We discuss the various strategies used in identifying specific features of the evolution. We show the importance of the flexibility of being able to use different numerical schemes, different slicing conditions, different formulations of the Einstein equations (standard ADM vs. first order hyperbolic), and different sets of equations (linearized vs. full Einstein equations). A non-linear scalar field equation is presented which captures some properties of the full Einstein equations, and has been useful in our understanding of the coupling between finite differencing errors and non-linearites. We present a set of monitoring devices which have been crucial in our studying of the waves, including Riemann invariants, pseudo-energy momentum tensor, hamiltonian constraint violation, and fourier spectrum analysis.