Testing gauge waves and linear waves using the Harmonic Almost Killing Equations through the Z4 formalism

Abstract

[eng] One of the most remarkable features of Einstein’s equations in general relativity is their coordinate independence, arising from their covariant nature. This property makes the choice of coordinates essential, as an appropriate system can simplify problem-solving, while a poor choice can introduce issues like coordinate singularities. In numerical relativity, this choice becomes even more critical, as inappropriate coordinates can lead to significant computational challenges. This thesis builds upon the Z4 formalism, a 3+1 reformulation of Einstein’s equations, by incorporating the Harmonic Almost Killing Equations (HAKE), a generalization of the Killing equations. These equations are designed to minimize the dynamics of fields and identify symmetries in nearly stationary spacetimes. We introduce modifications to the original HAKE by implementing Lie derivative terms to improve the system’s hyperbolicity, giving rise to a new system that should be strongly hyperbolic everywhere and aligns stationary observers with the normal lines. To assess the effectiveness of these modifications, we test the Z4 formalism using two standard benchmarks: gauge waves, generated by a coordinate transformation in the time and x coordinates, and linear waves, resulting from a sinusoidal wave perturbation applied to the y and z components of the spacetime metric. The system is numerically solved after applying a 1+1 decomposition to the Z4 formalism with the modified HAKE gauge, and we analyze the behavior of key components under different conditions, verifying the functionality and hyperbolicity of the system. Additionally, harmonic slicing and the original HAKE gauge are evaluated for comparison, providing a baseline to assess the effectiveness of the modifications. The results demonstrate how the modified HAKE gauge improves the system’s hyperbolicity and stability, particularly in spacetimes with underlying symmetries. These findings provide valuable insights into the role of the HAKE gauge conditions in minimizing field dynamics in numerical relativity