Gravitational waves provide a powerful enhancement to our understanding of fundamental physics. To make the most of their detection we need to accurately model the entire process of their emission ...and propagation toward interferometers. Cauchy-characteristic extraction and matching are methods to compute gravitational waves at null infinity, a mathematical idealization of detector location, from numerical relativity simulations. Both methods can in principle contribute to modeling by providing highly accurate gravitational waveforms. An underappreciated subtlety in realizing this potential is posed by the (mere) weak hyperbolicity of the particular PDE systems solved in the characteristic formulation of the Einstein field equations. This shortcoming results from the popular choice of Bondi-like coordinates. So motivated, we construct toy models that capture that PDE structure and study Cauchy-characteristic extraction and matching with them. Where possible we provide energy estimates for their solutions and perform careful numerical norm convergence tests to demonstrate the effect of weak hyperbolicity on Cauchy-characteristic extraction and matching. Our findings strongly indicate that, as currently formulated, Cauchy-characteristic matching for the Einstein field equations would provide solutions that are, at best, convergent at an order lower than expected for the numerical method, and may be unstable. In contrast, under certain conditions, the extraction method can provide properly convergent solutions. Establishing however that these conditions hold for the aforementioned characteristic formulations is still an open problem.
Class. Quantum Grav. 40 175009 (2023); Corrigendum: Class. Quantum
Grav. 40 249503 (2023) We construct a numerical relativity code based on the
Baumgarte-Shapiro-Shibata-Nakamura (BSSN) formulation ...for the gravitational
quadratic $f(R)$ Starobinsky model. By removing the assumption that the
determinant of the conformal 3-metric is unity, we first generalize the BSSN
formulation for general $f(R)$ gravity theories in the metric formalism to
accommodate arbitrary coordinates for the first time. We then describe the
implementation of this formalism to the paradigmatic Starobinsky model. We
apply the implementation to three scenarios: the Schwarzschild black hole
solution, flat space with non-trivial gauge dynamics, and a massless
Klein-Gordon scalar field. In each case, long-term stability and second-order
convergence is demonstrated. The case of the massless Klein-Gordon scalar field
is used to exercise the additional terms and variables resulting from the
$f(R)$ contributions. For this model, we show for the first time that
additional damped oscillations arise in the subcritical regime as the system
approaches a stable configuration.
The characteristic initial (boundary) value problem has numerous applications in general relativity (GR) involving numerical studies, and is often formulated using Bondi-like coordinates. Recently it ...was shown that several prototype formulations of this type are only weakly hyperbolic. Presently we examine the root cause of this result. In a linear analysis we identify the gauge, constraint and physical blocks in the principal part of the Einstein field equations in such a gauge, and show that the subsystem related to the gauge variables is only weakly hyperbolic. Weak hyperbolicity of the full system follows as a consequence in many cases. We demonstrate this explicitly in specific examples, and thus argue that Bondi-like gauges result in weakly hyperbolic free evolution systems under quite general conditions. Consequently the characteristic initial (boundary) value problem of GR in these gauges is rendered ill-posed in the simplest norms one would like to employ. The possibility of finding good alternative norms, in which well-posedness is achieved, is discussed. So motivated, we present numerical convergence tests with an implementation of full GR which demonstrate the effect of weak hyperbolicity in practice.
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