Characteristic formulations of general relativity and applications

Abstract:

General relativity can describe various gravitational systems of astrophysical relevance, like black holes and neutron stars, or even strongly coupled systems through the holographic duality. In addition, more formal aspects of the theory like the stability of spacetimes and the formation of singularities are still topics of active research. In several cases, solutions in closed analytic form are not known, and perturbative methods are inadequate, leading to the employment of numerical techniques.

The characteristic initial (boundary) value problem has numerous applications in general relativity involving numerical studies and is often formulated using Bondi-like coordinates. Well-posedness of the resulting systems of partial differential equations, however, remains an open question. The answer to this question affects the accuracy, and potentially the reliability of conclusions drawn from numerical studies based on such formulations.

A numerical approximation can converge to the continuum limit only for well-posed systems. The notion of well-posedness is tightly related to that of hyperbolicity and includes the specification of a norm. In the first part of this thesis, we expand our understanding of the hyperbolicity and well-posedness of Bondi-like free evolution systems. We show that several prototype Bondi-like formulations are only weakly hyperbolic and 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 we 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 general relativity in these gauges is rendered ill-posed in the simplest norms one would like to employ. We discuss the implications of this result in accurate gravitational waveform modeling methods and work towards the construction of alternative norms that might be more appropriate.

We also present numerical tests that demonstrate weak hyperbolicity in practice and highlight important features to perform them effectively. In the second part, we turn our attention to applications of these formulations to strongly coupled systems via holography. We expect these studies to shed more light on the qualitative behavior of strongly coupled plasmas, but due to weak hyperbolicity, we cannot perform rigorous error estimates to our satisfaction. We present Jecco, a newly developed characteristic code that allows us to simulate the dynamics of strongly coupled plasmas.

Representative examples of the simulations that can be achieved with this code are provided, namely the out-of-equilibrium dynamics of said plasmas that undergo phase transitions. This is a putative scenario of the early universe and such simulations might provide insights into questions of fundamental nature.