Hyperbolicity of general relativity in Bondi-like gauges

Abstract

Bondi-like (single-null) characteristic formulations of general relativity are used for numerical work in both asymptotically flat and anti-de Sitter spacetimes. Well-posedness of the resulting systems of partial differential equations, however, remains an open question. The answer to this question affects 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; for the initial value problem in the L 2 norm this is characterized by strong hyperbolicity. We find that, due to a shared pathological structure, the systems arising from the aforementioned formulations are however only weakly hyperbolic. We present numerical tests for toy models that demonstrate the consequence of this shortcoming in practice for the characteristic initial boundary value problem. Working with alternative norms in which our model problems may be well-posed we show that convergence may be recovered. Finally we examine well-posedness of a model for Cauchy-Characteristic-Matching in which model symmetric and weakly hyperbolic systems communicate through an interface, with the latter playing the role of GR in Bondi gauge on characteristic slices. We find that, due to the incompatibility of the norms associated with the two systems, the composite problem does not naturally admit energy estimates.

A symmetric hyperbolic formulation of the vacuum Einstein equations in affine-null coordinates

“…While single-null coordinates provide several conceptual and technical advantages over Cauchy-like evolution schemes, the most commonly used formulations of the Einstein equations in those coordinate systems were recently shown to be only weakly -but not strongly-hyperbolic [1]. We have shown that by using the tetrad-based Newman-Penrose formalism, it is relatively straightforward to construct a symmetric hyperbolic formulation of the Einstein equations in affine-null coordinates, which are example of a single null (or Bondi-like) coordinate system.…”

“…[17,18]). This being said, numerical evolution of weakly hyperbolic systems may only give non-convergent/pathological results with initial data that is sufficiently "rough", and at sufficiently high numerical resolution [1]. The stable, convergent evolution of many numerical schemes that have used single-null (and weakly hyperbolic) formulations of the Einstein equations could potentially be explained by the fact that those earlier studies worked with smooth initial data, and use numerical dissipation (which is typically required for stable numerical evolution of even strongly hyperbolic systems, see, e.g.…”

“…where we have introduced the function B(u, r, θ a ) (c.f. [1]). This form of the metric captures a wide variety of singlenull coordinate systems, e.g.…”

“…al. [1]. Surprisingly, those authors found that the Einstein Equations, when written as an evolution system for the metric in those Bondi and affine-null coordinates, formed only a weakly-but not strongly-hyperbolic system of partial differential equations for the metric components.…”

A Note on the Strong Hyperbolicity of $f(R)$ Gravity with Dynamical Shifts

“…So, the other is nonperturbative. It is based on the Arnowitt-Deser-Misner (ADM) decomposition of the evolution equations [17] with some suitable gauge conditions or the Bondi-Sachs formalism [18][19][20]. The key technology of strong hyperbolicity is to check whether the eigenvalues of the principal part are all real or not and check whether the eigenvectors of the principal part of the equations span the whole eigenspace or not [21][22][23].…”