Mathematical issues in a fully constrained formulation of the Einstein equations

Abstract

Bonazzola, Gourgoulhon, Grandclément, and Novak [Phys. Rev. D 70, 104007 (2004)] proposed a new formulation for 3+1 numerical relativity. Einstein equations result, according to that formalism, in a coupled elliptic-hyperbolic system. We have carried out a preliminary analysis of the mathematical structure of that system, in particular focusing on the equations governing the evolution for the deviation of a conformal metric from a flat fiducial one. The choice of a Dirac's gauge for the spatial coordinates guarantees the mathematical characterization of that system as a (strongly) hyperbolic system of conservation laws. In the presence of boundaries, this characterization also depends on the boundary conditions for the shift vector in the elliptic subsystem. This interplay between the hyperbolic and elliptic parts of the complete evolution system is used to assess the prescription of inner boundary conditions for the hyperbolic part when using an excision approach to black hole spacetime evolutions.A second-order fully-constrained evolution formalism for the Einstein equations has been proposed in Ref. [18]. This evolution scheme, that will be referred in the following as Fully-Constrained Formulation (FCF), is based on a conformal 3+1 formulation of General Relativity and makes use of an elliptic condition for the choice of spatial coordinates, a generalized Dirac gauge, and a maximal condition for the slicing. The enforcement of the constraints along the evolution together with the elliptic nature of the employed gauge conditions, translates the FCF formalism into a mixed elliptic-hyperbolic Partial Differential Equations (PDE) system, consisting in five quasi-linear elliptic equations coupled with a tensorial second-order in time and in space evolution equation for the conformal metric. In this article, we aim at gaining insight on some mathematical issues associated with this PDE system and, in particular, assessing the hyperbolicity of the tensorial evolution part. A good understanding of the mathematical structure of the system will be crucial in the context of full 3D numerical relativity simulations, since the choice of state-of-the-art numerical tools will be adapted to the specific structures of the whole system governing the evolution of matter fields in a dynamical space-time: spectral methods for the elliptic subsystem [32], and modern high-resolution shock-capturing techniques for the hyperbolic part [33,34]. The implementation of the scheme in [18] will naturally extend previous works -following the Conformal Flatness Condition (CFC) approach of Isenberg-Wilson-Mathews [19, 20]-devoted to the study of some relevant astrophysical sources of gravitational radiation [21,22,23,24]. A. Gauge reduction, PDE evolution systems and well-posednessThe gauge character of General Relativity (GR) strongly conditions any attempt of finding a solution by solving a Partial Differential Equations (PDE) problem. In its standard formulation through the Einstein equation

Improved constrained scheme for the Einstein equations: An approach to the uniqueness issue

“…The analysis in [18] concludes that the reason behind the failures in these axisymmetric formulations is in fact related to the presence of wrong signs or, more precisely, to the indefinite character of certain non-linear Helmholtzlike equations present in the scheme (see [18] for details and also for a parallel numerical discussion in terms of a class of relaxation methods for the convergence of the elliptic solvers). Regarding the full three-dimensional case, fully constrained formalisms have been presented in [23,24,25]. While the work in [23,24] includes an elliptic subsystem closely related to the XCTS equations and therefore suffers potentially from these nonuniqueness problems, the uniqueness properties of the scheme of [25] must yet be studied.…”

Trapping horizons as inner boundary conditions for black hole spacetimes

“…First, it is shown that imposing the Dirac gauge in (1) indeed guarantees the real character of the eigenvalues corresponding to matrices A i , and therefore the hyperbolicity of the evolution system. Of particular relevance for the present inner BC discussion is the explicit determination of the (non-vanishing) characteristic speeds associated with the vector s normal to the excision surface S t , resulting in [7] λ (s) ± = −β ⊥ ± N (each one of multiplicity 6). (10) Taking into account the inequality in (4), consequence of the choice of a coordinate system adapted to the DH H by enforcing condition (6), we conclude the absence of ingoing radiative modes into the integration domain Σ t at the excision surface.…”

“…For this reason, we rather adopt the methodological choice of only prescribing MOTS as inner BCs. Regarding a possible FOTH condition failure, and according with the characteristic analysis in [7], monitoring the sign of (β ⊥ − N ) determines if inner BCs must or must not be provided for the radiative modes. This work represents an intermediate step in the ongoing program [6] addressing fullyconstrained excised black hole numerical evolutions.…”

“…(3), has been carried out in Ref. [7] by writing down the evolution equations as a first-order system in conservative form, i.e.…”

“…[6]. This approach maximizes the number of elliptic equations to be solved during the evolution, resulting in a coupled elliptic-hyperbolic PDE system [7]. Spectral methods [8] are then employed both to solve the elliptic subsystem and to handle the spatial part of the relevant hyperbolic operators.…”