Generalized harmonic spatial coordinates and hyperbolic shift conditions

Abstract

We propose a generalization of the condition for harmonic spatial coordinates analogous to the generalization of the harmonic time slices introduced by Bona et al., and closely related to dynamic shift conditions recently proposed by Lindblom and Scheel, and Bona and Palenzuela. These generalized harmonic spatial coordinates imply a condition for the shift vector that has the form of an evolution equation for the shift components. We find that in order to decouple the slicing condition from the evolution equation for the shift it is necessary to use a rescaled shift vector. The initial form of the generalized harmonic shift condition is not spatially covariant, but we propose a simple way to make it fully covariant so that it can be used in coordinate systems other than Cartesian. We also analyze the effect of the shift condition proposed here on the hyperbolicity of the evolution equations of general relativity in 1 1 dimensions and 3 1 spherical symmetry, and study the possible development of blowups. Finally, we perform a series of numerical experiments to illustrate the behavior of this shift condition.

Black hole head-on collisions and gravitational waves with fixed mesh-refinement and dynamic singularity excision

“…Primary among these is the question of gauge conditions. While substantial progress has been made in the recent past in deriving successful gauge conditions for evolutions that anchor black holes to a fixed coordinate location (see e. g. [4,19,20,47,48,49,50]) the question as to suitable gauge conditions for strongly time varying scenarios, such as spacetimes with moving black holes, remains largely unanswered.…”

Regularization of spherical and axisymmetric evolution codes in numerical relativity

“…As already mentioned in [16], the quantities ∆ i mn are components of a well defined tensor, while the Γ i mn are not and in fact are not even regular in spherical coordinates. One must also remember that the contraction used to construct the vector ∆ i = g mn ∆ mn must be done with the full metric associated with the space under study, instead of the flat metric.…”

“…This condition implies that the momentum constraints (2.4) are identically satisfied. We then choose a specific form for the function q and solve the Hamiltonian constraint for Ψ, which for the metric (5.15) becomes 16) with ∆ δ the flat space Laplacian. The function q is quasiarbitrary, and must only satisfy the following boundary conditions…”

3D simulations of Einstein's equations: Symmetric hyperbolicity, live gauges, and dynamic control of the constraints

“…In the presence of boundaries the situation is even more complicated: there are examples in the context of Einstein's equations explicitly showing ill posedness of certain strongly hyperbolic equations which do have smooth symmetrizers, when maximally dissipative boundary conditions are used (while for symmetric systems such a problem is known to be well posed) [28]. While we use time harmonic slicing in this paper, the freedom to use other slicing conditions could prove useful in other scenarios [29].…”