Hyperbolic formulations of General Relativity with Hamiltonian structure

With the aim of deriving symmetric hyperbolic free-evolution systems for GR that possess Hamiltonian structure and allow for the popular puncture gauge condition we analyze the hyperbolicity of Hamiltonian systems. We develop helpful tools which are applicable to either the first order in time, second order in space or the fully second order form of the equations of motion. For toy models we find that the Hamiltonian structure can simplify the proof of symmetric hyperbolicity. In GR we use a special structure of the principal part to prove symmetric hyperbolicity of a formulation that includes gauge conditions which are very similar to the puncture gauge.Comment: Our mathematica scripts are available at http://na.mathematik.uni-tuebingen.de/~richter

The Initial Value Formulation of Dynamical Chern-Simons Gravity

We derive an initial value formulation for dynamical Chern-Simons gravity, a modification of general relativity involving parity-violating higher derivative terms. We investigate the structure of the resulting system of partial differential equations thinking about linearization around arbitrary backgrounds. This type of consideration is necessary if we are to establish well-posedness of the Cauchy problem. Treating the field equations as an effective field theory we find that weak necessary conditions for hyperbolicity are satisfied. For the full field equations we find that there are states from which subsequent evolution is not determined. Generically the evolution system closes, but the full field equations are in no sense hyperbolic. In a cursory mode analysis we find that the equations of motion contain terms that may cause ill-posedness of the initial value problem.Comment: 16 pages; matches published versio

Revisiting Hyperbolicity of Relativistic Fluids

Motivated by the desire for highly accurate numerical computations of compact binary spacetimes in the era of gravitational wave astronomy, we reexamine hyperbolicity and well-posedness of the initial value problem for popular models of general relativistic fluids. Our analysis relies heavily on the dual-frame formalism, which allows us to work in the Lagrangian frame, where computation is relatively easy, before transforming to the desired Eulerian form. This general strategy allows for the construction of compact expressions for the characteristic variables in a highly economical manner. General relativistic hydrodynamics (GRHD), ideal magnetohydrodynamics (GRMHD) and resistive magnetohydrodynamics (RGRMHD) are considered in turn. In the first case we obtain a simplified form of earlier expressions. In the second we show that the flux-balance law formulation used in typical numerical applications is only weakly hyperbolic and thus does not have a well-posed initial value problem. Newtonian ideal MHD is found to suffer from the same problem when written in flux-balance law form. An alternative formulation, closely related to that of Anile and Pennisi, is instead shown to be strongly hyperbolic. In the final case we find that the standard forms of RGRMHD, relying upon a particular choice of `generalized Ohm's law', are only weakly hyperbolic. The latter problem may be rectified by adjusting the choice of Ohm's law, but we do not do so here. Along the way weak hyperbolicity of the field equations for dust and charged dust is also observed. More sophisticated systems, such as multi-fluid and elastic models are also expected to be amenable to our treatment.Comment: 28 pages, 1 figure, published versio

The mass ratio and the orbital parameters of the sdOB binary AA Doradus

The time sequence of 105 spectra covering one full orbital period of AA Dor has been analyzed. Direct determination of Vsini for the sdOB component from 97 spectra outside of the eclipse for the lines MgII 4481 A and SiIV 4089 A clearly indicated a substantially smaller value than estimated before. Detailed modelling of line profile variations for 8 spectra during the eclipse for the MgII 4481 A line, combined with the out-of-eclipse fits, gave Vsini = 31.8+/-1.8 km/s. The previous determinations of Vsini, based on the HeII 4686 A line, appear to be invalid because of the large natural broadening of the line. With the assumption of the solid-body, synchronous rotation of the sdOB primary, the measured values of the semi-amplitude K1 and Vsini lead to the mass ratio q = 0.213+/-0.013 which in turn gives K2 and thus the masses and radii of both components. The sdOB component appears to be less massive than assumed before, M1 = 0.25+/-0.05 Msol, but the secondary has its mass-radius parameters close to theoretically predicted for a brown dwarf, M2 = 0.054+/-0.010 Msol and R2 = 0.089+/-0.005 Rsol. Our results do not agree with the recent determination of Vuckovic et al. 2008 based on a K2 estimate from line-profile asymmetries.Comment: Accepted by MNRA

Constraint damping for the Z4c formulation of general relativity

One possibility for avoiding constraint violation in numerical relativity simulations adopting free-evolution schemes is to modify the continuum evolution equations so that constraint violations are damped away. Gundlach et. al. demonstrated that such a scheme damps low amplitude, high frequency constraint violating modes exponentially for the Z4 formulation of General Relativity. Here we analyze the effect of the damping scheme in numerical applications on a conformal decomposition of Z4. After reproducing the theoretically predicted damping rates of constraint violations in the linear regime, we explore numerical solutions not covered by the theoretical analysis. In particular we examine the effect of the damping scheme on low-frequency and on high-amplitude perturbations of flat spacetime as well and on the long-term dynamics of puncture and compact star initial data in the context of spherical symmetry. We find that the damping scheme is effective provided that the constraint violation is resolved on the numerical grid. On grid noise the combination of artificial dissipation and damping helps to suppress constraint violations. We find that care must be taken in choosing the damping parameter in simulations of puncture black holes. Otherwise the damping scheme can cause undesirable growth of the constraints, and even qualitatively incorrect evolutions. In the numerical evolution of a compact static star we find that the choice of the damping parameter is even more delicate, but may lead to a small decrease of constraint violation. For a large range of values it results in unphysical behavior.Comment: 13 pages, 24 figure