Numerical investigations of the asymptotics of solutions to the evolutionary form of the constraints

Abstract

Systematic numerical investigations of the asymptotics of near Schwarzschild vacuum initial data sets is carried out by inspecting solutions to the parabolic–hyperbolic and to the algebraic–hyperbolic forms of the constraints, respectively. One of our most important findings is that the concept of near Schwarzschild configurations, applied previously in [, ], is far too restrictive. It is demonstrated that by relaxing the conditions on the freely specifiable part of the data a more appropriate notion of near Schwarzschild initial data configurations can be defined which allows us to generate asymptotically flat initial data configurations.

Lineáris perturbációk és a kényszeregyenletek megoldásainak numerikus vizsgálata az általános relativitáselméletben

“…A fenti geometriai képhez illeszkedő koordinátákat alkalmazva azt mondhatjuk, hogy a (Σ, h ab , K ab ) kezdőadat erősen aszimptotikusan sík [113,114], ha egy kompakt Σ-beli halmaz komplementerét diffeomorf módon leképezhetjük valamely R 3 -beli origó középpontú B gömb komplementerére úgy, hogy az R 3 -on értelmezett x α Descartes-koordinátákkal meghatározott r = x 1 2 + x 2 2 + x 2 3 függvény r → ∞ határátmenete esetén a…”

Asymptotically flat vacuum initial data sets from a modified parabolic-hyperbolic formulation of the Einstein vacuum constraint equations

“…In particular, if these conditions for the free data are met, the Cauchy problem in the increasing ρ-direction is well-posed. We briefly comment on the claim in [20] that it is sufficient to interpret our modified formulation Eqs. (2.24)-(2.26) of the vacuum constraints (introduced in [19]) as the special case of Rácz's "original" formulation Eqs.…”

“…(2.16)- (2.18) where the free field κ is chosen to be determined by Eq. (2.23) in terms of some given field R and the unknown q "on the fly" at each time step of the evolution; see Section 4.3.1 in [20]. While this claim is evident on the one hand (because both the modified and the original formulations represent the same Einstein vacuum constraints), it may also be misleading.…”

“…These new equations preserve the parabolichyperbolic character of the PDEs, but at the same time also yield solutions with stable asymptotically flat asymptotics. In a similar spirit, a completely different modification was suggested in [20] in an attempt to resolve the instabilities present in Rácz's algebraichyperbolic formulation.…”

Black hole initial data by numerical integration of the parabolic–hyperbolic form of the constraints

“…The spatial domain was foliated with 2-spheres, the equations were integrated outwards towards the spacelike infinity and apart from the construction of the data the asymptotic flatness of the solutions was discussed. The issue of asymptotic flatness of the near Schwarzschild vacuum solutions within the parabolichyperbolic formulation was further discussed in the case of boundary conditions imposed on an arbitrary 2-sphere located between the strong field region and spatial infinity [14]. Unrestricted single and binary black hole initial data sets were elaborated on using an explicit PDE solver of the parabolic-hyperbolic set of constraints in [15].…”