Abstract
We consider unconstrained evolution schemes for the hyperboloidal initial value problem in numerical relativity as a promising candidate for the optimally efficient numerical treatment of radiating compact objects. Here, spherical symmetry already poses nontrivial problems and constitutes an important first step to regularize the resulting singular PDEs. We evolve the Einstein equations in their generalized BSSN and Z4 formulations coupled to a massless self-gravitating scalar field. Stable numerical evolutions are achieved for black hole initial data, and critically rely on the construction of appropriate gauge conditions.
Highlights
For isolated systems, their energy loss and radiation are in general only well defined at null infinity (I +)
A smooth way of combining spacelike foliations in the strong field region and reaching future null infinity is offered by the hyperboloidal initial value approach, pioneered by Friedrich [11, 12, 13]
We have recently presented stable evolutions of regular initial data which do not form black holes [17], here we use our code to evolve a scalar field interacting with a black hole as a problem to test how our algorithm handles black holes, and we demonstrate that our code can track the expected power law tails
Summary
Their energy loss and radiation are in general only well defined at null infinity (I +). A smooth way of combining spacelike foliations in the strong field region and reaching future null infinity is offered by the hyperboloidal initial value approach, pioneered by Friedrich [11, 12, 13]. One evolves along hyperboloidal slices, i.e. spacelike slices that reach null infinity In this approach regularizing the singular equations (2) in a way that is not just numerically stable, and avoids instabilities arising from the continuum equations, has proven to be difficult, in particular for hyperbolic free. The source function L0 is calculated from flat spacetime initial data on the hyperboloidal foliation. Due to their properties, we are interested in having the 1+log slicing condition around the origin and the harmonic one near I +.
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