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We propose a method for numerical relativity in which the spatial grid is finite and no outer boundary condition is needed. As a ‘proof of concept’ we implement this method for the case of a self-gravitating, spherically symmetric scalar field.
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Recently, there has been much interest in black hole echoes, based on the idea that there may be some mechanism (e.g., from quantum gravity) that waves/fields falling into a black hole could partially reflect off of an interface before reaching the horizon. There does not seem to be a good understanding of how to properly model a reflecting surface in...
Citations
... Recently Bieri, Garfinkle and Yau [6] proposed a general method for Cauchy evolution in numerical relativity whereby the boundary of the finite spatial computational domain is expanded along a spacelike direction at each time step. Additional initial data must be specified on this surface. ...
... One can also start with a standard Cauchy evolution with timelike boundary (where of course boundary conditions must be imposed) and switch to the ingoing boundary method at a certain time. Combinations with the outgoing boundary method of Bieri et al. [6] are also possible. ...
We develop a numerical method suitable for gravitational collapse based on Cauchy evolution with an ingoing characteristic boundary. Unlike similar methods proposed recently (Ripley; Bieri et al. in Class Quantum Grav 37:045015, 2020), the numerical grid remains fixed during the evolution and no points need to be removed or added. Increasing coordinate refinement of the central region as the field collapses is achieved solely through the choice of spatial gauge and particularly its boundary condition. We apply this method to study critical collapse of a massless scalar field in spherical symmetry using maximal slicing and isotropic coordinates. Known results on mass scaling, discrete self-similarity and universality of the critical solution (Choptuik in Phys Rev Lett 70:9, 1993) are reproduced using this considerably simpler numerical method.
... Recently Bieri, Garfinkle and Yau [6] proposed a general method for Cauchy evolution in numerical relativity whereby the boundary of the finite spatial computational domain is expanded along a spacelike direction at each time step. Additional initial data must be specified on this surface. ...
... One can also start with a standard Cauchy evolution with timelike boundary (where of course boundary conditions must be imposed) and switch to the ingoing boundary method at a certain time. Combinations with the outgoing boundary method of Bieri et al. [6] are also possible. ...
We develop a numerical method suitable for gravitational collapse based on Cauchy evolution with an ingoing characteristic boundary. Unlike similar methods proposed recently (Ripley; Bieri, Garfinkle & Yau 2019/20), the numerical grid remains fixed during the evolution and no points need to be removed or added. Increasing coordinate refinement of the central region as the field collapses is achieved solely through the choice of spatial gauge and particularly its boundary condition. We apply this method to study critical collapse of a massless scalar field in spherical symmetry using maximal slicing and isotropic coordinates. Known results on mass scaling, discrete self-similarity and universality of the critical solution (Choptuik 1993) are reproduced using this considerably simpler numerical method.
... However, the BL co-ordinates are not well behaved, both for r → r ∞ and r → r horizon . Therefore, motivated by the advances in the numerical relativity for dealing with these boundary conditions, [13,14,15,18,16,17,8], we study the separation of the Dirac equation in Kerr Geometry that is subjected to a hyperboloidal formulation and gauge fixing using the height function technique [18]. This establishes a co-ordinate system that is well-behaved for both, through the horizon (i.e "horizon penetrating") as well as at null infinity, J + . ...
The Dirac equation governs the behaviour of spin-1/2 particles. The equation's separability into decoupled radial and angular differential equations is a crucial step in analytical and numerical computations of quantities like eigenvalues, quasinormal modes and bound states. However, this separation has been performed in co-ordinate systems that are not well-behaved in either limiting regions of $r \rightarrow r_{horizon}$, $r \rightarrow r_\infty$ or both. In particular, the extensively used Boyer-Lindquist co-ordinates contains unphysical features of spacetime geometry for both $r_{horizon}$ and $r_\infty$. Therefore, motivated by the recently developed compact hyperboloidal co-ordinate system for Kerr Black Holes that is well behaved in these limiting regions, we attempt the separation of the Dirac equation. We first construct a null tetrad suitable for the separability analysis under the Newman-Penrose formalism. Then, an unexpected result is shown that by using the standard separability procedure based on the mode ansatz under this tetrad, the Dirac equation does not decouple into radial and angular equationsPossible reasons for this behaviour as well as importance of proving separability for various computations are discussed.
