Hyperbolic slicings of spacetime: singularity avoidance and gauge shocks

@article{Alcubierre2002HyperbolicSO,
  title={Hyperbolic slicings of spacetime: singularity avoidance and gauge shocks},
  author={Miguel Alcubierre},
  journal={Classical and Quantum Gravity},
  year={2002},
  volume={20},
  pages={607-623}
}
  • M. Alcubierre
  • Published 15 October 2002
  • Mathematics
  • Classical and Quantum Gravity
I study the Bona–Masso family of hyperbolic slicing conditions, considering in particular its properties when approaching two different types of singularities: focusing singularities and gauge shocks. For focusing singularities, I extend the original analysis of Bona et al and show that both marginal and strong singularity avoidance can be obtained for certain types of behaviour of the slicing condition as the lapse approaches zero. For the case of gauge shocks, I rederive a condition found… 
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References

SHOWING 1-10 OF 26 REFERENCES
Appearance of coordinate shocks in hyperbolic formalisms of general relativity
I consider the appearance of shocks in hyperbolic formalisms of General Relativity. I study the particular case of the Bona-Masso formalism with zero shift vector and show how shocks associated with
NUMERICAL RELATIVITY: EVOLVING SPACETIME
The construction of numerical solutions of Einstein's General Relativity equations is formulated as an initial-value problem. The space-plus-time (3 + 1) decomposition of the spacetime metric tensor
Horizon boundary condition for black hole spacetimes.
TLDR
The details of the implementation of a horizon boundary condition scheme based on using a horizon locking coordinate which locks the coordinate system to the geometry, and a finite differencing scheme which respects the causal structure of the spacetime are reported.
Kinematical conditions in the construction of spacetime
We adopt the point of view that a solution of Einstein's equations is an evolution of given initial Cauchy data. Implementing the evolution equations necessarily requires a determination, not
Towards a singularity-proof scheme in numerical relativity.
TLDR
A scheme which excises a region inside an apparent horizon containing the singularity through the use of a horizon-locking coordinate and a finite differencing which respects the causal structure of the spacetime.
Coordinates and boundary conditions for the general relativistic initial data problem
Using York's method, the author discusses techniques for numerically constructing GR initial data on a Cauchy surface representing spacetimes containing arbitrary numbers of black holes, each with
New formalism for numerical relativity.
TLDR
A new formulation of the Einstein equations is presented that casts them in an explicitly first order, flux-conservative, hyperbolic form, which permits the application to the Einstein equation of advanced numerical methods developed to solve the fluid dynamic equations, for the first time.
Fixing Einstein's equations
Einstein's equations for general relativity, when viewed as a dynamical system for evolving initial data, have a serious flaw: they cannot be proven to be well-posed (except in special coordinates).
Three-dimensional numerical relativity: The evolution of black holes.
TLDR
It is shown how special treatment of the conformal factor can lead to more accurate evolution, and techniques to handle black hole spacetimes in the absence of symmetries are discussed, which can prevent the development of large gradients in the metric functions that result from singularity avoiding time slicings.
Black Hole Excision for Dynamic Black Holes
We extend the previous work on 3D black hole excision to the case of distorted black holes, with a variety of dynamic gauge conditions that respond naturally to the spacetime dynamics. We show that
...
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