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Einstein's Equations with Asymptotically Stable Constraint Propagation
General Relativity and Quantum Cosmology · Submitted 4 Sep 1998
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim This feature suggests that this system may be a useful alternative to Einstein's equations when obtaining numerical solutions to full, non-linear gravity.
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The paper's own sentences, arranged. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.
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The abstract, sentence by sentence
We introduce a proposal to modify Einstein's equations by embedding them in a larger symmetric hyperbolic system. The additional dynamical variables of the modified system are essentially first integrals of the original constraints. The extended system of equations reproduces the usual dynamics on the constraint surface of general relativity, and therefore naturally includes the solutions to Einstein gravity. The main feature of this extended system is that, at least for a linearized version of it, the constraint surface is an attractor of the time evolution. This feature suggests that this system may be a useful alternative to Einstein's equations when obtaining numerical solutions to full, non-linear gravity.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | We introduce a proposal to modify Einstein's equations by embedding them in a larger symmetric hyperbolic system. | contribution 0.99 | 0.91 | 0.07 | asserted 0.98 |
| s1 | The additional dynamical variables of the modified system are essentially first integrals of the original constraints. | method 0.72 | 0.72 | 0.06 | asserted 1.00 |
| s2 | The extended system of equations reproduces the usual dynamics on the constraint surface of general relativity, and therefore naturally includes the solutions to Einstein gravity. | result 0.40 | 0.53 | 0.09 | asserted 1.00 |
| s3 | The main feature of this extended system is that, at least for a linearized version of it, the constraint surface is an attractor of the time evolution. | result 0.92 | 0.77 | 0.68 | asserted 0.78 |
| s4 | This feature suggests that this system may be a useful alternative to Einstein's equations when obtaining numerical solutions to full, non-linear gravity. | interpretation 0.96 | 0.24 | 0.45 | hedged possibility 1.00 |
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The paper
Original abstract
We introduce a proposal to modify Einstein's equations by embedding them in a larger symmetric hyperbolic system. The additional dynamical variables of the modified system are essentially first integrals of the original constraints. The extended system of equations reproduces the usual dynamics on the constraint surface of general relativity, and therefore naturally includes the solutions to Einstein gravity. The main feature of this extended system is that, at least for a linearized version of it, the constraint surface is an attractor of the time evolution. This feature suggests that this system may be a useful alternative to Einstein's equations when obtaining numerical solutions to full, non-linear gravity.
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References
How the literature uses it
13 arXiv papers cite this in 14 sentences
Papers citing it · 9 sentences without a confident label
Uses its method or tool 2 papers
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In our implementation we follow Pretorius [ref] by adding constraint damping terms in a fashion inspired by studies of the so-called $$-systems [ref] .
Generalized harmonic formulation in spherical symmetry · Generalized harmonic formulation · 1.00
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This allows the coordinate conditions to settle before incorporating deviations from GR, inducing only smooth constraint violations (which are damped through the now standard use of constraint damping [ref] ) and by-passing the solution of initial data problem within the EFT theory, a task which in itself has received also limited formal and numerical attention.
Modelling self-consistently beyond General Relativity · Introduction · 0.91
Cites it as support 1 paper
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This desirable behavior can be rigorously shown to hold in perturbations off flat spacetime [ref] and also “experimentally” verified in simulations involving black holes and neutron stars (e.g. [ref] ).
Numerical Relativity and Astrophysics · Brief Review of Techniques, Methods and Information Obtainable from Gravitational Waves · 0.99
Cites it as background 2 papers
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Brodbeck, Frittelli, Hübner and Reula [ref] proposed a very general formalism for evolving systems of equations with constraints, known as the $λ$-system, in which the constraint surface forms a set of attractor solutions for the system.
Hypergraph Discretization of the Cauchy Problem in General Relativity via Wolfram Model Evolution · The Z4 Formulation with Constraint Damping · 0.80
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This includes the $λ-$system [ref] and a family of adjusted systems [ref] .
On the Hyperbolicity and Stability of $3+1$ Formulations of Metric $f(R)$ Gravity · Introduction · 0.81
Discussion
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