Finding Apparent Horizons in Numerical Relativity

Jonathan Thornburg

RECORD

Finding Apparent Horizons in Numerical Relativity

Jonathan Thornburg

General Relativity and Quantum Cosmology · Submitted 7 Aug 1995

abstract compiled · 2026-09-23 · no language-model review

The authors' main claim We find this symbolic differentiation method of computing the $H(h)$ Jacobian to be {\em much} more efficient than the usual numerical perturbation method, and also much easier to implement than is commonly thought.

how it is stated · asserted 1.00 read the record →

Comment (0) Read on arXiv ↗

Cite ▾

Claim paper

The compiled record

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.

Open the compiled record

Compiled by classifier-1.13-20260917 via OpenRouter, question bank abstract-p1-20260923, 2026-09-23: 1 request, 7,262 input tokens, $0.0003 reported.

What the abstract itself states, as probabilities: claims first 0.03 · states limitation 0.93 · human participants 0.01 · controlled comparison 0.16 · randomized assignment 0.01 · states quantitative result 0.97 · compares with prior results 0.90 · mentions code or data availability 0.02.

The abstract, sentence by sentence

arXiv listing abstract v4. Results and the main claim are underlined; show every role.

This paper presents a detailed discussion of the ``Newton's method'' algorithm for finding apparent horizons in 3+1 numerical relativity. We describe a method for computing the Jacobian matrix of the finite differenced $H(h)$ function by symbolically differentiating the finite difference equations, giving the Jacobian elements directly in terms of the finite difference molecule coefficients used in computing $H(h)$. Assuming the finite differencing scheme commutes with linearization, we show how the Jacobian elements may be computed by first linearizing the continuum $H(h)$ equations, then finite differencing the linearized (continuum) equations. We find this symbolic differentiation method of computing the $H(h)$ Jacobian to be {\em much} more efficient than the usual numerical perturbation method, and also much easier to implement than is commonly thought. When solving the discrete $H(h) = 0$ equations, we find that Newton's method generally converges very rapidly. However, if the initial guess for the horizon position contains significant high-spatial-frequency error components, Newton's method has a small (poor) radius of convergence. This is {\em not} an artifact of insufficient resolution in the finite difference grid; rather, it appears to be caused by a strong nonlinearity in the continuum $H(h)$ function for high-spatial-frequency error components in $h$. Robust variants of Newton's method can boost the radius of convergence by O(1) factors, but the underlying nonlinearity remains, and appears to worsen rapidly with increasing initial-guess-error spatial frequency. Using 4th~order finite differencing, we find typical accuracies for computed horizon positions in the $10^{-5}$ range for $\Delta\theta = \frac{\pi/2}{50}$.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#	sentence	role	the authors' own work	states the main conclusion	strength
s0	This paper presents a detailed discussion of the ``Newton's method'' algorithm for finding apparent horizons in 3+1 numerical relativity.	contribution 0.99	0.68	0.08	asserted 0.97
s1	We describe a method for computing the Jacobian matrix of the finite differenced $H(h)$ function by symbolically differentiating the finite difference equations, giving the Jacobian elements directly in terms of the finite difference molecule coefficients used in computing $H(h)$.	method 0.86	0.93	0.06	asserted 1.00
s2	Assuming the finite differencing scheme commutes with linearization, we show how the Jacobian elements may be computed by first linearizing the continuum $H(h)$ equations, then finite differencing the linearized (continuum) equations.	method 0.94	0.79	0.05	asserted 0.71
s3	We find this symbolic differentiation method of computing the $H(h)$ Jacobian to be {\em much} more efficient than the usual numerical perturbation method, and also much easier to implement than is commonly thought.	result 0.79	0.92	0.32	asserted 1.00
s4	When solving the discrete $H(h) = 0$ equations, we find that Newton's method generally converges very rapidly.	result 1.00	0.93	0.17	asserted 1.00
s5	However, if the initial guess for the horizon position contains significant high-spatial-frequency error components, Newton's method has a small (poor) radius of convergence.	limitation 0.70	0.91	0.09	asserted 0.98
s6	This is {\em not} an artifact of insufficient resolution in the finite difference grid; rather, it appears to be caused by a strong nonlinearity in the continuum $H(h)$ function for high-spatial-frequency error components in $h$.	interpretation 0.96	0.86	0.13	hedged possibility 0.51
s7	Robust variants of Newton's method can boost the radius of convergence by O(1) factors, but the underlying nonlinearity remains, and appears to worsen rapidly with increasing initial-guess-error spatial frequency.	result 0.62	0.76	0.17	asserted 0.75
s8	Using 4th~order finite differencing, we find typical accuracies for computed horizon positions in the $10^{-5}$ range for $\Delta\theta = \frac{\pi/2}{50}$.	result 1.00	0.96	0.08	asserted 1.00

What kind of paper the classifier reports this is: method or tool 1.00 · evidence from simulation or computation 0.93.

What was examined, and what was not

sources read

arXiv listing abstract v4

not examined

full text
figures and tables
references and citing sentences
relations to other papers

levels run

P abstract

not run

S sections
T body sentences
C citations
R cross-paper relations

Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

The paper

Original abstract

This paper presents a detailed discussion of the ``Newton's method'' algorithm for finding apparent horizons in 3+1 numerical relativity. We describe a method for computing the Jacobian matrix of the finite differenced $H(h)$ function by symbolically differentiating the finite difference equations, giving the Jacobian elements directly in terms of the finite difference molecule coefficients used in computing $H(h)$. Assuming the finite differencing scheme commutes with linearization, we show how the Jacobian elements may be computed by first linearizing the continuum $H(h)$ equations, then finite differencing the linearized (continuum) equations. We find this symbolic differentiation method of computing the $H(h)$ Jacobian to be {\em much} more efficient than the usual numerical perturbation method, and also much easier to implement than is commonly thought. When solving the discrete $H(h) = 0$ equations, we find that Newton's method generally converges very rapidly. However, if the initial guess for the horizon position contains significant high-spatial-frequency error components, Newton's method has a small (poor) radius of convergence. This is {\em not} an artifact of insufficient resolution in the finite difference grid; rather, it appears to be caused by a strong nonlinearity in the continuum $H(h)$ function for high-spatial-frequency error components in $h$. Robust variants of Newton's method can boost the radius of convergence by O(1) factors, but the underlying nonlinearity remains, and appears to worsen rapidly with increasing initial-guess-error spatial frequency. Using 4th~order finite differencing, we find typical accuracies for computed horizon positions in the $10^{-5}$ range for $\Delta\theta = \frac{\pi/2}{50}$.

Human review

How Pith checks papers → · Full audit of this paper →

References

Forward citations

Cited by 7 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

Dipoles and chains of solitons in the Friedberg-Lee-Sirlin model
gr-qc 2024-11 conditional novelty 7.0 of 10

Stable dipolar boson stars and flat-space Q-ball chains are constructed in the Friedberg-Lee-Sirlin model, and chains beyond the dipole are found unstable in tested cases.
Nonlinear interactions between black holes and Proca fields
gr-qc 2015-05 conditional novelty 6.0 of 10

Fully nonlinear 3+1D simulations of a massive vector field around a nonrotating black hole produce long-lived quasi-bound Proca condensates whose frequencies and late-time tails match perturbative predictions.
GRChombo : Numerical Relativity with Adaptive Mesh Refinement
gr-qc 2015-03 conditional novelty 6.0 of 10

A new numerical relativity code, GRChombo, couples BSSN/CCZ4 evolution to Berger-Rigoutsos block-structured adaptive mesh refinement and is shown to stably evolve black holes, mergers and scalar collapse.
Error-analysis and comparison to analytical models of numerical waveforms produced by the NRAR Collaboration
gr-qc 2013-07 conditional novelty 6.0 of 10

A cross-code audit of 25 numerical black-hole merger waveforms finds that the effective-one-body and phenomenological template families used in LIGO/Virgo searches recover them with over 99% overlap at total masses of...
Introduction to dynamical horizons in numerical relativity
gr-qc 2006-04 conditional novelty 6.0 of 10

Dynamical horizon diagnostics are implemented in three numerical spacetimes, showing how black holes grow, settle to Kerr, and form inner timelike membranes.

Show 2 more citing papers

Black hole hair formation in shift-symmetric generalised scalar-tensor gravity
gr-qc 2016-10 accept novelty 4.0 of 10

A scalar linearly coupled to the Gauss-Bonnet invariant, evolved on Schwarzschild and collapsing-dust backgrounds, relaxes to the known static hairy-black-hole profile for every initial condition tested.
Gravitational waves from neutron star mergers and their relation to the nuclear equation of state
astro-ph.HE 2019-07 accept novelty 2.0 of 10

A review arguing that gravitational-wave observations of neutron-star mergers, especially GW170817, already constrain the supranuclear equation of state, favoring a relatively soft equation of state and excluding seve...

How later papers used it

46 later papers cite it

Evidence: some. 46 of the citing papers share no author with it, and 43 of those carry a confident label from Pith's citation classifier.

Pith read 4 of the 54 citing sentences, choosing only those whose citation markers point at this paper alone, and wrote the paraphrases from them. The summary, the counts, the evidence level, the links and the quotes come from code; the summary counts the papers listed below. A later paper is listed under a label when Pith's citation classifier gives one of its sentences that label at probability 0.8 or more (0.95 for support), Pith's reading finds that the citing paper itself did it, and a second check confirms the paraphrase against the sentence. This section lists uses only. It does not list criticism or disagreement, so their absence here says nothing about how the paper has been received. Papers that share an author with this one are listed apart, under "By the same authors". A quote is the citing authors' own sentence, shown when their paper's licence is CC BY, CC BY-SA or CC0. JSON

1 later paper by other authors is listed below: 1 used its method or tool.

Uses its method or tool 1 of 41 papers with this label

Detected a black hole forming near the origin shortly after the chain's components merged using the thorn from this paper.

Dipoles and chains of solitons in the Friedberg-Lee-Sirlin model (2024) · 0.96 · paraphrased: the licence does not allow quoting

How the literature uses it

41 arXiv papers cite this in 47 sentences

Each citing sentence is taken from the citing paper's LaTeX source and typed by Pith's classifier. A label appears at probability 0.8 or more (0.95 for support), a contest at 0.7 or more; the quoted words are the citing authors'. How each label was checked · JSON

Papers citing it · 4 sentences without a confident label

Builds on it 1 paper

This is an extension of the widely used method developed in [ref] .

Horizons in a binary black hole merger I: Geometry and area increase · The optical scalars and marginally trapped surfaces · 0.81

Uses its method or tool 37 papers

We monitored the evolving system for the formation of an apparent horizon, a null trapped surface which implies the existence of an event horizon at some larger radius [ref] .

The evolution of conifolds · Apparent horizons · 0.87
It employs the BSSN formulation of the Einstein equations [ref] , uses the moving puncture method [ref] , the Carpet package for mesh refinement [ref] , the spectral solver described in [ref] for 3+1 initial data and Thornburg's AHFinderDirect [ref] .

Numerical relativity for D dimensional axially symmetric space-times: formalism and code tests · The numerical treatment · 0.99
The various horizons are located using the method described in [ref] .

The Shear at the Common Dynamical Horizon in Binary Black Hole Mergers and its Imprint in their Gravitational Radiation · Numerical Simulations · 1.00

Discussion

Know something about this paper? Leave a comment, or highlight any passage above to comment on it.