Hyperboloidal conformal Penrose diagrams
Author
Alex Vañó-Viñuales
Title
Hyperboloidal conformal Penrose diagrams
Description
Creation of Penrose diagrams of hyperboloidal slices in the Schwarzschild geometry, related to publication https://doi.org/10.1088/1361-6382/ad3aca
Category
Academic Articles & Supplements
Keywords
numerical relativity, conformal Carter-Penrose diagrams, hyperboloidal foliations, Schwarzschild black hole, gravitational collapse
URL
http://www.notebookarchive.org/2024-04-algm73g/
DOI
https://notebookarchive.org/2024-04-algm73g
Date Added
2024-04-23
Date Last Modified
2024-04-23
File Size
40.39 megabytes
Supplements
Rights
CC BY 4.0
Download
Open in Wolfram Cloud
Hyperboloidal conformal Penrose diagrams
Alex Vañó-Viñuales, October 2023.
Supplementary material for Alex Vañó-Viñuales 2024 Class. Quantum Grav. 41 105003 https://doi.org/10.1088/1361-6382/ad3aca (“Conformal diagrams for stationary and dynamical strong-field hyperboloidal slices”).
MaTeX installation - execute only if MaTeX not installed
MaTeX installation - execute only if MaTeX not installed
ResourceFunction["MaTeXInstall"]
Out[]=
Paclet[MaTeX,1.7.6,<>]
Set location of repository (data in compressed files needs to be unzipped)
In[]:=
repolocation="~/other_repos/paper_repos/";
Setup, basic conformal diagrams and data loader
Setup
Setup
Loading MaTeX and its preamble for his notebook
Loading MaTeX and its preamble for his notebook
Various formulas
Various formulas
Initialization
Initialization
Physical domain quantities
Physical domain quantities
Compactified domain quantities
Compactified domain quantities
Values of parameters - trumpets
Values of parameters - trumpets
Height function expression
Height function expression
Loader setup
Loader setup
Covers and lettering for Penrose diagrams
Covers and lettering for Penrose diagrams
Basic construction of conformal Carter-Penrose diagrams
Basic construction of conformal Carter-Penrose diagrams
Minkowski
Minkowski
Equations
Equations
Graphs
Graphs
Schwarzschild
Schwarzschild
Equations
Equations
Graphs
Graphs
Load code data
Load code data
Select code output data
Select code output data
Load data
Load data
Stationary
Minkowski (in terms of the compactified radial coordinate)
Minkowski (in terms of the compactified radial coordinate)
Common definitions and expressions
Common definitions and expressions
In[]:=
minparsubs={aconf->Ω,Ccmc->0,mass->0,Kcmc->Kcmcset,rscri->1};
Equations
Equations
CMC Minkowski slices from closed-form expressions
CMC Minkowski slices from closed-form expressions
Assign data
Assign data
Height function
Height function
Equations
Equations
Graphs
Graphs
Scalar-field perturbed Minkowski slices from numerical data (requires having loaded data with pretag “pentrevol_”)
Scalar-field perturbed Minkowski slices from numerical data (requires having loaded data with pretag “pentrevol_”)
Assign data from code’s output
Assign data from code’s output
Height function
Height function
Equations
Equations
Graphs
Graphs
Final
Final
Schwarzschild (including CMC and non-CMC stationary data)
Schwarzschild (including CMC and non-CMC stationary data)
Select data: evaluate either “Closed form [...]” or “Numerical [...]” subsections and the rest
Select data: evaluate either “Closed form [...]” or “Numerical [...]” subsections and the rest
In[]:=
schwsubs={Q->0,omega,CcmcCcmcset,massmassset,KcmcKcmcset,rscri->1,allvalg[{aconf[r]->lilu[r]}]}//Flatten;
Closed-form expressions for constant-mean-curvature (CMC) slices
Closed-form expressions for constant-mean-curvature (CMC) slices
Numerical (requires having loaded data with pretag “imitateoldoslong_”)
Numerical (requires having loaded data with pretag “imitateoldoslong_”)
Evaluate
Evaluate
Equations
Equations
Constant radius curves for diagrams and other bits (re-evaluate if the location of the trumpet changes or if a different colour is desired)
Constant radius curves for diagrams and other bits (re-evaluate if the location of the trumpet changes or if a different colour is desired)
Construction using the physical radial coordinate (only for closed-form expressions)
Construction using the physical radial coordinate (only for closed-form expressions)
Height function
Height function
Equations
Equations
Graphs
Graphs
Construction integrating for Δh = h - f in physical Kerr-Schild coordinates (only for closed-form expressions)
Construction integrating for Δh = h - f in physical Kerr-Schild coordinates (only for closed-form expressions)
Height function
Height function
Equations
Equations
Graphs
Graphs
Construction using the physical tortoise radial coordinate (only for closed-form expressions) -- in paper’s appendix
Construction using the physical tortoise radial coordinate (only for closed-form expressions) -- in paper’s appendix
Height function
Height function
Equations
Equations
Graphs
Graphs
Construction using the compactified radial coordinate
Construction using the compactified radial coordinate
Height function
Height function
Equations
Equations
Graphs
Graphs
Construction integrating for Δh = h - f in compactified Kerr-Schild coordinates
Construction integrating for Δh = h - f in compactified Kerr-Schild coordinates
Height function
Height function
Equations
Equations
Graphs
Graphs
Graphs (non CMC slices) for “imitateoldoslong” output data
Graphs (non CMC slices) for “imitateoldoslong” output data
Construction using the compactified tortoise radial coordinate -- in paper’s appendix
Construction using the compactified tortoise radial coordinate -- in paper’s appendix
Height function
Height function
Equations
Equations
Graphs (CMC)
Graphs (CMC)
Graphs (non CMC slices) for “imitateoldoslong” output data
Graphs (non CMC slices) for “imitateoldoslong” output data
Dynamical
Trumpet relaxation
Trumpet relaxation
Load data
Load data
Evolution plots
Evolution plots
Static plots
Static plots
Comparison between initial CMC and final stationary slices
Comparison between initial CMC and final stationary slices
Comparison between height-function calculated (using tortoise coordinate) and eikonal evolved non-CMC stationary data
Comparison between height-function calculated (using tortoise coordinate) and eikonal evolved non-CMC stationary data
Comparison between height-function calculated (using Δh + f height function integration) and eikonal evolved non-CMC stationary data
Comparison between height-function calculated (using Δh + f height function integration) and eikonal evolved non-CMC stationary data
Collapse
Collapse
Load data
Load data
Static plots
Static plots
Evolution plots
Evolution plots
Appendix A -- translating metric quantities from conformally compactified domain to physical one
Cite this as: Alex Vañó-Viñuales, "Hyperboloidal conformal Penrose diagrams" from the Notebook Archive (2024), https://notebookarchive.org/2024-04-algm73g
Download
