Schwarzschild BH Perturbations in GR
Author
Sergi Sirera Lahoz
Title
Schwarzschild BH Perturbations in GR
Description
Derivation of Regge-Wheeler and Zerilli equations for a Schwarzschild black hole in GR by applying the perturbations at the level of the action (https://arxiv.org/abs/2301.10272).
Category
Academic Articles & Supplements
Keywords
gravity, gravitational waves, dark energy, general relativity, einstein, horndeski, scalar-tensor, modified gravity, quasinormal modes, LIGO, LISA, black holes, perturbation theory
URL
http://www.notebookarchive.org/2023-08-dcn2fan/
DOI
https://notebookarchive.org/2023-08-dcn2fan
Date Added
2023-08-29
Date Last Modified
2023-08-29
File Size
2.4 megabytes
Supplements
Rights
Redistribution rights reserved



Schwarzschild BH Perturbations in GR
Schwarzschild BH Perturbations in GR
In this notebook we derive the Regge-Wheeler and Zerilli equations for a Schwarzschild black hole in GR by directly varying the action with respect to the odd and even perturbation functions.Note that this method contrasts with the historical way these equations were first derived, where the perturbations were applied at the level of the equations of motion.This notebook has hugely benefitted from resources provided by: - xAct ( http://www.xact.es/ ) - Johannes Noller - Reggie Bernardo’s public git-hub repository ( https://github.com/reggiebernardo/notebooks/tree/main/thesis_notebooks ). - Public notebooks by Vitor Cardoso et al. ( https://centra.tecnico.ulisboa.pt/network/grit/files/ringdown/ ).This notebook follows closely the arguments in section III of Tattersall et. al. ( https://arxiv.org/abs/1711.01992 ) and we provide more details on the calculation along the way. We also point out a typo in that paper.If you use any of the results or techniques in this notebook please cite the repository and [2301.10272] ( https://arxiv.org/abs/2301.10272 ).If you have any comments or feedback on this notebook, please let me know. Thanks!Sergi Sirera Lahozsergi.sirera-lahoz@port.ac.ukInstitute of Cosmology and Gravitation - University of Portsmouth
January 2022
0. Setup and definitions
0. Setup and definitions
1. xCoba setup & quadratic action
1. xCoba setup & quadratic action
2. Quadratic action in components
2. Quadratic action in components
3. Odd sector
3. Odd sector
4. Even sector
4. Even sector
5. Results
5. Results
Here we just quote the final results, i.e. the Regge-Wheeler and Zerilli equations
In[]:=
RWequationZequation
Out[]=
Q[R[r]]-+[R[r]]0
2
ω
B[r](-3r+lr+r+3B[r]r)
2
l
3
r
′′
Q
Out[]=
2
ω
2(2M-r)(9+9Lr+3M+(1+L))Z[r]
3
M
2
M
2
L
2
r
2
L
3
r
4
r
2
(3M+Lr)
′′
Z


Cite this as: Sergi Sirera Lahoz, "Schwarzschild BH Perturbations in GR" from the Notebook Archive (2023), https://notebookarchive.org/2023-08-dcn2fan

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