ABSTRACT (original article): We study black hole linear perturbation theory in a four-dimensional Schwarzschild (anti) de Sitter background. When dealing with a positive cosmological constant, the corresponding spectral problem is solved systematically via the Nekrasov-Shatashvili functions or, equivalently, classical Virasoro conformal blocks. However, this approach can be more complicated to implement for certain perturbations if the cosmological constant is negative. For these cases, we propose an alternative method to set up perturbation theory for both small and large black holes in an analytical manner. Our analysis reveals a new underlying recursive structure that involves multiple polylogarithms. We focus on gravitational, electromagnetic, and conformally coupled scalar perturbations subject to Dirichlet and Robin boundary conditions. The low-lying modes of the scalar sector of gravitational perturbations and its hydrodynamic limit are studied in detail. CITATION (original article): Aminov, G., Arnaudo, P., Bonelli, G. et al. Black hole perturbation theory and multiple polylogarithms. J. High Energ. Phys. 2023, 59 (2023). https://doi.org/10.1007/JHEP11(2023)059
Full list of needed files: https://github.com/GlebAminov/BH_PolyLog
Mathematica Program to analytically compute (A)dS QNM frequencies.
To compute the AdS QNM frequencies with negative real part - use “AdS General n,l,s negative”.
Inside the folder, there is the “Executable files” subfolder. To proceed with the computation, the following steps are recommended:
1. Copy all the files from the “Executable files” subfolder into a new folder (after downloading the program to your computer).
2. Open the “Nf4_exe.nb” file - it will run all other files for you (except the “SW_to_Heun.nb” file, see step 9).
3. Specify your choice of the quantum numbers n0, l, s using “nlSub” (remember that n>=0, l>=s>=0). For example, the choice n0=0, l=s=1 is given by nlSub={n0 -> 0, l -> 1, s -> 1}.
4. Specify the maximum order “tMax” to which the wave functions will be computed. The range of values for “tMax” is between 1 and 9.
5. The variable “MaxKernels” controls the number of kernels used during the ParallelTable computation. By default, it is set to one. You can increase it to maximize the efficiency for higher values of “tMax”.
6. After you defined all three variables (“nlSub”, “tMax”, “MaxKernels”), start executing each cell in the “Nf4_exe.nb” file step by step. You can Shift-select all the cells and press Shift-Enter to put them in a queue.
7. After each cell is executed, the computation log will be generated as an output. The logs generated after cells (Wave function in the Left Region) and (Wave function in the Right Region) can be memory-consuming, so feel free to delete them after the computation is done.
8. The final cell of the “Nf4_exe.nb” file is (QNM frequency computation). Here you can also choose the numerical value of the BH radius, R1. The final result for the QNM frequency will be in this last computation log. The other Mathematica files called from the “Nf4_exe.nb” file do not hold relevant results.
9. To convert the wave functions from the SW form to Heun one, run the “SW_to_Heun.nb” file with the appropriate definitions of “nlSub” and “tMax”.

Program

Requirements :
n>=0
,
l>=s>=0
SPack=NotebookDirectory[];​​nlSub={n0->0,l1,s1};​​tMax=7;​​MaxKernels=4;​​​​qMax=2tMax;​​If[qMax>18,​​qMax=18;​​];​​​​WFLnls=StringJoin["WFN4zl",ToString[n0/.nlSub],"_",ToString[l/.nlSub],"_",ToString[s/.nlSub],".m"];​​WFRnls=StringJoin["WFN4zr",ToString[n0/.nlSub],"_",ToString[l/.nlSub],"_",ToString[s/.nlSub],".m"];​​​​Print["n,l,s Defined"]
n,l,s Defined
Wave function in the Left Region
CFL=CreateFile[File[StringJoin[SPack,WFLnls]]];​​​​IfToString[CFL]=="$Failed",​​Print["WFL was already computed. Delete the following file to compute again: ",WFLnls],​​Bψl=0;​​qj=0;​​WhileBψl==0&&qj<=Floor
qMax
2
,​​qNum=2qj;​​FileToEvaluateL=NotebookOpen[StringJoin[SPack,"Nf4_left.nb"]];​​SelectionMove[FileToEvaluateL,All,Notebook];​​NotebookEvaluate[FileToEvaluateL];​​qj++;;​​;
Begin Evaluation for qNum = 0
Dictionary Check: {0,
2
O[q]
,0}
Log Functions: 0
ψl0 Saved
Begin Evaluation for qNum = 2
Dictionary Check: {
2
q
+
3
O[q]
,
3
O[q]
,
3
O[q]
}
Log Functions: 2
Wave function:
1
2
(2-z)
3/2
z
+
2
q
1
2
(2-z)
3/2
z
fl2[z]+
(1-2z)(fl1[z]+
c
1
)
3/2
z
Equation to Solve:
1
2
3/2
z
(-9+7z-4z
μ
2
+3
2
z
μ
2
)+
2(1-2z+2
2
z
)
′
fl1
[z]
z
-
5/2
z
(4-5z+2
2
z
)
′
fl2
[z]-(-1+
z
)(1+
z
)
z
(-1+2z)
′′
fl1
[z]-
1
2
(-1+
z
)(1+
z
)(-2+z)
7/2
z
′′
fl2
[z]
Full Solution Check: 0
CoeffRules for ψl: 3
ψl2[z] =
3
4
(-2+z)
3/2
z
Log[z]-
3
(-1+
z
)
3
(1+1)
1Log[1](3+2
μ
2
)
4
3/2
z
+
15-12z-27
2
z
+24
3
z
+24
c
1
-48z
c
1
-4
μ
2
+20z
μ
2
-18
2
z
μ
2
-2
3
z
μ
2
+4
4
z
μ
2
24
3/2
z
ψl2[z] Saved
Begin Evaluation for qNum = 4
Dictionary Check: {
2
q
+
5
O[q]
,
5
O[q]
,
5
O[q]
}
Log Functions: 3
Wave function:
1
2
(2-z)
3/2
z
+
4
q
1
2
(2-z)
3/2
z
fl2[z]+
(1-2z)(fl1[z]+
c
3
)
3/2
z
+
2
q
ψlz
2
[z]
Equation to Solve: -
3
4
3/2
z
Log[z](-9+7z-4z
μ
2
+3
2
z
μ
2
)+
2
(-1+
z
)
2
1
11(1)(3+3z+3+2
3
z
μ
2
+3
4
z
μ
2
)
4
5/2
z
+
1
1
+1-1-(-1+
z
)(1+
z
)
z
(-1+2z)
′′
fl1
[z]-
1
2
(-1+
z
)(1+
z
)(-2+z)
7/2
z
′′
fl2
[z]
CoefficientRules Length 1: 3
CoefficientRules Length 2: 3
CoefficientRules Check: 0
aVar Defined, Length: 27
CoeffRules for Eq1: 3
Eq1 Length: 26
Separation for Eq1: 8
Solution/Variables 1: {26,26}
Sol1 Check: {
c
1
,
μ
2
,
μ
4
}
bVar Defined, Length: 25
CoeffRules for Eq2: 3
Eq2 Length: 24
Separation for Eq2: 12
Solution/Variables 2: {24,24}
Sol2 Check: {
c
1
,
μ
2
,
μ
4
}
Length of FSolCR: 3
Full Solution Check: {0}
CoeffRules for ψl: 7
ψl4[z] = -
9
16
(-2+z)
3/2
z
2
Log[z]
+
3(-1+2z-4
3
z
+2
4
z
)Log[-1+z]Log[z](3+2
μ
2
)
8
3/2
z
+
3
8
3(3+21)-111-
(-1+
z
)2(34+1)
96
5/2
z
-
-270+63z+34+432
3
z
μ
4
+192
4
z
μ
4
-168
5
z
μ
4
576
5/2
z
ψl4[z] Saved
Begin Evaluation for qNum = 6
Dictionary Check: {
2
q
+
7
O[q]
,
7
O[q]
,
7
O[q]
}
Log Functions: 5
Wave function:
1
2
(2-z)
3/2
z
+
6
q
1
2
(2-z)
3/2
z
fl2[z]+
(1-2z)(fl1[z]+
c
5
)
3/2
z
+
2
q
ψlz
2
[z]+
4
q
ψlz
4
[z]
Equation to Solve:
9
16
3/2
z
2
Log[z]
(-9+7z-4z
μ
2
+3
2
z
μ
2
)-
3
8
3/2
z
PolyLog[2,1-z](3+2
μ
2
)(-9+7z-4z
μ
2
+3
2
z
μ
2
)+11
CoefficientRules Length 1: 7
CoefficientRules Length 2: 7
CoefficientRules Check: 0
aVar Defined, Length: 66
CoeffRules for Eq1: 7
Eq1 Length: 65
Separation for Eq1: 9
Solution/Variables 1: {65,65}
Sol1 Check: {
c
1
,
c
3
,
μ
2
,
μ
4
,
μ
6
}
bVar Defined, Length: 62
CoeffRules for Eq2: 7
Eq2 Length: 61
Separation for Eq2: 10
Solution/Variables 2: {61,61}
Sol2 Check: {
c
1
,
c
3
,
μ
2
,
μ
4
,
μ
6
}
Length of FSolCR: 7
Full Solution Check: {0}
Wave function in the Right Region
BC in the Left Region
Preparing WFs and its Derivatives
Continuity at x = 0
QNM frequency computation
Chose numerical value of R1 (R1N)

CITE THIS NOTEBOOK

Black hole perturbation theory and multiple polylogarithms​
by Gleb Aminov, Paolo Arnaudo, Giulio Bonelli, Alba Grassi and Alessandro Tanzini
Wolfram Community, STAFF PICKS, June 7, 2024
​https://community.wolfram.com/groups/-/m/t/3189403