(* Content-type: application/vnd.wolfram.mathematica *) (*** Wolfram Notebook File ***) (* http://www.wolfram.com/nb *) (* CreatedBy='Mathematica 12.1' *) (*CacheID: 234*) (* Internal cache information: NotebookFileLineBreakTest NotebookFileLineBreakTest NotebookDataPosition[ 158, 7] NotebookDataLength[ 13913658, 233387] NotebookOptionsPosition[ 13874226, 232732] NotebookOutlinePosition[ 13874690, 232750] CellTagsIndexPosition[ 13874647, 232747] WindowFrame->Normal*) (* Beginning of Notebook Content *) Notebook[{ Cell[CellGroupData[{ Cell["Getting Started with the Black Hole Perturbation Toolkit", "Title", CellChangeTimes->{{3.8081341673855886`*^9, 3.808134175789938*^9}, { 3.808134635488454*^9, 3.808134636515626*^9}},ExpressionUUID->"a3e637ce-c5b3-40e2-bf7e-\ 81ca9e367328"], Cell["\<\ In this tutorial, we will give an overview of some of the Mathematica \ components of the Black Hole Perturbation Toolkit.\ \>", "Text", CellChangeTimes->{{3.808134643912387*^9, 3.8081346640647583`*^9}, { 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$CellContext`\[Lambda]$] + 1.2721305490496584`*^-18 Sin[54.63755787635592 $CellContext`\[Lambda]$] + 9.445597616136064*^-6 Sin[58.28006173477965 $CellContext`\[Lambda]$] - 6.122553444624025*^-18 Sin[61.92256559320338 $CellContext`\[Lambda]$] + 2.795732377637535*^-6 Sin[65.56506945162711 $CellContext`\[Lambda]$] - 2.6112153375229833`*^-17 Sin[69.20757331005083 $CellContext`\[Lambda]$] + 8.378316908212204*^-7 Sin[72.85007716847457 $CellContext`\[Lambda]$] + 2.4864369822334233`*^-17 Sin[76.49258102689829 $CellContext`\[Lambda]$] + 2.536195783646744*^-7 Sin[80.13508488532202 $CellContext`\[Lambda]$] + 3.077248600863206*^-17 Sin[83.77758874374575 $CellContext`\[Lambda]$] + 7.741282092412997*^-8 Sin[87.42009260216948 $CellContext`\[Lambda]$] + 4.579669976578771*^-17 Sin[91.0625964605932 $CellContext`\[Lambda]$] + 2.379411001269346*^-8 Sin[94.70510031901694 $CellContext`\[Lambda]$] + 5.525415516074274*^-18 Sin[98.34760417744066 $CellContext`\[Lambda]$] + 7.357045818507964*^-9 Sin[101.99010803586438` $CellContext`\[Lambda]$] - 2.443567379091468*^-17 Sin[105.63261189428812` $CellContext`\[Lambda]$] + 2.286435280170196*^-9 Sin[109.27511575271184` $CellContext`\[Lambda]$] - 2.9098587338965794`*^-18 Sin[112.91761961113558` $CellContext`\[Lambda]$] + 7.13754510275064*^-10 Sin[116.5601234695593 $CellContext`\[Lambda]$] - 2.1500057626500428`*^-17 Sin[120.20262732798302` $CellContext`\[Lambda]$] + 2.2368588499542132`*^-10 Sin[123.84513118640676` $CellContext`\[Lambda]$] - 4.658971621194853*^-18 Sin[127.48763504483048` $CellContext`\[Lambda]$] + 7.034506879391753*^-11 Sin[131.13013890325422` $CellContext`\[Lambda]$] - 3.1178138149312886`*^-18 Sin[134.77264276167793` $CellContext`\[Lambda]$] + 2.219070297124788*^-11 Sin[138.41514662010167` $CellContext`\[Lambda]$] + 3.1803263726241462`*^-18 Sin[142.0576504785254 $CellContext`\[Lambda]$] + 7.0196164527781825`*^-12 Sin[145.70015433694914` $CellContext`\[Lambda]$] + 1.0154478883766677`*^-18 Sin[149.34265819537285` $CellContext`\[Lambda]$] + 2.22607435956812*^-12 Sin[152.98516205379659` $CellContext`\[Lambda]$] - 1.3050768476244119`*^-17 Sin[156.62766591222032` $CellContext`\[Lambda]$] + 7.075581243069153*^-13 Sin[160.27016977064403` $CellContext`\[Lambda]$] - 3.4867941867133825`*^-17 Sin[163.91267362906777` $CellContext`\[Lambda]$] + 2.253521946344003*^-13 Sin[167.5551774874915 $CellContext`\[Lambda]$] - 3.2129293315698097`*^-17 Sin[171.1976813459152 $CellContext`\[Lambda]$] + 7.192533063640297*^-14 Sin[174.84018520433895` $CellContext`\[Lambda]$] - 4.887317874665269*^-17 Sin[178.4826890627627 $CellContext`\[Lambda]$] + 2.2989578990495474`*^-14 Sin[182.1251929211864 $CellContext`\[Lambda]$] - 9.762071325594972*^-18 Sin[185.76769677961013` $CellContext`\[Lambda]$] + 7.364126335972755*^-15 Sin[189.41020063803387` $CellContext`\[Lambda]$] + 2.76573837207623*^-18 Sin[193.05270449645758` $CellContext`\[Lambda]$] + 2.3691985873153243`*^-15 Sin[196.69520835488132` $CellContext`\[Lambda]$] + 2.2456783907190666`*^-17 Sin[200.33771221330505` $CellContext`\[Lambda]$] + 7.593752016088473*^-16 Sin[203.98021607172876` $CellContext`\[Lambda]$] + 1.7636355339097538`*^-17 Sin[207.6227199301525 $CellContext`\[Lambda]$] + 2.4067792777905804`*^-16 Sin[211.26522378857624` $CellContext`\[Lambda]$] + 3.3136158600438335`*^-17 Sin[214.90772764699997` $CellContext`\[Lambda]$] + 6.406912037941007*^-17 Sin[218.55023150542368` $CellContext`\[Lambda]$] - 3.21350414402261*^-18 Sin[222.19273536384742` $CellContext`\[Lambda]$] + 8.393823270855518*^-18 Sin[225.83523922227116` $CellContext`\[Lambda]$]), Listable]}, "TrajectoryDeltas" -> <| "\[CapitalDelta]tr" -> Function[{$CellContext`phase$}, 2 ((-28.945907668791314`) Sin[$CellContext`phase$] + 5.959675895805969 Sin[2 $CellContext`phase$] - 1.5068593308898812` Sin[3 $CellContext`phase$] + 0.40840360269759374` Sin[4 $CellContext`phase$] - 0.11430744964470137` Sin[5 $CellContext`phase$] + 0.03256195272165735 Sin[6 $CellContext`phase$] - 0.00937565239084275 Sin[7 $CellContext`phase$] + 0.0027185610707425034` Sin[8 $CellContext`phase$] - 0.0007921012010424376 Sin[9 $CellContext`phase$] + 0.00023159905366765265` Sin[10 $CellContext`phase$] - 0.00006789216462367197 Sin[11 $CellContext`phase$] + 0.00001994176776309864 Sin[12 $CellContext`phase$] - 5.866521117661514*^-6 Sin[13 $CellContext`phase$] + 1.7279596858456512`*^-6 Sin[14 $CellContext`phase$] - 5.094720984940201*^-7 Sin[15 $CellContext`phase$] + 1.5033611096282318`*^-7 Sin[16 $CellContext`phase$] - 4.439173755385769*^-8 Sin[17 $CellContext`phase$] + 1.3115642084739044`*^-8 Sin[18 $CellContext`phase$] - 3.876929175355688*^-9 Sin[19 $CellContext`phase$] + 1.1464788507442593`*^-9 Sin[20 $CellContext`phase$] - 3.3915497923314455`*^-10 Sin[21 $CellContext`phase$] + 1.0036036648195726`*^-10 Sin[22 $CellContext`phase$] - 2.970471212000061*^-11 Sin[23 $CellContext`phase$] + 8.795325578958568*^-12 Sin[24 $CellContext`phase$] - 2.6073987413610667`*^-12 Sin[25 $CellContext`phase$] + 7.719722297303214*^-13 Sin[26 $CellContext`phase$] - 2.253588262503938*^-13 Sin[27 $CellContext`phase$] + 6.457532921087607*^-14 Sin[28 $CellContext`phase$] - 1.930256720744841*^-14 Sin[29 $CellContext`phase$] + 6.024810280299183*^-15 Sin[30 $CellContext`phase$] - 3.37364544902225*^-15 Sin[31 $CellContext`phase$] + 9.332812300755222*^-16 Sin[32 $CellContext`phase$]), Listable], "\[CapitalDelta]t\[Theta]" -> Function[{$CellContext`phase$}, 2 (0.020111842241185066` Sin[2 $CellContext`phase$] - 1.1564823173178713`*^-18 Sin[3 $CellContext`phase$] - 3.990791292517453*^-6 Sin[4 $CellContext`phase$] + 2.081668171172169*^-18 Sin[5 $CellContext`phase$] + 7.918924338605933*^-10 Sin[6 $CellContext`phase$] + 2.106449935114694*^-18 Sin[7 $CellContext`phase$] - 1.5712387987920629`*^-13 Sin[8 $CellContext`phase$] + 4.9632366118225315`*^-18 Sin[9 $CellContext`phase$] + 3.278627369596166*^-17 Sin[10 $CellContext`phase$] + 4.100255488672453*^-18 Sin[11 $CellContext`phase$] + 9.577119190288622*^-18 Sin[12 $CellContext`phase$]), Listable], "\[CapitalDelta]\[Phi]r" -> Function[{$CellContext`phase$}, 2 (0.020785138483086166` Sin[$CellContext`phase$] + 0.00002716072032776123 Sin[2 $CellContext`phase$] + 5.348859015976912*^-7 Sin[3 $CellContext`phase$] - 5.026419714856175*^-9 Sin[4 $CellContext`phase$] - 3.64184377099619*^-14 Sin[5 $CellContext`phase$] - 2.4399673543192463`*^-13 Sin[6 $CellContext`phase$] + 1.398682756916157*^-15 Sin[7 $CellContext`phase$] + 1.870248747537495*^-18 Sin[8 $CellContext`phase$] - 2.6502719771867884`*^-18 Sin[9 $CellContext`phase$] - 1.7347234759768072`*^-19 Sin[10 $CellContext`phase$]), Listable], "\[CapitalDelta]\[Phi]\[Theta]" -> Function[{$CellContext`phase$}, 2 ((-1.3877787807814457`*^-16) Sin[$CellContext`phase$] + 0.16649015020429747` Sin[2 $CellContext`phase$] - 4.163336342344337*^-17 Sin[3 $CellContext`phase$] + 0.02771898761942354 Sin[4 $CellContext`phase$] - 2.151057110211241*^-17 Sin[5 $CellContext`phase$] + 0.0061532531592311535` Sin[6 $CellContext`phase$] - 1.3382152528963939`*^-17 Sin[7 $CellContext`phase$] + 0.001536684549291882 Sin[8 $CellContext`phase$] - 6.1679056923619804`*^-18 Sin[9 $CellContext`phase$] + 0.00040934867553583145` Sin[10 $CellContext`phase$] + 4.100255488672453*^-18 Sin[11 $CellContext`phase$] + 0.00011358757332477869` Sin[12 $CellContext`phase$] - 9.474258984181024*^-18 Sin[13 $CellContext`phase$] + 0.00003241923105486576 Sin[14 $CellContext`phase$] + 1.2721305490496584`*^-18 Sin[15 $CellContext`phase$] + 9.445597616136064*^-6 Sin[16 $CellContext`phase$] - 6.122553444624025*^-18 Sin[17 $CellContext`phase$] + 2.795732377637535*^-6 Sin[18 $CellContext`phase$] - 2.6112153375229833`*^-17 Sin[19 $CellContext`phase$] + 8.378316908212204*^-7 Sin[20 $CellContext`phase$] + 2.4864369822334233`*^-17 Sin[21 $CellContext`phase$] + 2.536195783646744*^-7 Sin[22 $CellContext`phase$] + 3.077248600863206*^-17 Sin[23 $CellContext`phase$] + 7.741282092412997*^-8 Sin[24 $CellContext`phase$] + 4.579669976578771*^-17 Sin[25 $CellContext`phase$] + 2.379411001269346*^-8 Sin[26 $CellContext`phase$] + 5.525415516074274*^-18 Sin[27 $CellContext`phase$] + 7.357045818507964*^-9 Sin[28 $CellContext`phase$] - 2.443567379091468*^-17 Sin[29 $CellContext`phase$] + 2.286435280170196*^-9 Sin[30 $CellContext`phase$] - 2.9098587338965794`*^-18 Sin[31 $CellContext`phase$] + 7.13754510275064*^-10 Sin[32 $CellContext`phase$] - 2.1500057626500428`*^-17 Sin[33 $CellContext`phase$] + 2.2368588499542132`*^-10 Sin[34 $CellContext`phase$] - 4.658971621194853*^-18 Sin[35 $CellContext`phase$] + 7.034506879391753*^-11 Sin[36 $CellContext`phase$] - 3.1178138149312886`*^-18 Sin[37 $CellContext`phase$] + 2.219070297124788*^-11 Sin[38 $CellContext`phase$] + 3.1803263726241462`*^-18 Sin[39 $CellContext`phase$] + 7.0196164527781825`*^-12 Sin[40 $CellContext`phase$] + 1.0154478883766677`*^-18 Sin[41 $CellContext`phase$] + 2.22607435956812*^-12 Sin[42 $CellContext`phase$] - 1.3050768476244119`*^-17 Sin[43 $CellContext`phase$] + 7.075581243069153*^-13 Sin[44 $CellContext`phase$] - 3.4867941867133825`*^-17 Sin[45 $CellContext`phase$] + 2.253521946344003*^-13 Sin[46 $CellContext`phase$] - 3.2129293315698097`*^-17 Sin[47 $CellContext`phase$] + 7.192533063640297*^-14 Sin[48 $CellContext`phase$] - 4.887317874665269*^-17 Sin[49 $CellContext`phase$] + 2.2989578990495474`*^-14 Sin[50 $CellContext`phase$] - 9.762071325594972*^-18 Sin[51 $CellContext`phase$] + 7.364126335972755*^-15 Sin[52 $CellContext`phase$] + 2.76573837207623*^-18 Sin[53 $CellContext`phase$] + 2.3691985873153243`*^-15 Sin[54 $CellContext`phase$] + 2.2456783907190666`*^-17 Sin[55 $CellContext`phase$] + 7.593752016088473*^-16 Sin[56 $CellContext`phase$] + 1.7636355339097538`*^-17 Sin[57 $CellContext`phase$] + 2.4067792777905804`*^-16 Sin[58 $CellContext`phase$] + 3.3136158600438335`*^-17 Sin[59 $CellContext`phase$] + 6.406912037941007*^-17 Sin[60 $CellContext`phase$] - 3.21350414402261*^-18 Sin[61 $CellContext`phase$] + 8.393823270855518*^-18 Sin[62 $CellContext`phase$]), Listable]|>, "FourVelocity" -> { Function[{$CellContext`\[Lambda]$}, ((-18.353406718477792` + 13.333333333333334` JacobiSN[ 1.3972501102329256` $CellContext`\[Lambda]$, 0.13215572516370697`]^2)^(-2) (-122.35604478985194` + 21.95457708696275 JacobiSN[ 1.3972501102329256` $CellContext`\[Lambda]$, 0.13215572516370697`]^2)^2 + 0.6074999999999999 JacobiSN[ 1.0007938772180671` (Rational[1, 2] Pi + 3.642503858423728 $CellContext`\[Lambda]$), 0.003169844448933532]^2)^(-1) ( 1.6369577299778217` + ( 0.81 + (-18.353406718477792` + 13.333333333333334` JacobiSN[ 1.3972501102329256` $CellContext`\[Lambda]$, 0.13215572516370697`]^2)^(-2) (-122.35604478985194` + 21.95457708696275 JacobiSN[ 1.3972501102329256` $CellContext`\[Lambda]$, 0.13215572516370697`]^2)^2) (0.81 - 2 (-18.353406718477792` + 13.333333333333334` JacobiSN[ 1.3972501102329256` $CellContext`\[Lambda]$, 0.13215572516370697`]^2)^(-1) (-122.35604478985194` + 21.95457708696275 JacobiSN[ 1.3972501102329256` $CellContext`\[Lambda]$, 0.13215572516370697`]^2) + (-18.353406718477792` + 13.333333333333334` JacobiSN[ 1.3972501102329256` $CellContext`\[Lambda]$, 0.13215572516370697`]^2)^(-2) (-122.35604478985194` + 21.95457708696275 JacobiSN[ 1.3972501102329256` $CellContext`\[Lambda]$, 0.13215572516370697`]^2)^2)^(-1) (-1.6369577299778217` + 0.9647074451613549 ( 0.81 + (-18.353406718477792` + 13.333333333333334` JacobiSN[ 1.3972501102329256` $CellContext`\[Lambda]$, 0.13215572516370697`]^2)^(-2) (-122.35604478985194` + 21.95457708696275 JacobiSN[ 1.3972501102329256` $CellContext`\[Lambda]$, 0.13215572516370697`]^2)^2)) - 0.7814130305806974 (1 - 0.7499999999999999 JacobiSN[ 1.0007938772180671` (Rational[1, 2] Pi + 3.642503858423728 $CellContext`\[Lambda]$), 0.003169844448933532]^2)), Listable], Function[{$CellContext`\[Lambda]$}, 3432.96708545474 JacobiCN[ 1.3972501102329253` $CellContext`\[Lambda]$, 0.13215572516370697`] JacobiDN[ 1.3972501102329253` $CellContext`\[Lambda]$, 0.13215572516370697`] JacobiSN[ 1.3972501102329253` $CellContext`\[Lambda]$, 0.13215572516370697`] ((-18.353406718477792` + 13.333333333333334` JacobiSN[ 1.3972501102329253` $CellContext`\[Lambda]$, 0.13215572516370697`]^2)^(-2)/((-18.353406718477792` + 13.333333333333334` JacobiSN[ 1.3972501102329256` $CellContext`\[Lambda]$, 0.13215572516370697`]^2)^(-2) (-122.35604478985194` + 21.95457708696275 JacobiSN[ 1.3972501102329256` $CellContext`\[Lambda]$, 0.13215572516370697`]^2)^2 + 0.6074999999999999 JacobiSN[ 1.0007938772180671` (Rational[1, 2] Pi + 3.642503858423728 $CellContext`\[Lambda]$), 0.003169844448933532]^2)), Listable], Function[{$CellContext`\[Lambda]$}, (-3.1570051611566434`) JacobiCN[ 1.0007938772180671` (Rational[1, 2] Pi + 3.642503858423728 $CellContext`\[Lambda]$), 0.003169844448933532] JacobiDN[ 1.0007938772180671` (Rational[1, 2] Pi + 3.642503858423728 $CellContext`\[Lambda]$), 0.003169844448933532] ((1 - 0.7499999999999999 JacobiSN[ 1.0007938772180671` (Rational[1, 2] Pi + 3.642503858423728 $CellContext`\[Lambda]$), 0.003169844448933532]^2)^ Rational[-1, 2]/((-18.353406718477792` + 13.333333333333334` JacobiSN[ 1.3972501102329256` $CellContext`\[Lambda]$, 0.13215572516370697`]^2)^(-2) (-122.35604478985194` + 21.95457708696275 JacobiSN[ 1.3972501102329256` $CellContext`\[Lambda]$, 0.13215572516370697`]^2)^2 + 0.6074999999999999 JacobiSN[ 1.0007938772180671` (Rational[1, 2] Pi + 3.642503858423728 $CellContext`\[Lambda]$), 0.003169844448933532]^2)), Listable], Function[{$CellContext`\[Lambda]$}, ((-18.353406718477792` + 13.333333333333334` JacobiSN[ 1.3972501102329256` $CellContext`\[Lambda]$, 0.13215572516370697`]^2)^(-2) (-122.35604478985194` + 21.95457708696275 JacobiSN[ 1.3972501102329256` $CellContext`\[Lambda]$, 0.13215572516370697`]^2)^2 + 0.6074999999999999 JacobiSN[ 1.0007938772180671` (Rational[1, 2] Pi + 3.642503858423728 $CellContext`\[Lambda]$), 0.003169844448933532]^2)^(-1) (-0.8682367006452194 + 0.9 (0.81 - 2 (-18.353406718477792` + 13.333333333333334` JacobiSN[ 1.3972501102329256` $CellContext`\[Lambda]$, 0.13215572516370697`]^2)^(-1) (-122.35604478985194` + 21.95457708696275 JacobiSN[ 1.3972501102329256` $CellContext`\[Lambda]$, 0.13215572516370697`]^2) + (-18.353406718477792` + 13.333333333333334` JacobiSN[ 1.3972501102329256` $CellContext`\[Lambda]$, 0.13215572516370697`]^2)^(-2) (-122.35604478985194` + 21.95457708696275 JacobiSN[ 1.3972501102329256` $CellContext`\[Lambda]$, 0.13215572516370697`]^2)^2)^(-1) (-1.6369577299778217` + 0.9647074451613549 ( 0.81 + (-18.353406718477792` + 13.333333333333334` JacobiSN[ 1.3972501102329256` $CellContext`\[Lambda]$, 0.13215572516370697`]^2)^(-2) (-122.35604478985194` + 21.95457708696275 JacobiSN[ 1.3972501102329256` $CellContext`\[Lambda]$, 0.13215572516370697`]^2)^2)) + 1.8188419221975798`/(1 - 0.7499999999999999 JacobiSN[ 1.0007938772180671` (Rational[1, 2] Pi + 3.642503858423728 $CellContext`\[Lambda]$), 0.003169844448933532]^2)), Listable]}, "RadialRoots" -> {20., 6.666666666666666, 1.6465932815222062`, 0.5303096596280751}, "PolarRoots" -> {0.5235987755983001, 2.617993877991493}, "a" -> 0.9, "p" -> 10., "e" -> 0.5, "Inclination" -> 0.5, "Type" -> {"Bound", "Eccentric", "Inclined"}, "InitialPhases" -> {0, 0, 0, 0}|>], Editable->False, SelectWithContents->True, Selectable->False]], "Output", CellChangeTimes->{{3.799162428329857*^9, 3.799162452604965*^9}, 3.7991624994164*^9, 3.799162728828418*^9, 3.79916284064028*^9, 3.7991629183425283`*^9, 3.7991630084142013`*^9, 3.7993853445335455`*^9, { 3.799385659663477*^9, 3.799385697680896*^9}, {3.8091996357759933`*^9, 3.8091996559577837`*^9}, 3.9846008338133783`*^9}, CellLabel->"Out[18]=",ExpressionUUID->"10ea1b45-b590-4552-ab6c-e0a125ce48dc"] }, Open ]] }, Closed]], Cell[CellGroupData[{ Cell["Orbit properties", "Subsubsection", CellChangeTimes->{{3.809199822560129*^9, 3.8091998257771187`*^9}},ExpressionUUID->"1c3458da-4f81-4202-ac56-\ 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"+", "l"}], ")"}]}]]}], ")"}], " ", "\[Gamma]"}], "+", InterpretationBox[ SuperscriptBox[ RowBox[{"O", "[", "\[Gamma]", "]"}], "2"], SeriesData[$CellContext`\[Gamma], 0, {}, 0, 2, 1], Editable->False]}], SeriesData[$CellContext`\[Gamma], 0, {$CellContext`l + $CellContext`l^2 - $CellContext`s ( 1 + $CellContext`s), (-2) $CellContext`m - 2 $CellContext`l^(-1) ( 1 + $CellContext`l)^(-1) $CellContext`m $CellContext`s^2}, 0, 2, 1], Editable->False]], "Output", CellChangeTimes->{{3.799345271836396*^9, 3.799345285229518*^9}, { 3.799352676570538*^9, 3.7993527037179832`*^9}, 3.809200547149364*^9, 3.984601887499758*^9}, CellLabel->"Out[78]=",ExpressionUUID->"f326c31a-8014-43d3-ab87-83be3c86978f"] }, Open ]], Cell["\<\ It is quicker to perform the expansion for specific values of {s,l,m}\ \>", "Text", CellChangeTimes->{{3.799350206190496*^9, 3.799350226001315*^9}},ExpressionUUID->"94abe280-b4dc-4865-9baf-\ 5f69c9016c97"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"small\[Gamma]Expansion", "=", 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CellChangeTimes->{{3.809202047305975*^9, 3.809202053796762*^9}},ExpressionUUID->"505fc38e-88a5-4f8c-9615-\ c77851c12a23"], Cell[TextData[{ "These functions can be treated as a normal ", StyleBox["Mathematica", FontSlant->"Italic"], " functions that can be evaluated at any value of the radius. Let\ \[CloseCurlyQuote]s try with ", Cell[BoxData[ FormBox[ RowBox[{"r", "=", "10"}], TraditionalForm]], $CellContext`ExpressionUUID -> "9ef23c69-b200-4632-91fc-b7fe95ca3292", ExpressionUUID->"9a8ab5b9-711f-4fad-8627-c3ce7ceed4af"] }], "Text", CellChangeTimes->{{3.8092016440829763`*^9, 3.809201650886588*^9}, 3.809202028678767*^9}, CellID->607300058, $CellContext`ExpressionUUID -> "68d199c3-4d38-4771-9636-cb88f054e177",ExpressionUUID->"0f7d376f-3b9c-4eb6-\ bde8-e5a7a44f21c3"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{ RowBox[{"R", "[", "\"\\"", "]"}], "[", "10", "]"}]], "Input", CellLabel->"In[89]:=", CellID->2049784262, $CellContext`ExpressionUUID -> "d2e735e4-2c09-491f-960b-a5aa10dfe9e5",ExpressionUUID->"ed7bed1a-aac7-4dab-\ 8312-1bf8b4dc04e2"], Cell[BoxData[ RowBox[{"4314.829304922222688724`11.551565504803362", "+", RowBox[{ "3806.9915044018633448177`11.503843973551891", " ", "\[ImaginaryI]"}]}]], "Output", CellChangeTimes->{3.809201654795969*^9, 3.984601962373283*^9}, 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Tagoshi, Living Rev. Relativity, 6, 6 (2003)", BaseStyle->"Hyperlink", ButtonData->{ URL["https://doi.org/10.12942/lrr-2003-6"], None}, ButtonNote->"https://doi.org/10.12942/lrr-2003-6"], " for details). The MST method is the default, but can also be specified \ using the Method -> \[OpenCurlyDoubleQuote]MST\[CloseCurlyDoubleQuote] \ option. The method has a single sub-option \ \[OpenCurlyDoubleQuote]RenormalizedAngularMomentum\[CloseCurlyDoubleQuote], \ which is passed as the Method option to RenormalizedAngularMomentum. 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In practice, this is \ rarely done by direct integration. Instead, efficient methods are developed \ on a case-by-case basis.\ \>", "Text", CellChangeTimes->{3.809202753362606*^9}, CellID->1129518513, $CellContext`ExpressionUUID -> "1d92682b-1190-44b6-962a-8f177eaa5371",ExpressionUUID->"56ee669b-44b8-4f94-\ ac4c-90111356dcdd"], Cell[TextData[{ "Inhomogeneous solutions are represented by a TeukolskyMode (or a \ ReggeWheelerMode), which stores information about the radial functions, \ angular functions and normalisation coefficients. So far, the cases of a \ point mass (", Cell[BoxData[ FormBox[ RowBox[{"s", "=", RowBox[{"-", "2"}]}], TraditionalForm]], $CellContext`ExpressionUUID -> "4cb9ce27-35d1-4db5-b4ca-d5ea2eaeeac1", ExpressionUUID->"13172feb-691f-4dc9-9edd-cd8f039eed93"], "), electric (", Cell[BoxData[ FormBox[ RowBox[{"s", "=", RowBox[{"-", "1"}]}], TraditionalForm]], $CellContext`ExpressionUUID -> "89bd8f07-4ab3-4b25-85d2-e37fce988e96", ExpressionUUID->"ff069cef-0ce4-4f78-a5bd-6f7163079943"], ") or scalar (", Cell[BoxData[ FormBox[ RowBox[{"s", "=", "0"}], TraditionalForm]], $CellContext`ExpressionUUID -> "42be6123-ae11-43dc-82c5-1c29d3cb5399", ExpressionUUID->"4f8cd21b-53f8-4b79-8eaf-c65ab2aba807"], ") charge on generic orbit is supported directly, and the framework exists \ for supporting other inhomogeneous solutions of interest." }], "Text", CellChangeTimes->{{3.809202760908408*^9, 3.809202764756749*^9}, { 3.9846021078924847`*^9, 3.98460210926385*^9}, {3.9847632212362757`*^9, 3.984763226767478*^9}}, CellID->62208511, $CellContext`ExpressionUUID -> "3fef7646-5790-4269-b967-bf489803f397",ExpressionUUID->"f803b158-361c-4942-\ 827e-a49c7ac5c04c"], Cell[CellGroupData[{ Cell[TextData[{ Cell[BoxData[ FormBox[ RowBox[{"s", "=", RowBox[{"-", "2"}]}], TraditionalForm]], $CellContext`ExpressionUUID -> "9171a475-2716-4ede-b6b2-843f7e6c08e5", ExpressionUUID->"857cc42f-c487-4765-8089-51d58b1b2243"], ", generic orbit" }], "Subsubsection", CellChangeTimes->{{3.9846020845800533`*^9, 3.984602088228766*^9}}, CellID->1861391935, $CellContext`ExpressionUUID -> "29caa7c3-f2f0-49d1-9fe1-0fbc5e877ccd",ExpressionUUID->"b17eda12-c6d3-4583-\ be92-cab1b9f880e9"], Cell[TextData[{ "First, we construct an orbit using the KerrGeodesics package. We consider a \ circular orbit of radius ", Cell[BoxData[ FormBox[ RowBox[{ SubscriptBox["r", "0"], "=", "10"}], TraditionalForm]], $CellContext`ExpressionUUID -> "93d3e902-f7ab-4608-aabd-8db3c9fd8da4", ExpressionUUID->"14e0e8ae-4ce4-49be-a6f9-1e78cc68f153"], "M in Kerr spacetime with a=0.5 and compute the s=2, l=2, m=2 mode" }], "Text", CellID->2019965696, $CellContext`ExpressionUUID -> "0448923f-e4cd-4cf7-88b5-3a7296760929",ExpressionUUID->"164e9c6b-d9d5-4be3-\ 8d20-465da52bba86"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"With", "[", RowBox[{ RowBox[{"{", RowBox[{ RowBox[{"a", "=", "0.5`32"}], ",", RowBox[{"r0", "=", "10.`32"}], ",", RowBox[{"e", "=", "0"}], ",", RowBox[{"x", "=", "1"}], ",", " ", RowBox[{"s", "=", RowBox[{"-", "2"}]}], ",", RowBox[{"l", "=", "2"}], ",", RowBox[{"m", "=", "2"}], ",", RowBox[{"n", "=", "0"}], ",", RowBox[{"k", "=", "0"}]}], "}"}], ",", "\[IndentingNewLine]", RowBox[{ RowBox[{"orbit", "=", RowBox[{"KerrGeoOrbit", "[", RowBox[{"a", ",", "r0", ",", 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