Black holes as possible sources of closed and semiclosed worlds by Frolov, Markov & Mukhonov.nb

(* 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[     27493,        628]
NotebookOptionsPosition[     23569,        557]
NotebookOutlinePosition[     24035,        574]
CellTagsIndexPosition[     23992,        571]
WindowFrame->Normal*)

(* Beginning of Notebook Content *)
Notebook[{

Cell[CellGroupData[{
Cell["\<\
Notebook for :  
Black holes as possible sources of closed and semiclosed worlds by Frolov, \
Markov & Mukhanov\
\>", "Title",
 CellGroupingRules->{"GroupTogetherGrouping", 10001.},
 CellChangeTimes->{{3.8217294718661222`*^9, 3.821729502043838*^9}, 
   3.8217295683553457`*^9, 3.821732221863006*^9, {3.821811203295624*^9, 
   3.8218112036548367`*^9}, {3.822041768762134*^9, 3.822041781290854*^9}, {
   3.822048009752109*^9, 3.8220480569607773`*^9}, {3.822504877727007*^9, 
   3.8225048874039307`*^9}, {3.822565447028533*^9, 3.822565457908236*^9}, {
   3.822822107930128*^9, 3.822822113128089*^9}, {3.823176459993638*^9, 
   3.823176470345335*^9}, {3.823359338529508*^9, 3.823359361520629*^9}, {
   3.823437107739418*^9, 3.823437114378701*^9}, {3.823437155219769*^9, 
   3.823437162159172*^9}, {3.8260277436913977`*^9, 3.82602776480827*^9}, {
   3.826187165606697*^9, 3.826187195199024*^9}, {3.826203403964402*^9, 
   3.826203410507988*^9}, 3.826203768834175*^9, {3.826205862327466*^9, 
   3.826205873167017*^9}, {3.826205903951358*^9, 3.8262059044305353`*^9}, 
   3.8554095760391483`*^9},
 TextAlignment->Center,ExpressionUUID->"0f39b985-4f26-4b22-8739-25480527aba9"],

Cell["\<\
Geoff Cope
University of Utah
April 1st, 2021 \
\>", "Author",
 CellGroupingRules->{"GroupTogetherGrouping", 10001.},
 CellChangeTimes->{{3.821729507652328*^9, 3.821729529657844*^9}, 
   3.8220480097522783`*^9, {3.822048070235491*^9, 3.822048084689715*^9}, {
   3.8225654346519403`*^9, 3.8225654361641483`*^9}, {3.8231764728567343`*^9, 
   3.823176482313031*^9}, {3.8233640113399878`*^9, 3.823364023347041*^9}, {
   3.823422996286858*^9, 3.823422999926001*^9}, {3.8260349123952703`*^9, 
   3.826034914714463*^9}, {3.826187198702311*^9, 3.826187203958405*^9}},
 TextAlignment->Center,ExpressionUUID->"39c7f5b2-30c3-4718-82b6-37799179c246"]
}, Open  ]],

Cell[BoxData[
 RowBox[{
  RowBox[{"(*", " ", 
   RowBox[{
    RowBox[{
     RowBox[{
     "problem", " ", "appears", " ", "to", " ", "be", " ", "sign", " ", 
      "convetion", " ", "with", " ", "Riemmann", " ", "and", " ", "Ricci"}], 
     "..."}], " ", "combining", " ", "those", " ", "with", " ", "a", " ", 
    "Ricci", " ", "Scalar", " ", "squared", " ", "which", " ", "is", " ", 
    "correct"}], " ", "*)"}], " "}]], "Input",
 CellChangeTimes->{3.826185237350555*^9},
 CellLabel->"In[1]:=",ExpressionUUID->"c7460edd-db1c-4c71-9fd0-1057a0afc02b"],

Cell[CellGroupData[{

Cell["Hyperlink To Documentation", "Section",
 CellGroupingRules->{"GroupTogetherGrouping", 10000.},
 CellChangeTimes->{{3.820944675605734*^9, 
  3.8209446931410646`*^9}},ExpressionUUID->"f86330fb-db11-4cbc-8053-\
07e801bf4ef0"],

Cell[BoxData[
 RowBox[{"Hyperlink", "[", 
  RowBox[{
  "\"\<GeneralRelativityTensors Documentation and Download\>\"", ",", 
   "\[IndentingNewLine]", 
   "\"\<https://github.com/BlackHolePerturbationToolkit\>\""}], 
  "]"}]], "Input",
 CellGroupingRules->{"GroupTogetherGrouping", 10000.},
 CellChangeTimes->{{3.820944619768764*^9, 3.820944681268063*^9}},
 CellLabel->"In[2]:=",ExpressionUUID->"704dca9e-6aef-4850-81dc-7a7b0a531f87"],

Cell[BoxData[
 TemplateBox[{
  "\"GeneralRelativityTensors Documentation and Download\"", 
   "https://github.com/BlackHolePerturbationToolkit"},
  "HyperlinkURL"]], "Output",
 CellGroupingRules->{"GroupTogetherGrouping", 10000.},
 CellChangeTimes->{{3.820944653562152*^9, 3.820944681268403*^9}, 
   3.821729905862431*^9, 3.821732641380642*^9, 3.8218069931429567`*^9, 
   3.821811425912986*^9, 3.822042652654992*^9, {3.822048793766124*^9, 
   3.8220488009116488`*^9}, 3.82250502887521*^9, 3.822505578829629*^9, 
   3.822565490613895*^9, 3.8225662097813597`*^9, 3.822606717894772*^9, 
   3.822739433713368*^9, 3.822822210464344*^9, 3.8233594284652433`*^9, 
   3.8233607803938913`*^9, 3.823360880234103*^9, 3.823361224481102*^9, 
   3.82336209870968*^9, 3.823364030542309*^9, 3.823437204902042*^9, 
   3.8260279978867607`*^9, 3.82605154665021*^9, 3.82605224785146*^9, 
   3.82605422602782*^9, 3.826095790794064*^9, 3.826185687825083*^9, 
   3.82618579874909*^9, 3.826186039536813*^9, 3.826187231451871*^9, 
   3.8262034462511997`*^9},
 CellLabel->"Out[2]=",ExpressionUUID->"4ba230bc-b067-448f-8732-3027638d3d2d"]
}, Closed]],

Cell[CellGroupData[{

Cell["Hyperlink  To Article", "Section",
 CellGroupingRules->{"GroupTogetherGrouping", 10001.},
 CellChangeTimes->{
  3.820937794883616*^9, {3.820938266275806*^9, 3.82093829464357*^9}, 
   3.820943883376583*^9, {3.8210324492405148`*^9, 3.821032453951701*^9}, {
   3.8217297279218817`*^9, 3.8217297313782063`*^9}, {3.822593613363134*^9, 
   3.822593616181035*^9}, {3.822736144359599*^9, 3.8227361450547543`*^9}, {
   3.823359417697976*^9, 3.823359420672374*^9}, {3.826190945278699*^9, 
   3.82619094661141*^9}, {3.826203847916677*^9, 
   3.8262038562829933`*^9}},ExpressionUUID->"6d515bb2-1694-473d-a1e7-\
f6235e3e467f"],

Cell[BoxData[
 RowBox[{"Hyperlink", "[", 
  RowBox[{
  "\"\<Black holes as possible sources of closed and semiclosed worlds by \
Frolov, Markov & Mukhanov\>\"", ",", "\[IndentingNewLine]", 
   "\"\<https://journals.aps.org/prd/abstract/10.1103/PhysRevD.41.383\>\""}], 
  "]"}]], "Input",
 CellGroupingRules->{"GroupTogetherGrouping", 10001.},
 CellChangeTimes->{{3.8218867746254797`*^9, 3.821886810805148*^9}, 
   3.823176507207197*^9, {3.823472612814559*^9, 3.823472633860148*^9}, {
   3.826203812605998*^9, 3.826203832886978*^9}, {3.826205917136896*^9, 
   3.826205920062436*^9}, {3.826205976911607*^9, 3.8262059806383467`*^9}},
 CellLabel->
  "In[122]:=",ExpressionUUID->"8dc6bc9a-5ff8-492a-ad8e-f96df043da49"],

Cell[BoxData[
 TemplateBox[{
  "\"Black holes as possible sources of closed and semiclosed worlds by \
Frolov, Markov & Mukhanov\"", 
   "https://journals.aps.org/prd/abstract/10.1103/PhysRevD.41.383"},
  "HyperlinkURL"]], "Output",
 CellGroupingRules->{"GroupTogetherGrouping", 10001.},
 CellChangeTimes->{3.821886813557355*^9, 3.8231765072073307`*^9, 
  3.823359428608782*^9, 3.8233607804338617`*^9, 3.823360880363216*^9, 
  3.823361224626254*^9, 3.823362098931829*^9, 3.823364030688095*^9, 
  3.8234372051552353`*^9, 3.823472634251237*^9, 3.8260279980033703`*^9, 
  3.8260515467743177`*^9, 3.8260522480075808`*^9, 3.826054226145361*^9, 
  3.826095790927781*^9, 3.82618568784027*^9, 3.826185798762553*^9, 
  3.826186039549511*^9, 3.8261872314618464`*^9, 3.826203446261386*^9, 
  3.8262038346187763`*^9, 3.826205981585454*^9},
 CellLabel->
  "Out[122]=",ExpressionUUID->"c76297ad-1b76-4e7b-bd9d-cd55a14fb55e"],

Cell[BoxData[
 RowBox[{"Hyperlink", "[", 
  RowBox[{
  "\"\<Colliding Plane Waves by Griffiths - Book\>\"", ",", 
   "\"\<ftp://nozdr.ru/biblio/kolxoz/P/PGr/Griffiths%20J.B.%20Colliding%\
20plane%20waves%20in%20general%20relativity%20(OUP,%201991)(ISBN%200198532091)\
(254s)_PGr_.pdf\>\""}], "]"}]], "Input",
 CellGroupingRules->{"GroupTogetherGrouping", 10001.},
 CellChangeTimes->{{3.821971700891346*^9, 3.821971727288177*^9}, 
   3.8220418523671618`*^9, {3.8220419016581783`*^9, 3.822041910552957*^9}},
 CellLabel->"In[4]:=",ExpressionUUID->"0b7bccec-ffab-497a-839f-f5f28a891505"],

Cell[BoxData[
 TemplateBox[{
  "\"Colliding Plane Waves by Griffiths - Book\"", 
   "ftp://nozdr.ru/biblio/kolxoz/P/PGr/Griffiths%20J.B.%20Colliding%20plane%\
20waves%20in%20general%20relativity%20(OUP,%201991)(ISBN%200198532091)(254s)_\
PGr_.pdf"},
  "HyperlinkURL"]], "Output",
 CellGroupingRules->{"GroupTogetherGrouping", 10001.},
 CellChangeTimes->{
  3.8219717866333103`*^9, 3.822041852367313*^9, 3.822041913511463*^9, 
   3.8220426527016563`*^9, {3.822048793815063*^9, 3.8220488009576597`*^9}, 
   3.822505029060484*^9, 3.822505578941113*^9, 3.82256549066683*^9, 
   3.822566209832062*^9, 3.822606717963243*^9, 3.822739433796803*^9, 
   3.822822210610302*^9, 3.8233594286355753`*^9, 3.8233607804460297`*^9, 
   3.8233608803853903`*^9, 3.823361224649728*^9, 3.823362098967597*^9, 
   3.823364030715418*^9, 3.823437205194394*^9, 3.826027998036727*^9, 
   3.826051546789892*^9, 3.826052248058763*^9, 3.826054226161038*^9, 
   3.826095790945072*^9, 3.826185687859961*^9, 3.8261857987837877`*^9, 
   3.8261860395693913`*^9, 3.826187231480541*^9, 3.826203446277575*^9},
 CellLabel->"Out[4]=",ExpressionUUID->"fa45c8af-9aec-4a88-b5ae-5ce1622503a1"]
}, Closed]],

Cell[CellGroupData[{

Cell["Utilities and Package Load", "Section",
 CellGroupingRules->{"GroupTogetherGrouping", 10000.},
 CellChangeTimes->{{3.820937646765551*^9, 
  3.8209376725449123`*^9}},ExpressionUUID->"d9da0e4c-915d-44e2-b31a-\
630b88f04449"],

Cell[BoxData[{
 RowBox[{"ClearAll", "[", "\"\<Global`*\>\"", "]"}], "\[IndentingNewLine]", 
 RowBox[{"<<", "Utilities`CleanSlate`"}], "\[IndentingNewLine]", 
 RowBox[{"CleanSlate", "[", "]"}], "\[IndentingNewLine]", 
 RowBox[{
  RowBox[{"ClearInOut", "[", "]"}], 
  "\[IndentingNewLine]"}], "\[IndentingNewLine]", 
 RowBox[{
  RowBox[{"pdConv", "[", "f_", "]"}], ":=", 
  RowBox[{"TraditionalForm", "[", 
   RowBox[{"f", "/.", 
    RowBox[{
     RowBox[{
      RowBox[{
       RowBox[{"Derivative", "[", "inds__", "]"}], "[", "g_", "]"}], "[", 
      "vars__", "]"}], "\[RuleDelayed]", 
     RowBox[{"Apply", "[", 
      RowBox[{
       RowBox[{
        RowBox[{"Defer", "[", 
         RowBox[{"D", "[", 
          RowBox[{
           RowBox[{"g", "[", "vars", "]"}], ",", "##"}], "]"}], "]"}], "&"}], 
       ",", 
       RowBox[{
        RowBox[{"Transpose", "[", 
         RowBox[{"{", 
          RowBox[{
           RowBox[{"{", "vars", "}"}], ",", 
           RowBox[{"{", "inds", "}"}]}], "}"}], "]"}], "/.", 
        RowBox[{"{", 
         RowBox[{
          RowBox[{
           RowBox[{"{", 
            RowBox[{"var_", ",", "0"}], "}"}], "\[RuleDelayed]", 
           RowBox[{"Sequence", "[", "]"}]}], ",", 
          RowBox[{
           RowBox[{"{", 
            RowBox[{"var_", ",", "1"}], "}"}], "\[RuleDelayed]", 
           RowBox[{"{", "var", "}"}]}]}], "}"}]}]}], "]"}]}]}], 
   "]"}]}]}], "Input",
 CellGroupingRules->{"GroupTogetherGrouping", 10000.},
 CellChangeTimes->{
  3.8209376549030037`*^9, {3.821032517164062*^9, 3.821032518258361*^9}},
 CellLabel->"In[5]:=",ExpressionUUID->"2b2cae6d-61a1-4d1b-9192-e58d6dbb821d"],

Cell[BoxData[
 InterpretationBox[
  RowBox[{"\<\"  (CleanSlate) Contexts purged: \"\>", "\[InvisibleSpace]", 
   RowBox[{"{", "\<\"Global`\"\>", "}"}]}],
  SequenceForm["  (CleanSlate) Contexts purged: ", {"Global`"}],
  Editable->False]], "Print",
 CellGroupingRules->{"GroupTogetherGrouping", 10000.},
 CellChangeTimes->{
  3.820932754769992*^9, 3.820933427630714*^9, 3.820937654903143*^9, 
   3.821729906190301*^9, 3.821732641685704*^9, 3.8218069933778133`*^9, 
   3.82181142618342*^9, 3.8220426528487463`*^9, {3.822048793940605*^9, 
   3.822048801089246*^9}, 3.822505029191699*^9, 3.822505579049889*^9, 
   3.822565490838564*^9, 3.822566210008327*^9, 3.822606718180009*^9, 
   3.8226371008270607`*^9, 3.822739433887854*^9, 3.822761800089258*^9, 
   3.8228172997642097`*^9, 3.822822210725956*^9, 3.822988687208267*^9, 
   3.823359428732101*^9, 3.823360780582996*^9, 3.823360880477087*^9, 
   3.823361224742215*^9, 3.8233620990886593`*^9, 3.8233640308219624`*^9, 
   3.823437205329228*^9, 3.826027998129552*^9, 3.826051546851453*^9, 
   3.8260522481550007`*^9, 3.8260542262552233`*^9, 3.826095791058305*^9, 
   3.826185687983923*^9, 3.8261857988817873`*^9, 3.826186039665844*^9, 
   3.826187231626996*^9, 3.8262034463703527`*^9},
 CellLabel->
  "During evaluation of \
In[5]:=",ExpressionUUID->"2df103fc-fffa-4267-a539-42749566d01a"],

Cell[BoxData[
 InterpretationBox[
  RowBox[{"\<\"  (CleanSlate) Approximate kernel memory recovered: \"\>", 
   "\[InvisibleSpace]", "\<\"8 Kb\"\>"}],
  SequenceForm["  (CleanSlate) Approximate kernel memory recovered: ", "8 Kb"],
  Editable->False]], "Print",
 CellGroupingRules->{"GroupTogetherGrouping", 10000.},
 CellChangeTimes->{
  3.820932754769992*^9, 3.820933427630714*^9, 3.820937654903143*^9, 
   3.821729906190301*^9, 3.821732641685704*^9, 3.8218069933778133`*^9, 
   3.82181142618342*^9, 3.8220426528487463`*^9, {3.822048793940605*^9, 
   3.822048801089246*^9}, 3.822505029191699*^9, 3.822505579049889*^9, 
   3.822565490838564*^9, 3.822566210008327*^9, 3.822606718180009*^9, 
   3.8226371008270607`*^9, 3.822739433887854*^9, 3.822761800089258*^9, 
   3.8228172997642097`*^9, 3.822822210725956*^9, 3.822988687208267*^9, 
   3.823359428732101*^9, 3.823360780582996*^9, 3.823360880477087*^9, 
   3.823361224742215*^9, 3.8233620990886593`*^9, 3.8233640308219624`*^9, 
   3.823437205329228*^9, 3.826027998129552*^9, 3.826051546851453*^9, 
   3.8260522481550007`*^9, 3.8260542262552233`*^9, 3.826095791058305*^9, 
   3.826185687983923*^9, 3.8261857988817873`*^9, 3.826186039665844*^9, 
   3.826187231626996*^9, 3.82620344637246*^9},
 CellLabel->
  "During evaluation of \
In[5]:=",ExpressionUUID->"d0658f32-350c-4e79-834f-73a200ba789e"],

Cell[BoxData[
 RowBox[{"{", 
  RowBox[{"\<\"Utilities`CleanSlate`\"\>", ",", "\<\"System`\"\>", 
   ",", "\<\"Global`\"\>"}], "}"}]], "Output",
 CellGroupingRules->{"GroupTogetherGrouping", 10000.},
 CellChangeTimes->{
  3.820932754773623*^9, 3.820933427635436*^9, 3.8209376549033947`*^9, 
   3.821729906193787*^9, 3.821732641691049*^9, 3.821806993386058*^9, 
   3.821811426187071*^9, 3.822042652852829*^9, {3.822048793944077*^9, 
   3.822048801092909*^9}, 3.822505029195156*^9, 3.8225055790570173`*^9, 
   3.8225654908422747`*^9, 3.8225662100125237`*^9, 3.8226067181902227`*^9, 
   3.822637100839163*^9, 3.8227394339100733`*^9, 3.822761800103311*^9, 
   3.822817299776285*^9, 3.822822210739685*^9, 3.822988687217742*^9, 
   3.823359428736412*^9, 3.823360780587545*^9, 3.8233608804810534`*^9, 
   3.823361224746982*^9, 3.823362099093968*^9, 3.823364030838743*^9, 
   3.8234372053364553`*^9, 3.826027998133745*^9, 3.826051546860629*^9, 
   3.826052248166594*^9, 3.826054226260648*^9, 3.8260957910651817`*^9, 
   3.82618568799067*^9, 3.8261857988893957`*^9, 3.826186039672536*^9, 
   3.826187231630183*^9, 3.826203446373919*^9},
 CellLabel->"Out[0]=",ExpressionUUID->"d4a8d864-2fae-4bd8-8982-fafd2928cecf"],

Cell[BoxData[
 RowBox[{"<<", "GeneralRelativityTensors`"}]], "Input",
 CellGroupingRules->{"GroupTogetherGrouping", 10000.},
 CellChangeTimes->{3.8209376549035053`*^9},
 CellLabel->"In[2]:=",ExpressionUUID->"2b5a37ee-b19b-4a81-81b3-e48ee8ad19aa"]
}, Closed]],

Cell[CellGroupData[{

Cell["Custom Notation", "Section",
 CellGroupingRules->{"GroupTogetherGrouping", 10001.},
 CellChangeTimes->{{3.8225114042235403`*^9, 3.822511435665856*^9}, 
   3.823176577809782*^9},ExpressionUUID->"28cf6690-5ed1-4127-bc38-\
7d6baa9d8cb5"],

Cell[BoxData[{
 RowBox[{"Clear", "[", "dtReplace", "]"}], "\[IndentingNewLine]", 
 RowBox[{
  RowBox[{"dtReplace", " ", "=", " ", 
   RowBox[{"{", "\[IndentingNewLine]", 
    RowBox[{
     RowBox[{
      RowBox[{"Dt", "[", "r", "]"}], "\[Rule]", " ", "dr"}], " ", ",", " ", 
     "\[IndentingNewLine]", 
     RowBox[{
      RowBox[{"Dt", "[", "t", "]"}], "\[Rule]", " ", "dt"}], " ", ",", " ", 
     "\[IndentingNewLine]", 
     RowBox[{
      RowBox[{"Dt", "[", "u", "]"}], "\[Rule]", " ", "du"}], " ", ",", " ", 
     "\[IndentingNewLine]", 
     RowBox[{
      RowBox[{"Dt", "[", "v", "]"}], "\[Rule]", " ", "dv"}], " ", ",", " ", 
     "\[IndentingNewLine]", " ", 
     RowBox[{
      RowBox[{"Dt", "[", "\[Theta]", "]"}], " ", "\[Rule]", " ", 
      "d\[Theta]"}], " ", ",", " ", "\[IndentingNewLine]", 
     RowBox[{
      RowBox[{"Dt", "[", "\[Phi]", "]"}], " ", "\[Rule]", " ", "d\[Phi]"}], 
     "  ", ",", "\[IndentingNewLine]", 
     RowBox[{
      RowBox[{"Dt", "[", "\[Eta]", "]"}], " ", "\[Rule]", " ", "d\[Eta]"}], 
     "  ", ",", " ", "\[IndentingNewLine]", 
     RowBox[{
      RowBox[{"Dt", "[", "\[Chi]", "]"}], "\[Rule]", " ", "d\[Chi]"}], " ", 
     ",", " ", "\[IndentingNewLine]", 
     RowBox[{
      RowBox[{"Dt", "[", "\[Rho]", "]"}], " ", "\[Rule]", " ", "d\[Rho]"}], 
     " ", ",", "\[IndentingNewLine]", 
     RowBox[{
      RowBox[{"Dt", "[", "\[CurlyPhi]", "]"}], " ", "\[Rule]", " ", 
      "d\[CurlyPhi]"}], " ", ",", " ", "\[IndentingNewLine]", " ", 
     RowBox[{
      RowBox[{"Dt", "[", "\[ScriptT]", "]"}], " ", "\[Rule]", " ", 
      "d\[ScriptT]"}], " ", ",", " ", "\[IndentingNewLine]", 
     RowBox[{
      RowBox[{"Dt", "[", "\[ScriptZ]", "]"}], " ", "\[Rule]", " ", 
      "d\[ScriptZ]"}], ",", "\[IndentingNewLine]", 
     RowBox[{
      RowBox[{"Dt", "[", "U", "]"}], "\[Rule]", " ", "dU"}], " ", ",", " ", 
     "\[IndentingNewLine]", 
     RowBox[{
      RowBox[{"Dt", "[", "V", "]"}], "\[Rule]", " ", "dV"}], " ", ",", 
     "\[IndentingNewLine]", 
     RowBox[{
      RowBox[{"Dt", "[", "z", "]"}], "\[Rule]", " ", "dz"}], " ", ",", " ", 
     "\[IndentingNewLine]", 
     RowBox[{
      RowBox[{"Dt", "[", "\[ScriptR]", "]"}], "\[Rule]", " ", "d\[ScriptR]"}],
      "  ", ",", " ", "\[IndentingNewLine]", 
     RowBox[{
      RowBox[{"Dt", "[", "\[CapitalPhi]", "]"}], "\[Rule]", " ", 
      "d\[CapitalPhi]"}]}], "}"}]}], ";"}], "\[IndentingNewLine]", 
 RowBox[{"dtReplace", " ", "//", " ", "TableForm"}]}], "Input",
 CellGroupingRules->{"GroupTogetherGrouping", 10001.},
 CellChangeTimes->{{3.82317656063256*^9, 3.823176577809904*^9}, {
  3.823437608803*^9, 3.823437618694414*^9}},
 CellLabel->"In[3]:=",ExpressionUUID->"7722233b-bb1f-4ebd-8278-bb6d375c5d22"],

Cell[BoxData[
 TagBox[
  TagBox[GridBox[{
     {
      RowBox[{
       RowBox[{"Dt", "[", "r", "]"}], "\[Rule]", "dr"}]},
     {
      RowBox[{
       RowBox[{"Dt", "[", "t", "]"}], "\[Rule]", "dt"}]},
     {
      RowBox[{
       RowBox[{"Dt", "[", "u", "]"}], "\[Rule]", "du"}]},
     {
      RowBox[{
       RowBox[{"Dt", "[", "v", "]"}], "\[Rule]", "dv"}]},
     {
      RowBox[{
       RowBox[{"Dt", "[", "\[Theta]", "]"}], "\[Rule]", "d\[Theta]"}]},
     {
      RowBox[{
       RowBox[{"Dt", "[", "\[Phi]", "]"}], "\[Rule]", "d\[Phi]"}]},
     {
      RowBox[{
       RowBox[{"Dt", "[", "\[Eta]", "]"}], "\[Rule]", "d\[Eta]"}]},
     {
      RowBox[{
       RowBox[{"Dt", "[", "\[Chi]", "]"}], "\[Rule]", "d\[Chi]"}]},
     {
      RowBox[{
       RowBox[{"Dt", "[", "\[Rho]", "]"}], "\[Rule]", "d\[Rho]"}]},
     {
      RowBox[{
       RowBox[{"Dt", "[", "\[CurlyPhi]", "]"}], "\[Rule]", "d\[CurlyPhi]"}]},
     {
      RowBox[{
       RowBox[{"Dt", "[", "\[ScriptT]", "]"}], "\[Rule]", "d\[ScriptT]"}]},
     {
      RowBox[{
       RowBox[{"Dt", "[", "\[ScriptZ]", "]"}], "\[Rule]", "d\[ScriptZ]"}]},
     {
      RowBox[{
       RowBox[{"Dt", "[", "U", "]"}], "\[Rule]", "dU"}]},
     {
      RowBox[{
       RowBox[{"Dt", "[", "V", "]"}], "\[Rule]", "dV"}]},
     {
      RowBox[{
       RowBox[{"Dt", "[", "z", "]"}], "\[Rule]", "dz"}]},
     {
      RowBox[{
       RowBox[{"Dt", "[", "\[ScriptR]", "]"}], "\[Rule]", "d\[ScriptR]"}]},
     {
      RowBox[{
       RowBox[{"Dt", "[", "\[CapitalPhi]", "]"}], "\[Rule]", 
       "d\[CapitalPhi]"}]}
    },
    GridBoxAlignment->{"Columns" -> {{Left}}, "Rows" -> {{Baseline}}},
    GridBoxSpacings->{"Columns" -> {
        Offset[0.27999999999999997`], {
         Offset[0.5599999999999999]}, 
        Offset[0.27999999999999997`]}, "Rows" -> {
        Offset[0.2], {
         Offset[0.4]}, 
        Offset[0.2]}}],
   Column],
  Function[BoxForm`e$, 
   TableForm[BoxForm`e$]]]], "Output",
 CellGroupingRules->{"GroupTogetherGrouping", 10001.},
 CellChangeTimes->{{3.8231765606326637`*^9, 3.82317657781*^9}, 
   3.8233594287980833`*^9, 3.823360780614534*^9, 3.8233608805437098`*^9, 
   3.823361224810658*^9, 3.823362099159925*^9, 3.8233640309198627`*^9, 
   3.823437205444152*^9, 3.823437619154374*^9, 3.826027998201126*^9, 
   3.8260515469301558`*^9, 3.826052248254979*^9, 3.826054226322803*^9, 
   3.826095791132465*^9, 3.826185688050398*^9, 3.826185798943942*^9, 
   3.826186039726972*^9, 3.826187231653651*^9, 3.82620344644106*^9},
 CellLabel->
  "Out[5]//TableForm=",ExpressionUUID->"0e799f63-cdfb-48b4-b6ab-18be508dc7cc"]
}, Closed]],

Cell[CellGroupData[{

Cell["Line Element and Metric Functions", "Section",
 CellGroupingRules->{"GroupTogetherGrouping", 10003.},
 CellChangeTimes->{{3.820937548919169*^9, 3.8209375728447866`*^9}, 
   3.8209383129401073`*^9, 3.8220481344968433`*^9, {3.8225115057038183`*^9, 
   3.822511507783984*^9}, 
   3.822763734568725*^9},ExpressionUUID->"1c9718c4-1b9a-4790-ad33-\
d4a196d0fc9d"],

Cell[BoxData[{
 RowBox[{"Clear", "[", "lineToMetric", "]"}], "\[IndentingNewLine]", 
 RowBox[{
  RowBox[{
   RowBox[{"lineToMetric", "[", 
    RowBox[{"lineelement_", " ", ",", " ", "differentials_"}], "]"}], ":=", 
   " ", "\[IndentingNewLine]", 
   RowBox[{"Table", "[", 
    RowBox[{
     RowBox[{"If", "[", 
      RowBox[{
       RowBox[{"i", "\[Equal]", "j"}], " ", ",", 
       RowBox[{"Coefficient", "[", 
        RowBox[{"lineelement", ",", " ", 
         RowBox[{
          RowBox[{"differentials", "[", 
           RowBox[{"[", "i", "]"}], "]"}], " ", 
          RowBox[{"differentials", "[", 
           RowBox[{"[", "j", "]"}], "]"}]}]}], "]"}], ",", 
       "\[IndentingNewLine]", 
       RowBox[{
        RowBox[{"(", 
         RowBox[{"1", "/", "2"}], ")"}], "*", 
        RowBox[{"Coefficient", "[", 
         RowBox[{"lineelement", ",", " ", 
          RowBox[{
           RowBox[{"differentials", "[", 
            RowBox[{"[", "i", "]"}], "]"}], " ", 
           RowBox[{"differentials", "[", 
            RowBox[{"[", "j", "]"}], "]"}]}]}], "]"}]}]}], "]"}], ",", 
     "\[IndentingNewLine]", 
     RowBox[{"{", 
      RowBox[{"i", ",", "1", ",", 
       RowBox[{"Length", "[", "differentials", "]"}]}], "}"}], ",", 
     RowBox[{"{", 
      RowBox[{"j", ",", "1", ",", 
       RowBox[{"Length", "[", "differentials", "]"}]}], "}"}]}], "]"}]}], 
  "  ", ";"}]}], "Input",
 CellGroupingRules->{"GroupTogetherGrouping", 10003.},
 CellChangeTimes->{3.821567412203915*^9, 3.821641945174967*^9, 
  3.821641981470141*^9, 3.8220481344970007`*^9, 3.822763734569355*^9},
 CellLabel->"In[6]:=",ExpressionUUID->"1f423262-82a5-4ae6-ab9b-55ad4490a23f"],

Cell[BoxData[{
 RowBox[{"Clear", "[", "metricToLine", "]"}], "\[IndentingNewLine]", 
 RowBox[{
  RowBox[{"metricToLine", "[", 
   RowBox[{"metric_", ",", " ", "differentials_"}], "]"}], ":=", " ", 
  "\[IndentingNewLine]", 
  RowBox[{"Sum", "[", 
   RowBox[{
    RowBox[{
     RowBox[{"metric", "[", 
      RowBox[{"[", 
       RowBox[{"i", ",", "j"}], "]"}], "]"}], " ", 
     RowBox[{"differentials", "[", 
      RowBox[{"[", "i", "]"}], "]"}], " ", 
     RowBox[{"differentials", "[", 
      RowBox[{"[", "j", "]"}], "]"}]}], " ", ",", " ", 
    RowBox[{"{", 
     RowBox[{"i", ",", "1", ",", 
      RowBox[{"Length", "[", "differentials", "]"}]}], "}"}], ",", 
    RowBox[{"{", 
     RowBox[{"j", ",", "1", ",", 
      RowBox[{"Length", "[", "differentials", "]"}]}], "}"}]}], 
   "]"}]}]}], "Input",
 CellGroupingRules->{"GroupTogetherGrouping", 10003.},
 CellChangeTimes->{{3.822762557092309*^9, 3.8227626322008677`*^9}, 
   3.8227637345698223`*^9, 3.822988709740047*^9},
 CellLabel->"In[8]:=",ExpressionUUID->"90ba5eca-3c4e-4350-ab2b-34d7bdad7e79"]
}, Closed]]
},
WindowSize->{1280, 755},
WindowMargins->{{0, Automatic}, {Automatic, 0}},
FrontEndVersion->"13.0 for Mac OS X x86 (64-bit) (December 2, 2021)",
StyleDefinitions->FrontEnd`FileName[{"Report"}, "StandardReport.nb", 
  CharacterEncoding -> "UTF-8"],
ExpressionUUID->"6e6ac187-ac8f-4f72-b6ee-3ce7a5e109fd"
]
(* End of Notebook Content *)

(* Internal cache information *)
(*CellTagsOutline
CellTagsIndex->{}
*)
(*CellTagsIndex
CellTagsIndex->{}
*)
(*NotebookFileOutline
Notebook[{
Cell[CellGroupData[{
Cell[580, 22, 1177, 19, 221, "Title",ExpressionUUID->"0f39b985-4f26-4b22-8739-25480527aba9",
 CellGroupingRules->{"GroupTogetherGrouping", 10001.}],
Cell[1760, 43, 648, 12, 76, "Author",ExpressionUUID->"39c7f5b2-30c3-4718-82b6-37799179c246",
 CellGroupingRules->{"GroupTogetherGrouping", 10001.}]
}, Open  ]],
Cell[2423, 58, 552, 12, 41, "Input",ExpressionUUID->"c7460edd-db1c-4c71-9fd0-1057a0afc02b"],
Cell[CellGroupData[{
Cell[3000, 74, 228, 4, 69, "Section",ExpressionUUID->"f86330fb-db11-4cbc-8053-07e801bf4ef0",
 CellGroupingRules->{"GroupTogetherGrouping", 10000.}],
Cell[3231, 80, 433, 9, 78, "Input",ExpressionUUID->"704dca9e-6aef-4850-81dc-7a7b0a531f87",
 CellGroupingRules->{"GroupTogetherGrouping", 10000.}],
Cell[3667, 91, 1110, 18, 53, "Output",ExpressionUUID->"4ba230bc-b067-448f-8732-3027638d3d2d",
 CellGroupingRules->{"GroupTogetherGrouping", 10000.}]
}, Closed]],
Cell[CellGroupData[{
Cell[4814, 114, 619, 10, 55, "Section",ExpressionUUID->"6d515bb2-1694-473d-a1e7-f6235e3e467f",
 CellGroupingRules->{"GroupTogetherGrouping", 10001.}],
Cell[5436, 126, 713, 13, 63, "Input",ExpressionUUID->"8dc6bc9a-5ff8-492a-ad8e-f96df043da49",
 CellGroupingRules->{"GroupTogetherGrouping", 10001.}],
Cell[6152, 141, 910, 16, 43, "Output",ExpressionUUID->"c76297ad-1b76-4e7b-bd9d-cd55a14fb55e",
 CellGroupingRules->{"GroupTogetherGrouping", 10001.}],
Cell[7065, 159, 583, 10, 84, "Input",ExpressionUUID->"0b7bccec-ffab-497a-839f-f5f28a891505",
 CellGroupingRules->{"GroupTogetherGrouping", 10001.}],
Cell[7651, 171, 1148, 19, 43, "Output",ExpressionUUID->"fa45c8af-9aec-4a88-b5ae-5ce1622503a1",
 CellGroupingRules->{"GroupTogetherGrouping", 10001.}]
}, Closed]],
Cell[CellGroupData[{
Cell[8836, 195, 228, 4, 55, "Section",ExpressionUUID->"d9da0e4c-915d-44e2-b31a-630b88f04449",
 CellGroupingRules->{"GroupTogetherGrouping", 10000.}],
Cell[9067, 201, 1640, 44, 260, "Input",ExpressionUUID->"2b2cae6d-61a1-4d1b-9192-e58d6dbb821d",
 CellGroupingRules->{"GroupTogetherGrouping", 10000.}],
Cell[10710, 247, 1335, 23, 29, "Print",ExpressionUUID->"2df103fc-fffa-4267-a539-42749566d01a",
 CellGroupingRules->{"GroupTogetherGrouping", 10000.}],
Cell[12048, 272, 1344, 23, 29, "Print",ExpressionUUID->"d0658f32-350c-4e79-834f-73a200ba789e",
 CellGroupingRules->{"GroupTogetherGrouping", 10000.}],
Cell[13395, 297, 1204, 19, 51, "Output",ExpressionUUID->"d4a8d864-2fae-4bd8-8982-fafd2928cecf",
 CellGroupingRules->{"GroupTogetherGrouping", 10000.}],
Cell[14602, 318, 246, 4, 51, "Input",ExpressionUUID->"2b5a37ee-b19b-4a81-81b3-e48ee8ad19aa",
 CellGroupingRules->{"GroupTogetherGrouping", 10000.}]
}, Closed]],
Cell[CellGroupData[{
Cell[14885, 327, 240, 4, 55, "Section",ExpressionUUID->"28cf6690-5ed1-4127-bc38-7d6baa9d8cb5",
 CellGroupingRules->{"GroupTogetherGrouping", 10001.}],
Cell[15128, 333, 2714, 61, 546, "Input",ExpressionUUID->"7722233b-bb1f-4ebd-8278-bb6d375c5d22",
 CellGroupingRules->{"GroupTogetherGrouping", 10001.}],
Cell[17845, 396, 2589, 76, 430, "Output",ExpressionUUID->"0e799f63-cdfb-48b4-b6ab-18be508dc7cc",
 CellGroupingRules->{"GroupTogetherGrouping", 10001.}]
}, Closed]],
Cell[CellGroupData[{
Cell[20471, 477, 362, 6, 55, "Section",ExpressionUUID->"1c9718c4-1b9a-4790-ad33-d4a196d0fc9d",
 CellGroupingRules->{"GroupTogetherGrouping", 10003.}],
Cell[20836, 485, 1659, 41, 126, "Input",ExpressionUUID->"1f423262-82a5-4ae6-ab9b-55ad4490a23f",
 CellGroupingRules->{"GroupTogetherGrouping", 10003.}],
Cell[22498, 528, 1055, 26, 84, "Input",ExpressionUUID->"90ba5eca-3c4e-4350-ab2b-34d7bdad7e79",
 CellGroupingRules->{"GroupTogetherGrouping", 10003.}]
}, Closed]]
}
]
*)