(* Content-type: application/vnd.wolfram.mathematica *) (*** Wolfram Notebook File ***) (* http://www.wolfram.com/nb *) (* CreatedBy='Mathematica 13.1' *) (*CacheID: 234*) (* Internal cache information: NotebookFileLineBreakTest NotebookFileLineBreakTest NotebookDataPosition[ 158, 7] NotebookDataLength[ 609093, 15887] NotebookOptionsPosition[ 549263, 14959] NotebookOutlinePosition[ 550285, 14990] CellTagsIndexPosition[ 550139, 14983] WindowFrame->Normal*) (* Beginning of Notebook Content *) Notebook[{ Cell[BoxData[ RowBox[{"Exit", "[", "]"}]], "Input", CellChangeTimes->{{3.9152071588722963`*^9, 3.9152071616052*^9}}, CellLabel->"In[25]:=",ExpressionUUID->"ec383ef3-5761-4a89-b51d-dc10c48b8d7e"], Cell[CellGroupData[{ Cell["GR2 computations in mathematica", "Title", CellChangeTimes->{{3.882947538691614*^9, 3.8829475451117487`*^9}, { 3.882982907992347*^9, 3.882982911723648*^9}}, FontSlant->"Italic",ExpressionUUID->"4a7af336-a134-4985-880a-0a9353fdb85e"], Cell["\<\ In this file we will collate some computations used during this course which \ would be painful to work out by hand but are simple in mathematica with a bit \ of code. (There is a package to do this however since it is sometimes a pain \ to first set up the package I have just provided some brute force code that \ will do the same job)\ \>", "Text", CellChangeTimes->{{3.88294754745632*^9, 3.882947629203136*^9}}, Background->RGBColor[ 0.88, 1, 0.88],ExpressionUUID->"27b8ffa3-392f-4b0b-b008-056bc2dd6cfb"], Cell[CellGroupData[{ Cell["Schwarzschild solution EOM check: brute force", "Subtitle", CellChangeTimes->{{3.877599985515149*^9, 3.87760000197775*^9}},ExpressionUUID->"798c2d03-c2c9-44a1-ba49-\ 29e45a89a85a"], Cell["\<\ We first input the coordinates we will use (and define dim to be the \ spacetime dimension by working out how many coordinates there are.)\ \>", "Text", CellChangeTimes->{{3.877600003980526*^9, 3.877600011472641*^9}, { 3.8776002159324636`*^9, 3.877600247364307*^9}, {3.877609726464471*^9, 3.877609752887948*^9}},ExpressionUUID->"93b68ed6-02ff-4a8f-8aaf-\ c95ebb93bf6f"], Cell[BoxData[{ RowBox[{ RowBox[{"xIN", "=", RowBox[{"{", RowBox[{"t", ",", "r", ",", "\[Theta]", ",", "\[Phi]"}], "}"}]}], ";"}], "\[IndentingNewLine]", RowBox[{ RowBox[{"dim", "=", RowBox[{"Length", "[", "xIN", "]"}]}], ";"}]}], "Input", CellChangeTimes->{ 3.877599544310772*^9, {3.877599583297554*^9, 3.877599603382118*^9}, { 3.877599943901291*^9, 3.877599972163571*^9}, {3.877609717245466*^9, 3.877609741133548*^9}, {3.882948195772376*^9, 3.882948196347543*^9}}, CellLabel->"In[91]:=",ExpressionUUID->"60ffc379-0cb3-4429-b4d9-8c0504fe50d3"], Cell["\<\ Make sure to press shift and enter together so that xIN turns from blue to \ black. The semi-colon makes mathematica evaluate the expression (after \ pressing shift enter) but hides the result. \ \>", "Text", CellChangeTimes->{{3.877600249760886*^9, 3.877600300556264*^9}},ExpressionUUID->"48d61072-3a80-4faf-a3cf-\ e2ef5281d989"], Cell["Next we want to input the metric:", "Text", CellChangeTimes->{{3.877600013709251*^9, 3.877600022839692*^9}},ExpressionUUID->"52b91fe9-4920-467f-ba50-\ 21c1f74ba968"], Cell[BoxData[ RowBox[{ SubscriptBox["g", "\[Mu]\[Nu]"], "=", RowBox[{"(", RowBox[{GridBox[{ { RowBox[{"-", RowBox[{"(", RowBox[{"1", "-", FractionBox[ SubscriptBox["r", "s"], "r"]}], ")"}]}]}, {"0"}, {"0"}, {"0"} }], GridBox[{ {"0"}, { SuperscriptBox[ RowBox[{"(", RowBox[{"1", "-", FractionBox[ SubscriptBox["r", "s"], "r"]}], ")"}], RowBox[{"-", "1"}]]}, {"0"}, {"0"} }], GridBox[{ {"0"}, {"0"}, { SuperscriptBox["r", "2"]}, {"0"} }], GridBox[{ {"0"}, {"0"}, {"0"}, { RowBox[{ SuperscriptBox["r", "2"], SuperscriptBox["sin", "2"], "\[Theta]"}]} }]}], ")"}]}]], "DisplayFormulaNumbered", CellChangeTimes->{{3.8776000265452833`*^9, 3.877600115345433*^9}},ExpressionUUID->"6f2008ec-5a88-497f-bbb4-\ 5a43113d27a0"], Cell["\<\ To get the power you should press control with the ^. 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There is \ a pre-built in function for computing the determinant, Det[].\ \>", "Text", CellChangeTimes->{{3.8776003464055634`*^9, 3.877600360419662*^9}, { 3.8776004872533607`*^9, 3.8776005086287613`*^9}},ExpressionUUID->"f70d14c5-c45f-49c9-b342-\ e8f053f4270e"], Cell[CellGroupData[{ Cell[BoxData[{ RowBox[{"Simplify", "[", RowBox[{"Det", "[", "gdd", "]"}], "]"}], "\[IndentingNewLine]", RowBox[{ RowBox[{"Det", "[", "gdd", "]"}], "//", "Simplify"}]}], "Input", CellChangeTimes->{{3.877600361888378*^9, 3.877600369396488*^9}, { 3.8776007624261827`*^9, 3.877600770512642*^9}, {3.877605422283029*^9, 3.8776054228804483`*^9}}, CellLabel-> "In[113]:=",ExpressionUUID->"2cfb6e6e-de5c-4780-91ff-ddb925943acb"], Cell[BoxData[ RowBox[{ RowBox[{"-", SuperscriptBox["r", "4"]}], " ", SuperscriptBox[ RowBox[{"Sin", "[", "\[Theta]", "]"}], "2"]}]], "Output", CellChangeTimes->{{3.877600364737482*^9, 3.877600369786489*^9}, { 3.8776007655484247`*^9, 3.8776007710928802`*^9}, 3.877605423436298*^9, 3.877605920170188*^9, 3.882948404477663*^9, 3.8829784640287743`*^9}, CellLabel-> "Out[113]=",ExpressionUUID->"fc7d9a43-53de-492e-9c5e-e04097a5e79b"], Cell[BoxData[ RowBox[{ RowBox[{"-", SuperscriptBox["r", "4"]}], " ", SuperscriptBox[ RowBox[{"Sin", "[", "\[Theta]", "]"}], "2"]}]], "Output", CellChangeTimes->{{3.877600364737482*^9, 3.877600369786489*^9}, { 3.8776007655484247`*^9, 3.8776007710928802`*^9}, 3.877605423436298*^9, 3.877605920170188*^9, 3.882948404477663*^9, 3.8829784640327063`*^9}, CellLabel-> "Out[114]=",ExpressionUUID->"0127ebcc-e9d3-473a-882d-e1356ab23800"] }, Open ]], Cell["\<\ this is as we expect. The Simplify appearing does as one would guess and \ simplifies the results. One can use either method, wither inputting \ everything between two Simplify[] or putting //Simplify at the end. My \ preference is Simplify[] as it makes inputting assumptions easier. \ \>", "Text", CellChangeTimes->{{3.877600511048835*^9, 3.877600516327661*^9}, { 3.877600649876185*^9, 3.877600650301983*^9}, {3.877600744528269*^9, 3.87760082422997*^9}},ExpressionUUID->"188c60a2-0f07-4837-81b7-\ 30c1d6f8cffd"], Cell["Now compute the inverse and call this gUU", "Text", CellChangeTimes->{{3.877600827415165*^9, 3.87760083676794*^9}},ExpressionUUID->"465c2305-d6e3-465e-adfe-\ e876caca0d18"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"gUU", "=", RowBox[{"Simplify", "[", RowBox[{"Inverse", "[", "gdd", "]"}], "]"}]}]], "Input", CellChangeTimes->{{3.877600839051709*^9, 3.877600848428295*^9}}, CellLabel-> "In[115]:=",ExpressionUUID->"88f95112-0bc0-43c5-95c1-9487c2ac421b"], Cell[BoxData[ RowBox[{"{", RowBox[{ RowBox[{"{", RowBox[{ RowBox[{"-", FractionBox["r", RowBox[{"r", "-", "rs"}]]}], ",", "0", ",", "0", ",", "0"}], "}"}], ",", RowBox[{"{", RowBox[{"0", ",", RowBox[{"1", "-", FractionBox["rs", "r"]}], ",", "0", ",", "0"}], "}"}], ",", RowBox[{"{", RowBox[{"0", ",", "0", ",", FractionBox["1", SuperscriptBox["r", "2"]], ",", "0"}], "}"}], ",", RowBox[{"{", RowBox[{"0", ",", "0", ",", "0", ",", FractionBox[ SuperscriptBox[ RowBox[{"Csc", "[", "\[Theta]", "]"}], "2"], SuperscriptBox["r", "2"]]}], "}"}]}], "}"}]], "Output", CellChangeTimes->{3.877600849261726*^9, 3.877605425161313*^9, 3.877605922366514*^9, 3.882948411824533*^9, 3.8829784657456923`*^9}, CellLabel-> "Out[115]=",ExpressionUUID->"6243322e-6f78-4849-a8dd-8a49bd59f4d7"] }, Open ]], Cell["\<\ As a check we can see that this gives the identity matrix. There are two ways \ we can do this either by summing over the indices or using the built in \ matrix product in mathematica. Let us do both. With the inbuilt matrix \ product using . we have\ \>", "Text", CellChangeTimes->{{3.8776008866732607`*^9, 3.877600960783593*^9}, { 3.877601031518764*^9, 3.877601044076414*^9}},ExpressionUUID->"126e51f5-5aa1-48d8-858e-\ 8b6befc444d7"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"Simplify", "[", RowBox[{ RowBox[{"gUU", ".", "gdd"}], "==", RowBox[{"IdentityMatrix", "[", "4", "]"}]}], "]"}]], "Input", CellChangeTimes->{{3.8776009625261097`*^9, 3.877600976186408*^9}, { 3.877601386585602*^9, 3.8776013904406033`*^9}, {3.8829484188065147`*^9, 3.882948419781498*^9}}, CellLabel-> "In[116]:=",ExpressionUUID->"3869afa3-b40f-4d49-8fd2-f8b5eb6a3bd4"], Cell[BoxData["True"], "Output", CellChangeTimes->{{3.877600965565276*^9, 3.8776009767826242`*^9}, 3.8776013911658983`*^9, 3.8776054270571337`*^9, 3.8776059244845333`*^9, { 3.882948413555607*^9, 3.882948420303266*^9}, 3.882978467823463*^9}, CellLabel-> "Out[116]=",ExpressionUUID->"c658b0f0-cbb2-44be-8f0d-eab3f6b2be84"] }, Open ]], Cell["\<\ For summing the indices we need to learn a few things. We need to be able to \ extract out the entries of the matrix. To do this we write gdd[[ 1,3 ]], \ note the double brackets. This will extract out the 13 component of the \ metric. For us this we be the cross term dt d\[Theta], of course this is zero \ here. Next note that we should be obtaining a matrix since we have two free \ indices that are not summed over. To write this properly we need to use Table \ which arrays the values for each combination into a table (matrix here). Note below that we sum over the m2 indices and use table for the m1 and m3 \ indices. \ \>", "Text", CellChangeTimes->{{3.877601046841076*^9, 3.877601270068255*^9}, 3.91519976445214*^9},ExpressionUUID->"555995e5-52a8-4991-a4ab-\ 2a8f713f4f8f"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"Simplify", "[", RowBox[{ RowBox[{"Table", "[", RowBox[{ RowBox[{"Sum", "[", RowBox[{ RowBox[{ RowBox[{"gUU", "[", RowBox[{"[", RowBox[{"m1", ",", "m2"}], "]"}], "]"}], RowBox[{"gdd", "[", RowBox[{"[", RowBox[{"m2", ",", "m3"}], "]"}], "]"}]}], ",", RowBox[{"{", RowBox[{"m2", ",", "1", ",", "dim"}], "}"}]}], "]"}], ",", RowBox[{"{", RowBox[{"m1", ",", "1", ",", "dim"}], "}"}], ",", RowBox[{"{", RowBox[{"m3", ",", "1", ",", "dim"}], "}"}]}], "]"}], "==", RowBox[{"IdentityMatrix", "[", "4", "]"}]}], "]"}]], "Input", CellChangeTimes->{{3.87760097996716*^9, 3.8776010263911133`*^9}, { 3.87760139343246*^9, 3.87760139645409*^9}, {3.877609764543274*^9, 3.87760977103544*^9}}, CellLabel->"In[15]:=",ExpressionUUID->"52ba5eed-05bd-4460-8540-82f85ef9adba"], Cell[BoxData["True"], "Output", CellChangeTimes->{3.877601027152958*^9, 3.877601396882928*^9, 3.877605429187542*^9, 3.877605926507387*^9, 3.877609771572414*^9, 3.88294843103918*^9, 3.882978469862207*^9, 3.9151997828981457`*^9}, CellLabel->"Out[15]=",ExpressionUUID->"a9e700ba-f1c2-475d-bcb7-c2e18facd39c"] }, Open ]], Cell["We see that it is the identity and therefore we are done. ", "Text", CellChangeTimes->{{3.877601274632653*^9, 3.877601296812881*^9}},ExpressionUUID->"9fdbe1dd-04c4-49ff-964b-\ ccc1980a0859"], Cell[CellGroupData[{ Cell["Christoffel symbols", "Subsubsection", CellChangeTimes->{{3.877601404370286*^9, 3.8776014087221537`*^9}},ExpressionUUID->"4869f7fd-21e4-41d3-a61a-\ be4f310f4de3"], Cell["\<\ We can now compute the Christoffel symbols. Recall that they are given by\ \>", "Text", CellChangeTimes->{{3.877601410545577*^9, 3.877601427129437*^9}},ExpressionUUID->"5e6e5843-aee0-499c-bc78-\ 6001e9c0cf46"], Cell[BoxData[ RowBox[{ SubscriptBox[ SuperscriptBox["\[CapitalGamma]", "\[Rho]"], "\[Mu]\[Nu]"], "=", RowBox[{ FractionBox["1", "2"], SuperscriptBox["g", "\[Rho]\[Sigma]"], RowBox[{"(", RowBox[{ RowBox[{ SubscriptBox["\[PartialD]", "\[Mu]"], SubscriptBox["g", "\[Sigma]\[Nu]"]}], "+", RowBox[{ SubscriptBox["\[PartialD]", "\[Nu]"], SubscriptBox["g", "\[Sigma]\[Mu]"]}], "-", RowBox[{ SubscriptBox["\[PartialD]", "\[Sigma]"], SubscriptBox["g", "\[Mu]\[Nu]"]}]}], ")"}]}]}]], "DisplayFormulaNumbered", CellChangeTimes->{{3.8776032179808683`*^9, 3.877603263615623*^9}},ExpressionUUID->"a0781fb9-cc3f-4bdf-8411-\ efa74b3199fb"], Cell["\<\ We can code this into mathematic using: sum, table and derivative. To take a \ derivative we use D[ , ]. The first entry is what you want to take a \ derivative of and the second is what you want to take the derivative with \ respect to. We can take the derivative using the xIN above and picking the \ relevant component. So the Christoffel symbols would then be (make sure you \ do not use r as an index!)\ \>", "Text", CellChangeTimes->{{3.877603266328589*^9, 3.877603352854341*^9}, { 3.8776037383690987`*^9, 3.87760373935826*^9}, {3.877603770569165*^9, 3.87760378762915*^9}, {3.8776046690686483`*^9, 3.877604693351357*^9}},ExpressionUUID->"f7673559-7aa3-403e-bcc3-\ 696f4c8a59bd"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"\[CapitalGamma]Udd", "=", RowBox[{"Simplify", "[", RowBox[{"Table", "[", RowBox[{ RowBox[{ FractionBox["1", "2"], RowBox[{"Sum", "[", RowBox[{ RowBox[{ RowBox[{"gUU", "[", RowBox[{"[", RowBox[{"p1", ",", "s1"}], "]"}], "]"}], RowBox[{"(", RowBox[{ RowBox[{"D", "[", RowBox[{ RowBox[{"gdd", "[", RowBox[{"[", RowBox[{"s1", ",", "n1"}], "]"}], "]"}], ",", RowBox[{"xIN", "[", RowBox[{"[", "m1", "]"}], "]"}]}], "]"}], "+", RowBox[{"D", "[", RowBox[{ RowBox[{"gdd", "[", RowBox[{"[", RowBox[{"s1", ",", "m1"}], "]"}], "]"}], ",", RowBox[{"xIN", "[", RowBox[{"[", "n1", "]"}], "]"}]}], "]"}], "-", 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We see that this sets the conserved quantities associated to the J1 and \ J2 to vanish but gives a non-trivial conserved quantity for J3" }], "Text", CellChangeTimes->{{3.8829827108755503`*^9, 3.8829827647594624`*^9}, { 3.8829828296005898`*^9, 3.882982895566334*^9}},ExpressionUUID->"b5a36c06-568f-4dda-96d2-\ a19313916c86"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"Simplify", "[", RowBox[{ RowBox[{ RowBox[{"{", RowBox[{"J1cons", ",", "J2cons", ",", "J3cons"}], "}"}], "/.", RowBox[{ RowBox[{"\[Theta]", "[", "\[Lambda]", "]"}], "->", FractionBox["\[Pi]", "2"]}]}], "/.", RowBox[{ RowBox[{ RowBox[{"\[Theta]", "'"}], "[", "\[Lambda]", "]"}], "->", " ", "0"}]}], "]"}]], "Input", CellChangeTimes->{{3.882982119410569*^9, 3.88298213215884*^9}, { 3.8829822663898582`*^9, 3.882982304580846*^9}, {3.882982848837376*^9, 3.882982858122157*^9}}, CellLabel-> "In[303]:=",ExpressionUUID->"78d105bd-e6c5-49b1-ad10-d16ca77934ce"], Cell[BoxData[ RowBox[{"{", RowBox[{"0", ",", "0", ",", RowBox[{"2", " ", SuperscriptBox[ RowBox[{"r", "[", "\[Lambda]", "]"}], "2"], " ", RowBox[{ SuperscriptBox["\[Phi]", "\[Prime]", MultilineFunction->None], "[", "\[Lambda]", "]"}]}]}], "}"}]], "Output",\ CellChangeTimes->{{3.882982128522664*^9, 3.882982132651104*^9}, { 3.882982268570327*^9, 3.8829823051233664`*^9}, 3.8829828587331047`*^9}, CellLabel-> "Out[303]=",ExpressionUUID->"069ac81d-a8dc-4079-b12a-0841c0255c07"] }, Open ]] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell["Gravitational wave EOM check: brute force", "Subtitle", CellChangeTimes->{{3.877599985515149*^9, 3.87760000197775*^9}, { 3.9151995721707277`*^9, 3.915199575563587*^9}},ExpressionUUID->"4972f3c3-3438-4976-b3f5-\ 7729fafb0f12"], Cell["\<\ We first input the coordinates we will use (and define dim to be the \ spacetime dimension by working out how many coordinates there are.)\ \>", "Text", CellChangeTimes->{{3.877600003980526*^9, 3.877600011472641*^9}, { 3.8776002159324636`*^9, 3.877600247364307*^9}, {3.877609726464471*^9, 3.877609752887948*^9}},ExpressionUUID->"77e52de2-2ca3-4119-9f60-\ 8d166bb66643"], Cell[BoxData[{ RowBox[{ RowBox[{"xIN", "=", RowBox[{"{", RowBox[{"u", ",", "v", ",", "x", ",", "y"}], "}"}]}], ";"}], "\[IndentingNewLine]", RowBox[{ RowBox[{"dim", "=", RowBox[{"Length", "[", "xIN", "]"}]}], ";"}]}], "Input", CellChangeTimes->{ 3.877599544310772*^9, {3.877599583297554*^9, 3.877599603382118*^9}, { 3.877599943901291*^9, 3.877599972163571*^9}, {3.877609717245466*^9, 3.877609741133548*^9}, {3.882948195772376*^9, 3.882948196347543*^9}, { 3.9151995840626793`*^9, 3.9151995900379267`*^9}}, CellLabel->"In[1]:=",ExpressionUUID->"28890f60-41c0-436c-837c-b0d9ee6ba172"], Cell["\<\ Make sure to press shift and enter together so that xIN turns from blue to \ black. 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There is \ a pre-built in function for computing the determinant, Det[].\ \>", "Text", CellChangeTimes->{{3.8776003464055634`*^9, 3.877600360419662*^9}, { 3.8776004872533607`*^9, 3.8776005086287613`*^9}},ExpressionUUID->"46eaa293-841a-460f-9c04-\ d70f83c7f992"], Cell[CellGroupData[{ Cell[BoxData[{ RowBox[{"Simplify", "[", RowBox[{"Det", "[", "gdd", "]"}], "]"}], "\[IndentingNewLine]", RowBox[{ RowBox[{"Det", "[", "gdd", "]"}], "//", "Simplify"}]}], "Input", CellChangeTimes->{{3.877600361888378*^9, 3.877600369396488*^9}, { 3.8776007624261827`*^9, 3.877600770512642*^9}, {3.877605422283029*^9, 3.8776054228804483`*^9}}, CellLabel->"In[11]:=",ExpressionUUID->"a4dcb9e0-aa8f-4fd6-805b-ef5df30980d8"], Cell[BoxData[ RowBox[{ RowBox[{"-", FractionBox["1", "4"]}], " ", SuperscriptBox[ RowBox[{"f", "[", "u", "]"}], "2"], " ", SuperscriptBox[ RowBox[{"g", "[", "u", "]"}], "2"]}]], "Output", CellChangeTimes->{{3.877600364737482*^9, 3.877600369786489*^9}, { 3.8776007655484247`*^9, 3.8776007710928802`*^9}, 3.877605423436298*^9, 3.877605920170188*^9, 3.882948404477663*^9, 3.8829784640287743`*^9, 3.91519972166208*^9}, CellLabel->"Out[11]=",ExpressionUUID->"94dad161-1ab3-40d5-8f06-f55c1835c71a"], Cell[BoxData[ RowBox[{ RowBox[{"-", FractionBox["1", "4"]}], " ", SuperscriptBox[ RowBox[{"f", "[", "u", "]"}], "2"], " ", SuperscriptBox[ RowBox[{"g", "[", "u", "]"}], "2"]}]], "Output", CellChangeTimes->{{3.877600364737482*^9, 3.877600369786489*^9}, { 3.8776007655484247`*^9, 3.8776007710928802`*^9}, 3.877605423436298*^9, 3.877605920170188*^9, 3.882948404477663*^9, 3.8829784640287743`*^9, 3.915199721666597*^9}, CellLabel->"Out[12]=",ExpressionUUID->"65d6e3b3-52a0-4b09-b471-838cb0645e51"] }, Open ]], Cell["\<\ this is as we expect. The Simplify appearing does as one would guess and \ simplifies the results. One can use either method, wither inputting \ everything between two Simplify[] or putting //Simplify at the end. My \ preference is Simplify[] as it makes inputting assumptions easier. \ \>", "Text", CellChangeTimes->{{3.877600511048835*^9, 3.877600516327661*^9}, { 3.877600649876185*^9, 3.877600650301983*^9}, {3.877600744528269*^9, 3.87760082422997*^9}},ExpressionUUID->"5c6611ba-9637-4357-8c3a-\ a29fa5c88627"], Cell["Now compute the inverse and call this gUU", "Text", CellChangeTimes->{{3.877600827415165*^9, 3.87760083676794*^9}},ExpressionUUID->"fe5c9738-7823-464c-bbd8-\ 15b1946a0949"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"gUU", "=", RowBox[{"Simplify", "[", RowBox[{"Inverse", "[", "gdd", "]"}], "]"}]}]], "Input", CellChangeTimes->{{3.877600839051709*^9, 3.877600848428295*^9}}, CellLabel->"In[8]:=",ExpressionUUID->"fdb6a060-1305-4228-86e4-eea8baaded6f"], Cell[BoxData[ RowBox[{"{", RowBox[{ RowBox[{"{", RowBox[{"0", ",", RowBox[{"-", "2"}], ",", "0", ",", "0"}], "}"}], ",", RowBox[{"{", RowBox[{ RowBox[{"-", "2"}], ",", "0", ",", "0", ",", "0"}], "}"}], ",", RowBox[{"{", RowBox[{"0", ",", "0", ",", FractionBox["1", SuperscriptBox[ RowBox[{"f", "[", "u", "]"}], "2"]], ",", "0"}], "}"}], ",", RowBox[{"{", RowBox[{"0", ",", "0", ",", "0", ",", FractionBox["1", SuperscriptBox[ RowBox[{"g", "[", "u", "]"}], "2"]]}], "}"}]}], "}"}]], "Output", CellChangeTimes->{3.877600849261726*^9, 3.877605425161313*^9, 3.877605922366514*^9, 3.882948411824533*^9, 3.8829784657456923`*^9, 3.915199728617901*^9, 3.915207178166359*^9}, CellLabel->"Out[8]=",ExpressionUUID->"c386efaa-8cfc-4c96-b96d-829995bd8e3f"] }, Open ]], Cell["\<\ As a check we can see that this gives the identity matrix. There are two ways \ we can do this either by summing over the indices or using the built in \ matrix product in mathematica. Let us do both. With the inbuilt matrix \ product using . we have\ \>", "Text", CellChangeTimes->{{3.8776008866732607`*^9, 3.877600960783593*^9}, { 3.877601031518764*^9, 3.877601044076414*^9}},ExpressionUUID->"ed75d263-f29b-4494-8e8e-\ 41d1e63631f1"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"Simplify", "[", RowBox[{ RowBox[{"gUU", ".", "gdd"}], "==", RowBox[{"IdentityMatrix", "[", "4", "]"}]}], "]"}]], "Input", CellChangeTimes->{{3.8776009625261097`*^9, 3.877600976186408*^9}, { 3.877601386585602*^9, 3.8776013904406033`*^9}, {3.8829484188065147`*^9, 3.882948419781498*^9}}, CellLabel->"In[14]:=",ExpressionUUID->"1369f3bf-8a34-4d4a-b58a-2614ea37f33b"], Cell[BoxData["True"], "Output", CellChangeTimes->{{3.877600965565276*^9, 3.8776009767826242`*^9}, 3.8776013911658983`*^9, 3.8776054270571337`*^9, 3.8776059244845333`*^9, { 3.882948413555607*^9, 3.882948420303266*^9}, 3.882978467823463*^9, 3.915199731213648*^9}, CellLabel->"Out[14]=",ExpressionUUID->"7b9e5e37-a997-48f3-8399-42ed09fcd49c"] }, Open ]], Cell["\<\ For summing the indices we need to learn a few things. We need to be able to \ extract out the entries of the matrix. To do this we write gdd[[ 1,3 ]], \ note the double brackets. This will extract out the 13 component of the \ metric. For us this we be the cross term dt d\[Theta], of course this is zero \ here. Next note that we should be obtaining a matrix since we have two free \ indices that are not summed over. To write this properly we need to use Table \ which arrays the values for each combination into a table (matrix here). Note below that we sum over the m2 indices and use table for the m1 and m3 \ indices. \ \>", "Text", CellChangeTimes->{{3.877601046841076*^9, 3.877601270068255*^9}, 3.915199751782832*^9},ExpressionUUID->"0e64eb6c-0fdc-4dc9-aca0-\ c3045663632c"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"Simplify", "[", RowBox[{ RowBox[{"Table", "[", RowBox[{ RowBox[{"Sum", "[", RowBox[{ RowBox[{ RowBox[{"gUU", "[", RowBox[{"[", RowBox[{"m1", ",", "m2"}], "]"}], "]"}], RowBox[{"gdd", "[", RowBox[{"[", RowBox[{"m2", ",", "m3"}], "]"}], "]"}]}], ",", RowBox[{"{", RowBox[{"m2", ",", "1", ",", "dim"}], "}"}]}], "]"}], ",", RowBox[{"{", RowBox[{"m1", ",", "1", ",", "dim"}], "}"}], ",", RowBox[{"{", RowBox[{"m3", ",", "1", ",", "dim"}], "}"}]}], "]"}], "==", RowBox[{"IdentityMatrix", "[", "4", "]"}]}], "]"}]], "Input", CellChangeTimes->{{3.87760097996716*^9, 3.8776010263911133`*^9}, { 3.87760139343246*^9, 3.87760139645409*^9}, {3.877609764543274*^9, 3.87760977103544*^9}}, CellLabel-> "In[117]:=",ExpressionUUID->"66ff8a84-4ace-4694-a36b-eca0446a569d"], Cell[BoxData["True"], "Output", CellChangeTimes->{3.877601027152958*^9, 3.877601396882928*^9, 3.877605429187542*^9, 3.877605926507387*^9, 3.877609771572414*^9, 3.88294843103918*^9, 3.882978469862207*^9}, CellLabel-> "Out[117]=",ExpressionUUID->"1e34dd6f-2ba8-4c93-8847-b9f63a2439b9"] }, Open ]], Cell["We see that it is the identity and therefore we are done. ", "Text", CellChangeTimes->{{3.877601274632653*^9, 3.877601296812881*^9}},ExpressionUUID->"cbfbacc4-012e-412b-a795-\ a4e912059594"], Cell[CellGroupData[{ Cell["Christoffel symbols", "Subsubsection", CellChangeTimes->{{3.877601404370286*^9, 3.8776014087221537`*^9}},ExpressionUUID->"a3f16bc5-6d13-4beb-af9b-\ 9cd6f23b4c28"], Cell["\<\ We can now compute the Christoffel symbols. Recall that they are given by\ \>", "Text", CellChangeTimes->{{3.877601410545577*^9, 3.877601427129437*^9}},ExpressionUUID->"afb5ae48-7041-4a3d-985f-\ fefd9799c7ba"], Cell[BoxData[ RowBox[{ SubscriptBox[ SuperscriptBox["\[CapitalGamma]", "\[Rho]"], "\[Mu]\[Nu]"], "=", RowBox[{ FractionBox["1", "2"], SuperscriptBox["g", "\[Rho]\[Sigma]"], RowBox[{"(", RowBox[{ RowBox[{ SubscriptBox["\[PartialD]", "\[Mu]"], SubscriptBox["g", "\[Sigma]\[Nu]"]}], "+", RowBox[{ SubscriptBox["\[PartialD]", "\[Nu]"], SubscriptBox["g", "\[Sigma]\[Mu]"]}], "-", RowBox[{ SubscriptBox["\[PartialD]", "\[Sigma]"], SubscriptBox["g", "\[Mu]\[Nu]"]}]}], ")"}]}]}]], "DisplayFormulaNumbered", CellChangeTimes->{{3.8776032179808683`*^9, 3.877603263615623*^9}},ExpressionUUID->"2420bbaf-2557-4950-9467-\ f4ad3c2d29b1"], Cell["\<\ We can code this into mathematic using: sum, table and derivative. To take a \ derivative we use D[ , ]. The first entry is what you want to take a \ derivative of and the second is what you want to take the derivative with \ respect to. We can take the derivative using the xIN above and picking the \ relevant component. So the Christoffel symbols would then be (make sure you \ do not use r as an index!)\ \>", "Text", CellChangeTimes->{{3.877603266328589*^9, 3.877603352854341*^9}, { 3.8776037383690987`*^9, 3.87760373935826*^9}, {3.877603770569165*^9, 3.87760378762915*^9}, {3.8776046690686483`*^9, 3.877604693351357*^9}},ExpressionUUID->"d4c7b0ba-4e1c-4015-93ab-\ 330b2c497e98"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"\[CapitalGamma]Udd", "=", RowBox[{"Simplify", "[", RowBox[{"Table", "[", RowBox[{ RowBox[{ FractionBox["1", "2"], RowBox[{"Sum", "[", RowBox[{ RowBox[{ RowBox[{"gUU", "[", RowBox[{"[", RowBox[{"p1", ",", "s1"}], "]"}], "]"}], RowBox[{"(", RowBox[{ RowBox[{"D", "[", RowBox[{ RowBox[{"gdd", "[", RowBox[{"[", RowBox[{"s1", ",", "n1"}], "]"}], "]"}], ",", RowBox[{"xIN", "[", RowBox[{"[", "m1", "]"}], "]"}]}], "]"}], "+", RowBox[{"D", "[", RowBox[{ RowBox[{"gdd", "[", RowBox[{"[", RowBox[{"s1", ",", "m1"}], "]"}], "]"}], ",", RowBox[{"xIN", "[", RowBox[{"[", "n1", "]"}], "]"}]}], "]"}], "-", 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CellChangeTimes->{{3.877600117466735*^9, 3.877600128547988*^9}, { 3.8776001872685823`*^9, 3.877600188401208*^9}, {3.877600304793062*^9, 3.877600332379765*^9}, {3.882950482289422*^9, 3.8829504830884523`*^9}, { 3.8829505280881166`*^9, 3.882950543147928*^9}}, CellLabel-> "In[113]:=",ExpressionUUID->"29120fc6-989f-4ea9-ac6f-46325c48f30d"], Cell["\<\ Or we can use the code below to input the line element and obtain a matrix. \ \>", "Text", CellChangeTimes->{{3.882951898013541*^9, 3.882951913633079*^9}}, Background->RGBColor[ 0.88, 1, 0.88],ExpressionUUID->"c85f0c31-dcaf-4108-ad1c-647250f14555"], Cell[BoxData[ RowBox[{ RowBox[{"metfromlineelement", ":=", RowBox[{ RowBox[{"Simplify", "[", RowBox[{"Table", "[", RowBox[{ RowBox[{"Coefficient", "[", RowBox[{ RowBox[{ FractionBox["#", "2"], RowBox[{"(", RowBox[{"1", "+", RowBox[{"KroneckerDelta", "[", RowBox[{"m1", ",", "m2"}], "]"}]}], ")"}]}], ",", RowBox[{ RowBox[{"d", "[", RowBox[{"xIN", "[", RowBox[{"[", "m1", "]"}], "]"}], "]"}], 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0.88],ExpressionUUID->"46494c93-7e4e-4966-af57-cb451e4dc0f9"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"MatrixForm", "[", "gdd", "]"}]], "Input", CellChangeTimes->{{3.87760121135117*^9, 3.877601214802021*^9}}, CellLabel-> "In[117]:=",ExpressionUUID->"4761b446-eb20-48d5-9ebe-56b8932b27f8"], Cell[BoxData[ TagBox[ RowBox[{"(", "\[NoBreak]", GridBox[{ { RowBox[{"-", SuperscriptBox["\[ExponentialE]", RowBox[{"2", " ", RowBox[{"\[CapitalPhi]\[CapitalPhi]", "[", "r", "]"}]}]]}], "0", "0", "0"}, {"0", SuperscriptBox["\[ExponentialE]", RowBox[{"2", " ", RowBox[{"\[CapitalPsi]\[CapitalPsi]", "[", "r", "]"}]}]], "0", "0"}, {"0", "0", SuperscriptBox["r", "2"], "0"}, {"0", "0", "0", RowBox[{ SuperscriptBox["r", "2"], " ", SuperscriptBox[ RowBox[{"Sin", "[", "\[Theta]", "]"}], "2"]}]} }, GridBoxAlignment->{"Columns" -> {{Center}}, "Rows" -> {{Baseline}}}, GridBoxSpacings->{"Columns" -> { Offset[0.27999999999999997`], { Offset[0.7]}, Offset[0.27999999999999997`]}, "Rows" -> { Offset[0.2], { 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Recall that they are given by\ \>", "Text", CellChangeTimes->{{3.877601410545577*^9, 3.877601427129437*^9}}, Background->RGBColor[ 0.88, 1, 0.88],ExpressionUUID->"8084a6db-bce3-482b-b34e-da6616311dc5"], Cell[BoxData[ RowBox[{ SubscriptBox[ SuperscriptBox["\[CapitalGamma]", "\[Rho]"], "\[Mu]\[Nu]"], "=", RowBox[{ FractionBox["1", "2"], SuperscriptBox["g", "\[Rho]\[Sigma]"], RowBox[{"(", RowBox[{ RowBox[{ SubscriptBox["\[PartialD]", "\[Mu]"], SubscriptBox["g", "\[Sigma]\[Nu]"]}], "+", RowBox[{ SubscriptBox["\[PartialD]", "\[Nu]"], SubscriptBox["g", "\[Sigma]\[Mu]"]}], "-", RowBox[{ SubscriptBox["\[PartialD]", "\[Sigma]"], SubscriptBox["g", "\[Mu]\[Nu]"]}]}], ")"}]}]}]], "DisplayFormulaNumbered", CellChangeTimes->{{3.8776032179808683`*^9, 3.877603263615623*^9}}, Background->RGBColor[ 0.87, 0.94, 1],ExpressionUUID->"06d4d60a-d121-46a6-b9cd-7662ca90214d"], Cell["\<\ We can code this into mathematic using: sum, table and derivative. To take a \ derivative we use D[ , ]. The first entry is what you want to take a \ derivative of and the second is what you want to take the derivative with \ respect to. We can take the derivative using the xIN above and picking the \ relevant component. 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Recall that the second Bianchi identity implies that the Einstein tensor is \ conserved and therefore the Energy-Momentum tensor should also be conserved. \ This must therefore come out of the Einstein equations immediately so we may \ replace the final Einstein equation with the conservation of the Energy \ momentum tensor\ \>", "Text", CellChangeTimes->{{3.882953276139241*^9, 3.882953282602858*^9}, { 3.882953372336936*^9, 3.8829533797730637`*^9}, {3.8829539500362597`*^9, 3.882953960605811*^9}, {3.88295419475889*^9, 3.882954274767602*^9}}, Background->RGBColor[ 0.88, 1, 0.88],ExpressionUUID->"f37705ee-c5e5-4b44-ab04-027f5951fa4f"], Cell[BoxData[ RowBox[{ RowBox[{ SubscriptBox["\[Del]", "\[Mu]"], SubscriptBox[ SuperscriptBox["T", "\[Mu]"], "\[Nu]"]}], "=", "0"}]], "DisplayFormulaNumbered", CellChangeTimes->{{3.882954285382062*^9, 3.882954299411683*^9}},ExpressionUUID->"eb20cf44-f02d-459a-b42d-\ 3874cadba5c4"], Cell[TextData[{ "To proceed first raise the index on ", Cell[BoxData[ FormBox[ SubscriptBox["T", "\[Mu]\[Nu]"], TraditionalForm]], FormatType->TraditionalForm,ExpressionUUID-> "2be85c7c-21cd-4d02-9a86-2026a4b31adb"], " to get ", Cell[BoxData[ SubscriptBox[ SuperscriptBox["T", "\[Mu]"], "\[Nu]"]], CellChangeTimes->{{3.882954285382062*^9, 3.882954299411683*^9}}, ExpressionUUID->"2cff6f1a-d031-4132-8533-44d9153162d4"], ". 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Let us double check this. 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" }], "Text", CellChangeTimes->{{3.8844266082184877`*^9, 3.884426637710042*^9}},ExpressionUUID->"6024907e-e307-4937-a074-\ 5f03c92a8000"], Cell["\<\ We first input the coordinates we will use (and define dim to be the \ spacetime dimension by working out how many coordinates there are.)\ \>", "Text", CellChangeTimes->{{3.877600003980526*^9, 3.877600011472641*^9}, { 3.8776002159324636`*^9, 3.877600247364307*^9}, {3.877609726464471*^9, 3.877609752887948*^9}},ExpressionUUID->"e1fa25c2-7866-4a24-b61a-\ 4a085084ca8a"], Cell[BoxData[{ RowBox[{ RowBox[{"xIN", "=", RowBox[{"{", RowBox[{"t", ",", "r", ",", "\[Theta]", ",", "\[Phi]"}], "}"}]}], ";"}], "\[IndentingNewLine]", RowBox[{ RowBox[{"dim", "=", RowBox[{"Length", "[", "xIN", "]"}]}], ";"}]}], "Input", CellChangeTimes->{ 3.877599544310772*^9, {3.877599583297554*^9, 3.877599603382118*^9}, { 3.877599943901291*^9, 3.877599972163571*^9}, {3.877609717245466*^9, 3.877609741133548*^9}, {3.882948195772376*^9, 3.882948196347543*^9}}, CellLabel->"In[1]:=",ExpressionUUID->"70a04cc4-5cdd-46eb-ab08-929a1f4f7426"], Cell["\<\ Make sure to press shift and enter together so that xIN turns from blue to \ black. The semi-colon makes mathematica evaluate the expression (after \ pressing shift enter) but hides the result. \ \>", "Text", CellChangeTimes->{{3.877600249760886*^9, 3.877600300556264*^9}},ExpressionUUID->"c7a85080-08ef-4ad9-9552-\ 3b61f097550e"], Cell["Next we want to input the metric:", "Text", CellChangeTimes->{{3.877600013709251*^9, 3.877600022839692*^9}},ExpressionUUID->"2196f548-c950-41df-b983-\ cb94722466d5"], Cell[BoxData[ RowBox[{ SubscriptBox["g", "\[Mu]\[Nu]"], "=", RowBox[{"(", RowBox[{GridBox[{ { RowBox[{ RowBox[{"-", "f"}], RowBox[{"(", "r", ")"}]}]}, {"0"}, {"0"}, {"0"} }], GridBox[{ {"0"}, { RowBox[{"g", RowBox[{"(", "r", ")"}]}]}, {"0"}, {"0"} }], GridBox[{ {"0"}, {"0"}, { SuperscriptBox["r", "2"]}, {"0"} }], GridBox[{ {"0"}, {"0"}, {"0"}, { RowBox[{ SuperscriptBox["r", "2"], SuperscriptBox["sin", "2"], "\[Theta]"}]} }]}], ")"}]}]], "DisplayFormulaNumbered", CellChangeTimes->{{3.8776000265452833`*^9, 3.877600115345433*^9}, { 3.884423787768649*^9, 3.884423793362671*^9}},ExpressionUUID->"5408efe3-02f5-46c2-8ffa-\ e581e2c666de"], Cell["\<\ To get the power you should press control with the ^. 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There is \ a pre-built in function for computing the determinant, Det[].\ \>", "Text", CellChangeTimes->{{3.8776003464055634`*^9, 3.877600360419662*^9}, { 3.8776004872533607`*^9, 3.8776005086287613`*^9}},ExpressionUUID->"1d4b4267-7b33-4797-9809-\ c53c20ed107e"], Cell[CellGroupData[{ Cell[BoxData[{ RowBox[{"Simplify", "[", RowBox[{"Det", "[", "gdd", "]"}], "]"}], "\[IndentingNewLine]", RowBox[{ RowBox[{"Det", "[", "gdd", "]"}], "//", "Simplify"}]}], "Input", CellChangeTimes->{{3.877600361888378*^9, 3.877600369396488*^9}, { 3.8776007624261827`*^9, 3.877600770512642*^9}, {3.877605422283029*^9, 3.8776054228804483`*^9}}, CellLabel->"In[10]:=",ExpressionUUID->"3bb38455-748c-4a95-b898-b10babd5ab7e"], Cell[BoxData[ RowBox[{ RowBox[{"-", SuperscriptBox["r", "4"]}], " ", RowBox[{"f", "[", "r", "]"}], " ", RowBox[{"g", "[", "r", "]"}], " ", SuperscriptBox[ RowBox[{"Sin", "[", "\[Theta]", "]"}], "2"]}]], "Output", CellChangeTimes->{{3.877600364737482*^9, 3.877600369786489*^9}, { 3.8776007655484247`*^9, 3.8776007710928802`*^9}, 3.877605423436298*^9, 3.877605920170188*^9, 3.882948404477663*^9, 3.8829784640287743`*^9, 3.884423890568315*^9}, CellLabel->"Out[10]=",ExpressionUUID->"6fcc83ed-d22d-44d6-bbc8-e4308e3b5333"], Cell[BoxData[ RowBox[{ RowBox[{"-", SuperscriptBox["r", "4"]}], " ", RowBox[{"f", "[", "r", "]"}], " ", RowBox[{"g", "[", "r", "]"}], " ", SuperscriptBox[ RowBox[{"Sin", "[", "\[Theta]", "]"}], "2"]}]], "Output", CellChangeTimes->{{3.877600364737482*^9, 3.877600369786489*^9}, { 3.8776007655484247`*^9, 3.8776007710928802`*^9}, 3.877605423436298*^9, 3.877605920170188*^9, 3.882948404477663*^9, 3.8829784640287743`*^9, 3.884423890572297*^9}, CellLabel->"Out[11]=",ExpressionUUID->"1bc60faf-81c2-408d-b2ca-357ad69fe6dc"] }, Open ]], Cell["\<\ this is as we expect. The Simplify appearing does as one would guess and \ simplifies the results. One can use either method, wither inputting \ everything between two Simplify[] or putting //Simplify at the end. My \ preference is Simplify[] as it makes inputting assumptions easier. \ \>", "Text", CellChangeTimes->{{3.877600511048835*^9, 3.877600516327661*^9}, { 3.877600649876185*^9, 3.877600650301983*^9}, {3.877600744528269*^9, 3.87760082422997*^9}},ExpressionUUID->"43745a9b-e0df-4046-9351-\ 99e4cb82488d"], Cell["Now compute the inverse and call this gUU", "Text", CellChangeTimes->{{3.877600827415165*^9, 3.87760083676794*^9}},ExpressionUUID->"254302df-f716-4df8-ab1b-\ 5b4a739fb110"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"gUU", "=", RowBox[{"Simplify", "[", RowBox[{"Inverse", "[", "gdd", "]"}], "]"}]}]], "Input", CellChangeTimes->{{3.877600839051709*^9, 3.877600848428295*^9}}, CellLabel->"In[12]:=",ExpressionUUID->"74a281e7-184a-4f9d-987c-55534b24e325"], Cell[BoxData[ RowBox[{"{", RowBox[{ RowBox[{"{", RowBox[{ RowBox[{"-", FractionBox["1", RowBox[{"f", "[", "r", "]"}]]}], ",", "0", ",", "0", ",", "0"}], "}"}], ",", RowBox[{"{", RowBox[{"0", ",", FractionBox["1", RowBox[{"g", "[", "r", "]"}]], ",", "0", ",", "0"}], "}"}], ",", RowBox[{"{", RowBox[{"0", ",", "0", ",", FractionBox["1", SuperscriptBox["r", "2"]], ",", "0"}], "}"}], ",", RowBox[{"{", RowBox[{"0", ",", "0", ",", "0", ",", FractionBox[ SuperscriptBox[ RowBox[{"Csc", "[", "\[Theta]", "]"}], "2"], SuperscriptBox["r", "2"]]}], "}"}]}], "}"}]], "Output", CellChangeTimes->{3.877600849261726*^9, 3.877605425161313*^9, 3.877605922366514*^9, 3.882948411824533*^9, 3.8829784657456923`*^9, 3.884423893094698*^9}, CellLabel->"Out[12]=",ExpressionUUID->"202f1f87-54b6-415b-a36f-4829c390bf15"] }, Open ]], Cell["\<\ As a check we can see that this gives the identity matrix. There are two ways \ we can do this either by summing over the indices or using the built in \ matrix product in mathematica. Let us do both. With the inbuilt matrix \ product using . we have\ \>", "Text", CellChangeTimes->{{3.8776008866732607`*^9, 3.877600960783593*^9}, { 3.877601031518764*^9, 3.877601044076414*^9}},ExpressionUUID->"fe8a9b06-2d73-44b0-a84e-\ 3755048f7a1f"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"Simplify", "[", RowBox[{ RowBox[{"gUU", ".", "gdd"}], "==", RowBox[{"IdentityMatrix", "[", "4", "]"}]}], "]"}]], "Input", CellChangeTimes->{{3.8776009625261097`*^9, 3.877600976186408*^9}, { 3.877601386585602*^9, 3.8776013904406033`*^9}, {3.8829484188065147`*^9, 3.882948419781498*^9}}, CellLabel->"In[13]:=",ExpressionUUID->"caf6e1a8-8d23-4cb6-8088-44d71eb072ee"], Cell[BoxData["True"], "Output", CellChangeTimes->{{3.877600965565276*^9, 3.8776009767826242`*^9}, 3.8776013911658983`*^9, 3.8776054270571337`*^9, 3.8776059244845333`*^9, { 3.882948413555607*^9, 3.882948420303266*^9}, 3.882978467823463*^9, 3.8844238959575644`*^9}, CellLabel->"Out[13]=",ExpressionUUID->"cc428aaf-8931-4de9-bc18-1e75e16fd223"] }, Open ]], Cell["\<\ For summing the indices we need to learn a few things. We need to be able to \ extract out the entries of the matrix. To do this we write gdd[[ 1,3 ]], \ note the double brackets. This will extract out the 13 component of the \ metric. For us this we be the cross term dt d\[Theta], of course this is zero \ her. Next note that we should be obtaining a matrix since we have two free \ indices that are not summed over. To write this properly we need to use Table \ which arrays the values for each combination into a table (matrix here). Note below that we sum over the m2 indices and use table for the m1 and m3 \ indices. \ \>", "Text", CellChangeTimes->{{3.877601046841076*^9, 3.877601270068255*^9}},ExpressionUUID->"ccd2745c-2f90-46a4-8e38-\ 21e407b5226c"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"Simplify", "[", RowBox[{ RowBox[{"Table", "[", RowBox[{ RowBox[{"Sum", "[", RowBox[{ RowBox[{ RowBox[{"gUU", "[", RowBox[{"[", RowBox[{"m1", ",", "m2"}], "]"}], "]"}], RowBox[{"gdd", "[", RowBox[{"[", RowBox[{"m2", ",", "m3"}], "]"}], "]"}]}], ",", RowBox[{"{", RowBox[{"m2", ",", "1", ",", "dim"}], "}"}]}], "]"}], ",", RowBox[{"{", RowBox[{"m1", ",", "1", ",", "dim"}], "}"}], ",", RowBox[{"{", RowBox[{"m3", ",", "1", ",", "dim"}], "}"}]}], "]"}], "==", RowBox[{"IdentityMatrix", "[", "4", "]"}]}], "]"}]], "Input", CellChangeTimes->{{3.87760097996716*^9, 3.8776010263911133`*^9}, { 3.87760139343246*^9, 3.87760139645409*^9}, {3.877609764543274*^9, 3.87760977103544*^9}}, CellLabel->"In[14]:=",ExpressionUUID->"f264eb17-1f12-465c-9c71-2c8f973f5cef"], Cell[BoxData["True"], "Output", CellChangeTimes->{3.877601027152958*^9, 3.877601396882928*^9, 3.877605429187542*^9, 3.877605926507387*^9, 3.877609771572414*^9, 3.88294843103918*^9, 3.882978469862207*^9, 3.884423897741044*^9}, CellLabel->"Out[14]=",ExpressionUUID->"5609bfe1-2c26-4b20-a975-d1c120b148e1"] }, Open ]], Cell["We see that it is the identity and therefore we are done. ", "Text", CellChangeTimes->{{3.877601274632653*^9, 3.877601296812881*^9}},ExpressionUUID->"0e2571e0-24cf-49a6-bf00-\ 7c8d5d3ef41e"], Cell[CellGroupData[{ Cell["Christoffel symbols", "Subsubsection", CellChangeTimes->{{3.877601404370286*^9, 3.8776014087221537`*^9}},ExpressionUUID->"9a3a5279-e1a9-41bd-80d8-\ 5aef3a817052"], Cell["\<\ We can now compute the Christoffel symbols. Recall that they are given by\ \>", "Text", CellChangeTimes->{{3.877601410545577*^9, 3.877601427129437*^9}},ExpressionUUID->"71a062b6-d095-4261-ac06-\ 20d1f5f0607c"], Cell[BoxData[ RowBox[{ SubscriptBox[ SuperscriptBox["\[CapitalGamma]", "\[Rho]"], "\[Mu]\[Nu]"], "=", RowBox[{ FractionBox["1", "2"], SuperscriptBox["g", "\[Rho]\[Sigma]"], RowBox[{"(", RowBox[{ RowBox[{ SubscriptBox["\[PartialD]", "\[Mu]"], SubscriptBox["g", "\[Sigma]\[Nu]"]}], "+", RowBox[{ SubscriptBox["\[PartialD]", "\[Nu]"], SubscriptBox["g", "\[Sigma]\[Mu]"]}], "-", RowBox[{ SubscriptBox["\[PartialD]", "\[Sigma]"], SubscriptBox["g", "\[Mu]\[Nu]"]}]}], ")"}]}]}]], "DisplayFormulaNumbered", CellChangeTimes->{{3.8776032179808683`*^9, 3.877603263615623*^9}},ExpressionUUID->"ca46673e-851b-4389-ab9b-\ f2c25979d2f5"], Cell["\<\ We can code this into mathematic using: sum, table and derivative. To take a \ derivative we use D[ , ]. The first entry is what you want to take a \ derivative of and the second is what you want to take the derivative with \ respect to. We can take the derivative using the xIN above and picking the \ relevant component. 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The semi-colon makes mathematica evaluate the expression (after \ pressing shift enter) but hides the result. \ \>", "Text", CellChangeTimes->{{3.877600249760886*^9, 3.877600300556264*^9}},ExpressionUUID->"2a16565c-feda-4691-81c2-\ 9460910dbf38"], Cell["Next we want to input the metric:", "Text", CellChangeTimes->{{3.877600013709251*^9, 3.877600022839692*^9}},ExpressionUUID->"b5be9a88-aa20-406a-94a0-\ 8f95fe7bed6d"], Cell[BoxData[ RowBox[{ SubscriptBox["g", "\[Mu]\[Nu]"], "=", RowBox[{"(", RowBox[{GridBox[{ { RowBox[{ RowBox[{"-", "f"}], RowBox[{"(", "r", ")"}]}]}, {"0"}, {"0"}, {"0"} }], GridBox[{ {"0"}, { RowBox[{"g", RowBox[{"(", "r", ")"}]}]}, {"0"}, {"0"} }], GridBox[{ {"0"}, {"0"}, { SuperscriptBox["\[Rho]", "2"]}, {"0"} }], GridBox[{ {"0"}, {"0"}, {"0"}, { RowBox[{ SuperscriptBox["\[Rho]", "2"], SuperscriptBox["sin", "2"], "\[Theta]"}]} }]}], ")"}]}]], "DisplayFormulaNumbered", CellChangeTimes->{{3.8776000265452833`*^9, 3.877600115345433*^9}, { 3.884423787768649*^9, 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{{Center}}, "Rows" -> {{Baseline}}}, GridBoxSpacings->{"Columns" -> { Offset[0.27999999999999997`], { Offset[0.7]}, Offset[0.27999999999999997`]}, "Rows" -> { Offset[0.2], { Offset[0.4]}, Offset[0.2]}}], "\[NoBreak]", ")"}], Function[BoxForm`e$, MatrixForm[BoxForm`e$]]]], "Output", CellChangeTimes->{3.8776012155017967`*^9, 3.877605419888331*^9, 3.877605918023896*^9, 3.877609759475008*^9, 3.882948402673052*^9, 3.8829784615677423`*^9, 3.8844238879610577`*^9, 3.884426699745843*^9, 3.884427086282588*^9}, CellLabel-> "Out[47]//MatrixForm=",ExpressionUUID->"d549b60d-ec1e-4bdd-b59b-\ 1b7a6a93dd5d"] }, Open ]], Cell["\<\ We now have our metric, we can compute the determinant for example. There is \ a pre-built in function for computing the determinant, Det[].\ \>", "Text", CellChangeTimes->{{3.8776003464055634`*^9, 3.877600360419662*^9}, { 3.8776004872533607`*^9, 3.8776005086287613`*^9}},ExpressionUUID->"ee6b4159-e19d-4f95-a3bf-\ 53f1ebaf07ed"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"Simplify", "[", RowBox[{"Det", "[", "gdd", "]"}], "]"}]], "Input", CellChangeTimes->{{3.877600361888378*^9, 3.877600369396488*^9}, { 3.8776007624261827`*^9, 3.877600770512642*^9}, {3.877605422283029*^9, 3.8776054228804483`*^9}, {3.884427094261078*^9, 3.884427094509157*^9}},ExpressionUUID->"94a74247-85dd-4bbd-b6e5-\ 35dc981a666f"], Cell[BoxData[ RowBox[{ RowBox[{"-", SuperscriptBox["\[Rho]", "4"]}], " ", RowBox[{"f", "[", "r", "]"}], " ", RowBox[{"g", "[", "r", "]"}], " ", SuperscriptBox[ RowBox[{"Sin", "[", "\[Theta]", "]"}], "2"]}]], "Output", CellChangeTimes->{{3.877600364737482*^9, 3.877600369786489*^9}, { 3.8776007655484247`*^9, 3.8776007710928802`*^9}, 3.877605423436298*^9, 3.877605920170188*^9, 3.882948404477663*^9, 3.8829784640287743`*^9, 3.884423890568315*^9, 3.8844267016401367`*^9, 3.884427088694949*^9}, CellLabel->"Out[48]=",ExpressionUUID->"c247ba5c-4256-429f-b0be-ebad383eefa7"], Cell[BoxData[ RowBox[{ RowBox[{"-", SuperscriptBox["\[Rho]", "4"]}], " ", RowBox[{"f", "[", "r", "]"}], " ", RowBox[{"g", "[", "r", "]"}], " ", SuperscriptBox[ RowBox[{"Sin", "[", "\[Theta]", "]"}], "2"]}]], "Output", CellChangeTimes->{{3.877600364737482*^9, 3.877600369786489*^9}, { 3.8776007655484247`*^9, 3.8776007710928802`*^9}, 3.877605423436298*^9, 3.877605920170188*^9, 3.882948404477663*^9, 3.8829784640287743`*^9, 3.884423890568315*^9, 3.8844267016401367`*^9, 3.884427088698732*^9}, CellLabel->"Out[49]=",ExpressionUUID->"e3bd5b0a-1637-4493-ba0d-ca692cc24c27"] }, Open ]], Cell["\<\ this is as we expect. The Simplify appearing does as one would guess and \ simplifies the results. One can use either method, wither inputting \ everything between two Simplify[] or putting //Simplify at the end. My \ preference is Simplify[] as it makes inputting assumptions easier. \ \>", "Text", CellChangeTimes->{{3.877600511048835*^9, 3.877600516327661*^9}, { 3.877600649876185*^9, 3.877600650301983*^9}, {3.877600744528269*^9, 3.87760082422997*^9}},ExpressionUUID->"e636a5d7-6e62-43d9-9153-\ 03e200995a90"], Cell["Now compute the inverse and call this gUU", "Text", CellChangeTimes->{{3.877600827415165*^9, 3.87760083676794*^9}},ExpressionUUID->"fdf46b25-6f89-4086-b761-\ a3fec7bd80e3"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"gUU", "=", RowBox[{"Simplify", "[", RowBox[{"Inverse", "[", "gdd", "]"}], "]"}]}]], "Input", CellChangeTimes->{{3.877600839051709*^9, 3.877600848428295*^9}}, CellLabel->"In[7]:=",ExpressionUUID->"913b3592-842d-4e5b-ae47-f1b487120f8a"], Cell[BoxData[ RowBox[{"{", RowBox[{ RowBox[{"{", RowBox[{ RowBox[{"-", FractionBox["1", RowBox[{"f", "[", "r", "]"}]]}], ",", "0", ",", "0", ",", "0"}], "}"}], ",", RowBox[{"{", RowBox[{"0", ",", FractionBox["1", RowBox[{"g", "[", "r", "]"}]], ",", "0", ",", "0"}], "}"}], ",", RowBox[{"{", RowBox[{"0", ",", "0", ",", FractionBox["1", SuperscriptBox["\[Rho]", "2"]], ",", "0"}], "}"}], ",", RowBox[{"{", RowBox[{"0", ",", "0", ",", "0", ",", FractionBox[ SuperscriptBox[ RowBox[{"Csc", "[", "\[Theta]", "]"}], "2"], SuperscriptBox["\[Rho]", "2"]]}], "}"}]}], "}"}]], "Output", CellChangeTimes->{3.877600849261726*^9, 3.877605425161313*^9, 3.877605922366514*^9, 3.882948411824533*^9, 3.8829784657456923`*^9, 3.884423893094698*^9, 3.8844267037456636`*^9, 3.884426934139646*^9, 3.884427096590177*^9, 3.884428064866345*^9}, CellLabel->"Out[7]=",ExpressionUUID->"e00d8d59-7f9f-49fd-940b-db0eb6cc0214"] }, Open ]], Cell["\<\ As a check we can see that this gives the identity matrix. There are two ways \ we can do this either by summing over the indices or using the built in \ matrix product in mathematica. Let us do both. With the inbuilt matrix \ product using . we have\ \>", "Text", CellChangeTimes->{{3.8776008866732607`*^9, 3.877600960783593*^9}, { 3.877601031518764*^9, 3.877601044076414*^9}},ExpressionUUID->"6230d366-19de-47cb-bdfa-\ 9586ed12b8ca"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"Simplify", "[", RowBox[{ RowBox[{"gUU", ".", "gdd"}], "==", RowBox[{"IdentityMatrix", "[", "4", "]"}]}], "]"}]], "Input", CellChangeTimes->{{3.8776009625261097`*^9, 3.877600976186408*^9}, { 3.877601386585602*^9, 3.8776013904406033`*^9}, {3.8829484188065147`*^9, 3.882948419781498*^9}}, CellLabel->"In[8]:=",ExpressionUUID->"ddf81b38-82ac-40b3-8a7d-88a8e9fb6379"], Cell[BoxData["True"], "Output", CellChangeTimes->{{3.877600965565276*^9, 3.8776009767826242`*^9}, 3.8776013911658983`*^9, 3.8776054270571337`*^9, 3.8776059244845333`*^9, { 3.882948413555607*^9, 3.882948420303266*^9}, 3.882978467823463*^9, 3.8844238959575644`*^9, 3.884426705603058*^9, 3.884427098810519*^9, 3.8844280669091454`*^9}, CellLabel->"Out[8]=",ExpressionUUID->"0fed06fd-4b36-4022-b674-796c8dc827f8"] }, Open ]], Cell["\<\ For summing the indices we need to learn a few things. We need to be able to \ extract out the entries of the matrix. To do this we write gdd[[ 1,3 ]], \ note the double brackets. This will extract out the 13 component of the \ metric. For us this we be the cross term dt d\[Theta], of course this is zero \ her. Next note that we should be obtaining a matrix since we have two free \ indices that are not summed over. To write this properly we need to use Table \ which arrays the values for each combination into a table (matrix here). Note below that we sum over the m2 indices and use table for the m1 and m3 \ indices. \ \>", "Text", CellChangeTimes->{{3.877601046841076*^9, 3.877601270068255*^9}},ExpressionUUID->"2209c841-96a6-45f9-8ae9-\ 6ec296b6840f"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"Simplify", "[", RowBox[{ RowBox[{"Table", "[", RowBox[{ RowBox[{"Sum", "[", RowBox[{ RowBox[{ RowBox[{"gUU", "[", RowBox[{"[", RowBox[{"m1", ",", "m2"}], "]"}], "]"}], RowBox[{"gdd", "[", RowBox[{"[", RowBox[{"m2", ",", "m3"}], "]"}], "]"}]}], ",", RowBox[{"{", RowBox[{"m2", ",", "1", ",", "dim"}], "}"}]}], "]"}], ",", RowBox[{"{", RowBox[{"m1", ",", "1", ",", "dim"}], "}"}], ",", RowBox[{"{", RowBox[{"m3", ",", "1", ",", "dim"}], "}"}]}], "]"}], "==", RowBox[{"IdentityMatrix", "[", "4", "]"}]}], "]"}]], "Input", CellChangeTimes->{{3.87760097996716*^9, 3.8776010263911133`*^9}, { 3.87760139343246*^9, 3.87760139645409*^9}, {3.877609764543274*^9, 3.87760977103544*^9}}, CellLabel->"In[52]:=",ExpressionUUID->"58ea82a6-0275-4210-98c2-93c5f8c78ef1"], Cell[BoxData["True"], "Output", CellChangeTimes->{3.877601027152958*^9, 3.877601396882928*^9, 3.877605429187542*^9, 3.877605926507387*^9, 3.877609771572414*^9, 3.88294843103918*^9, 3.882978469862207*^9, 3.884423897741044*^9, 3.8844267074881268`*^9, 3.8844269378130827`*^9, 3.884427100661669*^9}, CellLabel->"Out[52]=",ExpressionUUID->"8e3f2ac6-5d72-4ed4-a2fe-25f7913ac7c7"] }, Open ]], Cell["We see that it is the identity and therefore we are done. ", "Text", CellChangeTimes->{{3.877601274632653*^9, 3.877601296812881*^9}},ExpressionUUID->"2efd76ad-ceba-40e2-ba17-\ 9a280ee6934a"], Cell[CellGroupData[{ Cell["Christoffel symbols", "Subsubsection", CellChangeTimes->{{3.877601404370286*^9, 3.8776014087221537`*^9}},ExpressionUUID->"4ef6f69a-661b-4405-901e-\ 9fb2cedd335e"], Cell["\<\ We can now compute the Christoffel symbols. Recall that they are given by\ \>", "Text", CellChangeTimes->{{3.877601410545577*^9, 3.877601427129437*^9}},ExpressionUUID->"f5e3473b-96fa-4166-8288-\ 078d9fc6f699"], Cell[BoxData[ RowBox[{ SubscriptBox[ SuperscriptBox["\[CapitalGamma]", "\[Rho]"], "\[Mu]\[Nu]"], "=", RowBox[{ FractionBox["1", "2"], SuperscriptBox["g", "\[Rho]\[Sigma]"], RowBox[{"(", RowBox[{ RowBox[{ SubscriptBox["\[PartialD]", "\[Mu]"], SubscriptBox["g", "\[Sigma]\[Nu]"]}], "+", RowBox[{ SubscriptBox["\[PartialD]", "\[Nu]"], SubscriptBox["g", "\[Sigma]\[Mu]"]}], "-", RowBox[{ SubscriptBox["\[PartialD]", "\[Sigma]"], SubscriptBox["g", "\[Mu]\[Nu]"]}]}], ")"}]}]}]], "DisplayFormulaNumbered", CellChangeTimes->{{3.8776032179808683`*^9, 3.877603263615623*^9}},ExpressionUUID->"653b4633-d0f5-463c-8b69-\ 71cf7333794d"], Cell["\<\ We can code this into mathematic using: sum, table and derivative. To take a \ derivative we use D[ , ]. The first entry is what you want to take a \ derivative of and the second is what you want to take the derivative with \ respect to. We can take the derivative using the xIN above and picking the \ relevant component. So the Christoffel symbols would then be (make sure you \ do not use r as an index!)\ \>", "Text", CellChangeTimes->{{3.877603266328589*^9, 3.877603352854341*^9}, { 3.8776037383690987`*^9, 3.87760373935826*^9}, {3.877603770569165*^9, 3.87760378762915*^9}, {3.8776046690686483`*^9, 3.877604693351357*^9}},ExpressionUUID->"6360f8b1-1b83-4b94-b380-\ 744ee52a5830"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"\[CapitalGamma]Udd", "=", RowBox[{"Simplify", "[", RowBox[{"Table", "[", RowBox[{ RowBox[{ FractionBox["1", "2"], RowBox[{"Sum", "[", RowBox[{ RowBox[{ RowBox[{"gUU", "[", RowBox[{"[", RowBox[{"p1", ",", "s1"}], "]"}], "]"}], RowBox[{"(", RowBox[{ RowBox[{"D", "[", RowBox[{ RowBox[{"gdd", "[", RowBox[{"[", RowBox[{"s1", ",", "n1"}], "]"}], "]"}], ",", RowBox[{"xIN", "[", RowBox[{"[", "m1", "]"}], "]"}]}], "]"}], "+", RowBox[{"D", "[", RowBox[{ RowBox[{"gdd", "[", RowBox[{"[", RowBox[{"s1", ",", "m1"}], "]"}], "]"}], ",", RowBox[{"xIN", "[", RowBox[{"[", "n1", "]"}], "]"}]}], "]"}], "-", 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There is \ a pre-built in function for computing the determinant, Det[].\ \>", "Text", CellChangeTimes->{{3.8776003464055634`*^9, 3.877600360419662*^9}, { 3.8776004872533607`*^9, 3.8776005086287613`*^9}},ExpressionUUID->"785582ba-f973-4f01-b176-\ cdad381c16d4"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"Simplify", "[", RowBox[{"Det", "[", "gdd", "]"}], "]"}]], "Input", CellChangeTimes->{{3.877600361888378*^9, 3.877600369396488*^9}, { 3.8776007624261827`*^9, 3.877600770512642*^9}, {3.877605422283029*^9, 3.8776054228804483`*^9}, {3.884427094261078*^9, 3.884427094509157*^9}}, CellLabel->"In[57]:=",ExpressionUUID->"25bcaa9f-45de-4d8d-84a7-09fb12a4eebe"], Cell[BoxData[ RowBox[{ RowBox[{"-", RowBox[{"f", "[", "r", "]"}]}], " ", RowBox[{"g", "[", "r", "]"}]}]], "Output", CellChangeTimes->{{3.877600364737482*^9, 3.877600369786489*^9}, { 3.8776007655484247`*^9, 3.8776007710928802`*^9}, 3.877605423436298*^9, 3.877605920170188*^9, 3.882948404477663*^9, 3.8829784640287743`*^9, 3.884423890568315*^9, 3.8844267016401367`*^9, 3.884427088694949*^9, 3.88442980899617*^9}, CellLabel->"Out[57]=",ExpressionUUID->"2554714c-bca1-457d-8e71-dd16b73e92a1"] }, Open ]], Cell["\<\ this is as we expect. The Simplify appearing does as one would guess and \ simplifies the results. One can use either method, wither inputting \ everything between two Simplify[] or putting //Simplify at the end. My \ preference is Simplify[] as it makes inputting assumptions easier. \ \>", "Text", CellChangeTimes->{{3.877600511048835*^9, 3.877600516327661*^9}, { 3.877600649876185*^9, 3.877600650301983*^9}, {3.877600744528269*^9, 3.87760082422997*^9}},ExpressionUUID->"a64a362c-8f2a-4f6c-827b-\ 490532473c40"], Cell["Now compute the inverse and call this gUU", "Text", CellChangeTimes->{{3.877600827415165*^9, 3.87760083676794*^9}},ExpressionUUID->"fc65061a-8a6b-452d-adcc-\ fe1585da7b3d"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"gUU", "=", RowBox[{"Simplify", "[", RowBox[{"Inverse", "[", "gdd", "]"}], "]"}]}]], "Input", CellChangeTimes->{{3.877600839051709*^9, 3.877600848428295*^9}}, CellLabel->"In[58]:=",ExpressionUUID->"ac69da1f-d071-4943-841d-e69179c0c372"], Cell[BoxData[ RowBox[{"{", RowBox[{ RowBox[{"{", RowBox[{ RowBox[{"-", FractionBox["1", RowBox[{"f", "[", "r", "]"}]]}], ",", "0"}], "}"}], ",", RowBox[{"{", RowBox[{"0", ",", FractionBox["1", RowBox[{"g", "[", "r", "]"}]]}], "}"}]}], "}"}]], "Output", CellChangeTimes->{3.877600849261726*^9, 3.877605425161313*^9, 3.877605922366514*^9, 3.882948411824533*^9, 3.8829784657456923`*^9, 3.884423893094698*^9, 3.8844267037456636`*^9, 3.884426934139646*^9, 3.884427096590177*^9, 3.884428064866345*^9, 3.884429810668722*^9}, CellLabel->"Out[58]=",ExpressionUUID->"0d9733a1-fa0a-4d7f-850e-7947f9a5ee8c"] }, Open ]], Cell["\<\ As a check we can see that this gives the identity matrix. There are two ways \ we can do this either by summing over the indices or using the built in \ matrix product in mathematica. Let us do both. With the inbuilt matrix \ product using . we have\ \>", "Text", CellChangeTimes->{{3.8776008866732607`*^9, 3.877600960783593*^9}, { 3.877601031518764*^9, 3.877601044076414*^9}},ExpressionUUID->"becff995-9b4b-4a04-a2fd-\ 15b2951f02b3"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"Simplify", "[", RowBox[{ RowBox[{"gUU", ".", "gdd"}], "==", RowBox[{"IdentityMatrix", "[", "2", "]"}]}], "]"}]], "Input", CellChangeTimes->{{3.8776009625261097`*^9, 3.877600976186408*^9}, { 3.877601386585602*^9, 3.8776013904406033`*^9}, {3.8829484188065147`*^9, 3.882948419781498*^9}, {3.8844298182670307`*^9, 3.8844298184422197`*^9}}, CellLabel->"In[60]:=",ExpressionUUID->"5f39d8c9-8320-46eb-8687-46eecce1845e"], Cell[BoxData["True"], "Output", CellChangeTimes->{{3.877600965565276*^9, 3.8776009767826242`*^9}, 3.8776013911658983`*^9, 3.8776054270571337`*^9, 3.8776059244845333`*^9, { 3.882948413555607*^9, 3.882948420303266*^9}, 3.882978467823463*^9, 3.8844238959575644`*^9, 3.884426705603058*^9, 3.884427098810519*^9, 3.8844280669091454`*^9, {3.884429813697405*^9, 3.8844298189972277`*^9}}, CellLabel->"Out[60]=",ExpressionUUID->"094c8aa4-81a5-420d-9a82-4832165d981b"] }, Open ]], Cell["\<\ For summing the indices we need to learn a few things. We need to be able to \ extract out the entries of the matrix. To do this we write gdd[[ 1,3 ]], \ note the double brackets. This will extract out the 13 component of the \ metric. For us this we be the cross term dt d\[Theta], of course this is zero \ her. Next note that we should be obtaining a matrix since we have two free \ indices that are not summed over. To write this properly we need to use Table \ which arrays the values for each combination into a table (matrix here). Note below that we sum over the m2 indices and use table for the m1 and m3 \ indices. \ \>", "Text", CellChangeTimes->{{3.877601046841076*^9, 3.877601270068255*^9}},ExpressionUUID->"bb1fbe79-4a4f-4e4c-a652-\ 85da9838c921"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"Simplify", "[", RowBox[{ RowBox[{"Table", "[", RowBox[{ RowBox[{"Sum", "[", RowBox[{ RowBox[{ RowBox[{"gUU", "[", RowBox[{"[", RowBox[{"m1", ",", "m2"}], "]"}], "]"}], RowBox[{"gdd", "[", RowBox[{"[", RowBox[{"m2", ",", "m3"}], "]"}], "]"}]}], ",", RowBox[{"{", RowBox[{"m2", ",", "1", ",", "dim"}], "}"}]}], "]"}], ",", RowBox[{"{", RowBox[{"m1", ",", "1", ",", "dim"}], "}"}], ",", RowBox[{"{", RowBox[{"m3", ",", "1", ",", "dim"}], "}"}]}], "]"}], "==", RowBox[{"IdentityMatrix", "[", "2", "]"}]}], "]"}]], "Input", CellChangeTimes->{{3.87760097996716*^9, 3.8776010263911133`*^9}, { 3.87760139343246*^9, 3.87760139645409*^9}, {3.877609764543274*^9, 3.87760977103544*^9}, {3.884429827498472*^9, 3.884429827615348*^9}}, CellLabel->"In[62]:=",ExpressionUUID->"c7928ad8-80fb-4c71-95d2-b99d29ac1875"], Cell[BoxData["True"], "Output", CellChangeTimes->{ 3.877601027152958*^9, 3.877601396882928*^9, 3.877605429187542*^9, 3.877605926507387*^9, 3.877609771572414*^9, 3.88294843103918*^9, 3.882978469862207*^9, 3.884423897741044*^9, 3.8844267074881268`*^9, 3.8844269378130827`*^9, 3.884427100661669*^9, {3.8844298252944107`*^9, 3.884429827923847*^9}}, CellLabel->"Out[62]=",ExpressionUUID->"78f68252-eded-402f-9996-3f6bd8456abf"] }, Open ]], Cell["We see that it is the identity and therefore we are done. ", "Text", CellChangeTimes->{{3.877601274632653*^9, 3.877601296812881*^9}},ExpressionUUID->"7c8f373d-832c-467d-81e5-\ 1d08f5759cd1"], Cell[CellGroupData[{ Cell["Christoffel symbols", "Subsubsection", CellChangeTimes->{{3.877601404370286*^9, 3.8776014087221537`*^9}},ExpressionUUID->"6b9869e5-fe00-4906-bb8a-\ 66fe7e9178c6"], Cell["\<\ We can now compute the Christoffel symbols. Recall that they are given by\ \>", "Text", CellChangeTimes->{{3.877601410545577*^9, 3.877601427129437*^9}},ExpressionUUID->"7b634b27-25bf-4d57-97d5-\ 587afc5fb01e"], Cell[BoxData[ RowBox[{ SubscriptBox[ SuperscriptBox["\[CapitalGamma]", "\[Rho]"], "\[Mu]\[Nu]"], "=", RowBox[{ FractionBox["1", "2"], SuperscriptBox["g", "\[Rho]\[Sigma]"], RowBox[{"(", RowBox[{ RowBox[{ SubscriptBox["\[PartialD]", "\[Mu]"], SubscriptBox["g", "\[Sigma]\[Nu]"]}], "+", RowBox[{ SubscriptBox["\[PartialD]", "\[Nu]"], SubscriptBox["g", "\[Sigma]\[Mu]"]}], "-", RowBox[{ SubscriptBox["\[PartialD]", "\[Sigma]"], SubscriptBox["g", "\[Mu]\[Nu]"]}]}], ")"}]}]}]], "DisplayFormulaNumbered", CellChangeTimes->{{3.8776032179808683`*^9, 3.877603263615623*^9}},ExpressionUUID->"d69e3dcf-81c0-472a-a93b-\ 3b3332fc8b81"], Cell["\<\ We can code this into mathematic using: sum, table and derivative. To take a \ derivative we use D[ , ]. The first entry is what you want to take a \ derivative of and the second is what you want to take the derivative with \ respect to. We can take the derivative using the xIN above and picking the \ relevant component. So the Christoffel symbols would then be (make sure you \ do not use r as an index!)\ \>", "Text", CellChangeTimes->{{3.877603266328589*^9, 3.877603352854341*^9}, { 3.8776037383690987`*^9, 3.87760373935826*^9}, {3.877603770569165*^9, 3.87760378762915*^9}, {3.8776046690686483`*^9, 3.877604693351357*^9}},ExpressionUUID->"55433ae3-2a1f-4cdf-b545-\ 97d566ad7bf4"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"\[CapitalGamma]Udd", "=", RowBox[{"Simplify", "[", RowBox[{"Table", "[", RowBox[{ RowBox[{ FractionBox["1", "2"], RowBox[{"Sum", "[", RowBox[{ RowBox[{ RowBox[{"gUU", "[", RowBox[{"[", RowBox[{"p1", ",", "s1"}], "]"}], "]"}], RowBox[{"(", RowBox[{ RowBox[{"D", "[", RowBox[{ RowBox[{"gdd", "[", RowBox[{"[", RowBox[{"s1", ",", "n1"}], "]"}], "]"}], ",", RowBox[{"xIN", "[", RowBox[{"[", "m1", "]"}], "]"}]}], "]"}], "+", RowBox[{"D", "[", RowBox[{ RowBox[{"gdd", "[", RowBox[{"[", RowBox[{"s1", ",", "m1"}], "]"}], "]"}], ",", RowBox[{"xIN", "[", RowBox[{"[", "n1", "]"}], "]"}]}], "]"}], "-", 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