(* 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[ 194839, 5337] NotebookOptionsPosition[ 172906, 4998] NotebookOutlinePosition[ 173900, 5028] CellTagsIndexPosition[ 173757, 5021] WindowFrame->Normal*) (* Beginning of Notebook Content *) Notebook[{ Cell[CellGroupData[{ Cell["GR2 computations in mathamtica", "Title", CellChangeTimes->{{3.882947538691614*^9, 3.8829475451117487`*^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 \ 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->"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[117]:=",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}, CellLabel-> "Out[117]=",ExpressionUUID->"d9218721-d811-45d3-bc76-bd640c434c20"] }, 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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Recall that we can always use the SO(3) isometry \ to organise for the initial condition ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{ RowBox[{ OverscriptBox["\[Theta]", "."], "(", "0", ")"}], "=", "0"}], ",", " ", RowBox[{ RowBox[{"\[Theta]", "(", "0", ")"}], "=", FractionBox["\[Pi]", "2"]}]}], TraditionalForm]], FormatType->TraditionalForm,ExpressionUUID-> "e2c76d75-5e21-4227-ac08-110a7184e198"], ". 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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}}, Background->RGBColor[ 0.88, 1, 0.88],ExpressionUUID->"1b6a0e94-35ab-494a-bc6c-389c5bc4cc77"], 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}, {3.882950582761072*^9, 3.882950583053865*^9}}, CellLabel->"In[35]:=",ExpressionUUID->"fcc15794-833f-4ce0-81be-b203c1a1b168"], Cell[BoxData[ RowBox[{ RowBox[{"-", SuperscriptBox["\[ExponentialE]", RowBox[{"2", " ", RowBox[{"(", RowBox[{ RowBox[{"\[CapitalPhi]\[CapitalPhi]", "[", "r", "]"}], "+", RowBox[{"\[CapitalPsi]\[CapitalPsi]", "[", "r", "]"}]}], ")"}]}]]}], " ", 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, {3.882950568680356*^9, 3.882950583609577*^9}}, CellLabel->"Out[35]=",ExpressionUUID->"74dfff4f-d4ae-4ee0-bf3f-2942fbf81b5e"] }, Open ]], Cell["Now compute the inverse and call this gUU", "Text", CellChangeTimes->{{3.877600827415165*^9, 3.87760083676794*^9}}, Background->RGBColor[ 0.88, 1, 0.88],ExpressionUUID->"d1c26f6e-c2ef-42bf-a43a-5486cd655da1"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"gUU", "=", RowBox[{"Simplify", "[", RowBox[{"Inverse", "[", "gdd", "]"}], "]"}]}]], "Input", CellChangeTimes->{{3.877600839051709*^9, 3.877600848428295*^9}}, CellLabel->"In[36]:=",ExpressionUUID->"7df1db2b-88b7-4e2a-95ae-850dba95ef19"], Cell[BoxData[ RowBox[{"{", RowBox[{ RowBox[{"{", RowBox[{ RowBox[{"-", SuperscriptBox["\[ExponentialE]", RowBox[{ RowBox[{"-", "2"}], " ", RowBox[{"\[CapitalPhi]\[CapitalPhi]", "[", "r", "]"}]}]]}], ",", "0", ",", "0", ",", "0"}], "}"}], ",", RowBox[{"{", RowBox[{"0", ",", SuperscriptBox["\[ExponentialE]", RowBox[{ RowBox[{"-", "2"}], " ", RowBox[{"\[CapitalPsi]\[CapitalPsi]", "[", "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, 3.882950585945198*^9}, CellLabel->"Out[36]=",ExpressionUUID->"8ad425d6-dc03-4014-bf50-99d0db288884"] }, Open ]], Cell["As a check we can see that this gives the identity matrix. 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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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We find that this is equivalent to equation ", CounterBox["DisplayFormulaNumbered", "pressurederiv"], " as it should be. 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