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For more information on notebooks and Mathematica-compatible applications, contact Wolfram Research: web: http://www.wolfram.com email: info@wolfram.com phone: +1-217-398-0700 (U.S.) Notebook reader applications are available free of charge from Wolfram Research. ***********************************************************************) (*CacheID: 232*) (*NotebookFileLineBreakTest NotebookFileLineBreakTest*) (*NotebookOptionsPosition[ 56437, 1760]*) (*NotebookOutlinePosition[ 57410, 1790]*) (* CellTagsIndexPosition[ 57366, 1786]*) (*WindowFrame->Normal*) Notebook[{ Cell[CellGroupData[{ Cell["\<\ Connection Coefficients for Spherical Polar Coordinates, Computed \ Via GRTensorM in Mathematica\ \>", "Title"], Cell["Example by Kip, for Ph136c, 25 April 2000", "Subtitle"], Cell[CellGroupData[{ Cell["Import GRTensor into Mathematica, and Set Directory", "Section"], Cell[CellGroupData[{ Cell[BoxData[ \(<< grii/grt.m\)], "Input"], Cell[BoxData[ \(" GRTensorM Version 1.2 for Mathematica 3.x"\)], "Print"], Cell[BoxData[ \(" July 20, 1998\n"\)], "Print"], Cell[BoxData[ \(" Copyright 1996-98 by P. Musgrave, D. Pollney and K. Lake"\)], "Print"], Cell[BoxData[ \(" Update by Mustapha Ishak B."\)], "Print"], Cell[BoxData[ \(" Use qload[] or grload[] to load a metric."\)], "Print"], Cell[BoxData[ \(" ?grtensor for help.\n"\)], "Print"], Cell[BoxData[ InterpretationBox[ RowBox[{"\<\" Metric directory = \"\>", "\[InvisibleSpace]", InterpretationBox[ StyleBox[ "\"\\"\ ", ShowStringCharacters->True, NumberMarks->True], InputForm[ "/usr/local/mathematica/AddOns/ExtraPackages/grii/metrics/"], Editable->True, AutoDelete->True]}], SequenceForm[ " Metric directory = ", InputForm[ "/usr/local/mathematica/AddOns/ExtraPackages/grii/metrics/"]], Editable->False]], "Print"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(SetOptions[grii, MetricPath \[Rule] \ "\"]\)], "Input"], Cell[BoxData[ \({DisplayLimit \[Rule] 5000, TimeStamp \[Rule] True, DefaultSimplification \[Rule] Together, Verbose \[Rule] True, MetricPath \[Rule] "/home/kip/courses/ph136/1999.00/chapters/chap23/"}\)], "Output"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["Define Coordinates, Basis Vectors, and Metric", "Section"], Cell["\<\ I shall use as my example, the connection coefficients for \ spherical polar coordinates in 3-dimensional Euclidean space. The \ coordinates are denoted r, th (for theta) and ph (for phi). The input of \ coordinates, basis vectors and metric is initiated via the command \ \>", \ "Text"], Cell[BoxData[ \(makeg[sphere3d]\)], "Input"], Cell["\<\ Here \"sphere3d\" is the name that I am giving to this metric and \ basis. This command brings up a series of prompt windows into which I type \ the appropriate information:\ \>", "Text"], Cell[CellGroupData[{ Cell["First Prompt Window: Choice of what to enter", "Subsubsection"], Cell["\<\ >Do you wish to enter a > 1) metric: g[dn,dn] > 2) line element [ds], > 3)non-holonomic basis e1..e4, or > 4) null tetrad [l,n,m,mbar]?\ \>", "Text"], Cell[TextData[StyleBox[" 3", FontWeight->"Bold"]], "Text"], Cell["\<\ [Note: The phrase \"non-holonomic basis\" means \"non-coordinate \ basis\". I plan to use the orthonormal basis associated with spherical \ coordinates, so I choose this.]\ \>", "Text"] }, Open ]], Cell[CellGroupData[{ Cell["Second Prompt Window: Names of coordinates", "Subsubsection"], Cell[">Enter coordinates as a LIST (e.g. {r,thetha,phi,t})", "Text"], Cell[TextData[StyleBox[" {r,th,ph}", FontWeight->"Bold"]], "Text"] }, Open ]], Cell[CellGroupData[{ Cell["\<\ Third Prompt Window: Choice of components of basis vectors to enter\ \ \>", "Subsubsection"], Cell["\<\ >Would you like to enter > 1) covariant components, > 2) contravariant components, or > 3) both.\ \>", "Text"], Cell[TextData[{ " ", StyleBox["2", FontWeight->"Bold"] }], "Text"], Cell[TextData[{ "[Note: If we write our orthonormal basis vectors as ", Cell[BoxData[ FormBox[ SubscriptBox[ StyleBox["e", FontWeight->"Bold"], \(\([j]\)\(\ \)\)], TraditionalForm]]], "= ", Cell[BoxData[ FormBox[ SubsuperscriptBox[ StyleBox["e", FontWeight->"Bold"], \([j]\), "k"], TraditionalForm]]], "\[PartialD]/\[PartialD]", Cell[BoxData[ \(TraditionalForm\`x\^k\)]], ", then ", Cell[BoxData[ FormBox[ SubsuperscriptBox[ StyleBox["e", FontWeight->"Bold"], \([j]\), "k"], TraditionalForm]]], "are the \"contravariant components\" (of the non-coordinate basis vector \ ", Cell[BoxData[ \(TraditionalForm\`e\_\((j)\)\)]], ") to which the above query refers. I have chosen to enter these \ contravariant components.]" }], "Text"] }, Open ]], Cell[CellGroupData[{ Cell["\<\ Fourth , Fifth and Sixth Prompt Windows: Values of components of \ basis vectors\ \>", "Subsubsection"], Cell["\<\ >Enter the contravariant components of basis vector e[1] as a List \ (e.g. {1,0,0,0})\ \>", "Text"], Cell[" {1,0,0}", "Text", FontWeight->"Bold"], Cell[TextData[{ "[Note: This says that ", Cell[BoxData[ FormBox[ SubscriptBox[ StyleBox["e", FontWeight->"Bold"], \([r]\)], TraditionalForm]]], " is equal to \[PartialD]/\[PartialD]r ]" }], "Text"], Cell["\<\ >Enter the contravariant components of basis vector e[2] as a List \ (e.g.{1,0,0,0})\ \>", "Text"], Cell[TextData[StyleBox[" {0,1/r,0}", FontWeight->"Bold"]], "Text"], Cell[TextData[{ "[Note: This says that ", Cell[BoxData[ FormBox[ SubscriptBox[ StyleBox["e", FontWeight->"Bold"], \([\[Theta]]\)], TraditionalForm]]], " is (1/r)\[PartialD]/\[PartialD]\[Theta] ]" }], "Text"], Cell["\<\ >Enter the contravariant components of basis vector e[2] as a List \ (e.g.{1,0,0,0}\ \>", "Text"], Cell[" {0,0,1/(r Sin[th])}", "Text", FontWeight->"Bold"], Cell[TextData[{ "[Note: This says that ", Cell[BoxData[ FormBox[ SubscriptBox[ StyleBox["e", FontWeight->"Bold"], \([\[Phi]]\)], TraditionalForm]]], " is 1/(r Sin[\[Theta]]) \[PartialD]/\[PartialD]\[Phi]]" }], "Text"] }, Open ]], Cell[CellGroupData[{ Cell["\<\ Remaining Prompt Windows: Metric components in the non-coordinate \ basis, i.e. inner product of the basis vectors with each other \ \>", \ "Subsubsection"], Cell["\<\ >Is the basis inner product > 1) Diagonal, or > 2) Symmetric?\ \>", "Text"], Cell[TextData[{ " ", StyleBox[" 1", FontWeight->"Bold"] }], "Text"], Cell[">Enter eta[1,1]", "Text"], Cell[TextData[StyleBox[" 1", FontWeight->"Bold"]], "Text"], Cell[TextData[{ "[Note: eta[j,k] is GRTensor's notation for the components of the metric in \ the non-coordinate basis. My response says that eta[1,1] = (the inner \ product of ", Cell[BoxData[ FormBox[ SubscriptBox[ StyleBox["e", FontWeight->"Bold"], \([r]\)], TraditionalForm]]], "with itself) is unity -- as is dictated by the fact that my chosen basis \ is orthonormal." }], "Text"], Cell[">Enter eta[2,2]", "Text"], Cell[TextData[StyleBox[" 1", FontWeight->"Bold"]], "Text"], Cell[">Enter eta[3,3]", "Text"], Cell[" 1", "Text", FontWeight->"Bold"] }, Open ]], Cell[CellGroupData[{ Cell["GRTensor's Response to this Input:", "Subsubsection"], Cell[BoxData[ \(\*"\<\"\"\>"\)], "Print"], Cell[BoxData[ \("Makeg 2.0; GRTensor metric/basis entry utility"\)], "Print"], Cell[BoxData[ \(\*"\<\"\"\>"\)], "Print"], Cell[BoxData[ \("To quit makeg, type 'Exit' at any prompt."\)], "Print"], Cell[BoxData[ \(\*"\<\"\"\>"\)], "Print"], Cell[BoxData[ \("The values you have entered are:"\)], "Print"], Cell[BoxData[ InterpretationBox[\("Coordinates ="\[InvisibleSpace]{r, th, ph}\), SequenceForm[ "Coordinates =", {r, th, ph}], 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Editable->False], "\[InvisibleSpace]", "\<\" = \"\>", "\[InvisibleSpace]", "1"}], SequenceForm[ eta, Subscript[ "(3)"], Subscript[ "(3)"], " = ", 1], Editable->False]], "Print"], Cell["\<\ [Note: At this point, in response to a window, I tell GRTensor to \ save the metric; it saves it in my chosen directory, so I can use it in the \ future. In response to another window, I tell GRTensor that I want to use \ this metric in a calculation. It then responds with the following summary of \ the metric and basis:]\ \>", "Text"], Cell[BoxData[ InterpretationBox[\("Information written to \ "\[InvisibleSpace]"/home/kip/courses/ph136/1999.00/chapters/chap23/sphere3d.\ g"\), SequenceForm[ "Information written to ", "/home/kip/courses/ph136/1999.00/chapters/chap23/sphere3d.g"], Editable->False]], "Print"], Cell[BoxData[ InterpretationBox[\("Default metric = "\[InvisibleSpace]sphere3d\), SequenceForm[ "Default metric = ", sphere3d], Editable->False]], "Print"], Cell[BoxData[ InterpretationBox[\("For the "\[InvisibleSpace]sphere3d\[InvisibleSpace]" \ metric."\), SequenceForm[ "For the ", sphere3d, " metric."], Editable->False]], "Print"], Cell[BoxData[ \(" "\)], "Print"], Cell[BoxData[ \("Coordinates"\)], "Print"], Cell[BoxData[ InterpretationBox[ RowBox[{"\<\" x\"\>", "\[InvisibleSpace]", InterpretationBox[\(\[Null]\^1\), Superscript[ 1], Editable->False], "\[InvisibleSpace]", "\<\" = \"\>", "\[InvisibleSpace]", "r", 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Cell[CellGroupData[{ Cell["\<\ Compute and Display Connection Coefficients in Non-Coordinate Basis\ \ \>", "Section"], Cell["\<\ I now compute the connection coefficients in my orthonormal basis \ using the following command:\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(grcalc[Chr[bdn, bdn, bup]]\)], "Input"], Cell[BoxData[ InterpretationBox[\("Created a definition for "\[InvisibleSpace]Chr[bdn, bdn, bup]\), SequenceForm[ "Created a definition for ", Chr[ bdn, bdn, bup]], Editable->False]], "Print"], Cell[BoxData[ InterpretationBox[\("For the "\[InvisibleSpace]sphere3d\[InvisibleSpace]" \ metric."\), SequenceForm[ "For the ", sphere3d, " metric."], Editable->False]], "Print"], Cell[BoxData[ InterpretationBox[\("Calculated "\[InvisibleSpace]e[bup, up]\[InvisibleSpace]"( "\[InvisibleSpace]0.010000000000000009`\ \[InvisibleSpace]" sec.)"\), SequenceForm[ "Calculated ", e[ bup, up], "( ", 0.010000000000000009, " sec.)"], Editable->False]], "Print"], Cell[BoxData[ 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sec.)"], Editable->False]], "Print"], Cell[BoxData[ InterpretationBox[\("Calculated "\[InvisibleSpace]Chr[dn, dn, dn]\[InvisibleSpace]"( "\[InvisibleSpace]0.009999999999999898`\ \[InvisibleSpace]" sec.)"\), SequenceForm[ "Calculated ", Chr[ dn, dn, dn], "( ", 0.0099999999999998979, " sec.)"], Editable->False]], "Print"], Cell[BoxData[ InterpretationBox[\("Calculated "\[InvisibleSpace]Chr[bdn, bdn, bup]\[InvisibleSpace]"( "\[InvisibleSpace]0.07000000000000006`\ \[InvisibleSpace]" sec.)"\), SequenceForm[ "Calculated ", Chr[ bdn, bdn, bup], "( ", 0.070000000000000062, " sec.)"], Editable->False]], "Print"], Cell[BoxData[ InterpretationBox[\("CPU Time = "\[InvisibleSpace]0.15000000000000002`\ \[InvisibleSpace]" sec."\), SequenceForm[ "CPU Time = ", 0.15000000000000002, " sec."], Editable->False]], "Print"] }, Open ]], Cell["\<\ [Note: \"Chr\" is GRTensor's name for the connection coefficients \ (which it calls \"Christoffel Symbols\"); GRTensor uses a different \ convention from mine (and MTW's) for the ordering of indices on the \ connection coefficients: it puts the up index last instead of first, i.e. it \ permutes the indices leftward by one slot. The notation \"bdn\" means the \ down (covariant) index in the non-coordinate basis (\"b\"); \"bup\" means the \ up (contravariant) index in the non-coordinate basis. I print out the \ connection coefficients for the non-coordinate basis with the command\ \>", \ "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(grdisplay[Chr[bdn, bdn, bup]]\)], "Input"], Cell[BoxData[ InterpretationBox[\("For the "\[InvisibleSpace]sphere3d\[InvisibleSpace]" \ metric."\), SequenceForm[ "For the ", sphere3d, " metric."], Editable->False]], "Print"], Cell[BoxData[ \(" "\)], "Print"], Cell[BoxData[ \("Chr[bdn, bdn, bup]"\)], "Print"], Cell[BoxData[ \(\*"\<\"\"\>"\)], "Print"], Cell[BoxData[ InterpretationBox[ RowBox[{ InterpretationBox[ RowBox[{ InterpretationBox[ RowBox[{ InterpretationBox[ RowBox[{"\<\"Chr \"\>", "\[InvisibleSpace]", "\<\" \"\>", "\[InvisibleSpace]", InterpretationBox[\(\[Null]\_"(r)"\), Subscript[ "(r)"], Editable->False]}], SequenceForm[ "Chr ", " ", Subscript[ "(r)"]], Editable->False], "\[InvisibleSpace]", "\<\" \"\>", "\[InvisibleSpace]", InterpretationBox[\(\[Null]\_"(th)"\), Subscript[ "(th)"], Editable->False]}], SequenceForm[ SequenceForm[ "Chr 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(XXXXX) in Chapter 23.]", "Text"] }, Open ]], Cell[CellGroupData[{ Cell["\<\ Compute and Display Connection Coefficients in Coordinate \ Basis\ \>", "Section"], Cell["\<\ GRTensor denotes the connectin coefficients in the coordinate basis \ by Chr[dn,dn,up] (i.e. the same as in the non-coordinate basis, but with the \ \"b\" omitted from bdn,bdn,bup). I compute them with the following \ command:\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(grcalc[Chr[dn, dn, up]]\)], "Input"], Cell[BoxData[ InterpretationBox[\("For the "\[InvisibleSpace]sphere3d\[InvisibleSpace]" \ metric."\), SequenceForm[ "For the ", sphere3d, " metric."], Editable->False]], "Print"], Cell[BoxData[ InterpretationBox[\("Calculated "\[InvisibleSpace]Chr[dn, dn, up]\[InvisibleSpace]"( "\[InvisibleSpace]0.020000000000000018`\ \[InvisibleSpace]" sec.)"\), SequenceForm[ "Calculated ", Chr[ dn, dn, up], "( ", 0.020000000000000018, " sec.)"], Editable->False]], "Print"], Cell[BoxData[ InterpretationBox[\("CPU Time = "\[InvisibleSpace]0.030000000000000027`\ \[InvisibleSpace]" sec."\), SequenceForm[ "CPU Time = ", 0.030000000000000027, " sec."], Editable->False]], "Print"] }, Open ]], Cell["I then print them out:", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(grdisplay[Chr[dn, dn, up]]\)], "Input"], Cell[BoxData[ InterpretationBox[\("For the 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