(************** Content-type: application/mathematica ************** Mathematica-Compatible Notebook This notebook can be used with any Mathematica-compatible application, such as Mathematica, MathReader or Publicon. The data for the notebook starts with the line containing stars above. To get the notebook into a Mathematica-compatible application, do one of the following: * Save the data starting with the line of stars above into a file with a name ending in .nb, then open the file inside the application; * Copy the data starting with the line of stars above to the clipboard, then use the Paste menu command inside the application. Data for notebooks contains only printable 7-bit ASCII and can be sent directly in email or through ftp in text mode. Newlines can be CR, LF or CRLF (Unix, Macintosh or MS-DOS style). NOTE: If you modify the data for this notebook not in a Mathematica- compatible application, you must delete the line below containing the word CacheID, otherwise Mathematica-compatible applications may try to use invalid cache data. 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[ 49035, 1366]*) (*NotebookOutlinePosition[ 49803, 1392]*) (* CellTagsIndexPosition[ 49759, 1388]*) (*WindowFrame->Normal*) Notebook[{ Cell[CellGroupData[{ Cell["0. Function Definitions", "Section"], Cell[BoxData[{ \(Christoffeldown[metric_, \ coord_] := \ Module[{dim\ = \ Length[coord], \ dg}, \ dg\ = \ Table[ D[metric\[LeftDoubleBracket]c, d\[RightDoubleBracket], coord\[LeftDoubleBracket]b\[RightDoubleBracket]], \ {b, 1, dim}, \ {c, 1, dim}, \ {d, 1, dim}]; \ 1/2\ \((\ Transpose[dg, \ {2, 3, \ 1}]\ + \ Transpose[dg, \ {3, 2, \ 1}]\ - \ dg)\)]\), "\n", \(\(Christoffel[metric_, \ coord_]\ := \ \ Inverse[metric] . Christoffeldown[metric, \ coord];\)\), "\n", \(\(Christoffel[chrisdown]\ := \ Inverse[metric] . chrisdown;\)\), "\[IndentingNewLine]", \(\(showgamma[\[CapitalGamma]_] := \ Table[MatrixForm[\[CapitalGamma]\[LeftDoubleBracket] i\[RightDoubleBracket]], \ {i, \ 1, \ Length[\[CapitalGamma]]}];\)\)}], "Input", CellLabel->"In[1]:="], Cell[BoxData[{ \(\(getricci[g_, \ coord_] := \ \ Module[{dim\ = \ Length[coord], \ rawricci}, \[IndentingNewLine]rawricci\ = \ Table[\n\t\t\t\ \ \ \ \ \ \ \ \ \ Sum[\[IndentingNewLine]D[ g\[LeftDoubleBracket]c, a, b\[RightDoubleBracket], \ coord\[LeftDoubleBracket]c\[RightDoubleBracket]] - \ D[g\[LeftDoubleBracket]c, a, c\[RightDoubleBracket], \ coord\[LeftDoubleBracket]b\[RightDoubleBracket]]\ + \ Sum[g\[LeftDoubleBracket]c, a, b\[RightDoubleBracket] g\[LeftDoubleBracket]d, c, d\[RightDoubleBracket] - g\[LeftDoubleBracket]d, a, c\[RightDoubleBracket] g\[LeftDoubleBracket]c, d, b\[RightDoubleBracket], \ {d, dim}], \ {c, \ dim}], \ \[IndentingNewLine]{a, \ dim}, \ {b, \ dim}]; \ rawricci // Simplify];\)\ \), "\n", \(\(ttrace[covtens_, \ metric_] := \ \ \ Module[{dim\ = \ Length[metric]}, \ Sum[\(Inverse[metric]\)\[LeftDoubleBracket]a, b\[RightDoubleBracket]\ covtens\[LeftDoubleBracket]a, b\[RightDoubleBracket], {a, \ dim}, \ {b, \ dim}] // Simplify];\)\), "\[IndentingNewLine]", \(\(getscalar[ric_, \ metric_] := \ ttrace[ric, \ metric];\)\), "\[IndentingNewLine]", \(\(geteinstein[ric_, \ metric_] := \ \((ric\ - \ getscalar[ric, \ metric] metric/2\ )\) // FullSimplify;\)\)}], "Input", CellLabel->"In[5]:="] }, Open ]], Cell[" ", "Text", Editable->False, Selectable->False, CellFrame->{{0, 0}, {0, 2}}, ShowCellBracket->False, CellMargins->{{0, 0}, {1, 1}}, CellElementSpacings->{"CellMinHeight"->1}, CellFrameMargins->False, CellFrameColor->RGBColor[0, 0, 1], CellSize->{Inherited, 4}], Cell[CellGroupData[{ Cell["1. 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\ call\ \[ExponentialE]\^C[1]\ the\ Schwarzschild\ radius\ rs, \ so\ \ \ \ B[ r]\ = \ \(r\/\(r - rs\)\ = \ \(\(\((1 - rs\/r)\)\^\(-1\) . \ Next\ the\ 1 - 1\ component\ should\ also\ equal\ 0\)\(:\)\)\)\)], "Text", FontFamily->"Courier New"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{ RowBox[{"DSolve", "[", RowBox[{ RowBox[{ RowBox[{\(A[r]\), "-", \(A[r]\ r\/\(\(-rs\) + r\)\), "+", RowBox[{"r", " ", RowBox[{ SuperscriptBox["A", "\[Prime]", MultilineFunction->None], "[", "r", "]"}]}]}], "\[Equal]", "0"}], ",", " ", \(A[r]\), ",", "r"}], "]"}], "//", "Simplify"}]], "Input", CellLabel->"In[35]:="], Cell[BoxData[ \({{A[r] \[Rule] \(\((r - rs)\)\ C[1]\)\/r}}\)], "Output", CellLabel->"Out[35]="] }, Open ]], Cell[BoxData[ \(Thus\ with\ constant\ C[1] = 1\ we\ get\ the\ Schwarzschild\ metric\ as\ hoped\ for . \ \ \ \[IndentingNewLine]Let' s\ \(check--\)\ the\ Einstein\ Tensor\ should\ now\ \(\(vanish\)\(:\)\ \)\)], "Text", FontFamily->"Courier New"], Cell[BoxData[ \(A[r_] := C \((1 - rs/r)\); 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