(*********************************************************************** Mathematica-Compatible Notebook This notebook can be used on any computer system with Mathematica 3.0, MathReader 3.0, or any compatible application. The data for the notebook starts with the line of 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. ***********************************************************************) (*NotebookFileLineBreakTest NotebookFileLineBreakTest*) (*NotebookOptionsPosition[ 35084, 1344]*) (*NotebookOutlinePosition[ 60993, 2280]*) (* CellTagsIndexPosition[ 60949, 2276]*) (*WindowFrame->Normal*) Notebook[{ Cell["\<\ Covariant and contravariant components of a generic metric, its determinant \ and Christoffel symbols\ \>", "Section", Evaluatable->False], Cell["\<\ Loading EinS... Be sure to change the next input line to the directory you \ placed file Eins.m\ \>", "SmallText", PageWidth->Infinity, Evaluatable->False], Cell["SetDirectory[\"c:/klioner/tensor\"];", "Input", PageWidth->Infinity], Cell[CellGroupData[{ Cell["<Infinity], Cell["\<\ EinS 2.6 (9 June 1997) Mathematica version: Silicon Graphics 3.0 (October 4, 1996) release number 0\ \>", "Print"] }, Open ]], Cell["\<\ Now EinS is ready to work. In this demo notebook we will show one way to \ program within EinS and Mathematica in general. This example is conceived to show some basic features of EinS, but by no \ means to show a preferable style of programming in EinS or Mathematica. Mathematica language is powerful enough to give user an \ opportunity to program in many styles: from procedure-oriented (PASCAL-like) or pure functional (LISP-like) to logical \ (PROLOG-like).\ \>", "SmallText", PageWidth->Infinity, Evaluatable->False], Cell[CellGroupData[{ Cell["\<\ Definition of a general expansion of covariant components of the metric \ tensor\ \>", "Subsubsection", Evaluatable->False], Cell["\<\ Here we define various objects entering in the expansion of the metric tensor \ with the aid of DefObject. Let us look on the online help message about this procedure\ \>", "SmallText", PageWidth->Infinity, Evaluatable->False], Cell[CellGroupData[{ Cell["?DefObject", "Input", PageWidth->Infinity], Cell["\<\ DefObject[object_name,valence,\"printname\", {i1,i2,...},{j1,j2,...}, {l1,...},ESIndicesLow -> True] defines the object of specified valence to be symmetric with respect to indices number i1, i2, ... and alternating with respect to indices number j1, j2, ... Both list are optional and empty by default. The procedure also defines output format (for PrintES and EinS2TeX for the object. The list {l1,l2,...} (empty by default) specified number of indices which will be printed as subscripts, while all other indices will be printed as superscripts. The 3rd parameter is optional. It could be a string representing print name of the object with specified name and valence The 6th parameter is optional. It could be ESIndicesLow -> True as shown above (ESIndicesLow -> False, by default) and makes all subscripts to be superscripts and vice verse. Note that objects with the same input name, but with different valences can have different symmetry properties, print names, and output and TeX formats.\ \>", "Print"] }, Open ]], Cell["Well. Let us try...", "SmallText", PageWidth->Infinity, Evaluatable->False], Cell[CellGroupData[{ Cell["DefObject[U,0]", "Input"], Cell["\<\ DefObject:: Object defined as follows: input name: U, valence 0\ \>", "Print"] }, Open ]], Cell[CellGroupData[{ Cell["DefObject[R,1]", "Input", PageWidth->Infinity], Cell["\<\ DefObject:: Object defined as follows: input name: R, valence 1 indices number(s) {1} are printed as superscripts\ \>", "Print"] }, Open ]], Cell[CellGroupData[{ Cell["DefObject[U,1]", "Input", PageWidth->Infinity], Cell["\<\ DefObject:: Object defined as follows: input name: U, valence 1 indices number(s) {1} are printed as superscripts\ \>", "Print"] }, Open ]], Cell[CellGroupData[{ Cell["DefObject[U,2,{1,2}]", "Input", PageWidth->Infinity], Cell["\<\ DefObject:: Object defined as follows: input name: U, valence 2 symmetric in indices: {1, 2} indices number(s) {1, 2} are printed as superscripts\ \>", "Print"] }, Open ]], Cell[CellGroupData[{ Cell["DefObject[W,1]", "Input", PageWidth->Infinity], Cell["\<\ DefObject:: Object defined as follows: input name: W, valence 1 indices number(s) {1} are printed as superscripts\ \>", "Print"] }, Open ]], Cell[CellGroupData[{ Cell["DefObject[W,2,{1,2}]", "Input", PageWidth->Infinity], Cell["\<\ DefObject:: Object defined as follows: input name: W, valence 2 symmetric in indices: {1, 2} indices number(s) {1, 2} are printed as superscripts\ \>", "Print"] }, Open ]], Cell[CellGroupData[{ Cell["DefObject[g,2,{1,2},ESIndicesLow -> True]", "Input", PageWidth->Infinity], Cell["\<\ DefObject:: Object defined as follows: input name: g, valence 2 symmetric in indices: {1, 2} indices number(s) {1, 2} are printed as subscripts\ \>", "Print"] }, Open ]], Cell[TextData[{ "Here is a rule defining convariant components of the metric. Rules are one \ of the ways to program within ", StyleBox["Mathematica", FontSlant->"Italic"], "." }], "SmallText", PageWidth->Infinity, Evaluatable->False], Cell["\<\ CovariantComponents={ g[0,0]:> 1+c1^2 U+c1^4 W+O[c1]^5, g[0,i_]:>c1 R[i]+c1^3 U[i]+c1^5 W[i]+O[c1]^6, g[i_,j_]:>-Delta[i,j]+c1^2 U[i,j]+ c1^4 W[i,j]+O[c1]^5};\ \>", "Input", PageWidth->Infinity], Cell["Now we can print covariant metric as symbols:", "SmallText", PageWidth->Infinity, Evaluatable->False], Cell[CellGroupData[{ Cell["PrintES[g[0,0]]", "Input", PageWidth->Infinity], Cell["\<\ g 00\ \>", "Print"] }, Open ]], Cell[CellGroupData[{ Cell["Print[g[0,i]]", "Input", PageWidth->Infinity], Cell["\<\ g 0i\ \>", "Print"] }, Open ]], Cell[CellGroupData[{ Cell["Print[g[i,j]]", "Input", PageWidth->Infinity], Cell["\<\ g ij\ \>", "Print"] }, Open ]], Cell["or substitute their expansions s defined above:", "SmallText", PageWidth->Infinity, Evaluatable->False], Cell[CellGroupData[{ Cell["PrintES[g[0,0] /. CovariantComponents]", "Input", PageWidth->Infinity], Cell["\<\ -2 -4 -1 5 1 + U c + W c + O[c ]\ \>", "Print"] }, Open ]], Cell[CellGroupData[{ Cell["PrintES[g[0,i] /. CovariantComponents]", "Input", PageWidth->Infinity], Cell["\<\ i -1 i -3 i -5 -1 6 R c + U c + W c + O[c ]\ \>", "Print"] }, Open ]], Cell[CellGroupData[{ Cell["PrintES[g[i,0] /. CovariantComponents]", "Input", PageWidth->Infinity], Cell["\<\ i -1 i -3 i -5 -1 6 R c + U c + W c + O[c ]\ \>", "Print"] }, Open ]], Cell[CellGroupData[{ Cell["PrintES[g[i,j] /. CovariantComponents]", "Input", PageWidth->Infinity], Cell[TextData[ " ij ij -2 ij -4 -1 5\n-\[Delta] + U c + W c + O[c \ ]"], "Print"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["Definition of contravariant components of the metric", "Subsubsection", Evaluatable->False], Cell["\<\ Defining contravariant components of the metric as EinS's object. Note that \ we want to output the contravariant components with the same name \"g\" as covariant ones. EinS in fact does not know anyhow \ that two object \"gc\" and \"g\" (as defined in the previous section) represent covariant and contravariant components of the \ same tensor. We want just to tell him to output the object \"gc\" with the print name \"g\".\ \>", "SmallText", PageWidth->Infinity, Evaluatable->False], Cell[CellGroupData[{ Cell["DefObject[gc,2,\"g\",{1,2}]", "Input", PageWidth->Infinity], Cell["\<\ DefObject:: Object defined as follows: input name: gc, valence 2 print name: g symmetric in indices: {1, 2} indices number(s) {1, 2} are printed as superscripts\ \>", "Print"] }, Open ]], Cell["\<\ In principle EinS could help to find expansions of the contravariant \ components of metric from the given expansions of the covariant ones, but here we prefer to specify rules for computing the \ expansions of the contravariant components \"manually\" (of course, they have been found with EinS!) and then check that the two \ matrices are really invert of each other.\ \>", "SmallText", PageWidth->Infinity, Evaluatable->False], Cell["\<\ ContravariantComponents={ gc[0,0]:>Module[{k,l}, 1+c1^2 (-U-DefES[R[k] R[k],{k}])+ c1^4 (U^2-W+2U DefES[R[k] R[k],{k}]+ DefES[R[k] R[k] R[l] R[l],{k,l}]- 2DefES[R[k] U[k],{k}]- DefES[U[k,l]R[k]R[l],{k,l}])+ O[c1]^5], gc[0,i_]:>Module[{k,l}, c1 R[i]+ c1^3 (U[i]-U R[i]- R[i] DefES[R[k] R[k],{k}]+ DefES[U[i,k]R[k],{k}])+ c1^5 (R[i] (U^2-W+ 2U DefES[R[k] R[k],{k}]- 2 DefES[R[k] U[k],{k}]+ DefES[R[k] R[k] R[l] R[l],{k,l}]- DefES[U[k,l]R[k]R[l],{k,l}])+ W[i]+DefES[R[k] W[i,k],{k}]- U U[i]- DefES[R[k] R[k],{k}] U[i]+ DefES[U[i,k]U[k],{k}]- U DefES[U[i,k]R[k],{k}]+ DefES[R[l]U[i,k]U[l,k],{k,l}]- DefES[U[i,k]R[k]R[l]R[l],{k,l}])+ O[c1]^6], gc[i_,j_]:>Module[{k}, -Delta[i,j]+ c1^2 (-U[i,j]+R[i] R[j])+ c1^4 (-W[i,j]-U R[i] R[j]- R[i]R[j] DefES[R[k] R[k],{k}]+ R[i]U[j]+R[j]U[i]- DefES[U[i,k]U[j,k],{k}]+ R[j]DefES[U[i,k]R[k],{k}]+ R[i]DefES[U[j,k]R[k],{k}])+ O[c1]^5]};\ \>", "Input", PageWidth->Infinity], Cell["Now we can print contravariant metric as symbols:", "SmallText", PageWidth->Infinity, Evaluatable->False], Cell[CellGroupData[{ Cell["Print[gc[0,0]]", "Input", PageWidth->Infinity], Cell["\<\ 00 g\ \>", "Print"] }, Open ]], Cell[CellGroupData[{ Cell["Print[gc[0,i]]", "Input", PageWidth->Infinity], Cell["\<\ 0i g\ \>", "Print"] }, Open ]], Cell[CellGroupData[{ Cell["Print[gc[i,j]]", "Input", PageWidth->Infinity], Cell["\<\ ij g\ \>", "Print"] }, Open ]], Cell["or substitute their expansions as defined above:", "SmallText", PageWidth->Infinity, Evaluatable->False], Cell[CellGroupData[{ Cell["PrintES[gc[0,0] /. ContravariantComponents]", "Input", PageWidth->Infinity], Cell["\<\ a a -2 2 a a b b a a a a 1 + (-U - R R ) c + (U - W + 2 U R R + R R R R - 2 R U - a b ab -4 -1 5 R R U ) c + O[c ]\ \>", "Print"] }, Open ]], Cell[CellGroupData[{ Cell["PrintES[gc[0,i] /. ContravariantComponents]", "Input", PageWidth->Infinity], Cell["\<\ i -1 a ia i a a i i -3 R c + (R U - U R - R R R + U ) c + a ia a ia b b a ia b ab ia a ia (-(U R U ) - R U R R + U U + R U U + R W + 2 a a b b a a a a a b ab i i (U - W + 2 U R R + R R R R - 2 R U - R R U ) R - U U - a a i i -5 -1 6 R R U + W ) c + O[c ]\ \>", "Print"] }, Open ]], Cell[CellGroupData[{ Cell["PrintES[gc[i,0] /. ContravariantComponents]", "Input", PageWidth->Infinity], Cell["\<\ i -1 a ia i a a i i -3 R c + (R U - U R - R R R + U ) c + a ia a ia b b a ia b ab ia a ia (-(U R U ) - R U R R + U U + R U U + R W + 2 a a b b a a a a a b ab i i (U - W + 2 U R R + R R R R - 2 R U - R R U ) R - U U - a a i i -5 -1 6 R R U + W ) c + O[c ]\ \>", "Print"] }, Open ]], Cell[CellGroupData[{ Cell["PrintES[gc[i,j] /. ContravariantComponents]", "Input", PageWidth->Infinity], Cell[TextData[ " ij i j ij -2 ia ja a ja i a ia j\n-\[Delta] \ + (R R - U ) c + (-(U U ) + R U R + R U R - \n \n i j \ a a i j j i i j ij -4 -1 5\n U R R - R R R R \ + R U + R U - W ) c + O[c ]"], "Print"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["Are the contravariant components correct? Various tests", "Subsubsection", Evaluatable->False], Cell["\<\ Let us check if the contravariant components as we specified them in the list \ of rules named ContravariantComponents are in agreement with the covariant components as we specified them in the list \ of rules named CovariantComponents. Namely we have to check that the two matrices defined by the covariant and \ contravariant components are inverse of each other. We can check this in a variety of ways. Two ways are described below. Note \ the use of EinS's function ToComponents allowing one to re-write all implicit summation in explicit form.\ \>", "SmallText", PageWidth->Infinity], Cell[CellGroupData[{ Cell["\<\ Way 1: we check that matrix product of g[i,j] and gc[k,l] is identity matrix \ within truncation errors of our expansions\ \>", "SmallText", PageWidth->Infinity, Evaluatable->False], Cell["\<\ unit=ExpandAll[Table[g[i-1,j-1] /. CovariantComponents, {i,4},{j,4}] . Table[gc[i-1,j-1] /. ContravariantComponents, {i,4},{j,4}]];\ \>", "Input", PageWidth->Infinity], Cell[CellGroupData[{ Cell["\<\ Do[Print[i-1,\":\",j-1,\":\", ToComponents[ComputeES[unit[[i,j]]]]],{i,4},{j,4}]\ \>", "Input", PageWidth->Infinity], Cell["\<\ -1 5 0:0:1 + O[c ] -1 6 0:1:O[c ] -1 6 0:2:O[c ] -1 6 0:3:O[c ] -1 6 1:0:O[c ] -1 5 1:1:1 + O[c ] -1 5 1:2:O[c ] -1 5 1:3:O[c ] -1 6 2:0:O[c ] -1 5 2:1:O[c ] -1 5 2:2:1 + O[c ] -1 5 2:3:O[c ] -1 6 3:0:O[c ] -1 5 3:1:O[c ] -1 5 3:2:O[c ] -1 5 3:3:1 + O[c ]\ \>", "Print"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["\<\ Way 2: We directly Invert matrix of covariant components and check that the \ inverse does coincide with matrix of contravariant components within the truncation errors of our expansions\ \>", "SmallText", PageWidth->Infinity, Evaluatable->False], Cell["\<\ inver=Inverse[Table[g[i,j] /. CovariantComponents,{i,0,3},{j,0,3}]];\ \>", "Input", PageWidth->Infinity], Cell[CellGroupData[{ Cell["\<\ Do[Print[i,\":\",j,\":\", ToComponents[ComputeES[gc[i,j] /. \ ContravariantComponents]]-inver[[i+1,j+1]]], {i,0,3},{j,0,3}]\ \>", "Input", PageWidth->Infinity], Cell["\<\ -1 5 0:0:O[c ] -1 6 0:1:O[c ] -1 6 0:2:O[c ] -1 6 0:3:O[c ] -1 6 1:0:O[c ] -1 5 1:1:O[c ] -1 5 1:2:O[c ] -1 5 1:3:O[c ] -1 6 2:0:O[c ] -1 5 2:1:O[c ] -1 5 2:2:O[c ] -1 5 2:3:O[c ] -1 6 3:0:O[c ] -1 5 3:1:O[c ] -1 5 3:2:O[c ] -1 5 3:3:O[c ]\ \>", "Print"] }, Open ]] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["Defining and checking the determinant of the metric", "Subsubsection", Evaluatable->False], Cell["\<\ We introduce the definition of the determinant and check that this definition \ really coincides with the determinant of the matrix of the covariant components and with the inverse of the determinant of the \ matrix of the contravariant components as defined above.\ \>", "SmallText", PageWidth->Infinity, Evaluatable->False], Cell["\<\ detg:=-1+c1^2(-U-DefES[R[i]R[i],{i}]+ DefES[U[i,i],{i}])+ c1^4(-W-2DefES[R[i]U[i],{i}]+ U DefES[U[i,i],{i}]+ DefES[W[i,i],{i}]+ 1/2 DefES[U[i,j]^2,{i,j}]- 1/2 DefES[U[i,i] U[j,j],{i,j}]- DefES[R[i]R[j]U[i,j],{i,j}]+ DefES[R[i]^2 U[j,j],{i,j}])+ O[c1]^6;\ \>", "Input", PageWidth->Infinity], Cell["\<\ determinant1=Det[Table[g[i,j] /. CovariantComponents, {i,0,3},{j,0,3}]];\ \>", "Input", PageWidth->Infinity], Cell[CellGroupData[{ Cell["Expand[ToComponents[ComputeES[detg-determinant1]]]", "Input", PageWidth->Infinity], Cell[OutputFormData["\<\ SeriesData[c1, 0, {}, 5, 5, 1]\ \>", "\<\ -1 5 O[c ]\ \>"], "Output"] }, Open ]], Cell["\<\ determinant2=Det[Table[gc[i,j] /. ContravariantComponents, {i,0,3},{j,0,3}]];\ \>", "Input", PageWidth->Infinity], Cell[CellGroupData[{ Cell["Expand[ToComponents[ComputeES[detg-1/determinant2]]]", "Input", PageWidth->Infinity], Cell[OutputFormData["\<\ SeriesData[c1, 0, {}, 5, 5, 1]\ \>", "\<\ -1 5 O[c ]\ \>"], "Output"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["Computing Christoffel symbols of the second kind", "Subsubsection", Evaluatable->False], Cell["\<\ We define object \"Gam\" with 3 indices. This object will represent \ Christoffel symbol of the second kind. \ \>", "SmallText", PageWidth->Infinity], Cell[CellGroupData[{ Cell[TextData["DefObject[Gam,3,\"\[CapitalGamma]\",{2,3},{},{2,3}]"], "Input", PageWidth->Infinity], Cell[TextData[{ "DefObject:: Object defined as follows:\n input name: Gam, \ valence 3\n print name: \[CapitalGamma]\n symmetric \ in indices: {2, 3}", "\n indices number(s) {1} are printed as superscripts\n \ indices number(s) {2, 3} are printed as subscripts" }], "Print"] }, Open ]], Cell["\<\ We define a set of coordinates. This automatically generate simplification \ rules as well as output formats for partial derivatives with respect to these coordinates.\ \>", "SmallText", PageWidth->Infinity, Evaluatable->False], Cell[CellGroupData[{ Cell["DefRS[x,t]", "Input", PageWidth->Infinity], Cell["\<\ -1 0 i DefRS:: reference system (t = c x ,x ) defined.\ \>", "Print"] }, Open ]], Cell["\<\ Now we are ready to define Christoffel symbols in terms of metric. Let us \ recall that we have defined objects \"g\" and \"gc\" above. \ \>", "SmallText", PageWidth->Infinity, Evaluatable->False], Cell[CellGroupData[{ Cell["\<\ withmetric ={ Gam[i_,k_,l_] :> Module[{m}, DefES[1/2 gc[i,m]* (PD[g[m,k],x[l]]+ PD[g[m,l],x[k]]- PD[g[k,l],x[m]]),{m}, ESRange -> $ESDimension]]};\ \>", "Input", PageWidth->Infinity], Cell["\<\ We can operate now with Christoffel symbols. We can treat them as symbols \ (below we work with Gam[i,0,0], but we could do the same for any Christoffel symbol):\ \>", "SmallText", PageWidth->Infinity, Evaluatable->False] }, Open ]], Cell[CellGroupData[{ Cell["Print[Gam[i,0,0]]", "Input", PageWidth->Infinity], Cell[TextData[" i\n\[CapitalGamma]\n 00"], "Print"] }, Open ]], Cell["\<\ We can compute them in terms of metric components to be hold as symbols (note \ that \"space-time\" indices are printed with apostroph \" ' \" ):\ \>", "SmallText", PageWidth->Infinity, Evaluatable->False], Cell[CellGroupData[{ Cell["PrintES[Gam[i,0,0] /. withmetric]", "Input", PageWidth->Infinity], Cell[TextData[ " i\[Alpha] -1\ng (-g + 2 c g )\n 00,\[Alpha] \ 0\[Alpha],t\n--------------------------\n 2"], "Print"] }, Open ]], Cell["\<\ We can separate zero (\"time\") value of the dummy indices and non-zero \ (\"spatial\") values with the aid of function SplitTime\ \>", "SmallText", PageWidth->Infinity, Evaluatable->False], Cell[CellGroupData[{ Cell["PrintES[SplitTime[Gam[i,0,0] /. withmetric]]", "Input", PageWidth->Infinity], Cell["\<\ ia -1 -1 0i g (-g + 2 c g ) c g g 00,a 0a,t 00,t -------------------------- + ------------- 2 2\ \>", "Print"] }, Open ]], Cell["\<\ and compute Christoffel symbols in terms of expansion of metric components \ \>", "SmallText", PageWidth->Infinity, Evaluatable->False], Cell[CellGroupData[{ Cell["\<\ PrintES[gi00=ComputeES[SplitTime[Gam[i,0,0] /. withmetric] /. CovariantComponents /. ContravariantComponents]]\ \>", "Input", PageWidth->Infinity], Cell["\<\ ia a i U U U W U R R ,i i -2 ,a a ia ,i i ,a (--- - R ) c + (------- - R U + --- - U - --------- + 2 ,t 2 ,t 2 ,t 2 i U R a a i ,t -4 -1 5 R R R + ------) c + O[c ] ,t 2\ \>", "Print"] }, Open ]], Cell["\<\ How many terms (monomials which may involve implicit summation) do we have in \ the result?\ \>", "SmallText", PageWidth->Infinity, Evaluatable->False], Cell[CellGroupData[{ Cell["SeriesLength[gi00]", "Input", PageWidth->Infinity], Cell["\<\ -2 2 terms at c -4 7 terms at c\ \>", "Print"] }, Open ]], Cell["\<\ EinS can produce LaTeX form of any expression with aligned form with \ specified number of terms per line\ \>", "SmallText", PageWidth->Infinity, Evaluatable->False], Cell[CellGroupData[{ Cell["\<\ $TermsPerLine=4; EinS2TeX[gi00]\ \>", "Input", PageWidth->Infinity], Cell[OutputFormData["\<\ \"\\\\begin{eqnarray}\\n&&\\\\biggl[ {1\\\\over 2}{U}_{,i}\\n-{R^{i}_{ \\ }}_{,t}\\\\biggr] c^{-2}\\n\\\\nonumber\\\\\\\\\\n&&+\\\\biggl[ {1\\\\over \\ 2}{U}_{,a}U^{ia}_{ }\\n-{R^{a}_{ }}_{,t}U^{ia}_{ }\\n+{1\\\\over \\ 2}{W}_{,i}\\n-{U^{i}_{ }}_{,t}\\n\\\\nonumber\\\\\\\\\\n&&-{1\\\\over \\ 2}{U}_{,a}R^{a}_{ }R^{i}_{ }\\n+{R^{a}_{ }}_{,t}R^{a}_{ }R^{i}_{ \\ }\\n+{1\\\\over 2}{U}_{,t}R^{i}_{ }\\\\biggr] \\ c^{-4}\\n\\\\nonumber\\\\\\\\\\n&&+O(c^{-5})\\\\nonumber\\\\\\\\\\n\\\\end{\ eqnarray}\"\ \>", "\<\ \\begin{eqnarray} &&\\biggl[ {1\\over 2}{U}_{,i} -{R^{i}_{ }}_{,t}\\biggr] c^{-2} \\nonumber\\\\ &&+\\biggl[ {1\\over 2}{U}_{,a}U^{ia}_{ } -{R^{a}_{ }}_{,t}U^{ia}_{ } +{1\\over 2}{W}_{,i} -{U^{i}_{ }}_{,t} \\nonumber\\\\ &&-{1\\over 2}{U}_{,a}R^{a}_{ }R^{i}_{ } +{R^{a}_{ }}_{,t}R^{a}_{ }R^{i}_{ } +{1\\over 2}{U}_{,t}R^{i}_{ }\\biggr] c^{-4} \\nonumber\\\\ &&+O(c^{-5})\\nonumber\\\\ \\end{eqnarray}\ \>"], "Output", LineSpacing->{1, 0}] }, Open ]], Cell["\<\ or save it directly to the file. For example, in plain TeX form (you should \ add only \"\\bye\" at the end of file, then run TeX and enjoy the publication quality printout of your expression in traditional \ notations):\ \>", "SmallText", PageWidth->Infinity, Evaluatable->False], Cell["\<\ $TeXDialect=$PlainTeX; SaveTeX[\"demo1tex.tex\",gi00]\ \>", "Input", PageWidth->Infinity], Cell["\<\ At last we can substitute more specific form of the metric by giving \ definitions for the potentials entering in the metric. Namely we want now to compute Christoffel symbols in a reference system \ rigidly rotating relative to harmonic ones. In this we will use lower precision by substituting zero for higher order \ potentials. We have first to define new object which will appear in our calculations: omega[i] represents the angular velocity of \ rotation of the rotating reference system relative to non-rotating one.\ \>", "SmallText", PageWidth->Infinity, Evaluatable->False], Cell[CellGroupData[{ Cell[TextData["DefObject[omega,1,\"\[Omega]\"]"], "Input", PageWidth->Infinity], Cell[TextData[{ "DefObject:: Object defined as follows:\n input name: omega, \ valence 1", "\n print name: \[Omega]\n indices number(s) {1} \ are printed as superscripts" }], "Print"] }, Open ]], Cell["\<\ Standard procedure DefObject does not allow to define separately output name \ and name for TeX output, but we want to have \\omega for our object omega in TeX output. It is easy. 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(*********************************************************************** End of Mathematica Notebook file. ***********************************************************************)