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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[ 897758, 17857]*) (*NotebookOutlinePosition[ 898517, 17883]*) (* CellTagsIndexPosition[ 898473, 17879]*) (*WindowFrame->Normal*) Notebook[{ Cell[CellGroupData[{ Cell[TextData[{ StyleBox["Geodesic Motion in the Ferrari-Iba\[NTilde]ez metrics", FontSize->40, FontColor->RGBColor[0.87892, 0.117189, 0.308598]], "\n", StyleBox["A particle followed from Flat Space to a GW collision background", FontFamily->"Verdana", FontSize->24, FontSlant->"Italic", FontVariations->{"CompatibilityType"->0}] }], "Title", TextAlignment->Center], Cell[TextData[{ StyleBox["Ref.: D. 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redefined as ", Cell[BoxData[ \(TraditionalForm\`K\_x\)]], ", ", Cell[BoxData[ \(TraditionalForm\`K\_y\)]], ", ", Cell[BoxData[ \(TraditionalForm\`K\_u\)]], " :" }], "Text", FontSize->24], Cell[CellGroupData[{ Cell[BoxData[ \(Solve[MakeSum[Metricg[li, lj] X[ui] U[uj]] \[Equal] Kx\ const, U[1]]\)], "Input", FontSize->24], Cell[BoxData[ \({{Ux[\[Tau]] \[Rule] \(-\(\(Kx\ \((1 + \[Sigma]\ Sin[ u[\[Tau]]])\)\)\/\(\(-1\) + \[Sigma]\ Sin[ u[\[Tau]]]\)\)\)}}\)], "Output", FontSize->24] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(Solve[MakeSum[Metricg[li, lj] Y[ui] U[uj]] \[Equal] Ky\ const, U[2]]\)], "Input", FontSize->24], Cell[BoxData[ \({{Uy[\[Tau]] \[Rule] \(Ky\ Sec[u[\[Tau]]]\^2\)\/\((1 + \[Sigma]\ Sin[u[\ \[Tau]]])\)\^2}}\)], "Output", FontSize->24] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(Solve[MakeSum[Metricg[li, lj] V[ui] U[uj]] \[Equal] Ku\ const, U[3]]\)], "Input", FontSize->24], Cell[BoxData[ \({{Uu[\[Tau]] \[Rule] \(-\(Ku\/\(2\ \((1 + \[Sigma]\ \ 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Proof:\ \>", "Text", FontSize->24], Cell[CellGroupData[{ Cell[BoxData[ \(a = Simplify[Eq[4]] /. ss\)], "Input", FontSize->24], Cell[BoxData[ RowBox[{ RowBox[{\(\(Ky\^2\ Sec[u[\[Tau]]]\^3\ \((\[Sigma]\ Cos[2\ u[\[Tau]]] - Sin[u[\[Tau]]])\)\)\/\(2\ \((1 + \[Sigma]\ \ Sin[u[\[Tau]]])\)\^5\)\), "+", RowBox[{ SuperscriptBox["v", "\[Prime]\[Prime]", MultilineFunction->None], "[", "\[Tau]", "]"}]}], "==", \(\(Kx\^2\ \[Sigma]\ Cos[ u[\[Tau]]]\)\/\(2\ \((\(-1\) + Sin[u[\[Tau]]]\^2)\)\^2\)\)}]], \ "Output", FontSize->24] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(b = Simplify[Eq[5]]\)], "Input", FontSize->24], Cell[BoxData[ RowBox[{ RowBox[{\((1 + \[Sigma]\ Sin[u[\[Tau]]])\), " ", RowBox[{"(", RowBox[{\(Kx\^2\/\(1 - \[Sigma]\ Sin[u[\[Tau]]]\)\), "+", \(\(Ky\^2\ Sec[u[\[Tau]]]\^2\)\/\((1 + \[Sigma]\ \ Sin[u[\[Tau]]])\)\^3\), "+", FractionBox[ RowBox[{"2", " ", "Ku", " ", RowBox[{ SuperscriptBox["v", "\[Prime]", MultilineFunction->None], "[", "\[Tau]", "]"}]}], \(1 + \[Sigma]\ 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Inherited}, {Inherited, Inherited}}, FontSize->24], Cell[CellGroupData[{ Cell[BoxData[ \(Table[ Eq[k] = \[PartialD]\_\[Tau]\ U[k] + Simplify[\((\[Sum]\+\(i = 1\)\%4\(\[Sum]\+\(j = 1\)\%4 AffineG[ k, \(-i\), \(-j\)] U[i] U[j]\))\) /. ss] \[Equal] 0, {k, 4}] // TableForm\)], "Input", FontSize->24], Cell[BoxData[ InterpretationBox[GridBox[{ { RowBox[{ RowBox[{ SuperscriptBox["Ux", "\[Prime]", MultilineFunction->None], "[", "\[Tau]", "]"}], "==", RowBox[{ "2", " ", "\[Sigma]", " ", \(Sec[t[\[Tau]]]\), " ", \(Ux[\[Tau]]\), " ", RowBox[{ SuperscriptBox["t", "\[Prime]", MultilineFunction->None], "[", "\[Tau]", "]"}]}]}]}, { RowBox[{ RowBox[{ RowBox[{ SuperscriptBox["Uy", "\[Prime]", MultilineFunction->None], "[", "\[Tau]", "]"}], "+", RowBox[{\(Uy[\[Tau]]\), " ", RowBox[{"(", RowBox[{ FractionBox[ RowBox[{ "2", " ", "\[Sigma]", " ", \(Cos[t[\[Tau]]]\), " ", RowBox[{ SuperscriptBox["t", "\[Prime]", MultilineFunction->None], "[", "\[Tau]", "]"}]}], \(1 + \[Sigma]\ Sin[t[\[Tau]]]\)], "-", 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"]"}], "2"], "+", SuperscriptBox[ RowBox[{ SuperscriptBox["z", "\[Prime]", MultilineFunction->None], "[", "\[Tau]", "]"}], "2"]}], ")"}]}]}], ")"}]}], \(\((1 + \[Sigma]\ Sin[t[\[Tau]]])\)\^4\)], "+", RowBox[{ SuperscriptBox["t", "\[Prime]\[Prime]", MultilineFunction->None], "[", "\[Tau]", "]"}]}], "==", "0"}]} }, RowSpacings->1, ColumnSpacings->3, RowAlignments->Baseline, ColumnAlignments->{Left}], TableForm[ { Equal[ Derivative[ 1][ Ux][ \[Tau]], Times[ 2, \[Sigma], Sec[ t[ \[Tau]]], Ux[ \[Tau]], Derivative[ 1][ t][ \[Tau]]]], Equal[ Plus[ Derivative[ 1][ Uy][ \[Tau]], Times[ Uy[ \[Tau]], Plus[ Times[ 2, \[Sigma], Cos[ t[ \[Tau]]], Power[ Plus[ 1, Times[ \[Sigma], Sin[ t[ \[Tau]]]]], -1], Derivative[ 1][ t][ \[Tau]]], Times[ -2, Tan[ z[ \[Tau]]], Derivative[ 1][ z][ \[Tau]]]]]], 0], Equal[ Plus[ Times[ Cos[ z[ \[Tau]]], Sin[ z[ \[Tau]]], Power[ Uy[ \[Tau]], 2]], Times[ 2, \[Sigma], Cos[ t[ \[Tau]]], Power[ Plus[ 1, Times[ \[Sigma], Sin[ t[ \[Tau]]]]], -1], Derivative[ 1][ t][ \[Tau]], 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Let's verify. This defines a generic symmetric tensor:\ \>", "Text", FontSize->24], Cell[BoxData[ \(DefineTensor[Kil, {{2, 1}, 1}]\)], "Input", FontSize->24], Cell[TextData[{ "\nThe Killing equation is ", Cell[BoxData[ StyleBox[\(\[Xi]\_\(\[Mu]; \[Nu]\) + \[Xi]\_\(\[Nu]; \[Mu]\) = 0\), FontSize->24, FontWeight->"Bold"]], FontSize->18], ". Let's just verify the left side is zero in both cases.\nThe first:" }], "Text", FontSize->24], Cell[BoxData[ \(\(Kil[li_, lj_] = MakeSum[CDtoOD[ CD[Metricg[li, la]\ V1[ua], lj] + CD[Metricg[lj, la]\ V1[ua], li]]];\)\)], "Input", FontSize->24], Cell[CellGroupData[{ Cell[BoxData[ \(Table[FullSimplify[Kil[\(-i\), \(-j\)]], {i, 4}, {j, 4}] // MatrixForm\)], "Input", FontSize->24], Cell[BoxData[ TagBox[ RowBox[{"(", "\[NoBreak]", GridBox[{ {"0", "0", "0", "0"}, {"0", "0", "0", "0"}, {"0", "0", "0", "0"}, {"0", "0", "0", "0"} }], "\[NoBreak]", ")"}], (MatrixForm[ #]&)]], "Output", FontSize->24] }, Open ]], Cell["\<\ The second one:\ \>", "Text", FontSize->24], Cell[BoxData[ \(\(Kil[li_, lj_] = MakeSum[CDtoOD[ CD[Metricg[li, la]\ V2[ua], lj] + CD[Metricg[lj, la]\ V2[ua], li]]];\)\)], "Input", FontSize->24], Cell[CellGroupData[{ Cell[BoxData[ \(Table[FullSimplify[Kil[\(-i\), \(-j\)]], {i, 4}, {j, 4}] // MatrixForm\)], "Input", FontSize->24], Cell[BoxData[ TagBox[ RowBox[{"(", "\[NoBreak]", GridBox[{ {"0", "0", "0", "0"}, {"0", "0", "0", "0"}, {"0", "0", "0", "0"}, {"0", "0", "0", "0"} }], "\[NoBreak]", ")"}], (MatrixForm[ #]&)]], "Output", FontSize->24] }, Open ]], Cell[TextData[{ "\nLet's write associated relations, where the constants are redefined as \ ", Cell[BoxData[ \(TraditionalForm\`K\_x\)]], ", ", Cell[BoxData[ \(TraditionalForm\`K\_y\)]], ", ", Cell[BoxData[ FormBox[ SubscriptBox["K", StyleBox["z1", FontSlant->"Italic"]], TraditionalForm]]], ", ", Cell[BoxData[ FormBox[ SubscriptBox["K", StyleBox["z2", FontSlant->"Italic"]], TraditionalForm]]], " (the two latter constants being related in some way to the angular \ momentum of the test particle).\nThe first two ones:" }], "Text", FontSize->24], Cell[CellGroupData[{ Cell[BoxData[ \(Solve[MakeSum[Metricg[li, lj] X[ui] U[uj]] \[Equal] Kx\ const, U[1]]\)], "Input", FontSize->24], Cell[BoxData[ \({{Ux[\[Tau]] \[Rule] \(-\(\(Kx\ \((1 + \[Sigma]\ Sin[ t[\[Tau]]])\)\)\/\(\(-1\) + \[Sigma]\ Sin[ t[\[Tau]]]\)\)\)}}\)], "Output", FontSize->24] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(Solve[MakeSum[Metricg[li, lj] Y[ui] U[uj]] \[Equal] Ky\ const, U[2]]\)], "Input", FontSize->24], Cell[BoxData[ \({{Uy[\[Tau]] \[Rule] \(Ky\ Sec[z[\[Tau]]]\^2\)\/\((1 + \[Sigma]\ Sin[t[\ \[Tau]]])\)\^2}}\)], "Output", FontSize->24] }, Open ]], Cell["\<\ and the corresponding re-definitions:\ \>", "Text", FontSize->24], Cell[BoxData[{ \(\(Ux[\[Tau]_] = Kx\ \(1 + \[Sigma]\ Sin[t[\[Tau]]]\)\/\(1 - \[Sigma]\ \ Sin[t[\[Tau]]]\);\)\), "\[IndentingNewLine]", \(\(Uy[\[Tau]_] = Ky\ Sec[z[\[Tau]]]\^2\/\((1 + \[Sigma]\ Sin[t[\[Tau]]])\)\^2;\)\)}], \ "Input", FontSize->24], Cell["\<\ Then the latter two, involving the new Killing vectors. The first one:\ \>", "Text", FontSize->24], Cell[CellGroupData[{ Cell[BoxData[ \(Simplify[ MakeSum[Metricg[li, lj] V1[ui] U[uj]] \[Equal] Kz1\ const]\)], "Input", FontSize->24], Cell[BoxData[ RowBox[{ RowBox[{\(const\ Ky\ Sin[y[\[Tau]]]\ Tan[z[\[Tau]]]\), "+", RowBox[{ "const", " ", \(Cos[y[\[Tau]]]\), " ", \(\((1 + \[Sigma]\ Sin[t[\[Tau]]])\)\^2\), " ", RowBox[{ SuperscriptBox["z", "\[Prime]", MultilineFunction->None], "[", "\[Tau]", "]"}]}]}], "==", \(const\ Kz1\)}]], "Output", FontSize->24] }, Open ]], Cell["\<\ so z'(\[Tau]) is, here:\ \>", "Text", FontSize->24], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"KilX", "=", RowBox[{"Simplify", "[", RowBox[{"Solve", "[", RowBox[{ RowBox[{"%", "/.", RowBox[{ RowBox[{ SuperscriptBox["z", "\[Prime]", MultilineFunction->None], "[", "\[Tau]", "]"}], "\[Rule]", "zp"}]}], ",", "zp"}], "]"}], "]"}]}]], "Input", FontSize->24], Cell[BoxData[ \({{zp \[Rule] \(Sec[y[\[Tau]]]\ \((Kz1 - Ky\ Sin[y[\[Tau]]]\ \ Tan[z[\[Tau]]])\)\)\/\((1 + \[Sigma]\ Sin[t[\[Tau]]])\)\^2}}\)], "Output", FontSize->24] }, Open ]], Cell["\<\ and the second one:\ \>", "Text", FontSize->24], Cell[CellGroupData[{ Cell[BoxData[ \(Simplify[ MakeSum[Metricg[li, lj] V2[ui] U[uj]] \[Equal] Kz2\ const]\)], "Input", FontSize->24], Cell[BoxData[ RowBox[{\(const\ Ky\ Cos[y[\[Tau]]]\ Tan[z[\[Tau]]]\), "==", RowBox[{"const", " ", RowBox[{"(", RowBox[{"Kz2", "+", RowBox[{\(\((1 + \[Sigma]\ Sin[t[\[Tau]]])\)\^2\), " ", \(Sin[y[\[Tau]]]\), " ", RowBox[{ SuperscriptBox["z", "\[Prime]", MultilineFunction->None], "[", "\[Tau]", "]"}]}]}], ")"}]}]}]], "Output", FontSize->24] }, Open ]], Cell["\<\ so z'(\[Tau]) is, this time:\ \>", "Text", FontSize->24], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"KilY", "=", RowBox[{"Simplify", "[", RowBox[{"Solve", "[", RowBox[{ RowBox[{"%", "/.", RowBox[{ RowBox[{ SuperscriptBox["z", "\[Prime]", MultilineFunction->None], "[", "\[Tau]", "]"}], "\[Rule]", "zp"}]}], ",", "zp"}], "]"}], "]"}]}]], "Input", FontSize->24], Cell[BoxData[ \({{zp \[Rule] \(\(-Kz2\)\ Csc[y[\[Tau]]] + Ky\ Cot[y[\[Tau]]]\ Tan[z[\ \[Tau]]]\)\/\((1 + \[Sigma]\ Sin[t[\[Tau]]])\)\^2}}\)], "Output", FontSize->24] }, Open ]], Cell["\<\ The two expressions must be the same:\ \>", "Text", FontSize->24], Cell[CellGroupData[{ Cell[BoxData[ \(Simplify[\(KilX[\([1, 1]\)]\)[\([2]\)] - \(KilY[\([1, 1]\)]\)[\([2]\)]] \[Equal] 0\)], "Input", FontSize->24], Cell[BoxData[ \(\(Kz1\ Sec[y[\[Tau]]] + Csc[y[\[Tau]]]\ \((Kz2 - Ky\ Sec[y[\[Tau]]]\ \ Tan[z[\[Tau]]])\)\)\/\((1 + \[Sigma]\ Sin[t[\[Tau]]])\)\^2 == 0\)], "Output", FontSize->24] }, Open ]], Cell["\<\ that means a prime integral of the motion is found:\ \>", "Text", FontSize->24], Cell[BoxData[ \(\(IntMot = Kz1\ Sec[y[\[Tau]]] + Csc[y[\[Tau]]]\ \((Kz2 - Ky\ Sec[y[\[Tau]]]\ Tan[z[\[Tau]]])\) == 0;\)\)], "Input", FontSize->24], Cell[TextData[{ "\nThe \"", StyleBox["z", FontSlant->"Italic"], "\" coordinate turns out to be an explicit function of one of the wavefront \ coordinates, regarding them as evolving with the proper time \[Tau] !" }], "Text", FontSize->24], Cell[CellGroupData[{ Cell[BoxData[ \(Solve[IntMot, z[\[Tau]]]\)], "Input", FontSize->24], Cell[BoxData[ \({{z[\[Tau]] \[Rule] ArcTan[\(Kz2\ Cos[y[\[Tau]]] + Kz1\ Sin[y[\[Tau]]]\)\/Ky]}}\)], \ "Output", FontSize->24] }, Open ]] }, Closed]], Cell[CellGroupData[{ Cell["Phase II - Reducing the system", "Section", CellFrame->{{0, 0}, {0.5, 0}}, FontSize->24], Cell["\<\ Again, two equations reduce to identities:\ \>", "Text", FontSize->24], Cell[CellGroupData[{ Cell[BoxData[ \(FullSimplify[Eq[1]] /. ss\)], "Input", FontSize->24], Cell[BoxData[ \(True\)], "Output", FontSize->24] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(FullSimplify[Eq[2]]\)], "Input", FontSize->24], Cell[BoxData[ \(True\)], "Output", FontSize->24] }, Open ]], Cell["\<\ and the others read as:\ \>", "Text", FontSize->24], Cell[CellGroupData[{ Cell[BoxData[ \(Simplify[Eq[3]]\)], "Input", FontSize->24], Cell[BoxData[ RowBox[{ RowBox[{\(\(Ky\^2\ Sec[z[\[Tau]]]\^2\ Tan[ z[\[Tau]]]\)\/\((1 + \[Sigma]\ Sin[t[\[Tau]]])\)\^4\), "+", FractionBox[ RowBox[{"2", " ", "\[Sigma]", " ", \(Cos[t[\[Tau]]]\), " ", RowBox[{ SuperscriptBox["t", "\[Prime]", MultilineFunction->None], "[", "\[Tau]", "]"}], " ", RowBox[{ SuperscriptBox["z", "\[Prime]", MultilineFunction->None], "[", "\[Tau]", "]"}]}], \(1 + \[Sigma]\ Sin[t[\[Tau]]]\)], "+", RowBox[{ SuperscriptBox["z", "\[Prime]\[Prime]", MultilineFunction->None], "[", "\[Tau]", "]"}]}], "==", "0"}]], "Output", FontSize->24] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(Simplify[Eq[4]]\)], "Input", FontSize->24], Cell[BoxData[ RowBox[{ RowBox[{ RowBox[{ RowBox[{"(", RowBox[{"\[Sigma]", " ", \(Cos[t[\[Tau]]]\), " ", RowBox[{"(", RowBox[{\(\(Ky\^2\ Sec[z[\[Tau]]]\^2\)\/\(1 + \[Sigma]\ Sin[ t[\[Tau]]]\)\), "-", \(\((Kx + Kx\ \[Sigma]\ \ Sin[t[\[Tau]]])\)\^2\/\((\(-1\) + \[Sigma]\ Sin[t[\[Tau]]])\)\^2\), "+", RowBox[{\(\((1 + \[Sigma]\ Sin[t[\[Tau]]])\)\^3\), " ", SuperscriptBox[ RowBox[{ SuperscriptBox["t", "\[Prime]", MultilineFunction->None], "[", "\[Tau]", "]"}], "2"]}], "+", RowBox[{\(\((1 + \[Sigma]\ Sin[t[\[Tau]]])\)\^3\), " ", SuperscriptBox[ RowBox[{ SuperscriptBox["z", "\[Prime]", MultilineFunction->None], "[", "\[Tau]", "]"}], "2"]}]}], ")"}]}], ")"}], "/", \(\((1 + \[Sigma]\ Sin[t[\[Tau]]])\)\^4\)}], "+", RowBox[{ SuperscriptBox["t", "\[Prime]\[Prime]", MultilineFunction->None], "[", "\[Tau]", "]"}]}], "==", "0"}]], "Output", FontSize->24] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(Simplify[Eq[5]]\)], "Input", FontSize->24], Cell[BoxData[ RowBox[{ RowBox[{\((1 + \[Sigma]\ Sin[t[\[Tau]]])\), " ", RowBox[{"(", RowBox[{\(Kx\^2\/\(1 - \[Sigma]\ Sin[t[\[Tau]]]\)\), "+", RowBox[{\((1 + \[Sigma]\ Sin[t[\[Tau]]])\), " ", RowBox[{"(", RowBox[{\(\(Ky\^2\ Sec[z[\[Tau]]]\^2\)\/\((1 + \[Sigma]\ \ Sin[t[\[Tau]]])\)\^4\), "-", SuperscriptBox[ RowBox[{ SuperscriptBox["t", "\[Prime]", MultilineFunction->None], "[", "\[Tau]", "]"}], "2"], "+", SuperscriptBox[ RowBox[{ SuperscriptBox["z", "\[Prime]", MultilineFunction->None], "[", "\[Tau]", "]"}], "2"]}], ")"}]}]}], ")"}]}], "==", "\[ScriptCapitalN]"}]], "Output", FontSize->24] }, Open ]], Cell["\<\ From the other 2 Killing vectors, the dependence from the z coordinate is \ formally eliminated from Eq[3], Eq[4]:\ \>", "Text", FontSize->24], Cell[BoxData[ \(\(z[\[Tau]_] = ArcTan[\(Kz2\ Cos[y[\[Tau]]] + Kz1\ Sin[y[\[Tau]]]\)\/Ky];\)\)], \ "Input", FontSize->24], Cell["\<\ and we get a further identity:\ \>", "Text", FontSize->24], Cell[CellGroupData[{ Cell[BoxData[ \(Simplify[\((Eq[ 3] /. \(y''\)[\[Tau]] \[Rule] \(Uy'\)[\[Tau]])\) /. \(y'\)[\ \[Tau]] \[Rule] Uy[\[Tau]]]\)], "Input", FontSize->24], Cell[BoxData[ \(True\)], "Output", FontSize->24] }, Open ]], Cell["\<\ The new system (of 2 equations) can be cast in the form:\ \>", "Text", FontSize->24], Cell[BoxData[ RowBox[{ RowBox[{\(Eq[1]\), "=", RowBox[{ RowBox[{ SuperscriptBox["t", "\[Prime]\[Prime]", MultilineFunction->None], "[", "\[Tau]", "]"}], "\[Equal]", RowBox[{\(1\/4\), " ", "\[Sigma]", " ", \(Cos[t[\[Tau]]]\), " ", RowBox[{"(", RowBox[{\(-\(\(4\ \((Ky\^2 + Kz1\^2 + Kz2\^2)\)\)\/\((1 + \[Sigma]\ \ Sin[t[\[Tau]]])\)\^5\)\), "+", \(\(4\ Kx\^2\)\/\((1 - Sin[t[\[Tau]]]\^2)\)\^2\), "-", FractionBox[ RowBox[{"4", " ", SuperscriptBox[ RowBox[{ SuperscriptBox["t", "\[Prime]", MultilineFunction->None], "[", "\[Tau]", "]"}], "2"]}], \(1 + \[Sigma]\ Sin[t[\[Tau]]]\)]}], ")"}]}]}]}], ";"}]], "Input", FontSize->24], Cell[BoxData[ RowBox[{ RowBox[{\(Eq[2]\), "=", RowBox[{ SuperscriptBox[ RowBox[{ SuperscriptBox["t", "\[Prime]", MultilineFunction->None], "[", "\[Tau]", "]"}], "2"], "\[Equal]", \(Kx\^2\/Cos[t[\[Tau]]]\^2 + \(Ky\^2 + Kz1\^2 + Kz2\^2 \ - \[ScriptCapitalN]\ \((1 + \[Sigma]\ Sin[t[\[Tau]]])\)\^2\)\/\((1 + \[Sigma]\ \ Sin[t[\[Tau]]])\)\^4\)}]}], ";"}]], "Input", FontSize->24], Cell["\<\ They are compatible:\ \>", "Text", FontSize->24], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{ RowBox[{"Simplify", "[", RowBox[{ RowBox[{ FractionBox[\(\[PartialD]\_\[Tau]\ Last[Eq[2]]\), RowBox[{"2", RowBox[{ SuperscriptBox["t", "\[Prime]", MultilineFunction->None], "[", "\[Tau]", "]"}]}]], "-", \(Last[Eq[1]]\)}], "/.", RowBox[{ SuperscriptBox[ RowBox[{ SuperscriptBox["t", "\[Prime]", MultilineFunction->None], "[", "\[Tau]", "]"}], "2"], "\[Rule]", \(Last[Eq[2]]\)}]}], "]"}], "/.", "ss"}]], "Input", FontSize->24], Cell[BoxData[ \(0\)], "Output", FontSize->24] }, Open ]], Cell["\<\ so, assuming that the observer's time is not controversial to the particle's \ one, we can write (just considering the positive square root of the right \ side of Eq[2]):\ \>", "Text", FontSize->24], Cell[BoxData[ FormBox[ RowBox[{ RowBox[{ SuperscriptBox["t", "\[Prime]", MultilineFunction->None], "(", "\[Tau]", ")"}], "=", SqrtBox[ RowBox[{\(K\_x\^2\/\(cos\^2\ \(t(\[Tau])\)\)\), "+", FractionBox[ RowBox[{\(K\_y\^2\), "+", SuperscriptBox[ SubscriptBox["K", StyleBox["z1", FontSlant->"Italic"]], "2"], "+", SuperscriptBox[ SubscriptBox["K", StyleBox["z2", FontSlant->"Italic"]], "2"], "-", \(\[ScriptCapitalN]\ [\[Sigma]\ sin\ \(t(\[Tau])\) + 1]\^\(\(\ \)\(2\)\)\)}], \(\([\[Sigma]\ sin\ \(t(\ \[Tau])\) + 1]\)\^\(\(\ \)\(4\)\)\)]}]]}], TraditionalForm]], "Text", TextAlignment->Center, FontSize->24] }, Closed]], Cell[CellGroupData[{ Cell["\<\ Phase III - The final forms in (y, t) first, then in (z, t) and (u, v)\ \>", "Section", CellFrame->{{0, 0}, {0.5, 0}}, FontSize->24], Cell[BoxData[{ RowBox[{ RowBox[{\(Eq[1]\), "=", RowBox[{ RowBox[{ SuperscriptBox["x", "\[Prime]", MultilineFunction->None], "[", "\[Tau]", "]"}], "==", \(Kx\ \(1 + \[Sigma]\ Sin[t[\[Tau]]]\)\/\(1 - \[Sigma]\ \ Sin[t[\[Tau]]]\)\)}]}], ";"}], "\[IndentingNewLine]", RowBox[{ RowBox[{\(Eq[2]\), "=", RowBox[{ RowBox[{ SuperscriptBox["y", "\[Prime]", MultilineFunction->None], "[", "\[Tau]", "]"}], "==", \(Ky\ Sec[z[\[Tau]]]\^2\/\((1 + \[Sigma]\ \ Sin[t[\[Tau]]])\)\^2\)}]}], ";"}], "\[IndentingNewLine]", \(z[\[Tau]_] = ArcTan[\(Kz2\ Cos[y[\[Tau]]] + Kz1\ Sin[y[\[Tau]]]\)\/Ky];\), "\ \[IndentingNewLine]", RowBox[{ RowBox[{\(Eq[3]\), "=", RowBox[{ RowBox[{ SuperscriptBox["t", "\[Prime]", MultilineFunction->None], "[", "\[Tau]", "]"}], "\[Equal]", \(\@\(Kx\^2\/Cos[t[\[Tau]]]\^2 + \(Ky\^2 + Kz1\^2 + Kz2\ \^2 - \[ScriptCapitalN]\ \((1 + \[Sigma]\ Sin[t[\[Tau]]])\)\^2\)\/\((1 + \ \[Sigma]\ Sin[t[\[Tau]]])\)\^4\)\)}]}], ";"}]}], "Input", FontSize->24], Cell[CellGroupData[{ Cell["\<\ Let's make the equations independent from y(\[Tau]):\ \>", "Subsubsection", CellMargins->{{54, Inherited}, {Inherited, Inherited}}, FontSize->24], Cell[TextData[{ "\nFrom Eq[2] and ", StyleBox["z", FontSlant->"Italic"], "(\[Tau]) the following expression of ", StyleBox["z", FontSlant->"Italic"], "'(\[Tau]) is:" }], "Text", FontSize->24], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"zp", "=", RowBox[{"FullSimplify", "[", RowBox[{\(\[PartialD]\_\[Tau]\ z[\[Tau]]\), "/.", RowBox[{ RowBox[{ SuperscriptBox["y", "\[Prime]", MultilineFunction->None], "[", "\[Tau]", "]"}], "\[Rule]", \(Last[Eq[2]]\)}]}], "]"}]}]], "Input", FontSize->24], Cell[BoxData[ \(\(Kz1\ Cos[y[\[Tau]]] - Kz2\ Sin[y[\[Tau]]]\)\/\((1 + \[Sigma]\ Sin[t[\ \[Tau]]])\)\^2\)], "Output", FontSize->24] }, Open ]], Cell[TextData[{ "\nKeeping ", StyleBox["z", FontSlant->"Italic"], "(\[Tau]) locally unexpressed:" }], "Text", FontSize->24], Cell[BoxData[ \(\(z[\[Tau]_] =. ;\)\)], "Input", FontSize->24], Cell["\<\ a subsystem of two equations can be written:\ \>", "Text", FontSize->24], Cell[BoxData[ RowBox[{ RowBox[{"syst", "=", RowBox[{"{", RowBox[{ RowBox[{ RowBox[{ SuperscriptBox["z", "\[Prime]", MultilineFunction->None], "[", "\[Tau]", "]"}], "==", \(\(Kz1\ Cos[y[\[Tau]]] - Kz2\ Sin[ y[\[Tau]]]\)\/\((1 + \[Sigma]\ \ Sin[t[\[Tau]]])\)\^2\)}], ",", \(z[\[Tau]] == ArcTan[\(Kz2\ Cos[y[\[Tau]]] + Kz1\ Sin[y[\[Tau]]]\)\/Ky]\)}], "}"}]}], ";"}]], "Input", FontSize->24], Cell[TextData[{ "\nfrom which the goniometric functions of ", StyleBox["y", FontSlant->"Italic"], "(\[Tau]) are derived as:" }], "Text", FontSize->24], Cell[CellGroupData[{ Cell[BoxData[ \(sc = FullSimplify[Solve[syst, {Cos[y[\[Tau]]], Sin[y[\[Tau]]]}]]\)], "Input",\ FontSize->24], Cell[BoxData[ RowBox[{"{", RowBox[{"{", RowBox[{ RowBox[{\(Cos[y[\[Tau]]]\), "\[Rule]", FractionBox[ RowBox[{\(Ky\ Kz2\ Tan[z[\[Tau]]]\), "+", RowBox[{ "Kz1", " ", \(\((1 + \[Sigma]\ Sin[t[\[Tau]]])\)\^2\), " ", RowBox[{ SuperscriptBox["z", "\[Prime]", MultilineFunction->None], "[", "\[Tau]", "]"}]}]}], \(Kz1\^2 + Kz2\^2\)]}], ",", RowBox[{\(Sin[y[\[Tau]]]\), "\[Rule]", FractionBox[ RowBox[{\(Ky\ Kz1\ Tan[z[\[Tau]]]\), "-", RowBox[{ "Kz2", " ", \(\((1 + \[Sigma]\ Sin[t[\[Tau]]])\)\^2\), " ", RowBox[{ SuperscriptBox["z", "\[Prime]", MultilineFunction->None], "[", "\[Tau]", "]"}]}]}], \(Kz1\^2 + Kz2\^2\)]}]}], "}"}], "}"}]], "Output", FontSize->24] }, Open ]], Cell[TextData[{ "\nNow, it will be sufficient to write ", Cell[BoxData[ \(TraditionalForm\`\(sin\^2\) y + \(cos\^2\) y = 1\)]] }], "Text", FontSize->24], Cell[CellGroupData[{ Cell[BoxData[ \(goniom = FullSimplify[ Last[sc[\([1, 1]\)]]\^2 + Last[sc[\([1, 2]\)]]\^2] \[Equal] 1\)], "Input", FontSize->24], Cell[BoxData[ RowBox[{ FractionBox[ RowBox[{\(Ky\^2\ Tan[z[\[Tau]]]\^2\), "+", RowBox[{\(\((1 + \[Sigma]\ Sin[t[\[Tau]]])\)\^4\), " ", SuperscriptBox[ RowBox[{ SuperscriptBox["z", "\[Prime]", MultilineFunction->None], "[", "\[Tau]", "]"}], "2"]}]}], \(Kz1\^2 + Kz2\^2\)], "==", "1"}]], "Output", FontSize->24] }, Open ]], Cell[TextData[{ "\nto have the form of ", StyleBox["z", FontSlant->"Italic"], "'(\[Tau]) which will appear in the final system:" }], "Text", FontSize->24], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"FullSimplify", "[", RowBox[{"Solve", "[", RowBox[{"goniom", ",", RowBox[{ SuperscriptBox["z", "\[Prime]", MultilineFunction->None], "[", "\[Tau]", "]"}]}], "]"}], "]"}]], "Input", FontSize->24], Cell[BoxData[ RowBox[{"{", RowBox[{ RowBox[{"{", RowBox[{ RowBox[{ SuperscriptBox["z", "\[Prime]", MultilineFunction->None], "[", "\[Tau]", "]"}], "\[Rule]", \(-\(\@\(1 - \(Ky\^2\ Tan[z[\[Tau]]]\^2\)\/\(Kz1\^2 + \ Kz2\^2\)\)\/\@\(\((1 + \[Sigma]\ Sin[t[\[Tau]]])\)\^4\/\(Kz1\^2 + \ Kz2\^2\)\)\)\)}], "}"}], ",", RowBox[{"{", RowBox[{ RowBox[{ SuperscriptBox["z", "\[Prime]", MultilineFunction->None], "[", "\[Tau]", "]"}], "\[Rule]", \(\@\(1 - \(Ky\^2\ Tan[z[\[Tau]]]\^2\)\/\(Kz1\^2 + Kz2\ \^2\)\)\/\@\(\((1 + \[Sigma]\ Sin[t[\[Tau]]])\)\^4\/\(Kz1\^2 + Kz2\^2\)\)\)}], "}"}]}], "}"}]], "Output", FontSize->24] }, Open ]], Cell[TextData[{ "\nsince the waves are not regressive, ", StyleBox["z", FontSlant->"Italic"], "'(\[Tau]) can be expressed by:" }], "Text", FontSize->24], Cell[CellGroupData[{ Cell[BoxData[ \(PowerExpand[%[\([2, 1, 2]\)]]\)], "Input", FontSize->24], Cell[BoxData[ \(\(\@\(Kz1\^2 + Kz2\^2\)\ \@\(1 - \(Ky\^2\ Tan[z[\[Tau]]]\^2\)\/\(Kz1\^2 \ + Kz2\^2\)\)\)\/\((1 + \[Sigma]\ Sin[t[\[Tau]]])\)\^2\)], "Output", FontSize->24] }, Open ]], Cell[BoxData[ FormBox[ RowBox[{"\[IndentingNewLine]", RowBox[{ "This", " ", "parametric", " ", "engineering", " ", "is", " ", "obviously", " ", RowBox[{"suitable", ":", "\[IndentingNewLine]", "\t", RowBox[{ SuperscriptBox[ StyleBox[ SubscriptBox["K", StyleBox["z1", FontSlant->"Italic"]], FontSlant->"Italic"], "2"], "+", RowBox[{ SuperscriptBox[ SubscriptBox["K", StyleBox["z2", FontSlant->"Italic"]], "2"], "\[LongRightArrow]", SuperscriptBox[ SubscriptBox["K", StyleBox["z", FontSlant->"Italic"]], "2"]}]}]}]}]}], TraditionalForm]], "Text", TextAlignment->Left, FontSize->24] }, Open ]], Cell[CellGroupData[{ Cell["The final system in (x, y, z, t)", "Subsection", CellMargins->{{54, Inherited}, {Inherited, Inherited}}, FontSize->24], Cell[BoxData[{ RowBox[{ RowBox[{\(Eq[1]\), "=", RowBox[{ RowBox[{ SuperscriptBox["x", "\[Prime]", MultilineFunction->None], "[", "\[Tau]", "]"}], "==", \(Kx\ \(1 + \[Sigma]\ Sin[t[\[Tau]]]\)\/\(1 - 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\[Pi]/2)", FontWeight->"Bold"], " for ", StyleBox["v", FontSlant->"Italic"], ", giving ", Cell[BoxData[ \(TraditionalForm\`K\_z\)]], " the value (calculated by the function ", StyleBox["CondKz", FontSlant->"Italic"], ") that makes the geodesics ", Cell[BoxData[ FormBox[ SuperscriptBox["C", StyleBox["1", FontSlant->"Italic"]], TraditionalForm]]], " in ", StyleBox["v", FontSlant->"Italic"], "=0; ", StyleBox["Lim", FontSlant->"Italic"], " is calculated to carry on and stop the drawing just at the line \ t=\[Pi]/2; the starting point for the numerical calculation is assumed to be \ ", StyleBox["(-\[Pi]/2, 0)", FontWeight->"Bold"], ": the test particle, emerging from flat space, enters the single-wave zone \ with different values of ", Cell[BoxData[ \(TraditionalForm\`K\_u\)]], ", whose value directly belongs to the projection onto ", StyleBox["z", FontSlant->"Italic"], " of its momentum." }], "Text", FontSize->24], Cell[BoxData[ RowBox[{\(NumericCalculation[\[ScriptCapitalN]_, \[Sigma]_, 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