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A. Wheeler e da T. Regge\ \>", "Subsection", CellMargins->{{70, Inherited}, {Inherited, Inherited}}], Cell[CellGroupData[{ Cell["\<\ La metrica spaziotemporale imperturbata \[EGrave] quella indotta da un buco \ nero neutro statico (segnatura adottata: - + + + ) \ \>", "Subsubsection", CellMargins->{{70, Inherited}, {Inherited, Inherited}}], Cell[BoxData[ FormBox[ RowBox[{\(g\_\(\[Mu]\ \[Nu]\)\), "=", TagBox[ RowBox[{"(", "\[NoBreak]", GridBox[{ {\(-\((1 - \(2\ m\)\/r)\)\), "0", "0", "0"}, {"0", \(1\/\(1 - \(2\ m\)\/r\)\), "0", "0"}, {"0", "0", \(r\^2\), "0"}, {"0", "0", "0", \(r\^2\ \(\(sin\^2\)(\[Theta])\)\)} }], "\[NoBreak]", ")"}], (MatrixForm[ #]&)]}], TraditionalForm]], "Text", TextAlignment->Center, TextJustification->0] }, Open ]], Cell[CellGroupData[{ Cell["\<\ L'analisi del problema perturbativo consiste nello scrivere le equazioni che \ governano il sistema attraverso il sistema di Einstein ridotto: \ \>", "Subsubsection", CellMargins->{{70, Inherited}, {Inherited, Inherited}}], Cell[BoxData[{ \(\[Delta]\ \(R\_\(\(\ \)\(\[Mu]\ \[Nu]\)\)\) \((g\&~)\) = 0\), "\[IndentingNewLine]", \(g\&~\_\(\(\ \)\(\[Mu]\ \[Nu]\)\) = g\_\(\(\ \)\(\[Mu]\ \[Nu]\)\) + h\_\(\(\ \)\(\[Mu]\ \[Nu]\)\)\)}], "Text", TextAlignment->Center, TextJustification->0], Cell["\<\ Per sfruttare le simmetrie del problema, la perturbazione \[EGrave] divisa in \ due parti che rimangono separate sotto l'azione di un qualunque operatore \ tensoriale:\ \>", "Text"] }, Open ]], Cell[CellGroupData[{ Cell["\<\ La parte \"dispari\", o \"assiale\", o \"magnetica\" (questa terminologia \ tradizionale risulta parzialmente impropria ed un po' fuorviante) \ \>", "Subsubsection", CellMargins->{{70, Inherited}, {Inherited, Inherited}}], Cell[BoxData[ FormBox[ RowBox[{\(\((h\^odd)\)\_\(\[Mu]\ \[Nu]\)\), "=", TagBox[ RowBox[{"(", "\[NoBreak]", GridBox[{ {"0", "0", RowBox[{"-", FractionBox[ RowBox[{\(\(h\_0\)(t, r)\), " ", RowBox[{ SubsuperscriptBox["Y", "LM", TagBox[\((0, 1)\), Derivative], MultilineFunction->None], "(", \(\[Theta], \[Phi]\), ")"}]}], \(sin(\[Theta])\)]}], RowBox[{\(\(h\_0\)(t, r)\), " ", \(sin(\[Theta])\), " ", RowBox[{ SubsuperscriptBox["Y", "LM", TagBox[\((1, 0)\), Derivative], MultilineFunction->None], "(", \(\[Theta], \[Phi]\), ")"}]}]}, {"0", "0", RowBox[{"-", FractionBox[ RowBox[{\(\(h\_1\)(t, r)\), " ", RowBox[{ SubsuperscriptBox["Y", "LM", TagBox[\((0, 1)\), Derivative], MultilineFunction->None], "(", \(\[Theta], \[Phi]\), ")"}]}], \(sin(\[Theta])\)]}], RowBox[{\(\(h\_1\)(t, r)\), " ", \(sin(\[Theta])\), " ", RowBox[{ SubsuperscriptBox["Y", "LM", TagBox[\((1, 0)\), Derivative], MultilineFunction->None], "(", \(\[Theta], \[Phi]\), ")"}]}]}, {"simm", "simm", "0", "0"}, {"simm", "simm", "0", "0"} }], "\[NoBreak]", ")"}], (MatrixForm[ #]&)]}], TraditionalForm]], "Text", TextAlignment->Center, TextJustification->0] }, Open ]], Cell[CellGroupData[{ Cell["\<\ La parte \"pari\", o \"polare\", o \"elettrica\" \ \>", "Subsubsection", CellMargins->{{70, Inherited}, {Inherited, Inherited}}], Cell[BoxData[ FormBox[ RowBox[{\(\((h\^even)\)\_\(\[Mu]\ \[Nu]\)\), "=", TagBox[ RowBox[{"(", "\[NoBreak]", GridBox[{ {\(\((1 - \(2\ m\)\/r)\)\ \(\(H\_0\)(t, r)\)\ \(\(Y\_LM\)(\[Theta], \[Phi])\)\), \(\(\(H\_1\)( t, r)\)\ \(\(Y\_LM\)(\[Theta], \[Phi])\)\), "0", "0"}, { "simm", \(\(\(\(H\_2\)(t, r)\)\ \(\(Y\_LM\)(\[Theta], \[Phi])\)\)\/\(1 - \(2\ \ m\)\/r\)\), "0", "0"}, {"0", "0", \(r\^2\ \(K(t, r)\)\ \(\(Y\_LM\)(\[Theta], \[Phi])\)\), "0"}, {"0", "0", "0", \(r\^2\ \(K(t, r)\)\ \(\(sin\^2\)(\[Theta])\)\ \(\(Y\_LM\)(\[Theta], \ \[Phi])\)\)} }], "\[NoBreak]", ")"}], (MatrixForm[ #]&)]}], TraditionalForm]], "Text", TextAlignment->Center, TextJustification->0] }, Open ]], Cell[CellGroupData[{ Cell["\<\ Nelle equazioni ottenute, vengono imposte le condizioni di: \ \>", "Subsubsection", CellMargins->{{70, Inherited}, {Inherited, Inherited}}], Cell[TextData[{ "\t- indipendenza dall'angolo azimutale \[Phi] : M = 0 , ", Cell[BoxData[ \(TraditionalForm\`Y\_\(\(\ \)\(LM\)\)\ \((\[Theta], \ \[Phi])\) \ \[Rule] \(P\_\(\(L\)\(\ \)\)\)(cos\ \[Theta])\)]], "\n\t- frequenza definita delle perturbazioni : dipendenza da t fissata \ nella forma ", Cell[BoxData[ \(TraditionalForm\`e\^\(\(\ \)\(\(-i\)\ \[Omega]\ t\)\)\)]] }], "Text"] }, Open ]], Cell[CellGroupData[{ Cell["\<\ Con lo sviluppo in armoniche sferiche tensoriali (appositamente definite), le \ parti angolari sono completamente separate; questo risultato si pu\[OGrave] \ ottenere pi\[UGrave] semplicemente attraverso l'applicazione di alcune \ propriet\[AGrave] delle derivate dei polinomi di Legendre: \ \>", "Subsubsection", CellMargins->{{70, Inherited}, {Inherited, Inherited}}], Cell[BoxData[{ FormBox[ RowBox[{"\t\t\t", RowBox[{\(\(d\ \(\(P\_L\)(cos\ \[Theta])\)\)\/\(d\ \[Theta]\)\), "=", RowBox[{"L", " ", "csc", " ", RowBox[{"\[Theta]", " ", "[", RowBox[{ RowBox[{"-", RowBox[{ SubscriptBox[ TagBox["P", LegendreP], \(L - 1\)], "(", \(cos\ \[Theta]\), ")"}]}], "+", RowBox[{"cos", " ", "\[Theta]", " ", RowBox[{ SubscriptBox[ TagBox["P", LegendreP], "L"], "(", \(cos\ \[Theta]\), ")"}]}]}], "]"}]}]}]}], TraditionalForm], "\n", FormBox[ RowBox[{"\t\t\t", RowBox[{\(\(d\^\(\(\ \)\(2\)\)\ \(\(P\_L\)( cos\ \[Theta])\)\)\/\(d\ \[Theta]\^\(\(\ \)\(2\)\)\)\), "=", RowBox[{ RowBox[{ "L", " ", "cot", " ", "\[Theta]", " ", "csc", " ", "\[Theta]", " ", RowBox[{ SubscriptBox[ TagBox["P", LegendreP], \(L - 1\)], "(", \(cos\ \[Theta]\), ")"}]}], "-", RowBox[{"L", " ", \((cot\^2\ \[Theta] + L + 1)\), " ", RowBox[{ SubscriptBox[ TagBox["P", LegendreP], "L"], "(", \(cos\ \[Theta]\), ")"}]}]}]}]}], TraditionalForm], "\n", FormBox[ RowBox[{"\t\t\t", RowBox[{\(\(d\^\(\(\ \)\(3\)\)\ \(\(P\_L\)( cos\ \[Theta])\)\)\/\(d\ \[Theta]\^\(\(\ \)\(3\)\)\)\), "=", RowBox[{"L", " ", "[", RowBox[{ RowBox[{"csc", " ", "\[Theta]", " ", RowBox[{"(", RowBox[{\(\(-cot\^2\)\ \[Theta]\), "-", \(csc\^2\ \[Theta]\), "+", FormBox[\(L\^2 + L\), "TraditionalForm"]}], ")"}], " ", RowBox[{ SubscriptBox[ TagBox["P", LegendreP], \(L - 1\)], "(", \(cos\ \[Theta]\), ")"}]}], "+", RowBox[{ "cot", " ", "\[Theta]", " ", \((\(cot\^2\) \[Theta] + \(csc\^2\) \[Theta] - L\^2 + 1)\), " ", RowBox[{ SubscriptBox[ TagBox["P", LegendreP], "L"], "(", \(cos\ \[Theta]\), ")"}]}]}], "]"}]}]}], TraditionalForm]}], "Text"] }, Open ]], Cell[CellGroupData[{ Cell[TextData[{ "Attraverso pi\[UGrave] sostituzioni funzionali, inclusa l'introduzione \ dalla coordinata \"tortoise\" [", Cell[BoxData[ \(TraditionalForm\`\(r\^*\) = r + 2 m\ Log\ \((r\/\(2 m\) - 1)\)\)]], "], i due sistemi si riducono ad una sola equazione di secondo grado, \ affine all'eq. di Schroedinger:\n" }], "Subsubsection", CellMargins->{{70, Inherited}, {Inherited, Inherited}}], Cell[BoxData[ FormBox[ RowBox[{ RowBox[{ FractionBox[\(\[PartialD]\^2\( f(\(r\^*\))\)\), \(\[PartialD]\(r\^*\)\^2\), MultilineFunction->None], "+", \(\([\[Omega]\^2 - V(r)]\) \(f(r)\)\)}], "=", "0"}], TraditionalForm]], "Text", TextAlignment->Center, TextJustification->0], Cell["\<\ in cui il potenziale V(r) vale, nei due casi (Regge-Wheeler, 1957 - \ Vishveshwara, Zerilli, 1969):\ \>", "Text"], Cell[BoxData[{ \(\t\t\t\t\t\t\(V\^odd\) \((r)\) = \((1 - \(2 m\)\/r)\) \((\(2 \((\ \[Lambda] + 1)\)\)\/r\^2 + \(2 m\)\/r\^3)\)\), "\[IndentingNewLine]", \(\t\t\t\t\t\t\(V\^even\) \((r)\) = \((1 - \(2 m\)\/r)\) \(2 \ \[Lambda]\^2\ \((1 + \[Lambda])\) r\^3 + 6\ m\ \(\[Lambda]\^2\) r\^2 + 18\ \ \(m\^2\) \[Lambda]\ r + 18\ m\^3\)\/\(\(r\^3\) \((\[Lambda]\ r + 3 m)\)\^2\)\ \)}], "Text", TextAlignment->Left, TextJustification->0], Cell[TextData[{ "con ", Cell[BoxData[ \(TraditionalForm\`\[Lambda] = \((L + 1)\) \((L - 2)\)/2\)]], " ." }], "Text"] }, Open ]], Cell["\<\ Dalle soluzioni particolari di queste equazioni, si pu\[OGrave] risalire alla \ forma delle perturbazioni originali della metrica di Schw. e di qui alle \ quantit\[AGrave] (ampiezza, potenza) collegate con la radiazione di \ quadrupolo (L=2), ovvero alle onde gravitazionali.\ \>", "Subsubsection", CellMargins->{{70, Inherited}, {Inherited, Inherited}}] }, Open ]], Cell[CellGroupData[{ Cell["\<\ 1970: il caso concreto di una particella di massa M che perturba il buco nero \ di Schwarzschild viene trattato da F. Zerilli\ \>", "Subsection", CellMargins->{{70, Inherited}, {Inherited, Inherited}}], Cell["\<\ La trattazione \[EGrave], naturalmente, resa pi\[UGrave] complessa dalla \ presenza di un contributo non nullo al tensore energia-impulso:\ \>", "Text"], Cell[BoxData[{ \(\[Delta]\ \(G\_\(\(\ \)\(\[Mu]\ \[Nu]\)\)\) \((g\&~)\) = 8 \[Pi]\ T\_\(\(\ \)\(\[Mu]\ \[Nu]\)\)\), "\[IndentingNewLine]", \(g\&~\_\(\(\ \)\(\[Mu]\ \[Nu]\)\) = g\_\(\(\ \)\(\[Mu]\ \[Nu]\)\) + h\_\(\(\ \)\(\[Mu]\ \[Nu]\)\)\)}], "Text", TextAlignment->Center, TextJustification->0], Cell[CellGroupData[{ Cell[TextData[{ "Si dimostra che \[EGrave] ancora possibile ridurre, per le due diverse ", Cell[BoxData[ \(TraditionalForm\`h\_\(\(\ \)\(\[Mu]\ \[Nu]\)\)\)]], " , l'intero sistema di Einstein ad una sola equazione di secondo grado, \ con un termine di sorgente\n" }], "Subsubsection", CellMargins->{{70, Inherited}, {Inherited, Inherited}}], Cell[BoxData[ FormBox[ RowBox[{ RowBox[{ FractionBox[\(\[PartialD]\^2\( R(\(r\^*\))\)\), \(\[PartialD]\(r\^*\)\^2\), MultilineFunction->None], "+", \(\([\[Omega]\^2 - V(r)]\) \(R(r)\)\)}], "=", \(S(r)\)}], TraditionalForm]], "Text", TextAlignment->Center, TextJustification->0], Cell[TextData[{ "Tuttavia la derivazione dei diversi contributi che costituiscono la S(r) \ risente di alcuni errori di calcolo, mai completamente corretti, e di un \ problema di fondo: i sistemi di equazioni che si ottengono nei due casi, \ magnetico ed elettrico, sono costituiti da equazioni differenziali di primo e \ secondo grado (rispettivamente, 2+1 e 5+2). ", StyleBox["La compatibilit\[AGrave] delle equazioni di secondo grado non \ \[EGrave] stata indagata in modo soddisfacente", FontVariations->{"Underline"->True}], ": viene invocata la conservazione del tensore energia impulso ( ", Cell[BoxData[ \(TraditionalForm\`T\_\(\(\ \ \ \)\(\[Alpha]\)\)\%\(\(\ \ \)\(\[Alpha]\)\) = 0\)]], " ), ma questa non risolve il problema, oltre a non poter essere banalmente \ estesa al caso di ", StyleBox["caduta non radiale", FontVariations->{"Underline"->True}], "." }], "Text"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["\<\ 1973-1974: si affrontano le perturbazioni della soluzione di \ Reissner-Nordstroem, relativa al caso di un buco nero statico non \ elettricamente neutro, ricavando alcune relazioni esistenti fra radiazione \ gravitazionale ed elettromagnetica mutuamente indotte\ \>", "Subsection", CellMargins->{{70, Inherited}, {Inherited, Inherited}}], Cell["\<\ Sostanzialmente, viene trattata la risoluzione del sistema accoppiato di \ Einstein-Maxwell:\ \>", "Text"], Cell[BoxData[{ \(\[Delta]\ \(G\_\(\(\ \)\(\[Mu]\ \[Nu]\)\)\) \((g\&~)\) = 8 \[Pi]\ \((T\_\(\(\ \)\(\[Mu]\ \[Nu]\)\) + \[Delta]\ \(E\_\(\(\ \)\(\ \[Mu]\ \[Nu]\)\)\) \((F)\))\)\), "\[IndentingNewLine]", \(\[Delta]\ \(\((F\^\(\(\ \)\(\[Mu]\ \[Nu]\)\))\)\_\(\(;\)\(\ \)\(\[Nu]\)\ \)\) \((g\&~)\) = 4 \[Pi]\ J\^\(\(\ \)\(\[Mu]\)\)\), "\[IndentingNewLine]", \(g\&~\_\(\(\ \)\(\[Mu]\ \[Nu]\)\) = g\_\(\(\ \)\(\[Mu]\ \[Nu]\)\) + h\_\(\(\ \)\(\[Mu]\ \[Nu]\)\)\)}], "Text", TextAlignment->Center, TextJustification->0], Cell["\<\ per il caso di una particella perturbante dotata di massa e carica non nulla.\ \ \>", "Text"], Cell[TextData[{ "Questa procedura, oltre a risentire degli stessi problemi della \ precedente, fa uso della stessa decomposizione delle perturbazioni adottata \ da Regge e Wheeler, nella forma ricordata, ottenuta attraverso l'imposizione \ di una particolare scelta di gauge: la ", StyleBox["gauge di Regge-Wheeler", FontWeight->"Bold"], ", appunto, che, tuttavia, ", StyleBox["era stata calibrata sul caso di Schwarzschild", FontVariations->{"Underline"->True}], ". In alcuni casi particolari, sorgono delle incompatibilit\[AGrave] nei \ sistemi di Einstein gi\[AGrave] evidenziate nel caso di perturbazioni della \ sola metrica scritte nel caso generale a simmetria sferica statica, con" }], "Text"], Cell[BoxData[ FormBox[ RowBox[{\(g\_\(\[Mu]\ \[Nu]\)\), "=", TagBox[ RowBox[{"(", "\[NoBreak]", GridBox[{ {\(-\[ExponentialE]\^\(\[Nu](t, r)\)\), "0", "0", "0"}, {"0", \(\[ExponentialE]\^\(\[Lambda](t, r)\)\), "0", "0"}, {"0", "0", \(r\^2\), "0"}, {"0", "0", "0", \(r\^2\ \(\(sin\^2\)(\[Theta])\)\)} }], "\[NoBreak]", ")"}], (MatrixForm[ #]&)]}], TraditionalForm]], "Text", TextAlignment->Center, TextJustification->0], Cell["\<\ L'intera trattazione va ripetuta in quest'ultimo caso, distinguendo i casi in \ cui pu\[OGrave] essere adottata la gauge di R-W anche quando Q \[NotEqual] 0.\ \ \>", "Subsubsection", CellMargins->{{70, Inherited}, {Inherited, Inherited}}] }, Open ]], Cell[CellGroupData[{ Cell["\<\ 1973: Saul A. Teukolsky (e, a seguire, Press, Bardeen, Detweiler...) usa il \ formalismo tetradico di Newman-Penrose per ricavare le perturbazioni di un \ buco nero a simmetria dinamica assiale (soluzione di Kerr)\ \>", "Subsection", CellMargins->{{70, Inherited}, {Inherited, Inherited}}], Cell[TextData[{ "Usare il formalismo tetradico di Newman-Penrose significa scrivere le \ equazioni soddisfatte dalle proiezioni delle quantit\[AGrave] vettoriali e \ tensoriali rilevanti di un sistema su di una base dello spazio-tempo \ costituita da quattro vettori nulli (", Cell[BoxData[ \(TraditionalForm\`v\^\(\(\ \)\(\[Mu]\)\)\ v\_\(\(\ \)\(\[Mu]\)\) = 0\)]], "), equamente divisi in due gruppi ortogonali fra loro: (", StyleBox["l", FontSlant->"Italic"], ", ", StyleBox["n) ", FontSlant->"Italic"], "ed", " (", StyleBox["m", FontSlant->"Italic"], ", ", Cell[BoxData[ \(TraditionalForm\`m\&_\)]], "), con i primi due reali e tali che ", Cell[BoxData[ \(TraditionalForm\`l\^\(\(\ \)\(\[Mu]\)\)\ n\_\(\(\ \)\(\[Mu]\)\) = 1\)]], " ed i secondi coniugati fra loro e tali che ", Cell[BoxData[ \(TraditionalForm\`m\^\(\(\ \)\(\[Mu]\)\)\ m\&_\_\(\(\ \)\(\[Mu]\)\) = \ \(-1\)\)]], " ." }], "Subsubsection", CellMargins->{{70, Inherited}, {Inherited, Inherited}}], Cell["\<\ Le componenti dei tensori di curvatura della metrica sono esprimibili tramite \ un set di complicate relazioni algebriche alle componenti della tetrade \ nulla.\ \>", "Subsubsection", CellMargins->{{70, Inherited}, {Inherited, Inherited}}], Cell[CellGroupData[{ Cell["Perturbare il sistema significa perturbare la tetrade", "Subsubsection", CellMargins->{{70, Inherited}, {Inherited, Inherited}}], Cell[BoxData[ RowBox[{ RowBox[{ StyleBox["l", FontWeight->"Bold", FontSlant->"Italic"], "=", RowBox[{ SuperscriptBox[ StyleBox["l", FontWeight->"Bold", FontSlant->"Italic"], "A"], "+", SuperscriptBox[ StyleBox["l", FontWeight->"Bold", FontSlant->"Italic"], "B"]}]}], ";", RowBox[{ StyleBox["n", FontWeight->"Bold", FontSlant->"Italic"], "=", RowBox[{ SuperscriptBox[ StyleBox["n", FontWeight->"Bold", FontSlant->"Italic"], "A"], "+", SuperscriptBox[ StyleBox["n", FontWeight->"Bold", FontSlant->"Italic"], "B"]}]}], ";", RowBox[{ StyleBox["m", FontWeight->"Bold", FontSlant->"Italic"], "=", RowBox[{ SuperscriptBox[ StyleBox["m", FontWeight->"Bold", FontSlant->"Italic"], "A"], "+", SuperscriptBox[ StyleBox["m", FontWeight->"Bold", FontSlant->"Italic"], "B"]}]}], ";", RowBox[{ OverscriptBox[ StyleBox["m", FontWeight->"Bold", FontSlant->"Italic"], "_"], "=", RowBox[{ SuperscriptBox[ OverscriptBox[ StyleBox["m", FontWeight->"Bold", FontSlant->"Italic"], "_"], "A"], "+", SuperscriptBox[ OverscriptBox[ StyleBox["m", FontWeight->"Bold", FontSlant->"Italic"], "_"], "B"]}]}]}]], "Text", TextAlignment->Center, TextJustification->0], Cell["\<\ Il problema di rendere efficace questa metodologia (ovvero scrivere delle \ equazioni separabili) consiste nell'effettuare una scelta oculata della \ tetrade nulla; questo problema \[EGrave] sostanzialmente analogo a quello \ consistente nell'effettuare un'opportuna scelta di gauge per semplificare la \ forma delle perturbazioni nell'approccio armonico tensoriale. \ \>", "Text"] }, Open ]], Cell["\<\ Teukolsky perviene ad un'unica equazione (Master Equation) per campi a carica \ nulla di spin qualsiasi (specializzabili dunque ai casi scalare s=0, di \ neutrino s=\[PlusMinus]1/2, elettromagnetico s=\[PlusMinus]1, gravitazionale \ s=\[PlusMinus]2)\ \>", "Subsubsection", CellMargins->{{70, Inherited}, {Inherited, Inherited}}] }, Open ]], Cell[CellGroupData[{ Cell["\<\ Le due vie sono compatibili fra loro. Esempio: le onde scalari (s = 0) nella \ formulazione di Teukolsky soddisfano l'eq. di Regge-Wheeler\ \>", "Subsection", CellMargins->{{70, 46}, {Inherited, Inherited}}], Cell[TextData[{ "L'operatore \"", Cell[BoxData[ \(TraditionalForm\`\[EmptySquare]\)]], "\" di Teukolsky che compare nella parte radiale della Master Equation, nel \ caso di Schwarzschild (a = 0) con spin nullo, si scrive ", Cell[BoxData[ FormBox[ RowBox[{"\[EmptySquare]", "=", StyleBox[\(\(\[PartialD]\/\[PartialD]\ r\) \[CapitalDelta] \[PartialD]\/\[PartialD]\ r - \ \(r\^4\/\[CapitalDelta]\) \[PartialD]\^2\/\[PartialD]\ t\^2 - L(L + 1)\), FontSize->16]}], TraditionalForm]]], " :" }], "Text"], Cell[BoxData[ \(\(\[EmptySquare][ f_] := \[PartialD]\_r\ \((\[CapitalDelta] \[PartialD]\_r\ f)\) - \(r\^4\/\[CapitalDelta]\) \[PartialD]\_t\ \(\ \[PartialD]\_t\ f\) - L \((L + 1)\)\ f;\)\)], "Input"], Cell["con", "Text"], Cell[BoxData[ \(\(\[CapitalDelta] = r\^2 - 2 m\ r;\)\)], "Input"], Cell[TextData[{ "L'equazione ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"\[EmptySquare]", " ", RowBox[{"(", StyleBox[\(\(\(u\_\(\(L\)\(\ \)\)\)(t, \ r)\)\/r\), FontSize->16], StyleBox[")", FontSize->16]}]}], StyleBox["=", FontSize->16], StyleBox["0", FontSize->16]}], TraditionalForm]]], " :" }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"TeukEq", "=", RowBox[{ StyleBox[ RowBox[{"Collect", "[", RowBox[{\(Expand[\[EmptySquare][u\_L[t, r]\/r]]\), ",", RowBox[{"{", RowBox[{\(u\_L[t, r]\), ",", RowBox[{ SubsuperscriptBox["u", "L", TagBox[\((_, _)\), Derivative], MultilineFunction->None], "[", \(t, r\), "]"}]}], "}"}], ",", "Simplify"}], "]"}], WindowSize->{712, 599}, WindowMargins->{{Automatic, 51}, {1, Automatic}}], StyleBox[" ", WindowSize->{712, 599}, WindowMargins->{{Automatic, 51}, {1, Automatic}}], "\[Equal]", "0"}]}]], "Input"], Cell[BoxData[ RowBox[{ RowBox[{\(-\(\(\((2\ m + L\ \((1 + L)\)\ r)\)\ u\_L[t, r]\)\/r\^2\)\), "+", FractionBox[ RowBox[{"2", " ", "m", " ", RowBox[{ SubsuperscriptBox["u", "L", TagBox[\((0, 1)\), Derivative], MultilineFunction->None], "[", \(t, r\), "]"}]}], "r"], "+", RowBox[{\((\(-2\)\ m + r)\), " ", RowBox[{ SubsuperscriptBox["u", "L", TagBox[\((0, 2)\), Derivative], MultilineFunction->None], "[", \(t, r\), "]"}]}], "+", FractionBox[ RowBox[{\(r\^2\), " ", RowBox[{ SubsuperscriptBox["u", "L", TagBox[\((2, 0)\), Derivative], MultilineFunction->None], "[", \(t, r\), "]"}]}], \(2\ m - r\)]}], "==", "0"}]], "Output"] }, Open ]], Cell["La coordinata \"tortoise\" \[EGrave] definita:", "Text"], Cell[BoxData[ \(\(R[r_] = r + 2 m\ Log[r\/\(2 m\) - 1];\)\)], "Input"], Cell[TextData[{ "Le sostituzioni da adottare per le derivate in r sono: ", Cell[BoxData[ FormBox[ StyleBox[\(\(d\ f\)\/\(d\ r\) = \(d\ R\)\/\(d\ r\)\ \(d\ f\)\/\(d\ \ R\)\), FontSize->16], TraditionalForm]]], " , ", Cell[BoxData[ FormBox[ StyleBox[\(\(\(d\^2\) f\)\/\(d\ r\^2\) = \(\(d\^2\) R\)\/\(d\ r\^2\)\ \ \ \(d\ f\)\/\(d\ R\) + \((\(d\ R\)\/\(d\ r\))\)\^2\ \(\(d\^2\) f\)\/\(d\ \ R\^2\)\), FontSize->16], TraditionalForm]]] }], "Text"], Cell[BoxData[ RowBox[{ RowBox[{"Tortoise", "=", RowBox[{"{", RowBox[{ RowBox[{ RowBox[{ SubsuperscriptBox["u", "L", TagBox[\((0, 2)\), Derivative], MultilineFunction->None], "[", \(t, r\), "]"}], "\[Rule]", \((\((\[PartialD]\_r\ \(\[PartialD]\_r\ R[r]\))\) \[PartialD]\_R\ u\_L[t, R] + \(\((\[PartialD]\_r\ R[r])\)\^2\) \[PartialD]\_R\ \(\[PartialD]\_R\ u\_L[t, R]\))\)}], ",", RowBox[{ RowBox[{ SubsuperscriptBox["u", "L", TagBox[\((0, 1)\), Derivative], MultilineFunction->None], "[", \(t, r\), "]"}], "\[Rule]", \(\((\[PartialD]\_r\ R[r])\)\ \[PartialD]\_R\ u\_L[t, R]\)}]}], "}"}]}], ";"}]], "Input"], Cell[TextData[{ "La trasformazione di Fourier ridefinisce le derivate in t: ", Cell[BoxData[ FormBox[ RowBox[{ StyleBox[\(\[PartialD]\^m f\/\[PartialD]t\^m\), FontSize->16], StyleBox["=", FontSize->16], RowBox[{ StyleBox[\(\((\(-\[ImaginaryI]\)\ \[Omega])\)\^m\), FontSize->16], StyleBox["f", FontSize->16], " "}]}], TraditionalForm]]] }], "Text"], Cell[BoxData[{ \(\(Unprotect[Derivative];\)\), "\[IndentingNewLine]", \(\(Derivative /: \(\(Derivative[m_, n_]\)[f_]\)[t, r] := \((\(-\[ImaginaryI]\)\ \[Omega])\)\^m\ \(\(Derivative[n]\)[ f]\)[r];\)\), "\[IndentingNewLine]", \(\(Protect[Derivative];\)\)}], "Input"], Cell["e rimuove la dipendenza da t:", "Text"], Cell[BoxData[ RowBox[{ RowBox[{"FouTransf", "=", RowBox[{"{", RowBox[{ RowBox[{ RowBox[{ SubsuperscriptBox["u", "L", TagBox[\((0, 2)\), Derivative], MultilineFunction->None], "[", \(t, R\), "]"}], "\[Rule]", RowBox[{ SubsuperscriptBox["u", "L", "\[Prime]\[Prime]", MultilineFunction->None], "[", "R", "]"}]}], ",", \(u\_L[t, r] \[Rule] u\_L[r]\)}], "}"}]}], ";"}]], "Input"], Cell["L'equazione di Teukolsky diviene:", "Text"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"TeukEq2", "=", RowBox[{ RowBox[{"Collect", "[", RowBox[{\(\((Expand[\(\[CapitalDelta]\/r\^3\) First[TeukEq]] /. Tortoise)\) /. FouTransf\), ",", RowBox[{"{", RowBox[{\(u\_L[_]\), ",", RowBox[{ SubsuperscriptBox["u", "L", TagBox[\((_)\), Derivative], MultilineFunction->None], "[", "_", "]"}]}], "}"}], ",", "Simplify"}], "]"}], "\[Equal]", "0"}]}]], "Input"], Cell[BoxData[ RowBox[{ RowBox[{\(\(\((4\ m\^2 + 2\ \((\(-1\) + L + L\^2)\)\ m\ r + r\^2\ \((\(-L\) - L\^2 + r\^2\ \[Omega]\^2)\))\)\ u\_L[ r]\)\/r\^4\), "+", RowBox[{ SubsuperscriptBox["u", "L", "\[Prime]\[Prime]", MultilineFunction->None], "[", "R", "]"}]}], "==", "0"}]], "Output"] }, Open ]], Cell["che equivale all'equazione di Regge-Wheeler:", "Text"], Cell[BoxData[ RowBox[{ RowBox[{"RWEq", "=", RowBox[{ RowBox[{ RowBox[{ SubsuperscriptBox["u", "L", "\[Prime]\[Prime]", MultilineFunction->None], "[", "R", "]"}], "+", \(\((\[Omega]\^2 - V[r])\) u\_L[r]\)}], "\[Equal]", "0"}]}], ";"}]], "Input"], Cell["con il suo potenziale:", "Text"], Cell[BoxData[ \(\(V[ r] = \((1 - \(2 m\)\/r)\) \((\(L \((L + 1)\)\)\/r\^2 + \(2 m\)\/r\ \^3)\);\)\)], "Input"], Cell["Verifica:", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(Simplify[First[TeukEq2] - First[RWEq]]\)], "Input"], Cell[BoxData[ \(0\)], "Output"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["\<\ La diffusione del calcolo elettronico neutralizza le differenze tecniche fra \ i due metodi: l'analisi alla RWVZ torna \"di moda\" e conquista nuove \ posizioni\ \>", "Subsection", CellMargins->{{70, Inherited}, {Inherited, Inherited}}], Cell[CellGroupData[{ Cell["RWVZ:", "Subsubsection", CellMargins->{{70, Inherited}, {Inherited, Inherited}}], Cell[BoxData[ \(g\_\(\[Mu]\ \[Nu]\)\( \[Rule] \&1\)\(h\_\(\[Mu]\ \[Nu]\)\( \[Rule] \ \&2\)G \((\(g\&~\)\_\(\[Mu]\ \[Nu]\))\)\), T \((\(g\&~\)\_\(\[Mu]\ \[Nu]\))\), F \((\(g\&~\)\_\(\[Mu]\ \[Nu]\))\)\( \[Rule] \&3\)Einstein\ \((\(+Maxwell\ \))\)\)], "Text", TextAlignment->Center, TextJustification->0] }, Open ]], Cell[CellGroupData[{ Cell["NP:", "Subsubsection", CellMargins->{{70, Inherited}, {Inherited, Inherited}}], Cell[BoxData[ \(g\_\(\(\ \)\(\[Mu]\ \[Nu]\)\)\( \[Rule] \&1\)\(\((l, n, m, m\&_)\)\( \[Rule] \&2\)\(\((D, \[CapitalDelta], \[Delta], \ \(\[Delta]\^*\))\) + \((\[Kappa], \[Sigma], \[Lambda], \[Nu], \[Rho], \[Mu], \ \[Tau], \[Pi], \[Epsilon], \[Gamma], \[Alpha], \[Beta])\) + \ \((\[CapitalPsi]\_0, \[CapitalPsi]\_1, \[CapitalPsi]\_2, \[CapitalPsi]\_3, \ \[CapitalPsi]\_4)\)\( \[Rule] \&3\)34\ \(\(eqn\)\(.\)\)\)\)\)], "Text", TextAlignment->Center, TextJustification->0] }, Open ]], Cell["\<\ Inoltre, come ricordato nella precedente relazione, molti approcci \ post-newtoniani alla dinamica dei sistemi binari fanno uso del modello ad un \ solo corpo orbitante in una metrica efficace di tipo Schwarzschild, tornando \ alle tecniche lineari non mediate dagli invarianti geometrici\ \>", "Subsubsection", CellMargins->{{70, Inherited}, {Inherited, Inherited}}] }, Open ]] }, Closed]], Cell[CellGroupData[{ Cell[TextData[{ Cell[BoxData[ \(TraditionalForm\`\[FilledCircle]\)]], " Una chiave di non-linearit\[AGrave]: il moto geodetico nella collisione \ di onde gravitazionali" }], "Section", CellMargins->{{49, 46}, {Inherited, Inherited}}], Cell[CellGroupData[{ Cell["Origini e definizioni ", "Subsection", CellMargins->{{70, Inherited}, {Inherited, Inherited}}], Cell[TextData[{ "Una \"", StyleBox["metrica di onda gravitazionale piana", FontWeight->"Bold"], "\" \[EGrave] una soluzione delle equazioni di Einstein nel vuoto con le \ stesse caratteristiche di simmetria possedute da un'onda elettromagnetica \ piana nello spazio piatto, ovvero (supponendo z la direzione di \ propagazione):\n\t\t- invarianza per traslazioni lungo x, y e lungo le \ ipersuperfici t - z = costante ;\n\t\t- invarianza per rotazione delle \ ipersuperfici t - z = costante ." }], "Text"], Cell["\<\ 1970-1972: Szekeres e Khan-Penrose ricavano, a partire dalla metrica di \ Minkowski, delle soluzioni che corrispondono all'interazione di due onde \ gravitazionali piane collineari\ \>", "Subsubsection", CellMargins->{{70, Inherited}, {Inherited, Inherited}}], Cell[CellGroupData[{ Cell[TextData[{ "1985-1986: Ferrari ed Iba", StyleBox["\[NTilde]ez generalizzano questo risultato utilizzando come punto \ di partenza la metrica di Kasner\n", FontFamily->"Times New Roman"] }], "Subsubsection", CellMargins->{{70, Inherited}, {Inherited, Inherited}}], Cell[BoxData[ \(TraditionalForm\`d\ s\^2 = \(-\ t\^\(\(\ \)\(2\ s\_1\)\)\) d\ x\^2 - \(t\^\(\(\ \)\(2\ s\_2\)\)\) d\ y\^2 + \((d\ t\^2 - d\ z\^2)\)\ t\^\(\(\ \)\(\(-2\)\ s\_1\ s\_2\)\)\)], "Text", TextAlignment->Center, TextJustification->0], Cell[TextData[{ "Riducendo al caso spazio-temporale (", Cell[BoxData[ \(TraditionalForm\`s\_1 = 1, \ s\_2 = 0\)]], ") le espressioni trovate, si ottiene la rappresentazione degenere della \ metrica d'interazione di due onde impulsive, in cui si ha formazione o di un \ ", StyleBox["orizzonte", FontWeight->"Bold"], " o di una ", StyleBox["singolarit\[AGrave]", FontWeight->"Bold"], "." }], "Text"] }, Open ]], Cell["\<\ Lo studio delle soluzioni collisionali termina agli inizi degli anni '90 \ senza che sia stato analizzato il moto geodetico (in altre parole, senza aver \ esaminato la possibilit\[AGrave] di una fenomenologia legata alle particelle \ immerse in questa metrica)\ \>", "Subsubsection", CellMargins->{{70, Inherited}, {Inherited, Inherited}}] }, Open ]], Cell[CellGroupData[{ Cell["Impostazione dello studio delle geodetiche", "Subsection", CellMargins->{{70, Inherited}, {Inherited, Inherited}}], Cell["\<\ La metrica di Ferrari-Iba\[NTilde]ez, rappresentata in un sistema di \ coordinate dipendenti da un parametro proprio (x(\[Tau]), y(\[Tau]), \ z(\[Tau]), t(\[Tau])), si scrive, nella zona d'interazione:\ \>", "Text"], Cell[BoxData[ FormBox[ RowBox[{\(\(\((g\^\((1)\))\)\_\(\(\[Mu]\)\(\ \)\(\[Nu]\)\(\ \)\)\)(\ \[Tau])\), "=", TagBox[ RowBox[{"(", "\[NoBreak]", GridBox[{ {\(\(1 - \[Sigma]\ sin\ \(t(\[Tau])\)\)\/\(\[Sigma]\ sin\ \ \(t(\[Tau])\) + 1\)\), "0", "0", "0"}, { "0", \(\(cos\^2\) \(z(\[Tau])\)\ \((\[Sigma]\ sin\ \(t(\ \[Tau])\) + 1)\)\^2\), "0", "0"}, {"0", "0", \(\((\[Sigma]\ sin\ \(t(\[Tau])\) + 1)\)\^2\), "0"}, {"0", "0", "0", \(-\((\[Sigma]\ sin\ \(t(\[Tau])\) + 1)\)\^2\)} }], "\[NoBreak]", ")"}], (MatrixForm[ #]&)]}], TraditionalForm]], "Text", TextAlignment->Center, TextJustification->0], Cell["con \[Sigma] = \[PlusMinus]1 .", "Text"], Cell["\<\ Mentre nella zona di onda singola, in un'utile formulazione alternativa, \ indotta dal cambio di coordinate\ \>", "Text"], Cell["\<\ z(\[Tau]) = v(\[Tau]) - u(\[Tau]) t(\[Tau]) = v(\[Tau]) + u(\[Tau])\ \>", "Text", TextAlignment->Center, TextJustification->0, FontFamily->"Courier New"], Cell["si scrive:", "Text"], Cell[BoxData[ FormBox[ RowBox[{\(\((g\^\((2)\))\)\_\(\[Mu]\ \[Nu]\)\ \((\[Tau])\)\), "=", TagBox[ RowBox[{"(", "\[NoBreak]", GridBox[{ {\(\(1 - \[Sigma]\ sin\ \(u(\[Tau])\)\)\/\(\[Sigma]\ sin\ \ \(u(\[Tau])\) + 1\)\), "0", "0", "0"}, { "0", \(cos\^2\ \(u(\[Tau])\)\ \((\[Sigma]\ sin\ \(u(\[Tau])\ \) + 1)\)\^2\), "0", "0"}, {"0", "0", "0", \(\(-2\)\ \((\[Sigma]\ sin\ \(u(\[Tau])\) + \ 1)\)\^2\)}, {"0", "0", \(\(-2\)\ \((\[Sigma]\ sin\ \(u(\[Tau])\) + 1)\)\^2\), "0"} }], "\[NoBreak]", ")"}], (MatrixForm[ #]&)]}], TraditionalForm]], "Text", TextAlignment->Center, TextJustification->0], Cell[CellGroupData[{ Cell["\<\ L'equazione delle geodetiche della metrica (4 eq. scalari) \ \>", "Subsubsection", CellMargins->{{70, Inherited}, {Inherited, Inherited}}], Cell[BoxData[ \(TraditionalForm\`\[PartialD]U\^\[Mu]\/\[PartialD]\[Tau] + \(\ \[CapitalGamma]\_\(\[Rho]\ \[Sigma]\)\%\[Mu]\) U\^\[Rho]\ U\^\[Sigma] = 0\)], "Text", TextAlignment->Center, TextJustification->0] }, Open ]], Cell[CellGroupData[{ Cell["\<\ La condizione di normalizzazione delle 4-velocit\[AGrave] \ \>", "Subsubsection", CellMargins->{{70, Inherited}, {Inherited, Inherited}}], Cell[BoxData[ \(TraditionalForm\`U\^\[Mu]\ U\_\[Mu] = \[ScriptCapitalN]\)], "Text", TextAlignment->Center, TextJustification->0], Cell["\<\ dove il parametro \[ScriptCapitalN] pu\[OGrave] assumere i valori -1 \ (geodetiche del genere tempo -punti materiali-), 0 (geodetiche nulle), +1 \ (geodetiche del genere spazio). Queste cinque equazioni, da sole, non permettono di scrivere un sistema di \ primo grado rispetto a \[Tau] nelle coordinate dello spazio-tempo.\ \>", "Text"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["Simmetrie di Killing", "Subsection", CellMargins->{{70, Inherited}, {Inherited, Inherited}}], Cell[CellGroupData[{ Cell["\<\ Condizioni aggiuntive si ricavano sfruttando le simmetrie della metrica: i \ vettori di Killing \ \>", "Subsubsection", CellMargins->{{70, Inherited}, {Inherited, Inherited}}], Cell[TextData[{ "Questi particolari vettori sono tali che ", Cell[BoxData[ \(TraditionalForm\`\[Xi]\_\(\(\ \)\(\[Mu]\ ; \ \[Nu]\)\) + \[Xi]\_\(\(\ \ \)\(\[Nu]\ ; \ \[Mu]\)\)\)]], " = 0 . Per ognuno di essi si pu\[OGrave] scrivere l'ulteriore equazione" }], "Text"], Cell[BoxData[ FormBox[ RowBox[{\(\[Xi]\_\[Mu]\ U\^\[Mu]\), "=", StyleBox["costante", FontSlant->"Italic"]}], TraditionalForm]], "Text", TextAlignment->Center, TextJustification->0] }, Open ]], Cell[CellGroupData[{ Cell["\<\ Per la metrica di onda singola, tre vettori sono banalmente ricavati dalla \ non-dipendenza esplicita dei coefficienti della metrica da x, y, v :\ \>", "Subsubsection", CellMargins->{{70, Inherited}, {Inherited, Inherited}}], Cell[BoxData[{ \(\[Zeta]\_\((1)\) = \((1, \ 0, \ 0, \ 0)\)\), "\[IndentingNewLine]", \(\[Zeta]\_\((2)\) = \((0, \ 1, \ 0, \ 0)\)\), "\[IndentingNewLine]", \(\[Zeta]\_\((3)\) = \((0, \ 0, \ 0, \ 1)\)\)}], "Text", TextAlignment->Center, TextJustification->0] }, Open ]], Cell[CellGroupData[{ Cell[TextData[{ "Per la metrica d'interazione, oltre a ", Cell[BoxData[ \(TraditionalForm\`\[Xi]\_\((1)\), \ \[Xi]\_\((2)\)\)]], ", altri vettori connessi con la conservazione del momento angolare del \ sistema sono stati trovati:" }], "Subsubsection", CellMargins->{{70, Inherited}, {Inherited, Inherited}}], Cell[BoxData[{ \(\[Zeta]\_\((3)\) = \((0, \ sin\ y\ \ tan\ z, cos\ y\ , \ 0)\)\), "\[IndentingNewLine]", \(\[Zeta]\_\((4)\) = \((0, \ cos\ y\ \ tan\ z, \ \(-sin\)\ y, \ 0)\)\)}], "Text", TextAlignment->Center, TextJustification->0] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["Equazioni del moto geodetico", "Subsection", CellMargins->{{70, Inherited}, {Inherited, Inherited}}], Cell[CellGroupData[{ Cell["\<\ Vengono ricavate separatamente nelle due regioni di onda singola e di \ collisione (rispett. \"zona 2\" e \"zona 1\"). In zona 2 si ha: \ \>", "Subsubsection", CellMargins->{{70, Inherited}, {Inherited, Inherited}}], Cell[BoxData[ FormBox[ RowBox[{"\t\t\t\t\t", RowBox[{ RowBox[{ SuperscriptBox["x", "\[Prime]", MultilineFunction->None], "(", "\[Tau]", ")"}], "=", \(\(Kx\ \((\[Sigma]\ sin\ \(u(\[Tau])\) + 1)\)\)\/\(1 - \[Sigma]\ sin\ \(u(\[Tau])\)\)\)}]}], TraditionalForm]], "Text", TextAlignment->Left, TextJustification->0], Cell[BoxData[ FormBox[ RowBox[{"\t\t\t\t\t", RowBox[{ RowBox[{ SuperscriptBox["y", "\[Prime]", MultilineFunction->None], "(", "\[Tau]", ")"}], "=", \(\(Ky\ \(sec\^2\) \(u(\[Tau])\)\)\/\((\[Sigma]\ sin\ \(u(\ \[Tau])\) + 1)\)\^2\)}]}], TraditionalForm]], "Text", TextAlignment->Left, TextJustification->0], Cell[BoxData[ FormBox[ RowBox[{"\t\t\t\t\t", RowBox[{ RowBox[{ RowBox[{ SuperscriptBox["u", "\[Prime]", MultilineFunction->None], "(", "\[Tau]", ")"}], "=", \(-\(Ku\/\(2\ \((\[Sigma]\ sin\ \(u(\[Tau])\) + \ 1)\)\^2\)\)\)}], ";"}]}], TraditionalForm]], "Text", TextAlignment->Left, TextJustification->0], Cell[BoxData[ FormBox[ RowBox[{"\t\t\t\t\t", RowBox[{ RowBox[{ SuperscriptBox["v", "\[Prime]", MultilineFunction->None], "(", "\[Tau]", ")"}], "=", \(\(\[ScriptCapitalN] - \(\((\[Sigma]\ sin\ \(u(\[Tau])\) + 1)\ \)\ Kx\^2\)\/\(1 - \[Sigma]\ sin\ \(u(\[Tau])\)\) - Ky\^2\ \((\(sec\ \(u(\[Tau])\)\)\/\(\[Sigma]\ sin\ \ \(u(\[Tau])\) + 1\))\)\^2\)\/\(2\ Ku\)\)}]}], TraditionalForm]], "Text", TextAlignment->Left, TextJustification->0], Cell["\<\ dove Kx, Ky, Ku sono le costanti associate alle simmetrie di Killing (Kx, Ky \ sono le proiezioni sul fronte d'onda del momento della particella, Ku \ \[EGrave] associato all'energia con cui la particella entra in zona 2 \ proveniendo dallo spazio piatto).\ \>", "Text"] }, Open ]], Cell[CellGroupData[{ Cell["In zona 1 si ha:", "Subsubsection", CellMargins->{{70, Inherited}, {Inherited, Inherited}}], Cell[BoxData[ FormBox[ RowBox[{"\t\t\t\t\t", RowBox[{ RowBox[{ SuperscriptBox["x", "\[Prime]", MultilineFunction->None], "(", "\[Tau]", ")"}], "=", \(\(Kx\ \((\[Sigma]\ sin\ \(t(\[Tau])\) + 1)\)\)\/\(1 - \[Sigma]\ sin\ \(t(\[Tau])\)\)\)}]}], TraditionalForm]], "Text"], Cell[BoxData[ FormBox[ RowBox[{"\t\t\t\t\t", RowBox[{ RowBox[{ SuperscriptBox["y", "\[Prime]", MultilineFunction->None], "(", "\[Tau]", ")"}], "=", \(\(Ky\ sec\^2\ \(z(\[Tau])\)\)\/\((\[Sigma]\ sin\ \(t(\[Tau])\ \) + 1)\)\^2\)}]}], TraditionalForm]], "Text"], Cell[BoxData[ FormBox[ RowBox[{"\t\t\t\t\t", RowBox[{ RowBox[{ SuperscriptBox["z", "\[Prime]", MultilineFunction->None], "(", "\[Tau]", ")"}], "=", \(\@\(Kz\^2 - Ky\^2\ tan\^2\ \(z(\[Tau])\)\)\/\((\[Sigma]\ sin\ \ \(t(\[Tau])\) + 1)\)\^2\)}]}], TraditionalForm]], "Text"], Cell[BoxData[ FormBox[ RowBox[{"\t\t\t\t\t", RowBox[{ RowBox[{ SuperscriptBox["t", "\[Prime]", MultilineFunction->None], "(", "\[Tau]", ")"}], "=", \(\@\(Kx\^2\/\(1 - \[Sigma]\^2\ \(sin\^2\) \(t(\[Tau])\)\) + \ \(Ky\^2 + Kz\^2 - \[ScriptCapitalN]\ \((\[Sigma]\ sin\ \(t(\[Tau])\) + \ 1)\)\^2\)\/\((\[Sigma]\ sin\ \(t(\[Tau])\) + 1)\)\^4\)\)}]}], TraditionalForm]], "Text"], Cell[TextData[{ "in cui Kz vale ", Cell[BoxData[ \(TraditionalForm\`\@\(Kz\_1\^2 + Kz\_2\^2\)\)]], " , dove ", Cell[BoxData[ \(TraditionalForm\`Kz\_1, \ Kz\_2\)]], " sono le costanti associate ai due nuovi vettori di Killing. Applicando la \ trasformazione (z, t) \[Rule] (u, v) citata, \[EGrave] possibile, una volta \ impostati gli algoritmi di raccordo fra le due zone (in particolare, quelli \ che calcolano i valori di Kz corrispondenti al Ku iniziale), ", StyleBox["tracciare il percorso completo delle geodetiche nel piano (u, v) \ dalla frontiera della zona 2 con lo spazio piatto all'orizzonte/singolarit\ \[AGrave] in zona 1", FontWeight->"Bold"], "." }], "Text"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["Tracciamento delle geodetiche", "Subsection", CellMargins->{{70, Inherited}, {Inherited, Inherited}}], Cell[CellGroupData[{ Cell["\<\ A titolo di esempio, riporto una serie di geodetiche ottenute con la seguente \ scelta di parametri:\ \>", "Subsubsection", CellMargins->{{70, Inherited}, {Inherited, Inherited}}], Cell["\<\ \[ScriptCapitalN] = -1 (genere tempo) \[Sigma] = 1 (metrica con orizzonte) Kx = 1 Ky = 0\ \>", "Text", TextAlignment->Center, TextJustification->0], 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