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Path of a radial light ray\ \>", "Section", CellChangeTimes->{{3.4740260215815086`*^9, 3.4740260263939466`*^9}, { 3.4740261374395914`*^9, 3.4740261485020914`*^9}, {3.474029230782051*^9, 3.474029237813796*^9}}], Cell[TextData[{ Cell[BoxData[ FormBox[ RowBox[{"e", "=", RowBox[{ RowBox[{"(", RowBox[{"1", "-", FractionBox[ RowBox[{"2", "M"}], "r"]}], ")"}], FractionBox[ RowBox[{"d", " ", "t"}], RowBox[{"d", " ", "\[Lambda]"}]]}]}], TraditionalForm]], FormatType->"TraditionalForm"], " and ", Cell[BoxData[ FormBox[ RowBox[{"l", "=", RowBox[{ SuperscriptBox["r", "2"], FractionBox[ RowBox[{"d", " ", "\[Phi]"}], RowBox[{"d", " ", "\[Lambda]"}]]}]}], TraditionalForm]], FormatType->"TraditionalForm"], ". What is ", Cell[BoxData[ FormBox["\[Lambda]", TraditionalForm]], FormatType->"TraditionalForm"], "?" }], "Text", CellChangeTimes->{{3.4740261526427164`*^9, 3.4740262688927164`*^9}, { 3.4740263065177164`*^9, 3.4740263081583414`*^9}, {3.4740293891943*^9, 3.474029407038507*^9}, {3.474029459055464*^9, 3.474029609902618*^9}, { 3.474029647809353*^9, 3.474029705591343*^9}, {3.474029736247985*^9, 3.474029790123675*^9}}], Cell[TextData[{ "We have differential equations for ", Cell[BoxData[ FormBox[ FractionBox[ RowBox[{"d", " ", "r"}], RowBox[{"d", " ", "\[Lambda]"}]], TraditionalForm]], FormatType->"TraditionalForm"], " and ", Cell[BoxData[ FormBox[ FractionBox[ RowBox[{"d", " ", "\[Phi]"}], RowBox[{"d", " ", "\[Lambda]"}]], TraditionalForm]], FormatType->"TraditionalForm"], ", which we can solve. Do the simple case of an almost radial light ray, for \ which ", Cell[BoxData[ FormBox[ RowBox[{"l", "\[LessLess]", "1"}], TraditionalForm]], FormatType->"TraditionalForm"], ". In that case ", Cell[BoxData[ FormBox[ RowBox[{ SuperscriptBox[ RowBox[{"(", FractionBox["e", "l"], ")"}], "2"], "=", RowBox[{ RowBox[{ FractionBox["1", SuperscriptBox["l", "2"]], SuperscriptBox[ RowBox[{"(", FractionBox[ RowBox[{"d", " ", "r"}], RowBox[{"d", " ", "\[Lambda]"}]], ")"}], "2"]}], "+", RowBox[{ SubscriptBox["W", "eff"], "(", "r", ")"}]}]}], TraditionalForm]], FormatType->"TraditionalForm"], " becomes\n\t", Cell[BoxData[ FormBox[ RowBox[{ SuperscriptBox["e", "2"], "=", SuperscriptBox[ RowBox[{"(", FractionBox[ RowBox[{"d", " ", "r"}], RowBox[{"d", " ", "\[Lambda]"}]], ")"}], "2"]}], TraditionalForm]], FormatType->"TraditionalForm"], ".\nThe solution is\n\t", Cell[BoxData[ FormBox[ RowBox[{"r", "=", RowBox[{"e", " ", RowBox[{"\[Lambda]", "."}]}]}], TraditionalForm]], FormatType->"TraditionalForm"] }], "Text", CellChangeTimes->{{3.4740261526427164`*^9, 3.4740262688927164`*^9}, { 3.4740263065177164`*^9, 3.4740263081583414`*^9}, {3.4740293891943*^9, 3.474029407038507*^9}, {3.474029459055464*^9, 3.474029609902618*^9}, { 3.474029647809353*^9, 3.474029705591343*^9}, {3.474029736247985*^9, 3.4740297912174387`*^9}}], Cell[TextData[{ "Surprise: the parameter \[Lambda] is not time. For a radial light ray, the \ parameter \[Lambda] is the radial coordinate divided by the energy at \ \[Infinity]. (Recall ", Cell[BoxData[ FormBox["e", TraditionalForm]]], " is the energy at ", Cell[BoxData[ FormBox[ RowBox[{"r", "\[Rule]", RowBox[{"\[Infinity]", "."}]}], TraditionalForm]]], ")" }], "Text", CellChangeTimes->{{3.474026899586835*^9, 3.4740270695577602`*^9}, { 3.474027118214633*^9, 3.474027169402316*^9}, {3.474027257839816*^9, 3.474027273136691*^9}, {3.474028870780021*^9, 3.4740288829380608`*^9}, 3.4740292985669804`*^9}], Cell[TextData[{ "Calculate the coordinate time.\n\t", Cell[BoxData[{ FormBox[ RowBox[{"e", "=", RowBox[{ RowBox[{"(", RowBox[{"1", "-", FractionBox[ RowBox[{"2", "M"}], "r"]}], ")"}], FractionBox[ RowBox[{"d", " ", "t"}], RowBox[{"d", " ", "\[Lambda]"}]]}]}], TraditionalForm], "\[IndentingNewLine]", FormBox[ RowBox[{"=", RowBox[{ RowBox[{"(", RowBox[{"1", "-", FractionBox[ RowBox[{"2", "M"}], "r"]}], ")"}], FractionBox[ RowBox[{"d", " ", "t"}], RowBox[{"d", " ", "r"}]], FractionBox[ RowBox[{"d", " ", "r"}], RowBox[{"d", " ", "\[Lambda]"}]]}]}], TraditionalForm]}]], "\nFor radial paths, ", Cell[BoxData[ FormBox[ RowBox[{ SuperscriptBox["e", "2"], "=", SuperscriptBox[ RowBox[{"(", FractionBox[ RowBox[{"d", " ", "r"}], RowBox[{"d", " ", "\[Lambda]"}]], ")"}], "2"]}], TraditionalForm]]], ". Substitute to get\n\t", Cell[BoxData[{ FormBox[ RowBox[{ RowBox[{"d", " ", "t"}], "=", RowBox[{ RowBox[{"\[PlusMinus]", "d"}], " ", "r", " ", SuperscriptBox[ RowBox[{"(", RowBox[{"1", "-", FractionBox[ RowBox[{"2", "M"}], "r"]}], ")"}], RowBox[{"-", "1"}]]}]}], TraditionalForm], "\[IndentingNewLine]", FormBox[ RowBox[{"=", RowBox[{ RowBox[{"\[PlusMinus]", "d"}], " ", "r", " ", RowBox[{"(", RowBox[{"1", "+", FractionBox[ RowBox[{"2", "M"}], RowBox[{"r", "-", RowBox[{"2", "M"}]}]]}], ")"}]}]}], TraditionalForm]}]], "\nUse + for outgoing paths and \[Dash] for incoming paths. Substitute ", Cell[BoxData[ FormBox[ RowBox[{"r", "=", RowBox[{"e", " ", "\[Lambda]"}]}], TraditionalForm]], FormatType->"TraditionalForm"], " to get\n\t", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"\[CapitalDelta]", " ", "t"}], "=", RowBox[{"e", " ", RowBox[{"(", RowBox[{ SubscriptBox["\[Lambda]", "2"], "-", SubscriptBox["\[Lambda]", "1"], "+", RowBox[{"2", "M", " ", "log", " ", FractionBox[ RowBox[{ SubscriptBox["\[Lambda]", "2"], "-", RowBox[{"2", RowBox[{"M", "/", "e"}]}]}], RowBox[{ SubscriptBox["\[Lambda]", "1"], "-", RowBox[{"2", RowBox[{"M", "/", "e"}]}]}]]}]}], ")"}]}]}], TraditionalForm]], FormatType->"TraditionalForm"] }], "Text", CellChangeTimes->{{3.4740264057989664`*^9, 3.4740265277989664`*^9}, { 3.474026566237114*^9, 3.4740266150193005`*^9}, {3.4740266499574714`*^9, 3.4740268459767733`*^9}, {3.474029941718933*^9, 3.4740299969380364`*^9}, { 3.4740300291724195`*^9, 3.4740300310630445`*^9}, {3.4741010605625*^9, 3.47410106125*^9}, {3.474101159671875*^9, 3.47410126775*^9}, { 3.47410141659375*^9, 3.4741015123125*^9}, {3.47410184415625*^9, 3.47410187296875*^9}}], Cell[TextData[{ "If the energy of the photon is bigger, the parameter \[Lambda] changes more \ slowly as ", Cell[BoxData[ FormBox["r", TraditionalForm]], FormatType->"TraditionalForm"], " and ", Cell[BoxData[ FormBox["t", TraditionalForm]], FormatType->"TraditionalForm"], " change. 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