(* Content-type: application/mathematica *) (*** Wolfram Notebook File ***) (* http://www.wolfram.com/nb *) (* CreatedBy='Mathematica 7.0' *) (*CacheID: 234*) (* Internal cache information: NotebookFileLineBreakTest NotebookFileLineBreakTest NotebookDataPosition[ 145, 7] NotebookDataLength[ 90273, 1964] NotebookOptionsPosition[ 86672, 1850] NotebookOutlinePosition[ 87756, 1885] CellTagsIndexPosition[ 87677, 1880] WindowFrame->Normal*) (* Beginning of Notebook Content *) Notebook[{ Cell[CellGroupData[{ Cell["", "SlideShowNavigationBar", CellTags->"SlideShowHeader"], Cell[CellGroupData[{ Cell["The universe", "Section", CellChangeTimes->{{3.474102520375*^9, 3.474102523671875*^9}}], Cell[TextData[{ "In the 1920's, V. M. Slipher measured velocities of nearby galaxies. Hubble \ estimated their distances. Hubble (Hubble, E., 1929, Proc. Nat. Acad. 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Consider Milky Way, galaxy A at \ distance 1, and galaxy B at distance 2.", "\n", "1. Expansion is by a scale factor. ", Cell[BoxData[ FormBox[ RowBox[{ SubscriptBox["v", "A"], "=", "1"}], TraditionalForm]]], ", and ", Cell[BoxData[ FormBox[ RowBox[{ SubscriptBox["v", "B"], "=", "2"}], TraditionalForm]]], ". Suppose some time passed, and A has moved by 0.5 to 1.5. Then B has moved \ by 1 to 3. B remains twice as far as A.", "\n", "2. Centerless expansion.\nA is ", Cell[BoxData[ FormBox[ SuperscriptBox["5", RowBox[{"1", "/", "2"}]], TraditionalForm]]], " from B. MW is 2 from B.\nAfter time passed, A is ", Cell[BoxData[ FormBox[ RowBox[{ SuperscriptBox[ RowBox[{"(", RowBox[{ SuperscriptBox["3", "2"], "+", SuperscriptBox["1.5", "2"]}], ")"}], RowBox[{"1", "/", "2"}]], "=", RowBox[{ FractionBox["3", "2"], SuperscriptBox["5", RowBox[{"1", "/", "2"}]]}]}], TraditionalForm]]], " from B, and MW is ", Cell[BoxData[ FormBox[ RowBox[{ FractionBox["3", "2"], "2"}], TraditionalForm]]], " from B.\nGalaxy B is the center of the expansion too.", "\n", "3. There exists a beginning, when the scale factor is 0. In this example, \ let time be ", Cell[BoxData[ FormBox[ RowBox[{"-", "1"}], TraditionalForm]]], ".", "\n", "4. Hubble did not find special directions. The universe is isotropic." }], "Text", CellChangeTimes->{{3.474111071796875*^9, 3.47411108553125*^9}, { 3.474111141328125*^9, 3.4741111501875*^9}, {3.4741112199375*^9, 3.47411126521875*^9}, {3.47411218871875*^9, 3.474112196203125*^9}, { 3.474115911328125*^9, 3.474115914640625*^9}}], Cell[TextData[{ ButtonBox["\[FilledLeftTriangle]\[ThickSpace]\[ThickSpace]\[ThickSpace]", BaseStyle->"SlidePreviousNextLink", Appearance->{Automatic, None}, ButtonFunction:>FrontEndExecute[{ FrontEndToken[ FrontEnd`ButtonNotebook[], "ScrollPagePrevious"]}], ButtonNote->FEPrivate`FrontEndResource[ "FEStrings", "SlideshowPrevSlideText"]], "\[ThickSpace]\[ThickSpace]|\[ThickSpace]\[ThickSpace]", ButtonBox["\[ThickSpace]\[ThickSpace]\[ThickSpace]\[FilledRightTriangle]", BaseStyle->"SlidePreviousNextLink", Appearance->{Automatic, None}, ButtonFunction:>FrontEndExecute[{ FrontEndToken[ FrontEnd`ButtonNotebook[], "ScrollPageNext"]}], ButtonNote->FEPrivate`FrontEndResource[ "FEStrings", "SlideshowNextSlideText"]] }], "PreviousNext"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["", "SlideShowNavigationBar", CellTags->"SlideShowHeader"], Cell[CellGroupData[{ Cell["Puzzle", "Section", CellChangeTimes->{{3.474110204671875*^9, 3.4741102060625*^9}}], Cell[TextData[{ "Can galaxies go faster than light? 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Therefore\n\t", Cell[BoxData[ FormBox[ RowBox[{ SuperscriptBox["ds", "2"], "=", RowBox[{ RowBox[{ SubsuperscriptBox["r", "0", "2"], "d", " ", SuperscriptBox["latitude", "2"]}], "+", RowBox[{ SuperscriptBox["r", "2"], "d", " ", RowBox[{ SuperscriptBox["longitude", "2"], "."}]}]}]}], TraditionalForm]], FormatType->"TraditionalForm"] }], "Text", CellChangeTimes->{{3.4741084959375*^9, 3.474108549375*^9}, { 3.474108611546875*^9, 3.474108691796875*^9}}], Cell[TextData[{ "Lesson: If ", Cell[BoxData[ FormBox[ RowBox[{ SubsuperscriptBox["r", "0", "2"], ">", "0"}], TraditionalForm]], FormatType->"TraditionalForm"], ", then ", Cell[BoxData[ FormBox[ SubscriptBox["r", "0"], TraditionalForm]], FormatType->"TraditionalForm"], " is the radius of curvature of the 3-d space. 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