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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[ 41085, 1435]*) (*NotebookOutlinePosition[ 72223, 2532]*) (* CellTagsIndexPosition[ 71772, 2512]*) (*WindowFrame->Normal*) Notebook[{ Cell[CellGroupData[{ Cell[TextData[StyleBox[ "\n8.033 Problem Set 2, More Kinematic Effects of Relativity"]], "Subtitle", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[StyleBox["doug "]], "Subsubtitle", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[StyleBox[ "Index\n\tFrench: 4-5\n\tFrench: 4-9\n\tFrench: 4-12\n\tR & H: 2-14 A \ slow plane\n\tR & H: 2-21 Travel to the galactic center\n\tR & H: 2-24 \ Decay in flight (II)\n\tR & H: 2-25 Decay in flight (III)\n\tR & H: 2-26 \ Decay in flight (IV)\n\tR & H: 2-28 Simultaneous - but to whom?\n\tR & H: \ 2-36 What time is it anyway?\n\tBaranger: The cat's life\n\tBaranger: The \ particle's life\n\tBaranger: Trains and clocks\n\tBaranger: Blow up the \ Earth\n Post ramble: Initialization functions"]], "Text", CellMargins->{{-4, Inherited}, {Inherited, Inherited}}], Cell[CellGroupData[{ Cell[TextData[StyleBox[" French: 4-5"]], "Subsection", CellMargins->{{1, Inherited}, {Inherited, Inherited}}, Evaluatable->False, AspectRatioFixed->True, CellTags->"f4-5"], Cell[TextData[StyleBox[ "Q: A rocketship of proper length d travels at constant velocity v relative \ to a frame S. The nose of the ship (A') passes the point A in S at t = t' = \ 0, and at this instant a light signal is sent out from A' to B' (the end of \ the ship). (a) When, by rocketship time (t'), does the signal reach the tail \ (B') of the ship? (b) At what time t1, as measured in S, does the signal \ reach the tail (B') of the ship?\n(c) At what time t2, as measured in S, does \ the tail of the ship (B') pass the point A?"]], "Text", CellMargins->{{0, Inherited}, {Inherited, Inherited}}, Evaluatable->False, AspectRatioFixed->True], Cell[TextData[StyleBox[ "A: (a) In the rocket's frame, the light is emitted a proper length d from \ the origin traveling at c, so t' = d/c."]], "Text", CellMargins->{{Inherited, 17}, {Inherited, Inherited}}, Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ StyleBox[ "(b) In the rocket's frame, the event of the signal reaching the tail is \ represented by the quaternion "], StyleBox[Cell[BoxData[ \(q[d/C, \ d/C, 0, 0]\)], "Input"]], StyleBox[ ". In frame S, the light is blueshifted because the rocket is approaching \ at a speed of -beta. "] }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"t1", " ", "=", " ", RowBox[{ StyleBox["Simplify", FontWeight->"Plain"], StyleBox["[", FontWeight->"Plain"], StyleBox["\n", FontWeight->"Plain"], RowBox[{ StyleBox[\((\[Gamma][\[Beta]]\ - \ \[Beta]\ \[Gamma][\[Beta]])\), FontWeight->"Plain"], StyleBox[ RowBox[{ StyleBox[" ", FontWeight->"Plain", FontColor->RGBColor[0, 0, 0.500008]], " "}]], StyleBox[\(q[d/C, \ d/C, 0, 0]\ .\ {1, 0, 0, 0}\), FontWeight->"Plain"]}], StyleBox["]", FontWeight->"Plain"]}]}]], "Input"], Cell[BoxData[ \({\(-\(\(d\ \((\(-1\) + \[Beta])\)\)\/\(C\ \@\(1 - \[Beta]\^2\)\)\)\), \(-\(\(d\ \((\(-1\) + \[Beta])\)\)\/\(C\ \@\(1 - \[Beta]\^2\)\)\)\), 0, 0}\)], "Output"] }, Open ]], Cell[TextData[{ "The time the signal arrives in frame S is ", Cell[BoxData[ \(t1\ = \ \@\(\(1\ - \ \[Beta]\)\/\(1\ + \ \[Beta]\)\)\ d\/c\)], "Input"], ". " }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ "(c) The length of the ship in frame S must be calculated first. Boost the \ ship's end at q[0, d, 0, 0] to frame S. The boost quaternion is ", Cell[BoxData[ \(L\ = \ q[\[Gamma], \ \[Gamma]\ \[Beta], 0, 0]\)], "Input"], "." }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[BoxData[ \(\(ship\_end = \ q[\[Gamma][\(-\[Beta]\)], \ \(-\[Beta]\)\ \[Gamma][\(-\[Beta]\)], 0, 0] \ .\ q[0, \ d, 0, 0]; \)\)], "Input"], Cell[TextData[{ "The start of the ship will move for a time equal to the first term of the \ boosted quaternion, and moved by a distance ", StyleBox["x = vt/c", FontFamily->"Courier"], "." }], "Text", Evaluatable->False, TextAlignment->Left, TextJustification->0, AspectRatioFixed->True], Cell[BoxData[ \(\(ship\_start\ = \ q[ship\_end[\([1, 1]\)], \ \[Beta]\ ship\_end[\([1, 1]\)], 0, 0]; \)\)], "Input"], Cell["\<\ The ship's length in frame S will be the difference at the same \ instant in this frame between the boosted ship end and translocated ship \ start.\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(ship\_length = \ \((ship\_end\ - \ ship\_start)\).{1, 0, 0, 0}\)], "Input", CellMargins->{{Inherited, -57}, {Inherited, Inherited}}], Cell[BoxData[ \({0, d\/\@\(1 - \[Beta]\^2\) - \(d\ \[Beta]\^2\)\/\@\(1 - \[Beta]\^2\), 0, 0}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(t2\ = \ Simplify[ship\_length[\([2]\)]\/\[Beta]]\)], "Input"], Cell[BoxData[ \(\(d - d\ \[Beta]\^2\)\/\(\[Beta]\ \@\(1 - \[Beta]\^2\)\)\)], "Output"] }, Open ]], Cell[TextData[{ "The time the rocketship's tail arrives is ", Cell[BoxData[ StyleBox[\(t2\ = \ d/\[Beta]\ \[Gamma]\), FontFamily->"Courier"]], "Input"], " in frame S." }], "Text", Evaluatable->False, AspectRatioFixed->True] }, Open ]], Cell[CellGroupData[{ Cell[TextData[StyleBox[" French: 4-9"]], "Subsection", CellMargins->{{0, Inherited}, {Inherited, Inherited}}, Evaluatable->False, AspectRatioFixed->True, CellTags->"f4-9"], Cell["\<\ Q: Two spaceships, each measuring 100 m in its own rest frame, \ pass by each other traveling in opposite directions. The instruments on \ spaceship A determine that the front end of spaceship B requires 5 \ microseconds to traverse the full length of A. (a) What is the relative \ velocity of the 2 spaceships? (b) A clock in the front end of B reads exactly \ one o'clock as it passes by the front end of A. What will the clock read as \ it passes by the rear end of A?\ \>", "Text", Evaluatable->False, AspectRatioFixed->True], Cell["\<\ A: (a) Given a length and a time, divide one by the other to get \ the relative velocity.\ \>", "Text", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{ Cell[BoxData[ \(\(q[5\ 10\^\(-6\)\ s, \ 100\ m, 0, 0]\)[\([2, 1]\)] \/\(q[5\ 10\^\(-6\)\ s, \ 100\ m, 0, 0]\)[\([1, 1]\)]\)], "Input"], Cell[BoxData[ \(\(20000000\ m\)\/s\)], "Output"] }, Open ]], Cell["The relative velocity is 2 x 10^7 m/s.", "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[ "(b) The proper time of the clock in rocketship B is the interval, which when \ using quaternions is the square root of the first term of the quaternion \ squared."], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{ Cell[BoxData[ \(\@\(( q[5.\ 10\^\(-6\)\ s, \ 100\ m/c, 0, 0]\ .\ \n\t q[5.\ 10\^\(-6\)\ s, \ 100\ m/c, 0, 0])\)[\([1, 1]\)]\)], "Input"], Cell[BoxData[ \(4.98887651569858851406`*^-6\ \@s\^2\)], "Output"] }, Open ]], Cell["\<\ The clock in rocket B reads one o'clock plus 4.99 \ microseconds.\ \>", "Text", Evaluatable->False, AspectRatioFixed->True] }, Open ]], Cell[CellGroupData[{ Cell[TextData[StyleBox[" French: 4-12"]], "Subsection", CellMargins->{{0, Inherited}, {Inherited, Inherited}}, Evaluatable->False, AspectRatioFixed->True, CellTags->"f4-12"], Cell["\<\ Q: At noon a rocketship passes the Earth with a velocity 0.8c. \ Observers on the ship and on the Earth agree that it is noon. (a) At 12:30 P.M. as read by a rocketship clock, the ship passes an \ interplanetary navigational station that is fixed relative to the Earth and \ whose clocks read Earth time. What time is it at the station? (b) How far \ from the Earth (in Earth coordinates) is the station? (c) At 12:30 P.M. rocketship time the ship reports by radio back to Earth. \ When (by Earth time) does the Earth receive the signal? (d) The station on Earth replies immediately When (by rocket time) is the \ reply received? Solve this problem TWICE, once using the Earth as a reference frame and then \ using the rocket at the reference frame.\ \>", "Text", Evaluatable->False, AspectRatioFixed->True], Cell["\<\ A: (a) From the Earth frame, we are given the proper time on the \ rocket clock as 30'. This interval is equal to the one seen by the Earth, \ which is calculated by squaring the quaternion and solving for t.\ \>", "Text",\ CellMargins->{{Inherited, 44}, {Inherited, Inherited}}, Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{ Cell[BoxData[ \(Solve[ \((q[t, \ \(-0.8\)\ t, 0, 0]\ .\ \n\t\t\t\t\t\t\t q[t, \ \(-0.8\)\ t, 0, 0])\)[\([1, 1]\)]\ == \ \((30\ min)\)\^2, \ t]\)], "Input"], Cell[BoxData[ \({{t \[Rule] \(-50.0000000000000000086`\)\ min}, { t \[Rule] 50.0000000000000000086`\ min}}\)], "Output"] }, Open ]], Cell["The time on the Earth clock is 50 min.", "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[ "(b) Multiply the time by the speed and get the units right."], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{ Cell[BoxData[ \(d\_\(station\ \) = \ 0.8\ c\ \ 50\ min\ \(60\ s\)\/min\)], "Input"], Cell[BoxData[ \(7.19999999999999999982`*^11\ m\)], "Output"] }, Open ]], Cell[TextData[{ "The distance is 7.2 x ", Cell[BoxData[ \(TraditionalForm\`10^11\)]], " meters." }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell["\<\ (c) The time of the rocket emitting the signal, 50', plus its \ travel time from that location, 50' v/c = 40, is 90', or 1:30.\ \>", "Text", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{ Cell[BoxData[ \(t\_earthreceives\ = \ 50\ min\ + \ 0.8\ \ 50\ min\)], "Input"], Cell[BoxData[ \(90.`\ min\)], "Output"] }, Open ]], Cell["\<\ (d) Find the intersection of the world line of the rocket, x/c = v \ t/c, and the world line of the light emitted from the Earth at 90 min, x/c = t - 90'.\ \>", "Text", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{ Cell[BoxData[ \(Solve[x/C == \ t\ - \ 90\ min\ /. \ x/C -> \ 0.8\ t, \ t]\)], "Input"], Cell[BoxData[ \({{t \[Rule] 450.000000000000000043`\ min}}\)], "Output"] }, Open ]], Cell["\<\ The position of the event is 450' v/c = 360'. We need the proper \ time of this interval, which will be the time on the rocket clock.\ \>", "Text", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{ Cell[BoxData[ \(\@\(( q[450\ min, \ 360\ min, 0, 0]\ .\n\ \ q[450\ min, \ 360\ min, 0, 0]) \)[\([1, 1]\)]\)], "Input"], Cell[BoxData[ \(270\ \@min\^2\)], "Output"] }, Open ]], Cell["\<\ At 4:30 rocket time, the light from the Earth will be received at \ the rocket.\ \>", "Text", Evaluatable->False, AspectRatioFixed->True], Cell["\<\ A': Now from the rocket frame... (a') From the rocket frame, we are given t=30', x = 0. We need to boost this \ proper time interval to the Earth's frame.\ \>", "Text", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{ Cell[BoxData[ \(q[\[Gamma][0.8], \ 0.8\ \[Gamma][0.8], 0, 0]\ .\ \n q[30\ min, 0, 0, 0]\ .\ {1, 0, 0, 0}\)], "Input"], Cell[BoxData[ \({50.0000000000000000043`\ min, 40.`\ min, 0, 0}\)], "Output"] }, Open ]], Cell["The time in the Earth frame is 50 min.", "Text", Evaluatable->False, AspectRatioFixed->True], Cell["\<\ (b') The distance from the Earth in the Earth's frame is the second \ term of the above quaternion. Convert 40 min to meters.\ \>", "Text", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{ Cell[BoxData[ \(40\ min\ \ \(60\ s\)\/min\ c\)], "Input"], Cell[BoxData[ \(720000000000\ m\)], "Output"] }, Open ]], Cell[TextData[{ "The distance is 7.2 x ", Cell[BoxData[ \(TraditionalForm\`10^11\)]], " meters." }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell["\<\ (c') Find the intersection of the world line of the Earth, x/c = -v \ t/c, and the light emitted at 30', x/c = -t + 30.\ \>", "Text", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{ Cell[BoxData[ \(Solve[x/C\ == \ \(-t\)\ + \ 30\ min\ /. \ x/C\ -> \ \(-.8\)\ t, t] \)], "Input"], Cell[BoxData[ \({{t \[Rule] 150.000000000000000021`\ min}}\)], "Output"] }, Open ]], Cell["\<\ The position of this event is 150' v/c = 120'. We need the proper \ time of this interval, which will be the time on the Earth clock.\ \>", "Text",\ Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{ Cell[BoxData[ \(\@\(( q[150\ min, \ 120\ min, 0, 0]\ .\n\ \ q[150\ min, \ 120\ min, 0, 0]) \)[\([1, 1]\)]\)], "Input"], Cell[BoxData[ \(90\ \@min\^2\)], "Output"] }, Open ]], Cell["\<\ At 1:30 Earth time, the light will be received from the \ rocket.\ \>", "Text", Evaluatable->False, AspectRatioFixed->True], Cell["\<\ (d') It is 150' in the rocket frame when the Earth emits the \ signal. It will take 120' for the signal to arrive. 150' + 120' = 270' or \ 4:30.\ \>", "Text", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{ Cell[BoxData[ \(\(q[150\ min, \ 120\ min, 0, 0]\)[\([1, 1]\)]\ + \ \n \(q[150\ min, \ 120\ min, 0, 0]\)[\([2, 1]\)]\)], "Input"], Cell[BoxData[ \(270\ min\)], "Output"] }, Open ]], Cell[TextData["The same answer again!"], "Text", Evaluatable->False, AspectRatioFixed->True] }, Open ]], Cell[CellGroupData[{ Cell[TextData[StyleBox[" R & H: 2-14 A slow airplane"]], "Subsection", CellMargins->{{-2, Inherited}, {Inherited, Inherited}}, Evaluatable->False, AspectRatioFixed->True, CellTags->"r&h2-14"], Cell["\<\ Q: An airplane whose rest length is 40.0 m is moving at a uniform \ velocity with respect to the Earth at a speed of 630 m/s. (a) By what \ fraction of its rest length will it appear to be shortened to an observer on \ Earth? (b) How long would it take by Earth clocks for the airplane's clock \ to fall behind by 1 microsecond, assuming that only special relativity \ applies?\ \>", "Text", Evaluatable->False, AspectRatioFixed->True], Cell["\<\ A: (a) Boost the plane's tail in the plane's frame to the Earth's \ frame by a speed of -630 m/s.\ \>", "Text", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{ Cell[BoxData[ \(\[Beta]\_plane\ = \ 630.\ \(m/s\)\ /c\)], "Input"], Cell[BoxData[ \(2.10000000000000000013`*^-6\)], "Output"] }, Open ]], Cell[BoxData[ \(\(d\_tail\ = \ q[\[Gamma][\(-\[Beta]\_plane\)], \ \(-\[Beta]\_plane\)\ \[Gamma][\(-\[Beta]\_plane\)], 0, 0]\ .\ \n \t\t\ \ \ q[0, 40\ m, 0, 0]; \)\)], "Input"], Cell["\<\ Calculate the distance traveled by the nose in this amount of time. \ \ \>", "Text", Evaluatable->False, AspectRatioFixed->True], Cell[BoxData[ \(\(d\_nose\ = \ q[d\_tail[\([1, 1]\)], \ \[Beta]\_plane\ d\_tail[\([1, 1]\)], 0, 0]; \)\)], "Input"], Cell["\<\ Subtract the distance traveled by the nose from the tail. Take the \ ratio of this difference with the rest length.\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(1\ - \ \((d\_tail\ - \ d\_nose)\)[\([2, 1]\)]\/\(40\ m\)\)], "Input"], Cell[BoxData[ \(2.20499990878988971765`*^-12\)], "Output"] }, Open ]], Cell["\<\ The ratio of lengths as seen on the Earth is 1 minus this small \ number.\ \>", "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[ "(b) We want to know the differential time between a boosted clock and one at \ rest. This is the first term of the difference between a boosted and \ unboosted clock."], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{ Cell[BoxData[ \(\(\(( q[\[Gamma][\[Beta]\_plane], \ \[Beta]\_plane\ \[Gamma][\[Beta]\_plane], 0, 0]\ .\ q[t, 0, 0, 0]\ - \n\t\t\ q[t, 0, 0, 0])\)[\([1, 1]\)]\n\)\)], "Input"], Cell[BoxData[ \(2.20500007142021559047`*^-12\ t\)], "Output"] }, Open ]], Cell["Set this equal to 1 microsecond and solve for t.", "Text", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{ Cell[BoxData[ \(NSolve[%\ == \ 10\^\(-6\)\ s, \ t]\)], "Input"], Cell[BoxData[ \({{t \[Rule] 453514.724539628389992`\ s}}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(453515.\ s\ \ min\/\(60\ s\)\ hr\/\(60\ min\)\ day\/\(24\ hr\)\)], "Input"], Cell[BoxData[ \(5.24901620370370370358`\ day\)], "Output"] }, Open ]], Cell["\<\ The plane must travel for 4.53x10^5 s to get out of sync by a \ microsecond with the Earth, or 5.25 days.\ \>", "Text", Evaluatable->False, AspectRatioFixed->True] }, Open ]], Cell[CellGroupData[{ Cell[TextData[StyleBox[ " R & H: 2-21 Travel to the galactic center!"]], "Subsection", CellMargins->{{0, Inherited}, {Inherited, Inherited}}, Evaluatable->False, AspectRatioFixed->True, CellTags->"r&h2-21"], Cell["\<\ Q: (a) Can a person, in principle, travel from Earth to the \ galactic center (which is about 28,000 lyr distant) in a normal lifetime? \ (b) What constant velocity would be needed to make the trip in 30 years \ (proper time)?\ \>", "Text", Evaluatable->False, AspectRatioFixed->True], Cell["\<\ A: (b) Boost the rocketeer up to the BIG speed B=1-e, set the \ distance to the destination d, and solve for e.\ \>", "Text", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{ Cell[BoxData[ \(Solve[\n\t \((q[\[Gamma][1 - e], \ \((1 - e)\)\ \[Gamma][1 - e], 0, 0]\ .\n\ \ \ \ q[t, 0, 0, 0])\)[\([2, 1]\)]\ == \ d, \ e]\)], "Input"], Cell[BoxData[ \({{e \[Rule] \(d\^2 + t\^2 - d\ \@\(d\^2 + t\^2\)\)\/\(d\^2 + t\^2\)}, { e \[Rule] \(d\^2 + t\^2 + d\ \@\(d\^2 + t\^2\)\)\/\(d\^2 + t\^2\)}} \)], "Output"] }, Open ]], Cell[TextData["Plug in numbers."], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{ Cell[BoxData[ \(N[%\ /. \ {d -> 28000, \ t -> 30}]\)], "Input"], Cell[BoxData[ \({{e \[Rule] 5.7397909765835379613`*^-7}, { e \[Rule] 1.99999942602090234168`}}\)], "Output"] }, Open ]], Cell["\<\ The constant speed required to make the trip in 30 years is 1 - 5.7 \ x 10^-7 less than c. The answer to (a) is that as a purely mathematical \ exercise, one could say yes. However, this does not account for the energy \ required to reach such a speed. An analysis which investigated the energy \ requirements would probably conclude that it cannot be done.\ \>", "Text", Evaluatable->False, AspectRatioFixed->True] }, Open ]], Cell[CellGroupData[{ Cell[TextData[StyleBox[" R & H: 2-24 Decay in flight (II)"]], "Subsection", CellMargins->{{0, Inherited}, {Inherited, Inherited}}, Evaluatable->False, AspectRatioFixed->True, CellTags->"r&h2-24"], Cell["\<\ Q: The mean lifetime of muons stopped in a lead block in the \ laboratory is measured to be 2.2 microseconds. The mean lifetime of \ high-speed muons in a burst of cosmic rays observed from the Earth is \ measured to be 16 microseconds. Find the speed of these cosmic ray \ muons.\ \>", "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData["A: Boost the muon from its rest frame to the lab."], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[BoxData[ \(\(muon\_lab\ = \ q[\[Gamma][\[Beta]], \ \[Beta]\ \[Gamma][\[Beta]], 0, 0]\ .\ q[2.2\ \[Mu]s, \ 0, 0, 0]; \)\)], "Input"], Cell["\<\ Set the time component of the quaternion equal to 16 \ microseconds.\ \>", "Text", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{ Cell[BoxData[ \(Solve[muon\_lab[\([1, 1]\)]\ == \ 16\ \[Mu]s, \ \[Beta]]\)], "Input"], Cell[BoxData[ \({{\[Beta] \[Rule] \(-0.990501766782876566508`\)}, { \[Beta] \[Rule] 0.990501766782876566508`}}\)], "Output"] }, Open ]], Cell[TextData["The muon is travelling 0.9905 c."], "Text", Evaluatable->False, AspectRatioFixed->True] }, Open ]], Cell[CellGroupData[{ Cell[TextData[StyleBox[" R & H: 2-25 Decay in flight (III)"]], "Subsection", CellMargins->{{-1, Inherited}, {Inherited, Inherited}}, Evaluatable->False, AspectRatioFixed->True, CellTags->"r&h2-25"], Cell[TextData[ "Q: An unstable high-energy particle enters a detector and leaves a track \ 1.05 mm long before it decays. Its speed relative to the detector was \ 0.992c. What is its proper lifetime?"], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ StyleBox["A: Boost the proper path of unknown length L by v/c"], StyleBox[" ", FontFamily->"Symbol"], StyleBox["= 0.992, solve for L given the lab length L'."] }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{ Cell[BoxData[ \(Solve[ \((q[\[Gamma][\(-0.992\)], \ \(-\((\(-0.992\))\)\)\ \[Gamma][\(-0.992\)], 0, 0]\ .\n \t\t\t\ \ \ \ \ \ \ \ q[0, L, 0, 0])\)[\([2, 1]\)]\ == \ .00105\ m/c, L]\)], "Input"], Cell[BoxData[ \({{L \[Rule] 4.41832547465666001639`*^-13\ s}}\)], "Output"] }, Open ]], Cell[TextData[{ StyleBox["The average lifetime is 4.4 x "], StyleBox[Cell[BoxData[ \(TraditionalForm\`10^\(-13\)\)]]], StyleBox[" s."] }], "Text", Evaluatable->False, AspectRatioFixed->True] }, Open ]], Cell[CellGroupData[{ Cell[TextData[StyleBox[" R & H: 2-26 Decay in flight (IV)"]], "Subsection", CellMargins->{{-1, Inherited}, {Inherited, Inherited}}, Evaluatable->False, AspectRatioFixed->True, CellTags->"r&h2-26"], Cell[TextData[StyleBox[ "Q: In the target area of an accelerator laboratory there is a straight \ evacuated tube 300 m long. A momentary burst of 1 million radioactive \ particles enters at one end of the tube, moving at a speed of 0.80c. Half of \ them arrive at the other end without having decayed. (a) How long is the tube \ as measured by an observer moving with the particles? (b) What is the \ half-life of the particles in this same reference frame? (c) With what speed \ is the tube measured to move in this frame?"]], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData["A: (a) Same as above."], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{ Cell[BoxData[ \(Solve[ \((q[\[Gamma][\(-0.8\)], \ \(-\((\(-0.8\))\)\)\ \[Gamma][\(-0.8\)], 0, 0]\ .\ \n\t\t\t\t\t\t\tq[0, L, 0, 0])\)[\([2, 1]\)]\ == \ 300\ m, L]\)], "Input"], Cell[BoxData[ \({{L \[Rule] 179.999999999999999982`\ m}}\)], "Output"] }, Open ]], Cell[TextData["The tube looks 180 m long to the moving particles."], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[ "(b) The length of the target is equal to one half life, t = L/v."], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{ Cell[BoxData[ \(\(180\ m\)\/\(0.8\ c\)\)], "Input"], Cell[BoxData[ \(7.49999999999999999956`*^-7\ s\)], "Output"] }, Open ]], Cell[TextData["The half life is 750 nanoseconds."], "Text", Evaluatable->False, AspectRatioFixed->True], Cell["(c) By symmetry, v = 0.8c. By calculation.", "Text", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{ Cell[BoxData[ \(\(180\ m\)\/\(c\ \ 7.5\ 10\^\(-7\)\ s\)\)], "Input"], Cell[BoxData[ \(0.800000000000000000086`\)], "Output"] }, Open ]], Cell[TextData[ "The tube looks like it is moving 0.8c in the rest frame of the particles."], "Text", Evaluatable->False, AspectRatioFixed->True] }, Open ]], Cell[CellGroupData[{ Cell[TextData[StyleBox[ " R & H: 2-28 Simultaneous - but to whom?"]], "Subsection", CellMargins->{{1, Inherited}, {Inherited, Inherited}}, Evaluatable->False, AspectRatioFixed->True, CellTags->"r&h2-28"], Cell["\<\ Q: An experimenter arranges to trigger two flashbulbs \ simultaneously, a blue flash located at the origin of his reference frame and \ a red flash at x = 30 km. A second observer is moving at a speed of 0.25c in \ the direction of increasing x, and also views these flashes. (a) What time \ interval between them does he find? (b) Which flash does he say occurs first?\ \>", "Text", Evaluatable->False, AspectRatioFixed->True], Cell["\<\ A: (a) For the first observer, the blue flash stays at the origin. \ The red flash is boosted to a new location in spacetime. \ \>", "Text", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{ Cell[BoxData[ \(\((q[\[Gamma][0.25], \ \((0.25)\)\ \[Gamma][0.25], 0, 0]\ .\ \n\t q[0, 30\ \ 10\^3\ m/c, 0, 0])\)[\([1, 1]\)]\)], "Input"], Cell[BoxData[ \(\(-0.000025819888974716112568`\)\ s\)], "Output"] }, Open ]], Cell["There will be 26 microseconds between the flashes.", "Text", Evaluatable->False, AspectRatioFixed->True], Cell["\<\ (b) The origin won't change under the boost. From part (a) the \ flash of red light event will be changed to -26 microseconds. Therefore the \ red light appears first to the rocketeer.\ \>", "Text", Evaluatable->False, AspectRatioFixed->True] }, Open ]], Cell[CellGroupData[{ Cell[TextData[StyleBox[ " R & H: 2-36 What time is it anyway?"]], "Subsection", CellMargins->{{0, Inherited}, {Inherited, Inherited}}, Evaluatable->False, AspectRatioFixed->True, CellTags->"r&h2-36"], Cell["\<\ Q: Observers S and S' stand at the origins of their respective \ frames, which are moving relative to each other with a speed of 0.6c. Each \ has a standard clock, which, as usual, they set to zero when the two origins \ coincide. Observer S keeps the S' clock visually in sight. (a) What time \ will the S' clock record when the S clock records 5 microseconds? (b) What \ time will observer S actually read on the S' clock when his own clock reads 5 \ microseconds?\ \>", "Text", Evaluatable->False, AspectRatioFixed->True], Cell["\<\ A: (a) We must determine the proper time for a clock with t = 5 \ microseconds, and x = v t, by taking the square root of the first term of the \ event quaternion squared.\ \>", "Text", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{ Cell[BoxData[ \(\@\(( q[5\ \[Mu]s, 0.6\ \ 5\ \[Mu]s, 0, 0]\ .\ \n\t q[5\ \[Mu]s, 0.6\ \ 5\ \[Mu]s, 0, 0])\)[\([1, 1]\)]\)], "Input"], Cell[BoxData[ \(4.`\ \@\[Mu]s\^2\)], "Output"] }, Open ]], Cell["\<\ The S' clock will record 4 microseconds when the clock in S reaches \ 5 microseconds.\ \>", "Text", Evaluatable->False, AspectRatioFixed->True], Cell["\<\ (b) The intersection of the worldline of the rocket, x/c = 0.6 t \ and a lightcone passing through t = 5 microseconds, x = 0 can be solved for \ t.\ \>", "Text", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{ Cell[BoxData[ \(\(Solve[x/C\ == \ \(-t\)\ + \ 5\ \[Mu]s\ /. \ x/C -> 0.6\ t, t]\ \)\)], "Input"], Cell[BoxData[ \({{t \[Rule] 3.125`\ \[Mu]s}}\)], "Output"] }, Open ]], Cell["\<\ The S' clock will read the interval of the quaternion at this \ intersection. Calculate the interval as in part (a).\ \>", "Text", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{ Cell[BoxData[ \(\@\(( q[3.125\ \[Mu]s, \ 0.6\ \ 3.125\ \[Mu]s, 0, 0]\ .\ \n\t q[3.125\ \[Mu]s, \ 0.6\ \ 3.125\ \[Mu]s, 0, 0])\)[\([1, 1]\)]\)], "Input"], Cell[BoxData[ \(2.5`\ \@\[Mu]s\^2\)], "Output"] }, Open ]], Cell["\<\ At 5 microseconds, the observer in frame S will actually see 2.5 \ microseconds on the S' clock.\ \>", "Text", Evaluatable->False, AspectRatioFixed->True] }, Open ]], Cell[CellGroupData[{ Cell[TextData[StyleBox[" Baranger: The cat's life"]], "Subsection", CellMargins->{{-1, Inherited}, {Inherited, Inherited}}, Evaluatable->False, AspectRatioFixed->True, CellTags->"cat"], Cell["\<\ Q: A newborn cat is put aboard a ship leaving Earth for Andromeda \ at speed v = 0.6c. The cat dies on the ship at age 7 years. (a) How far \ from the Earth in the Earth's frame is the ship when the cat dies? (b) A \ radio signal is sent from the ship when the cat dies. When does this signal \ get to the Earth by Earth time? (c) Bonus: What is the probability amplitude that Schrodinger killed the cat? \ \ \>", "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[ "A: (a) The proper time of the cat's life is 7 years. Boost it to the \ Earth's frame."], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{ Cell[BoxData[ \(q[\[Gamma][\(-0.6\)], \ \(-\((\(-0.6\))\)\)\ \[Gamma][\(-0.6\)], 0, 0] \ .\ \nq[7\ yr, 0, 0, 0]\ .\ {1, 0, 0, 0}\)], "Input"], Cell[BoxData[ \({8.75`\ yr, 5.25`\ yr, 0, 0}\)], "Output"] }, Open ]], Cell[TextData[ "In the Earth's frame, the cat died after traveling a distance equal to 5.25 \ years."], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[ "(b) It will take 5.25 years for the light to get back from the time when the \ cat died (8.75 years), so the signal reaches Earth in\n5.25 + 8.75 = 14 \ years."], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ "(c) Schrodinger posed the question as a joke. He is ", StyleBox["definitely", FontSlant->"Italic"], " still laughing." }], "Text", Evaluatable->False, AspectRatioFixed->True] }, Open ]], Cell[CellGroupData[{ Cell[TextData[StyleBox[" Baranger: A particle's life"]], "Subsection", CellMargins->{{0, Inherited}, {Inherited, Inherited}}, Evaluatable->False, AspectRatioFixed->True, CellTags->"particle"], Cell[TextData[ "Q: A particle moving with speed v = 0.99c goes on the average a distance \ 12.5 m before decaying. What is its proper lifetime?"], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[ "A: Take the lifetime of the particle in its own frame, boost it to the \ lab's frame."], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[BoxData[ \(\(particle\_lab\ = \n\t\t\t q[\[Gamma][0.99], \ 0.99\ \[Gamma][0.99], 0, 0]\ .\ q[t, 0, 0, 0]; \)\)], "Input"], Cell[TextData[ "In the lab, x = 12 m. Set them equal, solve for the lifetime."], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{ Cell[BoxData[ \(Solve[particle\_lab[\([2, 1]\)]\ == \ 12\ m/c, t]\)], "Input"], Cell[BoxData[ \({{t \[Rule] 5.69969130491550885798`*^-9\ s}}\)], "Output"] }, Open ]], Cell[TextData["The lifetime is 5.7 ns."], "Text", Evaluatable->False, AspectRatioFixed->True] }, Open ]], Cell[CellGroupData[{ Cell[TextData[StyleBox[" Baranger: Trains & clocks"]], "Subsection", CellMargins->{{-2, Inherited}, {Inherited, Inherited}}, Evaluatable->False, AspectRatioFixed->True, CellTags->"trains"], Cell[TextData[ "Q: The train is moving with a velocity v. At the head of the train, the \ engineer compares her clock C'1 with a stationary clock C1 outside as she \ passes it, and finds that both clocks read time zero. At the same moment \ (for the train frame) the conductor in the caboose compares his clock C'2 \ (which therefore also reads zero) with a stationary clock C2 he happens to be \ passing. What does C2 read? The distance between the clocks C'1 and C'2 \ measured by people on the train is L."], "Text", Evaluatable->False, AspectRatioFixed->True], Cell["\<\ A: The interval for both sets of clocks is L/C. For the observer \ on the ground, set the time to t, the distance to vt/c. Square this \ quaternion, set the first term equal to the square of the interval, and solve \ for t. \ \>", "Text", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{ Cell[BoxData[ \(Solve[\ \@\((q[t, \ v\ t/C, \ 0, \ 0]\ .\ \ \n\tq[t, \ v\ t/C, \ 0, \ 0])\)[ \([1, 1]\)]\ == \ L/C, \ t]\)], "Input"], Cell[BoxData[ \({{t \[Rule] \(-\(\(I\ L\)\/\@\(\(-C\^2\) + v\^2\)\)\)}, { t \[Rule] \(I\ L\)\/\@\(\(-C\^2\) + v\^2\)}}\)], "Output"] }, Open ]], Cell[TextData[{ "The clock will read ", Cell[BoxData[ RowBox[{"t", " ", "=", " ", RowBox[{ RowBox[{"-", " ", StyleBox["\[Gamma]", FontFamily->"Symbol"]}], " ", \(L/c\)}]}]], "Input"], ". Note that ", StyleBox["Mathematica", FontSlant->"Italic"], " has erroneously injected a factor of I into the \"solution\"." }], "Text", Evaluatable->False, AspectRatioFixed->True] }, Open ]], Cell[CellGroupData[{ Cell[TextData[StyleBox[" Baranger: Blow up the Earth"]], "Subsection", CellMargins->{{0, Inherited}, {Inherited, Inherited}}, Evaluatable->False, AspectRatioFixed->True, CellTags->"blow up"], Cell["\<\ Q: Some inhabitants of the Andromeda nebula are traveling through \ the Milky Way in a flying saucer whose constant velocity equals 0.8c. Going \ by the Earth, they find out that it is A. D. 1996 here and they synchronize \ their clocks with ours. In A. D. 2005, mankind blows up the Earth. At what \ time, on their clock, do the travellers in the flying saucer learn of this \ event, assuming that they have been watching us all along through a \ telescope. Try a few ways of doing this problem.\ \>", "Text", Evaluatable->False, AspectRatioFixed->True], Cell["\<\ A: From the frame of the Earth, find the intersection of the world \ line of the saucer, x/c = 0.8 t, and the light cone from the explosion of the \ Earth, x/c = t + 9 yr.\ \>", "Text", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{ Cell[BoxData[ \(Solve[x/C\ == \ t\ - \ 9\ yr\ /. \ x/C -> .8\ t, t]\)], "Input"], Cell[BoxData[ \({{t \[Rule] 45.0000000000000000043`\ yr}}\)], "Output"] }, Open ]], Cell["\<\ The saucer has travelled a distance d = v t. Calculate the \ interval which will give the saucer's proper time.\ \>", "Text", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{ Cell[BoxData[ \(\@\(( q[45\ yr, \ .8\ 45\ yr, 0, 0]\ .\ \n\tq[45\ yr, \ .8\ 45\ yr, 0, 0]) \)[\([1, 1]\)]\)], "Input"], Cell[BoxData[ \(27.0000000000000000004`\ \@yr\^2\)], "Output"] }, Open ]], Cell["\<\ In 27 years time, or 2023, the saucer will note the demise of \ Earth.\ \>", "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[ "A': Repeat the calculation from the saucer frame. We know the interval is \ 9 years."], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{ Cell[BoxData[ \(Solve[ \((q[t, \ \(-0.8\) t, 0, 0]\ .\ \n\t\t\t\t\t\t\t q[t, \ \(-0.8\) t, 0, 0])\)[\([1, 1]\)]\ == \ \((9\ yr)\)\^2, t]\)], "Input"], Cell[BoxData[ \({{t \[Rule] \(-15.0000000000000000008`\)\ yr}, { t \[Rule] 15.0000000000000000008`\ yr}}\)], "Output"] }, Open ]], Cell[TextData[{ "The position will be x = v", StyleBox[" ", FontFamily->"Symbol"], "t = 12 years, which will take another twelve years to return, for a total \ of 27 years." }], "Text", Evaluatable->False, AspectRatioFixed->True] }, Open ]], Cell[CellGroupData[{ Cell[TextData[StyleBox[ " Post Ramble: Initialization functions"]], "Subsection", CellMargins->{{0, Inherited}, {Inherited, Inherited}}, Evaluatable->False, AspectRatioFixed->True, CellTags->"post"], Cell[TextData[StyleBox[ "There are a few tools required to solve problems in special relativity \ using quaternions to characterize events in spacetime. The most basic are a \ round value for c and gamma."]], "Text"], Cell[BoxData[ \(\(c\ = \ 3\ \ 10\^8\ m/s; \)\)], "Input", InitializationCell->True], Cell[BoxData[ RowBox[{ RowBox[{"\[Gamma]", StyleBox["[", FontFamily->"Courier"], StyleBox[ RowBox[{ StyleBox["\[Beta]", FontFamily->"Courier"], StyleBox["_", FontFamily->"Symbol"]}]], StyleBox["]", FontFamily->"Courier"]}], StyleBox[" ", FontFamily->"Courier"], StyleBox[":=", FontFamily->"Courier"], StyleBox[" ", FontFamily->"Courier"], StyleBox[\(1\/\@\(1\ - \ \[Beta]\^2\)\), FontFamily->"Courier"]}]], "Input", InitializationCell->True], Cell["\<\ Define a function for quaternions using its matrix \ representation.\ \>", "Text", Evaluatable->False, AspectRatioFixed->True], Cell[BoxData[ RowBox[{\(q[t_, \ x_, \ y_, \ z_]\), " ", ":=", " ", RowBox[{"(", GridBox[{ {"t", \(-x\), \(-y\), \(-z\)}, {"x", "t", \(-z\), "y"}, {"y", "z", "t", \(-x\)}, {"z", \(-y\), "x", "t"} }], ")"}]}]], "Input", InitializationCell->True], Cell[TextData[{ "A quaternion L that transforms a quaternion (L q[", StyleBox["x", FontWeight->"Bold"], "] = q[", StyleBox["x'", FontWeight->"Bold"], "]) identical to how the Lorentz transformation acts on 4-vectors \n(Lambda \ ", StyleBox["x", FontWeight->"Bold"], " = ", StyleBox["x'", FontWeight->"Bold"], ") should exist. These are described in detail in the notebook \"A \ different algebra for boosts.\" For boosts along the x axis with y = z = 0, \ the general function for L is" }], "Text", CellMargins->{{Inherited, 30}, {Inherited, Inherited}}, Evaluatable->False, TextAlignment->Left, AspectRatioFixed->True], Cell[BoxData[ \(L[t_, \ x_, \ \[Beta]_]\ := \n\t 1\/\(t\^2 + \ x\^2\)\ q[\[Gamma][\[Beta]]\ t\^2\ - \ 2\ \[Gamma][\[Beta]] \[Beta]\ t\ x\ + \ \[Gamma][\[Beta]]\ x\^2, \n\t\t\t\t\t\t\t\t\t\ \(-\[Beta]\)\ \[Gamma][\[Beta]]\ \((t\^2\ - \ x\^2)\), \ 0, \ 0] \)], "Input", InitializationCell->True], Cell["\<\ Most of the problems here involve much simpler cases for L, where t \ or x is zero, or t is equal to x.\ \>", "Text"], Cell["If t = 0, then ", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(L[0, \ x, \[Beta]]\ .\ {1, 0, 0, 0}\)], "Input"], Cell[BoxData[ \({1\/\@\(1 - \[Beta]\^2\), \[Beta]\/\@\(1 - \[Beta]\^2\), 0, 0}\)], "Output"] }, Open ]], Cell["If x = 0, then ", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(L[t, \ 0, \[Beta]]\ .\ {1, 0, 0, 0}\)], "Input"], Cell[BoxData[ \({1\/\@\(1 - \[Beta]\^2\), \(-\(\[Beta]\/\@\(1 - \[Beta]\^2\)\)\), 0, 0} \)], "Output"] }, Open ]], Cell["If t = x, then", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(Simplify[L[t, \ t, \[Beta]]\ .\ {1, 0, 0, 0}]\)], "Input"], Cell[BoxData[ \({\(1 - \[Beta]\)\/\@\(1 - \[Beta]\^2\), 0, 0, 0}\)], "Output"] }, Open ]], Cell["\<\ Note: this is for redshifts. Blueshifts have a plus instead of the \ minus.\ \>", "Text"], Cell["\<\ The problems are from \"Basic Concepts in Relativity\" by Resnick \ and Halliday, \[Copyright]1992 by Macmillian Publishing, \"Special Relativity\ \" by A. P. French, \[Copyright] 1966, 1968 by MIT, and Prof. M. Baranger of \ MIT.\ \>", "Text", Evaluatable->False, AspectRatioFixed->True] }, Open ]] }, Open ]] }, FrontEndVersion->"Microsoft Windows 3.0", ScreenRectangle->{{0, 640}, {0, 451}}, AutoGeneratedPackage->None, WindowToolbars->{"RulerBar", "EditBar"}, CellGrouping->Automatic, WindowSize->{477, 240}, WindowMargins->{{39, Automatic}, {Automatic, 12}}, PrintingCopies->1, PrintingPageRange->{Automatic, Automatic}, PageHeaders->{{Cell[ TextData[ { CounterBox[ "Page"]}], "PageNumber"], Inherited, Cell[ TextData[ { ValueBox[ "FileName"]}], "Header"]}, {Cell[ TextData[ { ValueBox[ "FileName"]}], "Header"], Inherited, Cell[ TextData[ { CounterBox[ "Page"]}], "PageNumber"]}}, PageHeaderLines->{False, Inherited}, PrintingOptions->{"PrintingMargins"->{{72, 57.5625}, {57.5625, 72}}, "PrintCellBrackets"->False, "PrintRegistrationMarks"->False, "PrintMultipleHorizontalPages"->False, "FirstPageHeader"->False}, PrivateNotebookOptions->{"ColorPalette"->{RGBColor, 128}}, ShowCellTags->False, RenderingOptions->{"ObjectDithering"->True, "RasterDithering"->False}, CharacterEncoding->"MacintoshAutomaticEncoding", StyleDefinitions -> Notebook[{ Cell[CellGroupData[{ Cell[TextData[StyleBox["Style Definitions"]], "Subtitle"], Cell[TextData[StyleBox[ "Modify the definitions below to change the default appearance of all cells \ in a given style. 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