More with Special RelativityPHYS 1301Prof. T.E. Coanversion: 4 Aug ’99Introduction:This lab will continue the illustration of important relativistic effects. We will solvetwo important “paradoxes” that sometimes arise when special relativity is discussed.By understanding why these apparent paradoxes are not really paradoxes, you willbetter understand special relativity.The first of these apparent paradoxes is the so-called “Tsao paradox.” (It’s named afterthe guy who thought it up.) This “paradox” is not particularly difficult to crack andwill help to clarify the notion of relative reference frames. The setting is this. A traintravels at very high speed so that it has an appreciable γ (that’s a lower case gamma inthe Greek alphabet). A runner on the train sprints toward the back of the train with thesame speed (with respect to the train), as the train moves forward (with respect to theground). The so-called “paradox” is this. We know that clocks on the train run slowcompared to ground clocks. We saw this many times in the special relativityintroduction lab. We know also that the runner’s clock runs slow compared to thetrain’s clock. Therefore, the runner’s clock should run “doubly slow” with respect tothe ground. But wait, the runner is not moving with respect to the ground! Thereforethe runner’s clock should run at the same rate as the ground clocks. So, how can it bethat the runner’s clock runs both “doubly slow” with respect to the ground clock andalso runs at the same rate as the ground clock?The second “paradox” is related but a bit subtler. Its solution emphasizes theimportance of keeping reference frames straight when discussing special relativity. Ifyou compare two objects, one moving with respect to the other, then you must takeinto account any change from one reference frame to another that an object mightmake in its motion. There is nothing to guarantee that during the motion of an objectthat object always remains in the same reference frame. To properly record clocktimes and rod lengths, you must account for all changes between reference frames.The setting for our second, so-called “twin paradox” is this. Twins are born on earth.One of them decides to travel to a distance star, take some photographs and returnhome to earth. Her twin stays at home. When the traveling twin returns home, bothtwins stand side by side and compare ages. The traveled twin is younger than the stay-at-home twin. This is not a paradox. This is a true statement. (If the Earth boundobserver says she has aged by 10 years and that her twin traveled with a = 9/16 cduring the trip to the star, the star bound twin will have aged only 8 years when the1twins stand side by side and compare ages.)The supposed paradox is sometimes stated this way. Relativity says that all motion isrelative. Therefore, who is to say who stayed still and who did the traveling? From thepoint of view of the star bound twin, the Earth twin zooms away and later returns.Hence, the Earth twin should be younger. But when the twins stand side by side, bothcannot be younger. We have a paradox, so goes the (faulty) reasoning.We will crack both paradoxes by using Spacetime comparing clock times in differentreference frames.Procedure:1. Locate the folder Spacetime. The instructor will help you if you can’t locate it.Double click (left mouse button) on the icon Spacetime. Click OK when queried.2. To refresh your memory, play for 10-15 minutes or so. Create rods and clocks atdifferent values of β and at different x positions. Make sure you can transformfrom one reference frame to another. Refer to the documentation in the labIntroduction to Special Relativity if necessary.3. We will solve the Tsao non-paradox first. First, create a moving clock string torepresent the train. (Select clock string from the Objects menu.) Make this clockstring have a β=0.9. Think of the various clocks as being located at the car ends.4. Advance the reference clock (that’s the clock along the β=0 strip and at x=0.00)until this clock reads t=7.0. Question 1: What time does the central train clocknow read?5. Create a new clock, to represent the runner, at the x-position of the central trainclock. Do this by first jumping to the train’s reference frame. (Do you see why?) 6. Now step the reference clock. You will probably have to select it using the Selectmenu. Question 2: What happens to the runner’s clock? Are the times differentbetween the reference clock and the runner’s clock? Question 3: Explain theresolution to the alleged Tsao paradox. Why is the “paradox” not a paradox?7. Delete the train clocks and the runner clock. You can do this by executing thecommand Clear All found under the Objects menu.8. Now we will examine the twin “paradox.” Suppose there is a star called Oasis adistance of 10 light-years away from us, as measured in the Earth’s referenceframe. (There isn’t really such a star, we are pretending.) We will let the Earth berepresented by one clock and Oasis by a second. We will use a shuttle andprogram it for a round trip. You can imagine that the earth bound twin has theEarth clock on her wrist and that the shuttle clock is attached to the wrist of thetraveling twin. The Oasis clock is there for clarity.9. Create a shuttle at x=0 and with β=0.6. Create the Oasis clock at x=6.0 and β=0.0.10. For now, the earth bound clock will be the reference clock. Advance this referenceclock until the shuttle is at x=6.0. You will see that the shuttle and the Oasis clockat the same position. Next, program the shuttle to reverse its direction at this point.The shuttle should merely change the sign but not the magnitude of its β. Thisoperation represents the traveling twin changing direction after she reaches Oasis.11. For maximum amusement, decrement (count down) the reference clock until theshuttle backs up and is aligned with the Earth again, just as if the traveling twin isabout to start her journey. Now advance the reference clock and the shuttle. Stopwhen the shuttle just reaches Oasis. (You will know this when you see 2 shuttlesdisplayed.) Question 4: When the shuttle reaches Oasis, what time does thereference clock read and what time does the shuttle clock read? 12. Continue stepping the clock until the shuttle arrives back at Earth. Question 5:What time does the reference clock read and what time does the shuttle clock read?Decrement the reference clock until it reads t=0.0.