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  1. Please go to View, then select Slide Show. To progress through a slide, hit the down arrow key. There are some animations in this example. If you’d like to see something again, hit the up arrow followed by the down arrow. The first slide in a set asks the questions, while the following slide(s) provides a detailed example. Answer the questions as best you can before moving to the next slide(s). You should have printed out a spacetime diagram. This spacetime diagram has the standard axes for the rest frame of the Sun and the Earth (assume an inertial frame). The Sun is located at x = 0; the Earth is located at x = 8 lt-min. The Jupiter 2 is a spaceship.

  2. Spacetime Diagram Worksheet This worksheet gives you practice constructing and interpreting a spacetime diagram • Graphically determining if the interval between pairs of events is time-like, light-like, or space-like • Determining if you can reverse the time ordering or space ordering of pairs of events • Problems for Practice • Experience with Events • Working with Worldlines • Finding the t’ and x’ axes for different reference frames • Finding the time and space order of events

  3. Jupiter 2 1) Experience with Events • Label an event C that occurs at x = 15 lt-min and t = 6 min. • Event A is the launch of a satellite from the Earth to the Sun. Where and when is the satellite launched, as measured by observers in the Sun/Earth (unprimed) frame? • Five minutes after launch, the satellite malfunctions at a distance of 2 lt-min from its launch point (still measured in the unprimed frame). Label this event B on the spacetime diagram. A

  4. Jupiter 2 B C • Shown on diagram. • Experience with Events (answers) • Draw line through event A parallel to x and find intersection with t. tA= 5 min Draw line through event A parallel to t and find intersection with x xA= 8 lt-min • Five minutes after launch, tB= tA+ 5 min = 10 min a distance of 2 lt-min from its launch point A xB= xA – 2 lt-min = 6 lt-min Why subtract? Recall the satellite is moving towards the Sun.

  5. Jupiter 2 B • Working with Worldlines • Draw and label the worldline of the Earth, located at x = 8 lt-min. • Determine the speed of the Jupiter 2, as measured in the Sun/Earth (unprimed) reference frame. • At the same time as event B, a comet passes by the Earth at 0.5c(as measured in the unprimed frame), headed away from the Sun. Draw and label the worldline of the comet. C A

  6. Dx = 4 lt-min Earth Dt = 5 min Jupiter 2 B Comet • Shown on diagram. • Working with Worldlines (answers) • Jupiter 2 trajectory shown on diagram is Jupiter 2’s worldline. Slope of Jupiter 2’s worldline is Dt/Dx = 5 min/4 lt-min = 1.25 min/lt-min. Recall that speed is Dx/Dt, or that slope of worldline is inverse of velocity to get speed = 0.8c. Note: magnitude of slope > 1 as required. If mag. of slope < 1, object moves faster than light! • “Same time as event B” means 10 min; “passes by Earth” means x = 8 lt-min, so know the Comet passes through that point. Slope of worldline is 1/v = 1/(0.5) = 2 min/lt-min. Comet moves away from Sun. Put together for worldline, shown. C A

  7. Jupiter 2 B • Finding the t’ and x’ axes for different reference frames • Draw/label the t’ axis for the comet. Earth • Draw/label the x’ axis for the comet. C A Comet

  8. Jupiter 2 x’ B • Worldline IS t’ axis! • Finding the t’ and x’ axes for different reference frames • The x’ axis is “mirror image” of t’ axis (about +45o). Or, since t’ line is “2 up, 1 over”, then x’ line is “1 up, 2 over.” In other words, the magnitude of the numerical value of the slope of the x’ line is the speed of the object. t’ Earth Note: Doesn’t matter where x’, t’ intersect (unless origin in primed reference frame indicated.) C A Comet

  9. Jupiter 2 B • Order the events A, B, and C according to observers in the Sun/Earth reference frame, from earliest to latest. Be clear in indicating if any events occur at the same time. • Finding the time and space order of events t’ Earth x’ • Order the events A, B, and C according to observers at rest with respect to the Comet, from earliest to latest. Be clear in indicating if any events occur at the same time. C A Comet

  10. Jupiter 2 B • To find the time order of events in the unprimed frame, draw a line THROUGH each event PARALLEL to the x axis, and see where it intersects the t axis. • Finding the time and space order of events (answers) t’ Earth A, then C, then B x’ • To find the time order of events in the primed frame, draw a line THROUGH each event PARALLEL to the x’ axis, and see where it intersects the t’ axis. C, then A, then B C • To find the space (left-right) order of events in a frame, draw a line THROUGH each event PARALLEL to that frame’s time axis, and see where the line intersects that frame’s position axis. A Comet

  11. Jupiter 2 B • Graphically determining if the interval between pairs of events is time-like, light-like, or space-like • Determine if the spacetime interval between the events A and B is space-like, time-like, or light-like. • Determine if the spacetime interval between the events A and C is space-like, time-like, or light-like. C A

  12. Jupiter 2 B • Graphically determining if the interval between pairs of events is time-like, light-like, or space-like • Can calculate the value for the spacetime interval, using the definition: (Ds)2 = (Dt)2 – (Dx)2. Here, DtAB = 5 min, and DxAB = 2 lt-min, so (Ds)2 = 21 min2. By definition, a positive (Ds)2 is a time-like event. Also, (Ds)2= 0 is light-like, and (Ds)2 < 0 is space-like. Note the units. (answers) However, can be done graph-ically without any calculations: Just draw a line that connects the two events. Compare the numerical value of the magnitude of the slope of this line to 1: If mag. slope > 1  time-like If mag. slope = 1  light-like If mag. slope < 1  space-like C A

  13. Graphically determining if the interval between pairs of events is time-like, light-like, or space-like (cont.) So if numerical value of magnitude of the slope of the line that connects the two events = 1, the interval is LIGHT-LIKE. Why does this work? Go back to definition of interval: (Ds)2 = (Dt)2 – (Dx)2. Divide by (Dx)2: So if numerical value of magnitude of the slope of the line that connects the two events > 1, the interval is TIME-LIKE. As (Dx)2 > 0, this won’t change the sign of I2. So if numerical value of magnitude of the slope of the line that connects the two events < 1, the interval is SPACE-LIKE. (cDt/Dx) IS the numerical value of the slope of the line connecting the two events. Since it is squared, doesn’t matter if slope is positive or negative. Alternate explanation in part 6)

  14. Jupiter 2 B • Graphically determining if the interval between pairs of events is time-like, light-like, or space-like (answers) t’ • Draw line connecting events A and B. Earth x’ |Slope of line| > 1  Time-like. • Draw line connecting events A and C. C |Slope of line| < 1  Space-like. A Comet

  15. Jupiter 2 B • Determining if you can reverse the time ordering or space ordering of pairs of events • Consider the events A and B. Can you reverse the time order or the space order of these events? • Consider the events A and C. Can you reverse the time order or the space order of these events? C A

  16. Jupiter 2 B • Determining if you can reverse the time ordering or space ordering of pairs of events (answers) • In part 5), showed that the interval between A and B was time-like. Time-like means that the time order of events is important. The time order of time-like events CANNOT be reversed. • This means that in ALL reference frames, the order of these events remains the same (though the time between the events of course may change.) • However, the space (left to right) order of time-like events CAN be changed. There exist reference frames where the left to right order is reversed. C A • Opposite holds for space-like events. A and C are space-like, so CANNOT change space order, but CAN change time order. Light-like events preserve their time AND space order in all frames.

  17. Jupiter 2 B • Determining if you can reverse the time ordering or space ordering of pairs of events (further explanation) How does this come about? As before, when classifying intervals, draw a line connecting the two events: This line COULD be a world-line, as the magnitude of its slope > 1. So this could be the t’ axis of some reference frame. Events A and B occur on a line parallel to the t’ axis (they actually occur on the t’ axis). Events which occur on lines parallel to the time axis of a reference frame must occur at the SAME PLACE in that frame. C A Event B occurs to the LEFT of A in the unprimed frame. In the proposed frame, they occur in the SAME place. Space order can be changed by switching reference frames.

  18. Jupiter 2 B • Determining if you can reverse the time ordering or space ordering of pairs of events (more explanation) What about time order? (similar argument as before) A and C are space-like, for which the time order is not unique. Again, draw a line connecting the two events: This line can NOT be a world-line, as the magnitude of its slope < 1. But this could be the x’ axis of some reference frame. Events A and C occur on a line parallel to the x’ axis. Events which occur on lines parallel to the position axis of a reference frame must occur at the SAME TIME in that frame. C A Event A occurs BEFORE event C in the unprimed frame. In the proposed frame, they occur at the SAME time. Time order can be changed by switching reference frames.

  19. Problems for Practice (gives you practice with most concepts and calculations related to our work in special relativity) • In the Sun/Earth frame, the event D occurs before event A. According to observers zipping by in the Millenium Falcon, A occurs before D. What is the minimum speed of the Millenium Falcon (as measured in the unprimed frame) for this to occur? • According to observers on board Jupiter 2, how fast is the comet moving? • What is the time between events A and B, according to an atomic clock on board the satellite? • According to observers on board Jupiter 2, is the interval between events A and C space-like, light-like, or time-like? • According to an alien hitching a ride on the comet, what is the distance between events A and B? (Tough question. Hints: find relative speed between comet and satellite. Use answer to question g) • Event D occurs on the Comet when it is halfway between the Sun and the Earth (in the unprimed frame). Label the event D. What time does event D occur, in the unprimed frame? • Assume the satellite moves with constant velocity from its launch point, and even after the malfunction. Draw/label the worldline of the satellite, and determine its speed, as measured in the unprimed frame. • Order the events A, B, C, and D in time according to observers on board the Jupiter 2. Assume the Jupiter 2 always travels with the constant velocity shown in the diagram. • Determine whether you can reverse the time order or the space order of events B and C. a) D occurs at 2 min, 4 lt-min; b) Worldline of satellite starts at A, goes through B. Speed is 0.4c; c) D, A, B, C; d) Space-like, so time order can be reversed; e) v > 0.75c; f) 0.9286; g) 4.58 min; h) Space-like; i) 5.19 lt-min

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