2. Lecture 5: PHYS344 Homework #1 Due in class Wednesday, Sept 9 th Read Chapters 1 and 2 of Krane, Modern Physics Problems: Chapter 2: 3, 5, 7, 8, 10, 14, 16, 17, 19, 20

3. We must think about how we measure space and time. In order to measure an object’s length in space, we must measure its leftmost and rightmost points at the same time if it’s not at rest. In order to measure an event’s duration in time, the start and stop measurements can occur at different positions, as long as the clocks are synchronized. If the positions are different, we need two people.

4. Proper Time To measure a duration, it’s best to use what’s called Proper Time. The Proper Time, T 0, is the time between two events (here two explosions) occurring at the same position (i.e., at rest) in a system as measured by a clock at that position. Same location Proper time measurements are in some sense the most fundamental measurements of a duration. But observers in moving systems, where the explosions’ positions differ, will also make such measurements. What will they measure?

5. Time Dilation and Proper Time If Mary and Melinda are careful to time and compare their measurements, what duration will they observe? Frank’s clock is stationary in K where two explosions occur. Mary, in moving K ’, is there for the first, but not the second. Fortunately, Melinda, also in K ’, is there for the second. K’ MaryMelinda K Frank Mary and Melinda are doing the best measurement that can be done. Each is at the right place at the right time.

6. Time Dilation Mary and Melinda measure the times for the two explosions in system K ’ as and. By the Lorentz transformation: This is the time interval as measured in the frame K ’. This is not proper time due to the motion of K ’ :. Frank, on the other hand, records x 2 – x 1 = 0 in K with a (proper) time: T 0 = t 2 – t 1, so we have:

7. 1) T ’ > T 0 : the time measured between two events at different positions is greater than the time between the same events at one position: this is time dilation. 2) The events do not occur at the same space and time coordinates in the two systems. 3) System K requires 1 clock and K ’ requires 2 clocks for the measurement. 4) Because the Lorentz transformation is symmetrical, time dilation is reciprocal: observers in K see time travel faster than for those in K ’. And vice versa! Time Dilation

8. Mirror FrankMary Time Dilation Example: Reflection Fred L K’K’ v K cT/2 Let T be the round- trip time in K vT/2

9. Reflection (continued) The time in the rest frame, K, is: Butor So the event in its rest frame (K ’ ) occurs faster than in the frame that’s moving compared to it (K).

10. Time stops for a light wave Because: And, when v approaches c : For anything traveling at the speed of light: In other words, any finite interval at rest appears infinitely long at the speed of light.

11. Proper Length When both endpoints of an object (at rest in a given frame) are measured in that frame, the resulting length is called the Proper Length. We’ll find that the proper length is the largest length observed. Observers in motion will see a contracted object.

12. Length Contraction ← Proper length where Mary’s and Melinda’s measured length is: Moving objects appear thinner! L0L0 Frank Sr. Frank Sr., at rest in system K, measures the length of his somewhat bulging waist: L 0 = x r  x ℓ Now, Mary and Melinda measure it, too, making simultaneous measurements ( ) of the left,, and the right endpoints, Frank Sr.’s measurement in terms of Mary’s and Melinda’s:

13. Length contraction is also reciprocal. So Mary and Melinda see Frank Sr. as thinner than he is in his own frame. But, since the Lorentz transformation is symmetrical, the effect is reciprocal: Frank Sr. sees Mary and Melinda as thinner by a factor of  also. Length contraction is also known as Lorentz contraction. Also, Lorentz contraction does not occur for the transverse directions, y and z.

14. Lorentz Contraction A fast- moving plane at different speeds. v = 10% c v = 80% c v = 99.9% c v = 99% c

15. 2.6: Addition of Velocities Taking differentials of the Lorentz transformation [here between the rest frame (K) and the space ship frame (K ’ )], we can compute the shuttle velocity in the rest frame ( u x = dx/dt ): Suppose a shuttle takes off quickly from a space ship already traveling very fast (both in the x direction). Imagine that the space ship’s speed is v, and the shuttle’s speed relative to the space ship is u ’. What will the shuttle’s velocity ( u ) be in the rest frame? v

16. The Lorentz Velocity Transformations Defining velocities as: u x = dx/dt, u y = dy/dt, u ’ x = dx ’ /dt ’, etc., we find: with similar relations for u y and u z : Note the  ’s in u y and u z.

17. The Inverse Lorentz Velocity Transformations If we know the shuttle’s velocity in the rest frame, we can calculate it with respect to the space ship. This is the Lorentz velocity transformation for u ’ x, u ’ y, and u ’ z. This is done by switching primed and unprimed and changing v to –v :

18. Relativistic velocity addition Speed, u ’ 0.25c Speed, u 0.50c0.75c v = 0.75c 1.0c 0.9c 0.8c 1.1c Galilean velocity addition Relativistic velocity addition 0