1. Special Relativity Physics 102: Lecture 28

6. Consequences: 1. Time Dilation t 0 is call the “proper time”. Here it is the time between two events that occur at the same place, in the rest frame. D

7. Time Dilation t 0 is proper time Because it is rest frame of event D D L=v  t ½ v  t

8. Importantly, when deriving time dilation the two events, 1) photon leaving right side of train and 2) photon arriving back at right side of train, occur at same point on the train .

10. Time Dilation Example v/c  0.1 1.005 0.2 1.021 0.5 1.155 0.9 2.294 0.99 7.089 0.999 22.366 0.9999 70.712 0.99999 223.607 0.999999 707.107 0.9999999 2236.068

11. Derive length contraction using the postulates of special relativity and time dilation. Consequences: 2. Length contraction

12. (ii) Train traveling to right with speed v  t 0 = 2  x 0 /c send photon to end of train and back the observer on the ground sees : forward backward where  t 0 = time for light to travel to front and back of train for the observer on the train.  x 0, = length of train according to the observer on the train. note: see D. Griffiths’ Introduction to Electrodynamics (i) Aboard train  t forward = (  x + v  t forward )/c =  x/(c-v)  t backward = (  x  v  t backward )/c =  x/(c+v)  t total =  t forward +  t backward = (2  x/c )  2 =  t 0   where we have used time dilation:  t tota l =  t 0  2  x  = c  t 0 = 2  x   x =  x  /  the length of the moving train is contracted!

13. Space Travel Alpha Centauri is 4.3 light-years from earth. (It takes light 4.3 years to travel from earth to Alpha Centauri). How long would people on earth think it takes for a spaceship traveling v=0.95c to reach Alpha Centauri? How long do people on the ship think it takes? People on ship have ‘proper’ time. They see earth leave, and Alpha Centauri arrive in  t 0  t 0 = 1.4 years Example

14. Length Contraction People on ship and on earth agree on relative velocity v = 0.95 c. But they disagree on the time (4.5 vs 1.4 years). What about the distance between the planets? Earth/Alpha L 0 = v t = .95 (3x10 8 m/s) (4.5 years) = 4x10 16 m (4.3 light years) Ship L = v t 0 = .95 (3x10 8 m/s) (1.4 years) = 1.25x10 16 m (1.3 light years) Example Length in moving frame Length in object’s rest frame

15. Length Contraction Notice that even though the proper time clock is on the space ship, the length they are measuring is not the proper length. They see a “moving stick of length L” with Earth at one end and Alpha-Centauri at the other. To calculate the proper length they multiply their measured length by  . Length in object’s rest frame Example Length in moving frame

16. Length Contraction Sue is carrying a pole 10 meters long. Paul is on a barn which is 8 meters long. If Sue runs quickly v=0.8 c, can she ever have the entire pole in the barn? Paul: The barn is 8 meters long, and the pole is only Sue: No way! This pole is 10 meters long and that barn is only Example Who is right? A) Paul B) Sue C) Both

17. ACT / Preflight 28.3 You’re at the interstellar café having lunch in outer space - your spaceship is parked outside. A speeder zooms by in an identical ship at half the speed of light. From your perspective, their ship looks: (1) longer than your ship (2) shorter than your ship (3) exactly the same as your ship

19. Consequence: Simultaneity depends on reference frame (i) Aboard train Light is turned on and arrives at the front and back of the train at the same time. The two events, 1) light arriving at the front of the train and 2) light arriving at the back of the train, are simultaneous . (ii) Train traveling to right with speed v Light arrives first at back of the train and then at the front of the train, because light travels at the same speed in all reference frames. The same two events, 1) light arriving at the front of the train and 2) light arriving at the back of the train, are NOT simultaneous . the observer on the ground sees : note: see D. Griffiths’ Introduction to Electrodynamics

20. Therefore whether two events occur at the same time -simultaneity- depends on reference frame.

21. Relativistic Momentum Relativistic Momentum Note: for v<<c p=mv Note: for v=c p=infinity Relativistic Energy Note: for v=0 E = mc 2 Objects with mass always have v<c! Note: for v<<c E = mc 2 + ½ mv 2 Note: for v=c E = infinity (if m is not 0)