Spacetime Invariance Spacetime Invariance Relativity and Astrophysics Lecture 05 Terry Herter Outline Simultaneity of events Length measurement Proof of invariance of Spacetime Interval Time Dilation Reading A2290 05 A2290 05 Lorentz contraction in the direction of motion Invariance of the transverse direction Spacetime Physics Chapter 3 Spacetime Invariance 2 1 Spacetime Invariance Length measurement How do we measure the length of a moving rod that is the distance from one end to the other end A2290 05 We can use the lattice of clocks to mark the location of the two ends at the same time BUT when the rod lies along the direction of motion someone riding with the rod doesn t agree that the marking of the positions occurred at the same time Therefore two different observers will disagree about the validity of the measurement Spacetime Invariance 3 Length measuring events Consider a train car moving along for which two lightning bolts strike the ends of the car at the same time according an observer on the ground Now consider the observer on the train She measures the front lightning bolt to strike first and the rear bolt later A2290 05 A2290 05 For him the char marks on the ground are a valid measurement of the distance She concludes that the ground observer made the front mark before the rear and the train has moved forward by the time the rear mark is made Therefore she concludes that the ground observers measurement of the length of the car too small and concludes that the length is really longer than he the ground observer measured Spacetime Invariance 4 2 Spacetime Invariance Lorentz Contraction The result is that the space separation between the ends of a rod is less in a frame in which the rod is moving than on in which it is at rest This effect is called Lorentz contraction Proper length A moving rod shrinks in the direction of motion The length of the rod is largest in the frame in which the rod is at rest The rest length of the rod measured in a frame in which the rod is at rest Note These are measurements that are made not what we see with our eyes A2290 05 Spacetime Invariance 5 Invariance of Transverse Dimension The rod contracts along the direction of motion longitudinal contraction There is no change in the transverse dimension The rod does NOT get thinner or fatter Thought experiment train moving on a track Suppose the ground observer measures the train to get thinner wheels would fall inside the track The train rider however would see the track shrink wheels would fall outside the track But the we can t have both no change in width of train Logically inconsistent can t change width in transverse direction Hypothetical Earth view A2290 05 A2290 05 Hypothetical Train view Spacetime Invariance 6 3 Spacetime Invariance Invariance of Transverse Events Consider two equal diameter cylinders moving towards each other There is no change in size in the transverse dimension Neither cylinder passes through the other Again this would be logically inconsistent which one would shrink Wrong Wrong Events separated only in the transverse direction that are simultaneous in one frame are simultaneous in both Suppose in the frame of one cylinder a set of simultaneous explosions occur around the rim These explosions will also be simultaneous in the other frame No preferred direction in the transverse direction so no explosion can go first or last A2290 05 Spacetime Invariance 7 Proof of Spacetime Interval Invariance Consider a train moving with a velocity v mirror v d A2290 05 A2290 05 Inside the train is light pulse is sent from the bottom of the train car to the top and is reflected back down by a mirror The photon travels a time t 2d c Spacetime Invariance 8 4 Spacetime Invariance View from outside Consider a train moving with a velocity v mirror v L d vt 2d Outside the train an observer sees the light pulse proceed along the above path The train moves a distance vt 2d before the photon returns to where it started And the photon travels a distance 2L ct A2290 05 Spacetime Invariance 9 Invariance We want to construct the spacetime interval for each frame mirror L d d 2 L2 d 2 And we have L ct 2 d L2 d 2 d 2 Now d ct 2 d vt 2 Putting this altogether we have c 2t 2 4d 2 c 2t 2 4 L2 4d 2 c 2t 2 c 2t 2 c 2t 2 Earth STI c 2t 2 Train STI A2290 05 A2290 05 Spacetime Invariance The result is independent of the velocity so we have proved the spacetime interval is constant 10 5 Spacetime Invariance Faster than light No Consider a rocket moving a velocity v relative to our laboratory frame The rocket emits two flashes of light separated by time t as measured in the rocket frame Using the space time interval we can figure out the time separation t in our lab frame t 2 x 2 t 2 0 2 t 2 x 2 But x vt Substituting this into the spacetime interval and solving for t we have t t 1 v 2 1 2 Now this expression only makes sense if v 1 otherwise we have imaginary time t Note the rocket can travel as close to the speed of light as we wish but can t exceed it The time difference is know as Time Dilation A2290 05 Spacetime Invariance 11 Time Dilation another proof Consider a rocket moving with a velocity v d A2290 05 A2290 05 Mirrors v Inside the rocket is a clock which consists of two mirrors separated by a distance d with a light beam bouncing back and forth Every time the photon hits a mirror we get a tick of the clock Spacetime Invariance 12 6 Spacetime Invariance Time Dilation To a person sitting on the rocket the time between ticks is t d c d To be clear we are leaving time in units of seconds and distance in meters What does a person outside the rocket in the lab frame see A2290 05 Spacetime Invariance 13 Time Dilation To the person in the lab frame the distance between the mirrors will have changed when a tick occurs d d vt A2290 05 A2290 05 If t is the time between ticks as seen by the person in the lab frame then the mirror will have moved a distance vt Spacetime Invariance 14 7 Spacetime Invariance Time Dilation Then d d 2 d 2 v 2t 2 d Or vt c 2 t 2 c 2 t 2 v 2t 2 d ct and d ct So that t t 1 v 2 c2 1 2 Or t t 1 v c 2 Since v c the denominator is always less than one and t t Thus the time between tickets is always longer in the lab frame than the rocket frame 2 A2290 05 Spacetime Invariance 15 Time Dilation So we have t t 1 v2 c2 t …