1. 1 1.Time Dilation 2.Length Contraction 3. Velocity transformation Einstein’s special relativity: consequences

2. 2 Consequences of the Lorentz transformation 1.1 Time dilation ZYX O Z’Z’Z’Z’ Y’Y’Y’Y’ X’X’X’X’ O’O’O’O’ The flash starts at time t’ 1, and goes off at time t’ 2, A light source fixed in S’  a flash of light lasting  t’ sec.  t’ = t’ 2 – t’ 1 Fixed in space:  x’ = x’ 2 – x’ 1 = 0 as measured by a clock in S’ How does an observer in S views the light on and off events? Light on time: Light off time:

3. 3 Light duration time measured by a clock in S:  x’ = 0 we have --S’ is the rest frame w.r.t the Strobe flash --  t’ is called  t rest (the proper time)  t moving >  t rest, this effect is called time dilation of a moving clock, or moving clocks run slower, i.e., Observer in S sees the clock in S’ running slower because the clock in S’ is moving w.r.t to him/her.  t rest is also called the proper time (the shortest) of the two event periods., -- S is the moving frame w.r.t the strobe light --  t is called  t moving

4. 4 1.2 Experimental demonstration of time dilation effect: Cosmic Ray evidence for ‘time dilation’ http://www.jlab.org/~cecire/cosmic.jpg  Mesons are formed at heights > 10 km in atmosphere. Observations found that most of them manage to survive down to sea level –despite their half-life being only Even moving at c, half should have decayed in a distance of: But: as they move so fast their clocks (proper time) run slower due to “ time dilation”. If v=0.999c,  =22.37

5. 5 1.3 The Twin Paradox http://www.phys.vt.edu/~takeuchi/relativity/notes/section15.html Right after their 20 th birthday, L blasts off in a rocket ship for a space trip, travelling at a speed 0.99c to a nearby star 30 light years away, then comes back with same speed, while M stay on Earth.. -- In the view of M: The journey will take time T = 2*30*c*year/0.99c = 60 year, so L will return when M is 20+60=80 yr. How much will L aged over the same period? L was travelling at a high speed and L’s clock, including her internal biological clock, were running slowly compared to M’s. Therefore when L reunites with M, L will have aged by T’ = T/  = 60/7 < 9 yr. So L is much younger than M. Conclusion: In one frame of reference, L is younger while from the other frame of reference, M is younger. This is the paradox. -- In the view of L: M was travelling away at high speed and M’s clock, including her internal biological clock, were running slowly compared to L’s. So when M reunites with L, M is younger than L.

6. 6 M:  x=0 the proper time for her is  t DR L: moves quickly, so (  x’)  0, so her proper time out to event T and back again will be much smaller by factor of  than  t DR. x ct M L T D R Worldlines in M’s or the Earth’s frame) The invariant between two events D and R: Let's draw this now in L's frame: A problem: just what frame do we choose? Frame S’ that is L's rest frame on her way out to the space? OR Frame S’ that is L's rest frame on her way back? L changes frames at event T This breaks the symmetry and resolves the paradox: M travels from event D to event R in a single frame with no changes, while L changes frames. L's worldline is crooked (non-inertial) while M's is straight (inertial)! Therefore: M’s point of view is right, L will be younger than M

7. 7 2 Length contraction O x S y zx O’ S’ y’ z’ x’ u - A rod lies on the x’ axis in S’, at rest relative to S’ -its two ends measured as x’ 1 and x’ 2 The length of the rod in S’ is What is the length of the rod measured in S? Because the rod is moving relative to S, we should measure the x-coordinates x 1 and x 2 of the ends of the rod at the same time, i.e.,  t=t 2 -t 1 =0, L = x 2 –x 1 Using Eq 6

8. 8 Call L’ as L rest, since the rod is at rest to S’ L is L moving since it moves with velocity u relative to S, -The length or the distance is measured differently by two observers in relative motion which shows the effect of length contraction on a moving rod. - One observer will measure a shorter length when the object is moving relative to him/her. -The longest length is measured when the rod is at rest relative to the observer---proper length. -Only lengths or distances parallel to the direction of the relative motion are affected by length contraction.

9. 9 v = 0.8c v = 0.9c v = 0.99c v = 0.9999c Animation: Chris Prior, RAL

10. 10 3. Velocity addition A) Velocity Transformation (these are components of 3-velocity) How velocities are transformed from one Ref. Frame to another? Differentiating L.T. equations: --In order to avoid confusion, we now use for the speed of the reference frame S’ w.r.t. S in x direction. Suppose a particle has a velocity in S’ in S: Differentiate Eq. (5)

11. 11 L5

12. 12 The inverse velocity transformation equation is The velocity transformation equation from S’ to S is From Eq(7) and (8) we have: i). When v and u x, u x ’ << C,, the L.T  G.T u x = u x ’ + v ii). When u x ‘ = C,  U x = C, and when u x =C  u x ’=C L.T. includes the constancy of the speed of light, as well as G.T. for the low speed world. Example #3

13. 13 Problem: Two space ships approach each other with velocities of 0.9c. According to an observer on the space ship, what is the velocity of the other ship. Solution: Use the velocity addition formula. Both u x and v are 0.9c. v' = 0.9944c < c Exercise: Suppose a spaceship is equipped with a series of one-shot rockets, each of which can accelerate the ship to c/2 from rest. It uses one rocket to leave the solar system (ignore gravity here) and is then traveling at c/2 (relative to us) in deep space. It now fires its second rocket, keeping the same direction. Find how fast it is moving relative to us. It now fires the third rocket, keeping the same direction. Find its new speed. Can you draw any general conclusions from your results? 4. Example: Velocity addition