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1 5. Consequences of the Lorentz transformation 5.1 Time dilation ZYX O Z Y X O The flash starts at time t 1, and goes off at time t 2, A light source in S a flash of light lasting t seconds t = t 2 – t 1 Also: 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:

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2 Light duration time measured by a clock in S: x = 0 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 on a moving clock, or moving clock runs slower, i.e., Observer in S sees the clock in S runs slower because the clock in S is moving w.r.t to him. t rest is also called the proper time (the shortest) of the two events. -- S is the moving frame w.r.t the strobe light -- t is called t moving

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3 Experimental demonstration of time dilation effect: Cosmic Ray evidence for time dilation 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

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4 The Twin Paradox 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 Ls clock, including her internal biological clock, were running slowly compared to Ms, 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 a high speed and Ms clock, including her internal biological clock, were running slowly compared to Ls, therefore when M reunites with L, M is younger than L.

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5 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 Ms or the Earths 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: Ms point of view is right, L will be younger than M

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6 5.2 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

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7 Call L as the L rest, since the rod is at rest to S L the 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

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8 6. Velocity addition A) Velocity Transformation 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)

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9 L5

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10 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.

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