# O’ O X’ X Z’ Z 5. Consequences of the Lorentz transformation

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O’ O X’ X Z’ Z 5. Consequences of the Lorentz transformation
5.1 Time dilation A light source in S’ a flash of light lasting t’ seconds Z Y X O Z’ Y’ X’ O’ The flash starts at time t’1, and goes off at time t’2, as measured by a clock in S’ t’ = t’2 – t’1 A strobe light located at rest in S’, produces a flash of light lasting t’ seconds. We take the flash on and off as two events. Also: x’ = x’2 – x’1 = 0 How does an observer in S views the light on and off events? Light on time: Light off time:

x’ = 0 Light duration time measured by a clock in S:
-- S is the moving frame w.r.t the strobe light -- t is called tmoving --S’ is the rest frame w.r.t the Strobe flash -- t’ is called trest(the proper time) The clock in S’ is at rest relative to the strobe light, so the frame S’ is called the clock’s rest frame and t’ is called trest, or the proper time. Similarly the clock in the frame S is moving relative to the strobe light, t is called tmoving we have tmoving > trest, this effect is called time dilation on a moving clock, or moving clock runs slower, i.e., Observer in S see’s the clock in S’ runs slower because the clock in S’ is moving w.r.t to him. trest is also called the proper time (the shortest) of the two events.

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 Muons are sub-atomic particles generated when cosmic rays strike the upper levels of our atmosphere. They have a half lifetime of about 2 microseconds (µs) meaning that every 2 µs, their population will reduce by half. The slowing of clocks in a moving system is a real effect and it applies equally to all kinds of time, for example biological, chemical reaction rates, even to the rate of growth of a cancer in a cancer patient! 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

The Twin Paradox Right after their 20th 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. There are two twins M and L who were born at exactly the same moment (a biological impossibility). There are numerous ways of trying to resolve the paradox. Here we shall briefly mention one of the most common and simple one. We know that we can only do physics in inertial frames of reference. However, L is not in such a frame, the principal of inertia is violated for L when she blasts off from Earth, again when she fires her thrusters to turn around, and once again, when she lands on Earth. If you were riding with L, you would “feel” these non-inertial force when you are pushed into your seat or lifted from it when M’s rocket is firing. -- In the view of L: M was travelling away at a high speed and M’s clock, including her internal biological clock, were running slowly compared to L’s, therefore when M reunites with L, M is younger than L. Conclusion: In one frame of reference, L is younger while from the other frame of reference, M is younger. This is the paradox.

Worldlines in M’s or the Earth’s frame)
The invariant between two events D and R: x ct M L T D R M: x=0 the proper time for her is tDR 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 tDR. Let's draw this now in L's frame: We can explain this paradox in worldlines of L and M. The worldlines of L and M are plotted in the rest frame of the Earth (frame S), with L's departure marked as event D, L's turnaround at the distant star as T and her return home as R. ……… We cannot choose both because they are different frames: L changes frames at event T. The time dilation effect is reciprocal: as observed from the point of view of any two clocks which are in motion with respect to each other, it will be the other party's clock that is time dilated. A problem: just what frame do we choose? Frame S’ that is L's rest frame on her way out to the space? Frame S’ that is L's rest frame on her way back? OR 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

What is the length of the rod measured in S?
5.2 Length contraction - A rod lies on the x’ axis in S’, at rest relative to S’ O x S y zx O’ S’ y’ z’ x’ u -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 x1 and x2 of the ends of the rod at the same time, i.e., t=t2-t1=0, L = x2 –x1 Using Eq 6

Call L’ as the Lrest, since the rod is at rest to S’
L the Lmoving since it moves with velocity u relative to S, which shows the effect of length contraction on a moving rod. -The length or the distance is measured differently by two observers in relative motion - One observer will measure a shorter length when the object is moving relative to him/her Note: the ladder paradox involves a long ladder travelling near the speed of light and being contained within a smaller garage -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

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)

L5

The velocity transformation equation from S’ to S is
The inverse velocity transformation equation is From Eq(7) and (8) we have: i). When v and ux, ux’ << C, ux = ux’ + v , the L.T G.T ii). When ux‘ = C,  Ux = C, and when ux=Cux’=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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