1 Relativity  H3: Relativistic kinematics  Time dilation  Length contraction

2 Light clock  The light clock is an imaginary device for measuring time in an absolute way  It does not have any moving parts, but simply relies on a pulse of light being continually reflected backwards and forwards between two mirrors

3 Proper time  Proper time- a time interval measured in a frame of reference between two events which occur at the same point in space

4 Time Dilation: The notion that time can be stretched  Suppose this light clock is "0 inside a transparent high- speed spaceship.  An observer who travels along with the ship and watches the light clock sees the flash reflecting straight up and down between the two mirrors, just as it would if the spaceship were at rest.

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6 Compare inside the space ship and outside the space ship  Suppose now that we are standing on the ground as the spaceship whizzes by us at high speed-say, half the speed of light.  Things are quite different from our reference frame, for we do not see the light path as being simple up-and-down motion.  Because each flash moves horizontally while it moves vertically between the two mirrors, we see the flash follow a diagonal path.  Earthbound frame of reference the flash travels a longer distance round trip between the mirrors.

7 Because the speed of light is the same in all reference frames (Einstein's second postulate)  The flash must travel for a corresponding longer time between the mirrors in our frame than in the reference frame of the on-board observer.  The longer diagonal distance must be divided by a correspondingly longer time interval to yield an unvarying value for the speed of light.  This stretching out of time is called time dilation.

8 The light clock is shown in three successive positions

9  Diagonal lines represent the path of the light flash as it starts from the lower mirror at position 1, moves to the upper mirror at position 2, and then back to the lower mirror at position 3.  Distances on the diagonal are marked ct, vt, and cto, which follows from the fact that the distance traveled by a uniformly moving object is equal to its speed multiplied by the time. The light clock is shown in three successive positions

10Times  t o = time it takes for the flash to move between the mirrors as measured from a frame of reference fixed to the light clock.  This is the time for straight up or down motion.  Speed of light = c,  Path of light is seen to move a vertical distance ct o. This distance between mirrors is at right angles to the motion of the light clock and is the same in both reference frames.  t = the time it takes the flash to move from one mirror to the other as measured from a frame of reference in which the light clock moves with speed v.  Speed of the flash is c and the time it takes to go from position 1 to position 2 is t, the diagonal distance traveled is ct.  During this time t, the clock (which travels horizontally at speed v) moves a horizontal distance vt from position 1 to position 2.

11 Three distances make up a right triangle

12 Rel ative time  The relationship between the time t o (call it proper time) in the frame of reference moving with the clock and the time t measured in another frame of reference (call it the relative time ) is:  v =speed of the clock relative to the outside observer (the same as the relative speed of the two observers)  c = speed of light

13 Express the time dilation equation more simply:

14 Lorentz factor . (gamma)

15 Reading the Lorentz Curve  Substitute 0.5c for v in the time-dilation equation and after some arithmetic find that  = 1.15; so t = 1.15 t o.  This means that if we viewed a clock on a space- ship traveling at half the speed of light, we would see the second hand take 1.15 minutes to make a revolution, whereas an observer riding with the clock would see it take 1 minute.

16 Reading the Lorentz Curve Cont:  If the spaceship passes us at 87% the speed of light,  = 2 and t = 2t o.  We would measure time events on the spaceship taking twice the usual intervals, for the hands of a clock on the ship would turn only half as fast as those on our own clock. Events on the ship would seem to take place in slow motion.  At 99.5% the speed of light,  = 10 and t = 10 t o ; we would see the second hand of the spaceship's clock take 10 minutes to sweep through a revolution requiring 1 minute on our clock.  At c, the moving clock would appear to run a tenth of our rate; it would tick only 6 seconds while our clock ticks 60 seconds. At 0.87 c, the moving clock ticks at half rate and shows 30 seconds to our 60 seconds; at 0.50 c, the moving clock ticks 1/1.15 as fast and ticks 52 seconds to our 60 seconds.  Moving clocks run slow.

17 Length Contraction  As objects move through space-time, space as well as time changes  In a nutshell, space is contracted, making the objects look shorter when they move by us at relativistic speeds.  What contracts is space itself.

18 Proper length  Proper length-The length of an object, or the distance between two points whose positions are measured at the same time as measured by an observer who as at rest with respect to it.

19 Lorenz contraction  v = relative velocity between the observed object and the observer  c = speed of light  L = the measured length of the moving object  L o = the measured length of the object at rest.

20 We can express this as L = ( 1 /  ) L o  At 87% of c, an object would be contracted to half its original length.  At 99.5% of c, it would contract to one- tenth its original length.  If the object were somehow able to move at c, its length would be zero.

21 Contraction takes place only in the direction of motion. If an object is moving horizontally, no contraction takes place vertically.

22 Length contraction should be of considerable interest to space voyagers.  The center of our Milky Way galaxy is 25,000 light-years away.  Does this mean that if we traveled in that direction at the speed of light it would take 25,000 years to get there?  From an Earth frame of reference, yes, but to the space voyagers, decidedly not!  At the speed of light, the 25,000-light-year distance would be contracted to no distance at all.  Space voyagers would arrive there instantly!

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