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RelativityRelativity

H3: Relativistic kinematicsH3: Relativistic kinematics Time dilationTime dilation Length contractionLength contraction

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Light clockLight clock

The light clock is an imaginary device for The light clock is an imaginary device for measuring time in an absolute way measuring time in an absolute way

It does not have any moving parts, but It does not have any moving parts, but simply relies on a pulse of light being simply relies on a pulse of light being continually reflected backwards and continually reflected backwards and forwards between two mirrors forwards between two mirrors

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Proper timeProper time

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

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Time Dilation: Time Dilation: The notion that time can The notion that time can be stretchedbe stretched

Suppose this light Suppose this light clock is "0 inside a clock is "0 inside a transparent high-transparent high-speed spaceship. speed spaceship.

An observer who An observer who travels along with the travels along with the ship and watches the ship and watches the light clock sees the light clock sees the flash reflecting straight flash reflecting straight up and down between up and down between the two mirrors, just the two mirrors, just as it would if the as it would if the spaceship were at spaceship were at rest. rest.

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Compare inside the space Compare inside the space ship and outside the ship and outside the

space shipspace ship Suppose now that we are standing on the ground Suppose now that we are standing on the ground

as the spaceship whizzes by us at high speed-say, as the spaceship whizzes by us at high speed-say, half the speed of light.half the speed of light.

Things are quite different from our reference Things are quite different from our reference frame, for we do not see the light path as being frame, for we do not see the light path as being simple up-and-down motion.simple up-and-down motion.

Because each flash moves horizontally while it Because each flash moves horizontally while it moves vertically between the two mirrors, we see moves vertically between the two mirrors, we see the flash follow a diagonal path. the flash follow a diagonal path.

Earthbound frame of reference the flash travels a Earthbound frame of reference the flash travels a longer distance round trip between the mirrors. longer distance round trip between the mirrors.

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Because the speed of light is the same in Because the speed of light is the same in all reference frames (Einstein's second all reference frames (Einstein's second

postulate)postulate) The flash must travel for a corresponding The flash must travel for a corresponding longer time between the mirrors in our frame longer time between the mirrors in our frame than in the reference frame of the on-board than in the reference frame of the on-board observer.observer.

The longer diagonal distance must be divided The longer diagonal distance must be divided by a correspondingly longer time interval to by a correspondingly longer time interval to yield an unvarying value for the speed of light.yield an unvarying value for the speed of light.

This stretching out of time is called This stretching out of time is called time time dilation. dilation.

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The light clock is shown in The light clock is shown in three successive positionsthree successive positions

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Diagonal lines Diagonal lines represent the path represent the path of the light flash as it of the light flash as it starts from the lower starts from the lower mirror at position 1, mirror at position 1, moves to the upper moves to the upper mirror at position 2, mirror at position 2, and then back to the and then back to the lower mirror at lower mirror at position 3.position 3.

Distances on the Distances on the diagonal are marked diagonal are marked ct, vt, ct, vt, and and cto, cto, which which follows from the fact follows from the fact that the distance that the distance traveled by a uniformly traveled by a uniformly moving object is equal moving object is equal to its speed multiplied to its speed multiplied by the time.by the time.

The light clock is shown in The light clock is shown in three successive positionsthree successive positions

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TimesTimes ttoo = time it takes for the flash to = time it takes for the flash to

move between the mirrors as move between the mirrors as measured from a frame of measured from a frame of reference fixed to the light reference fixed to the light clock. clock.

This is the time for straight up This is the time for straight up or down motion. or down motion.

Speed of light = c, Speed of light = c, Path of light is seen to move a Path of light is seen to move a

vertical distance vertical distance ctctoo. . This This

distance between mirrors is at distance between mirrors is at right angles to the motion of the right angles to the motion of the light clock and is the same in light clock and is the same in both reference frames. both reference frames.

t t = the time it takes the flash = the time it takes the flash to move from one mirror to to move from one mirror to the other as measured from the other as measured from a frame of reference in a frame of reference in which the light clock moves which the light clock moves with speed with speed v. v.

Speed of the flash is c and Speed of the flash is c and the time it takes to go from the time it takes to go from position 1 to position 2 is position 1 to position 2 is t, t, the diagonal distance the diagonal distance traveled is traveled is ct. ct.

During this time During this time t, t, the clock the clock (which travels horizontally at (which travels horizontally at speed speed v) v) moves a horizontal moves a horizontal distance distance vt vt from position 1 to from position 1 to position 2. position 2.

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Three distances make up a right Three distances make up a right

triangletriangle

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RelRelative timeative time The relationship between the time tThe relationship between the time too (call it (call it

proper time) in the frame of reference moving proper time) in the frame of reference moving with the clock and the time with the clock and the time t t measured in measured in another frame of reference (call it the another frame of reference (call it the relrelative ative timetime ) ) is:is:

v =v =speed of the clock relative to the outside speed of the clock relative to the outside observer (the same as the relative speed of the observer (the same as the relative speed of the two observers) two observers)

c = speed of lightc = speed of light 2

2

0

1c

v

tt

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Express the time dilation equation more Express the time dilation equation more simply: simply:

0tt

2

2

1

1

c

v

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Lorentz factor Lorentz factor . (gamma). (gamma)

2

2

1

1

c

v

1515

Reading the Lorentz Reading the Lorentz CurveCurve

Substitute Substitute 0.5c 0.5c for for v v in the time-dilation in the time-dilation equation and after some arithmetic find that equation and after some arithmetic find that = 1.15; so = 1.15; so t t = 1.15 = 1.15 ttoo. .

This means that if we viewed a clock on a This means that if we viewed a clock on a space- ship traveling at half the speed of space- ship traveling at half the speed of light, we would see the second hand take light, we would see the second hand take 1.15 minutes to make a revolution, whereas 1.15 minutes to make a revolution, whereas an observer riding with the clock would see an observer riding with the clock would see it take 1 minute. it take 1 minute.

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Reading the Lorentz Reading the Lorentz Curve Cont:Curve Cont: If the spaceship passes us at 87% the speed of light, If the spaceship passes us at 87% the speed of light, = 2 and = 2 and t t

= = 2t2too. . We would measure time events on the spaceship taking twice We would measure time events on the spaceship taking twice

the usual intervals, for the hands of a clock on the ship would 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 turn only half as fast as those on our own clock. Events on the ship would seem to take place in slow motion. ship would seem to take place in slow motion.

At 99.5% the speed of light, At 99.5% the speed of light, = 10 and = 10 and t t = 10 = 10 tto o ; ; we would see we would see the second hand of the spaceship's clock take 10 minutes to the second hand of the spaceship's clock take 10 minutes to sweep through a revolution requiring 1 minute on our clock.sweep through a revolution requiring 1 minute on our clock.

At 0.995 At 0.995 c, c, the moving clock would appear to run a tenth of our the moving clock would appear to run a tenth of our rate; it would tick only 6 seconds while our clock ticks 60 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 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 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 clock ticks 1/1.15 as fast and ticks 52 seconds to our 60 seconds. seconds.

Moving clocks run slow.Moving clocks run slow.

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Length ContractionLength Contraction

As objects move through space-time, As objects move through space-time, space as well as time changesspace as well as time changes

In a nutshell, space is contracted, making In a nutshell, space is contracted, making the objects look shorter when they move the objects look shorter when they move by us at relativistic speedsby us at relativistic speeds..

What contracts is space itselfWhat contracts is space itself..

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Proper lengthProper length

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

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Lorenz contractionLorenz contraction

v v = relative velocity between the = relative velocity between the observed object and the observerobserved object and the observer

c = speed of lightc = speed of light L L = the measured length of the moving = the measured length of the moving

objectobject LLoo = = the measured length of the object the measured length of the object

at rest.at rest.

2

2

1c

vLL o

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We can express this as We can express this as L L = ( 1 / = ( 1 / ) ) LLoo At 87% of c, an object At 87% of c, an object

would be contracted would be contracted to half its original to half its original length. length.

At 99.5% of c, it At 99.5% of c, it would contract to one-would contract to one-tenth its original tenth its original length. length.

If the object were If the object were somehow able to somehow able to move at c, its length move at c, its length would be zero.would be zero.

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

vertically.vertically.

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Length contraction should be of Length contraction should be of considerable interest to space considerable interest to space

voyagers.voyagers. The center of our Milky Way galaxy is 25,000 light-The center of our Milky Way galaxy is 25,000 light-

years away. years away. Does this mean that if we traveled in that direction Does this mean that if we traveled in that direction

at the speed of light it would take 25,000 years to at the speed of light it would take 25,000 years to get there? get there?

From an Earth frame of reference, yes, but to the From an Earth frame of reference, yes, but to the space voyagers, decidedly not! space voyagers, decidedly not!

At the speed of light, the 25,000-light-year At the speed of light, the 25,000-light-year distance would be contracted to no distance at all. distance would be contracted to no distance at all.

Space voyagers would arrive there instantly! Space voyagers would arrive there instantly!

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