Special RelativitySpecial Relativity

Einstein messes with space and timeEinstein messes with space and time

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How Fast Are You Moving Right Now?How Fast Are You Moving Right Now?

• 0 m/s 0 m/s relative torelative to your chair your chair• 400 m/s 400 m/s relative torelative to earth center (rotation) earth center (rotation)• 30,000 m/s 30,000 m/s relative torelative to the sun (orbit) the sun (orbit)• 220,000 m/s 220,000 m/s relative torelative to the galaxy center (orbit) the galaxy center (orbit)

Relative to What??Relative to What??• This is the gist of special relativityThis is the gist of special relativity

– it’s the exploration of the physics of relative motion– only relative velocities matter: no absolute frame– very relevant comparative velocity is c = 300,000,000

m/s

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A world without etherA world without ether

• For most of the 19th century, physicists thought that For most of the 19th century, physicists thought that space was permeated by “space was permeated by “luminiferous etherluminiferous ether””– this was thought to be necessary for light to propagate

• Michelson and Morley performed an experiment to Michelson and Morley performed an experiment to measure earth’s velocity through this substancemeasure earth’s velocity through this substance– first result in 1887– Michelson was first American to win Nobel Prize in physics

• Found that light waves Found that light waves don’tdon’t bunch up in direction of bunch up in direction of earth motionearth motion– shocked the physics world: no ether!!– speed of light is not measured relative to fixed medium– unlike sound waves, water waves, etc.

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Einstein’s Two Postulates of Special Einstein’s Two Postulates of Special RelativityRelativity

• The Speed of Light is Constant for any The Speed of Light is Constant for any observer, regardless of the motion of the observer, regardless of the motion of the observer or the motion of the light source.observer or the motion of the light source.

• The laws of physics are always the same in The laws of physics are always the same in all inertial ( non-accelerating) frames of all inertial ( non-accelerating) frames of reference.reference.

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Simultaneity is relative, not absoluteSimultaneity is relative, not absoluteObserver riding in spaceship atconstant velocity sees a flash oflight situated in the center of theship’s chamber hit both ends atthe same time

But to a stationary observer (or anyobserver in relative motion), thecondition that light travels each wayat the same speed in their own framemeans that the events will not be simultaneous. In the case pictured,the stationary observer sees the flashhit the back of the ship before thefront

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One person’s space is another’s timeOne person’s space is another’s time

• If simultaneity is broken, no one can agree on a If simultaneity is broken, no one can agree on a universal timeuniversal time that suits all that suits all– the relative state of motion is important

• Because the speed of light is constant (and finite) for Because the speed of light is constant (and finite) for all observers, space and time are all observers, space and time are unavoidablyunavoidably mixed mixed– we’ve seen an aspect of this in that looking into the distance

is the same as looking back in time

• Space and time mixing promotes unified view of Space and time mixing promotes unified view of spacetimespacetime– “events” are described by three spatial coordinates plus a

time

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The Lorentz TransformationThe Lorentz Transformation

• These Equations Relate the Time and These Equations Relate the Time and Positions of two Different Frames of Positions of two Different Frames of ReferenceReference

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The gamma factorThe gamma factor

• Gamma (Gamma () is a measure of how relativistic you are:) is a measure of how relativistic you are:• When When vv = 0, = 0, = 1.0 = 1.0

– and things are normal

• At At vv = 0.6 = 0.6cc, , = 1.25 = 1.25– a little strange

• At At vv = 0.8 = 0.8cc, , = 1.67 = 1.67– Very strange

• As As vvcc, ,

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What does What does do? do?

• Time dilationTime dilation: clocks on a moving platform appear to : clocks on a moving platform appear to tick slower by the factor tick slower by the factor – standing on platform, you see the clocks on a fast-moving

train tick slowly: people age more slowly, though to them, all is normal

• Length contractionLength contraction: moving objects appear to be : moving objects appear to be “compressed” along the direction of travel by the “compressed” along the direction of travel by the factor factor – standing on a platform, you see a shorter train slip past,

though the occupants see their train as normal length

Time DilationTime Dilation

T= T= TT00

TT00 = proper time. It’s measured by the = proper time. It’s measured by the

observer who is at rest with respect to observer who is at rest with respect to the clock the clock

T = time measured by the observer that T = time measured by the observer that

sees the clock moving sees the clock moving Note: only look at Note: only look at oneone clock in each problem. clock in each problem.

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Time Dilation ExampleTime Dilation Example

A rocket ship moves by you at .866c. The astronaut A rocket ship moves by you at .866c. The astronaut measures his heart rate at 1beat / second. What measures his heart rate at 1beat / second. What do you measure his heart rate to be?do you measure his heart rate to be?

TToo= 1.00s = 1.00s = 1/ (1-.866 = 1/ (1-.86622))1/2 1/2 = 2.00 = 2.00

T = T = TT00 = 2.00s = 2.00s

One beat every 2 seconds – he’ll live One beat every 2 seconds – he’ll live twice as long as you, all things being equal….twice as long as you, all things being equal….

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Time Dilation ExampleTime Dilation ExampleA muon has a lifetime 2 microseconds when it is at A muon has a lifetime 2 microseconds when it is at

rest. What does an observer measure the lifetime rest. What does an observer measure the lifetime to be, when the muons are moving at 99%c?to be, when the muons are moving at 99%c?

TToo= 2= 2s s = 1/ (1-.99 = 1/ (1-.9922))1/2 1/2 = 7.10 = 7.10

T = T = TT00 = 14.2 = 14.2 ss

This has been confirmed experimentally. Muons This has been confirmed experimentally. Muons produced in the upper atmosphere should not get produced in the upper atmosphere should not get to the Earth’s surface – but they do. to the Earth’s surface – but they do.

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Length Contraction Length Contraction

L= LL= L0 0 / /

LL00 = (proper length) measured by an = (proper length) measured by an

observer that is at rest wrt to the object observer that is at rest wrt to the object

L = length measured by an observer watching L = length measured by an observer watching

the object movethe object move

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Length Contraction ExampleLength Contraction Example

An astronaut builds a ship 20m long. He passesAn astronaut builds a ship 20m long. He passes

by you moving at .866c.by you moving at .866c.

How long do you say the ship is?How long do you say the ship is?

LL00 = 20m = 20m = 1/ (1-.866 = 1/ (1-.86622))1/2 1/2 = 2.00 = 2.00

L = 20/(2.00) = 10.0mL = 20/(2.00) = 10.0m

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Why don’t we see relativity every day?Why don’t we see relativity every day?

• We’re We’re soooo slowsoooo slow (relative to (relative to cc), that length ), that length contraction and time dilation don’t amount to contraction and time dilation don’t amount to muchmuch– 30 m/s freeway speed has v/c = 10-7

= 1.000000000000005

– 30,000 m/s earth around sun has v/c = 10-4

= 1.000000005• but precise measurements see this clearly

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Velocity AdditionVelocity Addition

• Also falling out of the requirement that the speed of Also falling out of the requirement that the speed of light is constant for all observers is a new rule for light is constant for all observers is a new rule for adding velocitiesadding velocities

• Galilean addition had that someone traveling at Galilean addition had that someone traveling at vv11

throwing a ball forward at throwing a ball forward at vv22 would make the ball go at would make the ball go at

vv11++vv22

• In relativity, In relativity,

– reduces to Galilean addition for small velocities

– can never get more than c if v1 and v2 are both c

– if either v1 OR v2 is c, then vrel = c: light always goes at c

Velocity Addition ExampleVelocity Addition Example

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A spaceship moving at .5c, fires a projectile at .6c in the direction it’s moving. What is the speed of the projectile measured by an observer at rest with respect to the ship?

V1 = .5c V2 = .6c

Vrel = (.5c + .6c) / ( 1 + (.5c)(.6c)/c2 ) = 1.1c / ( 1 + .3)

= (1.1)c/(1.3) = .85c NOT 1.1 c !

No correct calculation will ever give you any speed greater than c.

Velocity Addition ExampleVelocity Addition Example

A spaceship moving at .5c, fires a laser in the A spaceship moving at .5c, fires a laser in the direction it’s moving. What is the speed of the direction it’s moving. What is the speed of the laser light measured by an observer at rest laser light measured by an observer at rest with respect to the ship?with respect to the ship?

VV11 = .5c V = .5c V22 = c = c

VVrelrel = (.5c + c) / ( 1 + (.5c)(c)/c = (.5c + c) / ( 1 + (.5c)(c)/c22 ) = .6c / .6 ) = .6c / .6

= c ( it had to be c - the speed of light is = c ( it had to be c - the speed of light is

invariant. )invariant. )

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Classic ParadoxesClassic Paradoxes

• The twin paradox:The twin paradox:– one twin (age 30) sets off in rocket at high speed, returns to

earth after long trip– if v = 0.6c, 30 years will pass on earth while only 24 will pass

in high speed rocket– twin returns at age 54 to find sibling at 60 years old– why not the other way around?

– The moving twin is NOT in an inertial system – there is no paradox.

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What would I experience at light speed?What would I experience at light speed?

• It is It is impossibleimpossible to get a massive thing to to get a massive thing to travel truly at the speed of lighttravel truly at the speed of light– energy required is mc2, where as vc– so requires infinite energy to get all the way to c

• But if you are a massless photon…But if you are a massless photon…– to the outside, your clock is stopped– so you arrive at your destination in the same

instant you leave your source (by your clock)• across the universe in a perceived instant

– makes sense, if to you the outside world’s clock has stopped: you see no “ticks” happen before you hit

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EEtotal total = = mcmc2 2 = mc= mc22 + 1/2mv + 1/2mv22

Total Energy = Rest energy + Kinetic energy

Example: what is the rest energy of a 1kg rock?

mc2 = 1(300,000,000)2 =90,000,000,000,000,000 Joules

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Experimental ConfirmationExperimental Confirmation• We see time dilation in particle lifetimesWe see time dilation in particle lifetimes

– in accelerators, particles live longer at high speed• their clocks are running slowly as seen by us

• seen daily in particle accelerators worldwide

– cosmic rays make muons in the upper atmosphere• these muons only live for about 2 microseconds

• if not experiencing time dilation, they would decay before reaching the ground, but they do reach the ground in abundance

• We see length contraction of the lunar orbitWe see length contraction of the lunar orbit– squished a bit in the direction of the earth’s travel around the

sun

• EE = = mcmc22 extensively confirmed extensively confirmed– nuclear power/bombs– sun’s energy conversion mechanism