Special Relativity

1
Special Relativity

• Einstein messes with space and time

2
How Fast Are You Moving Right Now?

• 0 m/s relative to your chair
• 400 m/s relative to earth center (rotation)
• 30,000 m/s relative to the sun (orbit)
• 220,000 m/s relative to the galaxy center (orbit)
• Relative to What??
• This is the gist of special relativity
• its 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

3
A world without ether

• For most of the 19th century, physicists thought
  that space was permeated by luminiferous ether
• this was thought to be necessary for light to
  propagate
• Michelson and Morley performed an experiment to
  measure earths velocity through this substance
• first result in 1887
• Michelson was first American to win Nobel Prize
  in physics
• Found that light waves dont bunch up in
  direction of earth motion
• shocked the physics world no ether!!
• speed of light is not measured relative to fixed
  medium
• unlike sound waves, water waves, etc.

4
Einsteins Two Postulates of Special Relativity

• The Speed of Light is Constant for any observer,
  regardless of the motion of the observer or the
  motion of the light source.
• The laws of physics are always the same in all
  inertial ( non-accelerating) frames of reference.

5
Simultaneity is relative, not absolute
Observer riding in spaceship at constant velocity
sees a flash of light situated in the center of
the ships chamber hit both ends at the same time
But to a stationary observer (or any observer in
relative motion), the condition that light
travels each way at the same speed in their own
frame means that the events will not be
simultaneous. In the case pictured, the
stationary observer sees the flash hit the back
of the ship before the front
6
One persons space is anothers time

• If simultaneity is broken, no one can agree on a
  universal time that suits all
• the relative state of motion is important
• Because the speed of light is constant (and
  finite) for all observers, space and time are
  unavoidably mixed
• weve 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
  spacetime
• events are described by three spatial
  coordinates plus a time

7
The Lorentz Transformation

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

8
The gamma factor

• Gamma (?) is a measure of how relativistic you
  are
• When v 0, ? 1.0
• and things are normal
• At v 0.6c, ? 1.25
• a little strange
• At v 0.8c, ? 1.67
• Very strange
• As v?c, ???

9
What does ? do?

• Time dilation clocks on a moving platform appear
  to 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 contraction moving objects appear to be
  compressed along the direction of travel by the
  factor ?
• standing on a platform, you see a shorter train
  slip past, though the occupants see their train
  as normal length

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Time Dilation

• T gT0
• T0 proper time. Its measured by the
  observer who is at rest with respect to the clock
  
• T time measured by the observer that
• sees the clock moving
• Note only look at one clock in each problem.

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Time Dilation Example

• A rocket ship moves by you at .866c. The
  astronaut measures his heart rate at 1beat /
  second. What do you measure his heart rate to be?
• To 1.00s g 1/ (1-.8662)1/2
  2.00
• T gT0 2.00s
• One beat every 2 seconds hell live
  twice as long as you, all things being equal.

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Time Dilation Example

• A muon has a lifetime 2 microseconds when it is
  at rest. What does an observer measure the
  lifetime to be, when the muons are moving at
  99c?
• To 2ms g 1/ (1-.992)1/2 7.10
• T gT0 14.2 ms
• This has been confirmed experimentally.
  Muons produced in the upper atmosphere should not
  get to the Earths surface but they do.

13
Length Contraction

• L L0 / g
• L0 (proper length) measured by an
• observer that is at rest wrt to the object
  
• L length measured by an observer watching
• the object move

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Length Contraction Example

• An astronaut builds a ship 20m long. He passes
• by you moving at .866c.
• How long do you say the ship is?
• L0 20m g 1/ (1-.8662)1/2
  2.00
• L 20/(2.00) 10.0m

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Why dont we see relativity every day?

• Were soooo slow (relative to c), that length
  contraction and time dilation dont amount to
  much
• 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

16
Velocity Addition

• Also falling out of the requirement that the
  speed of light is constant for all observers is a
  new rule for adding velocities
• Galilean addition had that someone traveling at
  v1 throwing a ball forward at v2 would make the
  ball go at v1v2
• 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

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Velocity Addition Example
A spaceship moving at .5c, fires a projectile at
.6c in the direction its 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.
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Velocity Addition Example

• A spaceship moving at .5c, fires a laser in the
  direction its moving. What is the speed of the
  laser light measured by an observer at rest with
  respect to the ship?
• V1 .5c V2 c
• Vrel (.5c c) / ( 1 (.5c)(c)/c2 ) .6c
  / .6
• c ( it had to be c - the speed of
  light is
• invariant. )

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Classic Paradoxes

• 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?

• It is impossible to get a massive thing to travel
  truly at the speed of light
• energy required is ?mc2, where ??? as v?c
• so requires infinite energy to get all the way to
  c
• 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 worlds clock
  has stopped you see no ticks happen before you
  hit

21
Etotal gmc2 mc2 1/2mv2

• 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 Confirmation

• We 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 orbit
• squished a bit in the direction of the earths
  travel around the sun
• E mc2 extensively confirmed
• nuclear power/bombs
• suns energy conversion mechanism