Special Relativity

1
PH103
Special Relativity
Dr. James van Howe Lecture 14
April 16, 2008
Albert Einstein when he was a young patent clerk
Some slides courtesy of Prof. Vogel AKA Ms.
Relativity
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What first baseman was famous for saying Im not
an athlete. Im a professional baseball player.

• Keith Hernandez
• John Kruk
• Gorgeous George Sisler
• Lou Gherig

3

• Pretend you had a spaceship that could travel at
  nearly the speed of light. A laser beam is
  aimed directly toward you. Which of the
  following will produce the highest speed for the
  laser beam, in your reference frame. I.e., in
  which situation will the laser beam reach you the
  fastest, as you measure it.
• Driving spaceship toward laser
• Driving spaceship away from laser
• Staying stationary relative to laser
• None of the above

4

• You are traveling from Earth to a distant planet
  at near-light speed. I am left on the Earth.
• I measure your heart rate to be ___, you measure
  my heart rate to be ___.
• slow, slow
• slow, fast
• fast, slow
• fast, fast

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Prelude Define Events

• ?An event is something that occurs at
• One point in space
• One instant in time
• A party is not an event. Your arrival at the
  party is (approx).
• NOTE
• ?Events happen in all frames!
• Events dont move!
• Events dont have a frame!

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You Can Hide But You Cant Run

• Speed of Light is Measured to be c3X108 m/s by
  all.
• Can you catch up? NO!. No matter how fast you
  go, light still outruns you by c.
• Can you run away? NO! No matter how fast you go,
  light still catches up by c.
• What if the source moves? A lamp coming at you
  (or away from you), light still reaches you at c.

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Some Consequences

• Can be derived from constancy of speed of light
• Time interval between two events is different
  measured by different observers.
• Length of object or length of a trip is different
  measured by different observers.

8
Recall Classical Relativity
Plane as Reference Frame
Earth as Reference Frame
Inside the plane cabin, tpretzel5.0s
From view of the Earth, the pretzel goes...farther

• At any point, let the velocity of the pretzel
  measured by the plane observer be vp.
• The velocity measured by Earth observer is vplane
  vp. Earth observer measures faster speed
• pretzel goes farther, faster, but same time (5s)

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Einsteins Relativity
Light beam now instead of pretzel
Inside the plane cabin, tligth5.0 ms
From view of the Earth, the beam goes...farther

• At any point, the velocity of the light measured
  by the plane observer is c (the same)
• And the velocity measured by Earth observer is c,
  so Earth observer measures
• Light goes farther, same speed

So longer time!
tgt5ms
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Time Dilation

• Generalize result

Eq 26-1a

• Dt and Dto are both the time between same events
• measured by different observers
• v is relative speed of the two observers
• Notice that if vltltc, the two times are ?.
• Hard part which time is which?

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Proper time

• Whats the difference between Dt and Dto?
• Dto is the proper time
• It is always less than any other measured time.
• Def proper time is time in frame in which events
  occur at same place
• For example
• If the two events happen to someone or something,
  their frame measures proper time
• your heartbeat

• particles lifetime

• If someone is at both events, (or same distance
  away) their frame measures proper time
• person taking trip

• you and the clock on the wall

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Proper Time

• You are traveling from Earth to a distant planet
  at near-light speed. I am left on the Earth.
• Who measures the proper time of your heartbeat?
  Of my heartbeat? Of your trip to the planet?
• Me, me, me
• You, me, me
• You, me, you
• You, you, you

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Problem-solving Tips

• A light-year is distance light travels in a yr
• d vt 1 light-year (c)(1yr) 1c-yr
• Consistent units
• distance in light-years,
• speed as fraction of c,
• time in years

Ex 12c-yr/24yr 0.5c

• If a problem does not say what frame given values
  are relative to ,
• assume they are relative to Earth

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Names______________________________________
PH103, Time Dilation, Length Contraction
1. Ernie travels on a rocket to a planet 12
light-years away (measured from the earth) at a
speed of 0.6 c. Bert stays on Earth. Each
measures the trip to take a different amount of
time. How long does Ernie measure the trip? How
about Bert? Who measures the proper time?
2. Einstein tells us that for special relativity
we dont necessarily agree on measurements of
intervals of time or distance, but we will agree
on the relative speed between two inertial
frames. If Ernie and Bert agree on the speed they
are traveling away from each other, what is the
distance between planets that Ernie measures? How
about Bert?
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Solution

• Ernie travels to a planet 12 light-years away at
  a speed of 0.6 c. Bert stays on Earth. Each
  measures the trip to take a different amount of
  time. What do each measure?

• The values in example are relative to Earth
• In that frame (in any one frame), the laws of
  physics hold, including d vt
• This gives us time relative to Bert on Earth.
• To find the time relative to Ernie on the ship
• Use time dilation eqn
• Which is proper time?

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Bert
The length between planets measured from the
earth by Bert is 12 c-year as given by the
problem. Bert and Ernie agree that the Ship or
the Earth (depending on your point of view) is
traveling away at 0.6c.
So Bert measures
Ernie
Ernie measures the duration of his trip (start
and stop) in the same place (in the ship), so
Ernie measures the proper time, which is less
than Bert. Recall Einstein thought about the
clock standing still as he sped away from it at
light speed. Berts measurement of Ernies trip
therefore should be longer.
Time Dilation
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Length Contraction

• The problem tells us that from the Earth, Bert
  measures the distance Ernie travels between
  planets as 12 c-years
• Ernie sees the alien planet getting closer at
  0.6c for 16 years.
• d vt (0.6c)(16y) 9.6 c-yr
• Both are measuring the distance between Earth and
  the other planet, yet the distances are different!

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Length Contraction

• Generalize result

Eq 26-3

• L and Lo are both lengths of same thing (or
  trip)
• measured by different observers
• v is relative speed of the two observers
• Notice that if vltltc , the two lengths are ?.
• Hard part which length is which?

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Proper Length

• Whats the difference between L and Lo?
• Lo is the proper length
• Its always longer than other measured length.
• Def proper length is length in frame in which
  object (or ends of trip) is at rest
• For example
• Object, or anyone at rest relative to it,
  measures objects proper length.
• Your own height

• Length of ship you are riding on

• Someone measures the proper length between two
  objects, if both are at rest relative to them
• person on either planet, for a trip between
  planets

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• Can we find the trip length according to Ernie
  from length contraction?
• Yes.
• Who measures proper length?

• Bert

Wow -- it agrees!
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Just How Proper is it?

• If there is a proper time and proper length,
• is there a proper reference frame?
• NO!!!!!!!
• Proper time of trip in example Ernie
• Proper length of trip in example Bert
• Proper time of astronauts heartbeat Astronaut
• How does astronauts heartbeat look to you?
• Proper time of your heartbeat You
• How does your heartbeat look to astronaut?

22

• Pretend you had a spaceship that could travel at
  nearly the speed of light. A laser beam is
  aimed directly toward you. Which of the
  following will produce the highest speed for the
  laser beam, in your reference frame. I.e., in
  which situation will the laser beam reach you the
  fastest, as you measure it.
• Driving spaceship toward laser
• Driving spaceship away from laser
• Staying stationary relative to laser
• None of the above

23

• You are traveling from Earth to a distant planet
  at near-light speed. I am left on the Earth.
• I measure your heart rate to be ___, you measure
  my heart rate to be ___.
• slow, slow
• slow, fast
• fast, slow
• fast, fast

24
Proper Time

• You are traveling from Earth to a distant planet
  at near-light speed. I am left on the Earth.
• Who measures the proper time of your heartbeat?
  Of my heartbeat? Of your trip to the planet?
• Me, me, me
• You, me, me
• You, me, you
• You, you, you