PH 103

1
PH 103

• Dr. Cecilia Vogel
• Lecture 12

2
Review

• Consequences of Einsteins postulates
• constancy of speed of light
• time dilation
• length contraction
• NOTE ?Events happen in all frames.
• Events dont move, they dont have a frame.

Outline

• More consequences
• length contraction
• velocity transformation
• not vector addition

3
PAL

• Nikos travels to a planet 12 light-years away at
  a speed of 0.6 c. Juan stays on Earth.
• d vt in Juans frame 12c-yr(0.6c)t, t20yr
• Who measures the proper time? Nikos. To him,
  both events (takeoff and arrival) happen right
  here, so at the same place.
• time as measured by Nikos is Dto.

4
Length Contraction

• Generalize result

Eq 29-3

• 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?

5
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

6
Just How Proper is it?

• If there is a proper time and a proper length,
• is there a proper reference frame?
• NO!!!!!!!
• Proper time of trip in example Nikos
• Proper length of trip in example Juan
• Proper time of astronauts heartbeat
• Astronaut
• Astronauts heartbeat looks SLOW to you.
• Proper time of your heartbeat

You

• Your heartbeat looks SLOW to astronaut.

7
Velocity Transformation

• Velocity is distance/time
• Since time and distance are different in
  Einsteins relativity than classical relativity,
  expect velocity transformation is different, too
• Velocity transformation
• Differs from text

8
Velocity Transformation

• Note
• Speed will never be bigger than c
• If v12and v23 are lt c, then v13 lt c
• If v12 or v23 c, then v13 c
• speed of light the same for both observers
• Pay attention to the sign
• Pay attention to order of subscripts

9
Speed of Light is c!!

• Suppose 1 is a beam of light. Tan measures the
  light to travel at c (i.e. v12 c). If Lou
  travels at 0.5c relative to Tan, use the velocity
  transformation eqn below to prove that the light
  moves at speed c relative to Lou, too.

10
Using Velocity Transformation

• Step 1 Let v13 answer you seek.
• Step 2 Identify frames 1 and 3 with person or
  object.
• Step 3 Identify frame 2 -- whats left?
• Step 4 Determine value of v12 and v23
• If you have v21 or v32 change sign switch
• Step 5 Plug in v12 and v23 to get v13
• Step 6 Check that your answer makes sense!
• Big or small? Forward or backward?

11
EXAMPLE

• A spaceship is approaching the planet Zorgon at
  a speed of 0.85c. A diplomatic shuttle is sent
  ahead to arrive at the planet earlier. How fast
  should the shuttle be launched relative to the
  ship, in order to approach the planet at a rate
  of 0.95c?

Step 4 Determine value of v12 and v23 v12 is
v of shuttle relative to Zorgon v12
0.95c v23 is v of Zorgon relative to ship
-(ship relative to Zorgon) v23 -0.85c

• Step 1 Let v13 answer you seek.
• v13 shuttle relative to ship
• Step 2 Identify 1 and 3 with person or object.
• 1 is shuttle, 3 is ship
• Step 3 Whats left? 2 is Zorgon

12
Step 5 Plug in
Step 6 Check Small? Just need to add a bit
more to make it head toward planet
faster. Positive? Should be launched forward