Length Distance and Time

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Length (Distance) and Time
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According to Buzz
Zippy Slow Motion and squished
Buzz Regular Speed, Regular Size
V 0.95 C
V 0
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According to Zippy
Buzz Slow Motion and Squished, And the universe
is smaller
Zippy Regular Speed, Regular Size
V - 0.95 C
V 0
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Not faster than the speed of light

• The nearest star is 4 light years away, so it
  takes 1460 days for light to get there (earth
  perspective). If you wanted to get there in one
  day (pilots perspective), how fast would your
  ship have to go?
• Use length contraction and time dilation. Give
  you answer in the form of C with 8 decimals.
• What does the trip look like from the pilots
  perspective?
• What does the trip look like from the earths
  perspective?

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Buzz Perspective
When Zippy arrives at alpha centauri, Buzz has
counted Zippys heart beat 86,400 times (1day)
and his own has beat a few hundred million times
(4 years).
B
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Zippys Perspective
Z
When Zippy arrives at alpha centauri, Zippy has
counted his own hear beat 86,400 times (1 day)
and Buzz heart beat about 10 times (10 seconds).
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Using t and t0

• Earth (Buzz) Perspective
• t0 is 4 years. Earthings time the trip using
  stationary earth clocks.
• t is 1 day. Earthlings time the trip using
  moving ship clocks.

• Ship (Zippy) Perspective
• t0 is 1 day. Passengers time the trip using
  stationary ship clocks.
• t is 10 seconds. Passengers time the trip using
  moving earth clocks.

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Finding t with vd/t

• Earth (Buzz) Perspective
• Use vd/t to find t0 not t.
• d is distance ship travels/earth (normal
  length).
• V is velocity of ship/earth.
• Gives you t of trip according to an earth clock
  4 years.
• Find the 1 day time with time dilation
  equation.

• Ship (Zippy) Perspective
• Use vd/t to find t0 not t.
• d is distance earth travels/ship (length
  contracted).
• V is velocity of earth/ship
• (same as velocity ship/earth, so find V in the
  earth perspective where its easy and use it
  here).
• Gives you t of trip according to a ship clock 1
  day.
• Find the 10 second time with time dilation.

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Warm Up

• Which of the following are a matter of
  perspective and which are the same from all
  perspectives.
• The speed of the earth
• The speed of light
• The force exerted by a jet engine
• Size of the earth
• Accelerations
• The duration of 1 minute
• What are at least two ways that special
  relativity has been tested.

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The Twin Paradox

• Zippys situation makes since, everyone agrees,
  he aged 1 day.
• Buzz is the paradox. Buzz says he aged 4 year,
  Zippy says he aged 10 seconds.

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The thing we havent accounted for

• Zippy changes speed, Buzz doesnt.
• Velocity, distance, (and now we realize time)
  depend on your perspective.
• Buzz says zippy is moving.
• Zippy says Buzz is moving.
• Forces and accelerations do not depend on
  perspective.
• Buzz says there is a force on Zippy
• Zippy says there is a force on Zippy

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The Mental Short Cut

• When Zippy stops, he and Buzz are now in the same
  frame of reference.
• Zippy joined Buzz perspective.
• So Zippy has to agree with Buzz.
• Zippy aged 1 day and Buzz aged 4 years.
• As for how that comes about

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Remember the key idea of General Relativity
Acceleration without gravity and gravity without
moving produce the same effects
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Everything you do Is in slow motion
Effects of Gravity
Everything you do Is in fast motion
Neutron Star
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Effects of acceleration
Everything you do Is in slow motion
Everything you do Is in fast motion
g 10 negligible
a 1000x g
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What the trip really looks like for Zippy

• Zippy and Buzz are both 32 years old.
• Zippy has some crazy acceleration.
• Zippy is 32years10 minutes.
• Buzz went into fast motion and is now 34years
• Zippy cruises
• Zippy is 32years23 hours
• Buzz is in slow motion and is 34years3seconds
• Zippy decelerates
• Zippy is 32 years1 day
• Buzz went into fast motion again and is now 36
  years old.

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• The paradox is resolved when zippy stops.
• But while zippy is moving, the conflicting
  numbers are real.
• Zippy is really taller than Buzz
• Buzz is really taller than Zippy

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The Magic School Bus
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• A bus is 18 meters long.
• A garage is 9 meters long with open doors in
  front and back.
• For a stunt, you are going to put the bus in the
  garage and have blades of doom swing down at the
  entrance and exit simultaneously.
• You plan to use relativity to length contract the
  bus so that it will fit between the blades.

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Earth Perspective

• Gamma bus 2.
• vB/E 0.866C
• Garage 9m
• Bus 9m
• Bus fits between the doors for a microsecond,
  both blades come down at the same time then get
  out of the way, the bus passes through unharmed.

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Bus Perspective

• Gamma earth 2.
• vE/B 0.866C
• Bus 18m
• Garage 4.5m
• The blades swing and
• So do you have a happy stunt bus driver or a
  mangled bus?

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Resolving the Paradox

• Things that happen at the same time, (but at
  different locations) according to Buzz DONT
  happen at the same time according to Zippy.

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Example Buzz perspective
Zippy
Buzz sees light go left at C and light go right
at C Mike and Sulley receive the light at the
same time
Buzz
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• The previous description is what actually
  happens.
• Its what needs to happen in order to avoid
  paradoxes.
• You can write the equations to preserve symmetry
• Buzz sees Zippy going slow-mo, Zippy sees Buzz
  going slow-mo
• Or you can write them to preserve simultaneity
• Mike and Sulley get signals at same time
  according to Zippy and according to Buzz
• But you cant have both.

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Back to the Bus

• To tell if the bus fits, you have to look at the
  front of the bus and the back of the bus at the
  same time.
• At the same time will mean different things
  from different perspectives.

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Earth Perspective

• Blades are simultaneous
• Front of bus is right next to blade
• Back of bus is just past blade
• Bus is unharmed

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Bus Perspective

• Exit blade swings early before the front of the
  bus reaches it. Back of the bus is still outside
  the garage.
• Entrance blade swings late. Back of the bus has
  passed the entrance and the front is already out
  the exit.

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One Last Tidbit

• If its any comfort, not everything is relative.
• 2D distance
• 3D distance
• Space-Time distance
• This number is the same from all perspectives.

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Space vs Time

• If we had 4 spatial dimensions we would use the
  left equation.
• Since we have 3 spatial and 1 time dimension we
  use the right equation.
• So whats the difference between a spatial and
  temporal dimension?
• w2 and (ct)2 would be the same thing if there
  were an imaginary number stuck in there.

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The End