Twin Paradox

1
Twin Paradox

• The Theory of Relativity

2
About Relativity

• As an object approaches the speed of light, time
  slows down.
• (Moving clocks are slow)
• (Moving rulers are short)

3
A trip from Earth to Planet Hollywood

• Homer stays on Earth.
• Loner travels 10 light-years at 80 of the speed
  of light. (speed of light c)
• Beta (ß) is the velocity of the object compared
  to the speed of light. (ß 4/5c 0.8c)
• Gamma (?) is the effect of traveling at speeds
  close to light speed (c) has on time (t) or
  distance (x).

4
Homer
Planet Hollywood
Loner
Earth
5
Effect of Time on Spaceship

• Velocity (v) is v 4/5c 0.8c therefore
• ß 4/5 0.8

1
?
(ß² 0.8² 0.64)
1 - ß²
1 - ß² 1 - 0.64 0.36 and the v of 0.36 is 0.6
!!
? 1 0.6 1²/³ 5/3
6
As Viewed From Earth

• Without Relativity..
• x vt or t x/v
• x is light years traveled
• v is velocity.
• t is time.
• t 10/0.8c 12.5 years each way.
• There and back makes the trip
• 12.5 x 2 or 25 years!!

7
As Viewed From Earth

• With Relativity
• Homer sees Loners clock is running slow by
• ? 5/3 !!
• Therefore Loners clock reads 25 years ?
• 25 5/3 15 years!!

8
Physical Results of Trip

• Homer on Earth ages 25 years!!

• Loner, traveling at 80 the speed of light ages
  15 years!!

9
As Viewed From Spaceship

• Loner sees distance of planets contracted by ?
  5/3
• In Loners frame distance is 10 light years 5/3
  10 5/3 6 light years.
• Therefore t x/v 6/0.8 7.5 years each way.
• There and back is 7.5 x 2 15 year trip for
  Loner!!

10
Cosmic Muons

• Factor

1
?
1 - v²/c² (ß² v²/c²)
11
Muon Lifetime

• d vt
• t 2 x 10(-6) seconds
• v 3 x 10(8) m/s
• d 2 x 10(-6) 3 x 10(8) 600m 0.6km
• A muon only travels 0.6 km into the atmosphere!

12
Question??

• Muons are created in the upper atmosphere.
• Detectors tell us they reach the Earths surface.
• Suppose the distance needed to travel is
  approximately 10 km.
• If muons travel only 0.6 km, how do they reach
  the Earths surface??

13
With Relativity

• suppose v 0.999c

1
?
22
1 - v²/c²
14
Earths Reference Frame

• Muons clock runs slow by ?.
• Therefore the muon lives 22x longer and travels
  22x further!
• So d (0.6 km) (22) 13.2 km
• 13.2 km gt 10.0 km allowing for the muon to reach
  the Earths surface!!

15
Earths Reference Frame
Muon heading towards Earth at 0.999c
Homer notices that the muons clock is running
slow
16
Muons Reference Frame

• Earths atmosphere moves towards the muon at v
  0.999c, therefore the atmosphere is contracted by
  ?.
• Thickness of atmosphere is 10km/22 or 0.45 km to
  the muon.
• The muon travels 0.6 km and 0.6 gt 0.45 so the
  muon reaches Earth!!

17
Muons Reference Frame
Traveling at 0.999c The muons ruler is
shortened, while The astronauts,
barely Moving in comparison, See a thicker
atmosphere.
Astronauts atmosphere 10km
Muons atmosphere 0.45km
Earth
18
The End!!