Twin Paradox

The Theory of Relativity

About Relativity

• As an object approaches the speed of light, time slows down.

• (Moving clocks are slow)

• (Moving rulers are short)

A trip from A trip from EarthEarth to to PlanetPlanet HollywoodHollywood

►Homer stays on Earth.Homer stays on Earth.►Loner travels 10 light-years at 80% of the Loner travels 10 light-years at 80% of the

speed of light. (speed of light = c)speed of light. (speed of light = c)►Beta (Beta (ββ) is the velocity of the object ) is the velocity of the object

compared to the speed of light. (compared to the speed of light. (ββ = 4/5c = = 4/5c = 0.8c) 0.8c)

►Gamma (Gamma (γγ) is the effect of traveling at ) is the effect of traveling at speeds close to light speed (c) has on time speeds close to light speed (c) has on time (t) or distance (x).(t) or distance (x).

Earth

Homer

Loner

Planet Hollywo

od

Effect of Time on Spaceship

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

• β = 4/5 = 0.8

γ = 1

1 - β²1 - β² = 1 - 0.64 = 0.36 and the √ of 0.36 is 0.6 !!

(β² = 0.8² = 0.64)

γ = 1 ÷ 0.6 = 1²/³ = 5/3

As Viewed From EarthAs Viewed From Earth

Without Relativity…..Without Relativity….. xx = = vvt t or or tt = = xx//vv x is light years traveledx is light years traveled v is velocity.v is velocity. t is time.t is time. t t = = 1010//0.8c0.8c = 12.5 years each way. = 12.5 years each way. There and back makes the trip There and back makes the trip

12.5 x 2 or 25 years!!12.5 x 2 or 25 years!!

As Viewed From Earth As Viewed From Earth

•With RelativityWith Relativity• Homer sees Loner’s Homer sees Loner’s

clock is running slow clock is running slow by…by…

• γγ = 5/3 !! = 5/3 !!

• Therefore Loner’s clock Therefore Loner’s clock reads 25 years ÷ reads 25 years ÷ γγ

• 25 ÷25 ÷ 5/3 = 5/3 = 15 years!!15 years!!

Physical Results of Trip

Homer on Earth ages 25 years!!

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

As Viewed From Spaceship• Loner sees distance of planets contracted by

γ = 5/3

• In Loner’s 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!!

Cosmic MuonsCosmic MuonsFactor:Factor:

γ =1

1 - v²/c² (β² = v²/c²)

Muon LifetimeMuon Lifetime

d = d = vvtt t = t = 2 x 10(-6)2 x 10(-6) seconds secondsv = v = 3 x 10(8)3 x 10(8) m/s m/sd = d = 2 x 10(-6)2 x 10(-6) • • 3 x 10(8)3 x 10(8) = 600m = 0.6km = 600m = 0.6km

A muon only travels 0.6 km A muon only travels 0.6 km into the atmosphere!into the atmosphere!

Question??Question??

• Muons are created in the upper atmosphere.

• Detectors tell us they reach the Earth’s surface.

• Suppose the distance needed to travel is approximately 10 km.

• If muons travel only 0.6 km, how do they reach the Earth’s surface??

With Relativity suppose v = 0.999c

γ =1

1 - v²/c² = 22

Earth’s Reference FrameEarth’s Reference Frame

• Muon’s clock runs slow by γ.

• Therefore the muon lives 22x longer and travels 22x further!

• So… d = (0.6 km) (22) = 13.2 km

• 13.2 km > 10.0 km allowing for the muon to reach the Earth’s surface!!

Earth’s Reference FrameEarth’s Reference Frame

Muon heading towards Earth at

0.999c

Homer notices that the muon’s “clock” is running slow

Muon’s Reference FrameMuon’s Reference Frame

Earth’s atmosphere moves towards the Earth’s atmosphere moves towards the muon at v = 0.999c, therefore the muon at v = 0.999c, therefore the atmosphere is contracted by atmosphere is contracted by γγ..

Thickness of atmosphere is 10km/Thickness of atmosphere is 10km/22 22 or or 0.45 km to the muon.0.45 km to the muon.

The muon travels 0.6 km and 0.6 > 0.45 The muon travels 0.6 km and 0.6 > 0.45 so the muon reaches Earth!!so the muon reaches Earth!!

Muon’s Reference Frame

Earth

Muon’s atmosphere 0.45km

Astronaut’s atmosphere 10km

Traveling at 0.999cThe muon’s ruler is “shortened,” while The astronauts, barelyMoving in comparison, See a thicker atmosphere.

The End!!