Important Terms

Proper time Dt' :  The length of time of some event as observed by a person at rest with respect to the event. Proper length LP :   The length of an object observed by a person at rest with respect to the object. Dilated time Dt :   The length of time of some event as observed by a person moving with respect to the event.

Contracted length L :  The length of an object observed by a person moving with respect to the object. ------------------------------------------------------------------------ Simple rule: If there's zero movement relative to the event or object being measured, use a superscript prime for time and subscript P for length on the quantities. ------------------------------------------------------------------------ contraction: becoming smaller dilation: spreading out

 Important Equations and Concepts

Length Contraction L = LP (1-v2 /c2)1/2 If the object is moving with respect to an observer in an inertial reference system, it's smaller for that observer. ------------------------------------ Time Dilation Dt = Dt' / (1 - v2/c2)1/2 If the event is moving with respect to an observer in an inertial reference system, it takes longer to happen for that observer.

Mass Transformed into Energy DE = Dmc2 Mass gained or lost is equal to a gain or loss in energy. -------------------------------------- Energy Transformed into Mass Dm = DE / c2 Energy E gained or lost is equivalent to a gain or loss of mass.

Total Energy E = mc2 / (1 – v2/c2)1/2 ----------------------------- Rest Energy E0 = mc2 ----------------------------- Kinetic Energy KE = E – E0 The kinetic energy is not 1/2 mv2.

 Speed of Light Postulate

If a source of light is not accelerating with respect to an observer, the speed of light is always 300,000 km/s or 300m/ms in a vacuum, no matter what the relative speed between source and observer may be.

  Inertial Systems

Law of Inertia: If no net force is acting on an object, it either remains at rest, or continues to move with constant velocity. Inertial reference system: Any place in which Newton’s law of inertia is valid. If the airplane is not accelerating, it’s an inertial reference system.

 Laws of Physics are the Same in All Inertial Systems

Both observers say that the ball’s motion is exactly as expected from the acceleration caused by the pull of the earth. Both observers see the ball behaving as Newton’s laws and gravitational force predict it should.

 Speed of Light in Vacuum is the Same in All Inertial Systems

Both obervers measure the speed of light to be the same; the                       speed of light is seen to be the same in both inertial systems.

   Laser Light Clock 

 Time Dilation

Astronaut and earth observer each are in an inertial system, so each measures the same value for the speed of light, 300,000 km/s. Which person reports the longer time interval between the two events (departure of light from source, arrival at the detector) ?

 Time Dilation Explained: Part One

Each observer sees light propagate at the same speed: c = 300,000 km/s ---------------------------------- Since each observer sees the light travel a different distance, each observer measures a different time interval. Astronaut:  Dt' = 2D / c Observer:     Dt = 2s / c

   Time Dilation Explained:  Part Two

Pythagoras: s2 = L2 + D2      (1)

Dt = time interval         observed by         earth observer s = c Dt                  (2) ----------------------------- Dt' = time interval          observed          by astronaut D = cDt'                 (3)

Distance traveled by spacecraft: L = v Dt                        (4) ---------------------------------- Substituting into (1): c2 (Dt)2  = v2 (Dt)2 +                  c2 (Dt')2       (5) Solve this for (Dt)2: Dt = Dt' / (1 - v2/c2)1/2    (6)

    The Lifetime of a Muon

Muons are unstable particles produced when protons streaming in from the sun are absorbed in the atmosphere.  About one muon strikes each one cm2 of the earth's surface each second. Muons travel at a little less than the speed of light, v = 0.99 c Observers at rest with respect to these muons measure their average lifetime to be 2.2 ms; this is the proper time interval between creation in the upper atmosphere, and disintegration.

Dt'= 2.2 ms Dt = Dt' / (1 - v2/c2)1/2                  v / c = 0.99                 [1 – (.99)2]1/2 = 0.14 Dt = Dt' / 0.14     = 2.2 ms / 0.14     = 15.6 ms Earth observer measures the lifetime of the moving muon to be 15.6 ms.

  Time Dilation

Question: The star Alpha Centauri is 4.3 light years away. If a spacecraft could travel at a speed v = 0.95 c, how long would an earth observer say a trip to that star would it take? What would the traveler say? Answer: The traveler is stationary with respect to the events which define the time interval: the departure from earth, and arrival at the star. Thus, the traveler measures the proper time, Dt', while the earth observer measures the dilated time, Dt.

The earth observer says the trip takes Dt  = 4.3 / 0.95       = 4.5 years ----------------------------- Dt = Dt' / (1 - v2/c2)1/2 Rearranging: Dt' = Dt (1 - v2/c2)1/2        = 1.4 years

 The Twin Problem: An Apparent Paradox

Person on earth remained in an inertial reference system. -------------------------------- The space- traveler accelerated, and thus wasn't in an inertial reference system. ------------------------------- The special theory of relativity is valid for the person on earth, but not for the space traveler because he was in a non-inertial reference system.

 Length Contraction

Earth observer measures LP: The earth observer is at rest with respect to the earth-star line, so he measures the proper length LP. Astronaut measures  Dt': Since the astronaut's clock is at the beginning and end of the event--at his departure from earth, and at his arrival at the star-- he measures the proper time Dt'.

Each observer measures the same relative speed: v = v LP / Dt = L / Dt' L = LP (Dt'/Dt)     Dt =Dt'(1- v2/c2)1/2             Substituting: L = LP (1- v2 /c2)1/2

  Length Contraction

A rod is 20 meters long when observed while stationary with respect to an observer. How long is it to an observer moving at a speed v = 0.98 c with respect to the rod?

LP = 20 m L = LP (1 - v2/c2)1/2    = 20 [ 1 - (0.98)2 ]1/2    = 3.98 m

  Time Dilation and Length Contraction

Both observers measure the same relative speed, v.

Event begins with birth of muon and ends with its disintegration. -------------------------------------------- An observer on earth is moving with respect to the event, so he measures the dilated (longer) time, Dt = 5 x 10-5 s. Same observer measures the proper (longer) distance, LP. -------------------------------------------- Observer riding with the muon measures the proper (shorter) time Dt' = 2 x 10-6 s, and the contracted (shorter) distance L.

   Energy-Mass Equivalence 

Mass is energy:  E = mc2    Energy is mass:  m = E /c2

When a flashlight radiates a quantity of light energy, it loses mass. --------------------------------------- When a flower absorbs sunlight, its mass increases. --------------------------------------- When a uranium nucleus splits, the mass of the remnants is less than the original mass. The difference appears as light, heat, and kinetic energy.

  Energy to Mass Transformation 

From the sun:  0.10 watts /cm2

By how much would the mass of a butterfly increase after one hour facing the sun, assuming all of the energy is absorbed? -------------------------------------- Assume area of butterfly is 75 cm2. Time = 3600 seconds

DE = (0.10)(3600)(75)     = 27,000 Joules Dm = DE / c2 c = 3 x 108 m/s Dm = 3 x 10-13 kg

  Mass to Energy Transformation

What is the energy-equivalent of one gram* of matter?  E = mc2     = (0.001 kg)(3.0 x 108 m/s)2       = 9 x 1013 J --------------------------------------------- A penny has a mass of about 3 g.

One joule is the work done in lifting a one newton weight one meter. 10,000 newtons = 2500 pounds   1,000 meters = 3280 feet (0.6 mile) (10,000 N) (1,000 m) = 107 J ---------------------------------------------------------- About how many one-ton cars could be blasted about 1/2 mile upward if one gram of matter were completely converted to energy?

  Total Energy              Rest Energy           Kinetic Energy

What is the total energy of a particle of mass m = 2 kg moving with speed v = 0.9 c? ---------------------------------------- E = mc2 / (1-v2/c2)1/2     = 4.13 x 1017 J

What is the particle’s rest energy? ------------------------------------ E0 = mc2     = 1.80 x 1017 J

What is the particle’s kinetic energy? -------------------------------------- Kinetic Energy = E - E0 KE = 2.33 x 1017 J