Relativity 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 ofsome event as observed by a personmoving with respect to the event. Contracted length L :  The length of anobject observed by a person movingwith 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 smallerdilation: spreading out

Important Equations and Concepts
 Length Contraction L = LP (1-v2 /c2)1/2If the object is moving with respectto an observer in an inertial referencesystem, it's smaller for that observer.------------------------------------Time Dilation Dt = Dt' / (1 - v2/c2)1/2 If the event is moving with respectto an observer in an inertial referencesystem, it takes longer to happen for that observer. Mass Transformed into EnergyDE = Dmc2 Mass gained or lost is equal to a gain or loss in energy.--------------------------------------Energy Transformedinto MassDm = 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 – E0The kinetic energyis 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 anobject, it either remains at rest,or continues to move withconstant velocity. Inertial reference system: Any place in which Newton’slaw 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 thatthe ball’s motion is exactlyas expected from theacceleration caused bythe pull of the earth. Both observers see theball behaving as Newton’s laws and gravitationalforce 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 earthobserver each are inan inertial system, so each measures the samevalue for the speed oflight, 300,000 km/s.Which person reports the longer time interval between the two events(departure of light fromsource, arrival at thedetector) ?

Time Dilation Explained: Part One Each observer sees lightpropagate at the samespeed: 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 byspacecraft: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)

 Muons are unstable particles produced when protons streaming in from the sun are absorbed in the atmosphere.  Aboutone muon strikes each one cm2 of theearth'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 timeinterval 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 = 0.14Dt = 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 aspacecraft could travel at a speed v = 0.95 c, howlong would an earth observer say a trip to that starwould it take? What would the traveler say? Answer: The traveler is stationary with respect to the eventswhich define the time interval: the departure fromearth, and arrival at the star. Thus, the travelermeasures the proper time, Dt', while the earthobserver measures the dilated time, Dt. The earth observersays 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 earthremained in an inertialreference system.--------------------------------The space- traveler accelerated, and thus wasn't in an inertialreference system.-------------------------------The special theory ofrelativity is valid for theperson on earth, but notfor the space travelerbecause he was ina non-inertial referencesystem.

Length Contraction Earth observer measures LP:The earth observer is at restwith respect to the earth-starline, so he measures theproper length LP.Astronaut measures  Dt':Since the astronaut's clock is at the beginning and end of the event--at his departure fromearth, and at his arrival at thestar-- he measures the propertime Dt'. Each observer measuresthe 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 whenobserved while stationary withrespect to an observer. How long is it to an observermoving 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 movingwith respect to the event, so hemeasures the dilated (longer) time, Dt = 5 x 10-5 s. Same observer measures the proper (longer) distance, LP.--------------------------------------------Observer riding with the muonmeasures 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 aquantity of light energy, itloses mass.---------------------------------------When a flower absorbssunlight, its mass increases.---------------------------------------When a uranium nucleussplits, the mass of theremnants is less than theoriginal mass. The differenceappears as light, heat, andkinetic energy.

Energy to Mass Transformation From the sun:  0.10 watts /cm2 By how much would themass of a butterfly increaseafter one hour facing thesun, assuming all of theenergy is absorbed?--------------------------------------Assume area of butterflyis 75 cm2.Time = 3600 seconds DE = (0.10)(3600)(75)     = 27,000 JoulesDm = DE / c2c = 3 x 108 m/sDm = 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 onenewton 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 beblasted about 1/2 mile upward if onegram of matter were completely convertedto energy?

Total Energy              Rest Energy           Kinetic Energy
 What is the total energy ofa particle of mass m = 2 kgmoving with speed v = 0.9 c?----------------------------------------E = mc2 / (1-v2/c2)1/2     = 4.13 x 1017 J What is the particle’s restenergy?------------------------------------E0 = mc2     = 1.80 x 1017 J What is the particle’s kineticenergy?--------------------------------------Kinetic Energy = E - E0KE = 2.33 x 1017 J