Chapter 1 Relativity Special Relativity all motion is

Chapter 1: Relativity Special Relativity: “all motion is relative” All motion is described relative to an inertial frame of reference Inertial frame: frame of reference in which Newton’s law of inertia holds Inertial frames are (mathematically) simpler in Special Relativity, and are related in a (mathematically simple) manner. Accelerations and forces are accounted for in S. R. General Relativity = Special Relativity + Gravity (“curved space time” = more complex relations between inertial reference frames) Postulates of Special Relativity The laws of physics are the same in all inertial frames of reference. (mathmatical form, conservation laws, etc. ) The speed of light in free space has the same value for all inertial frames of reference. v ~ 3. 00 E 8 m/s cp 351 c 1: 1

The First Postulate and “everyday” relativity: The laws of physics are the same in all inertial frames of reference. y y’ x’ = x - vt y’ = y x x’ v z z’ z’ = z t’ = t Conservation of momentum, kinetic energy, etc But waves like sound waves have “special” inertial frame, the frame at rest w. r. t. medium cp 351 c 1: 2

Second Postulate: The speed of light in free space has the same value for all inertial frames of reference need a new relationship between coordinates of different inertial frames linear, invertible “transformation” = Lorentz Transformations cp 351 c 1: 3

Time Dialation: the time interval between events depends upon the inertial observer proper time t 0: the time interval between two events in a frame of reference where the events occur at the same place. timing with a moving mirror c t 0/2 c t/2 L 0 v v t/2 v cp 351 c 1: 4

Time Dialation and the passage of time t 0 = time interval for clock at rest relative to an observer = proper time t = time interval for clock in motion relative to an observer, t > t 0 v = speed of relative motion c = speed of light Example 1. 1: A spacecraft is moving relative to the earth. An observer finds that, according to her clock, 3601 s elapse between 1 PM and 2 PM on the spacecrafts’s clock. What is the spacecraft’s speed relative to the earth? cp 351 c 1: 5

Doppler Effect: frequency shifts due to moving source/observer doppler effect for sound n 0 = emitted frequency n = detected frequency v = speed of observer V = speed of source c = speed of sound But, light does not require a medium! cp 351 c 1: 6

Doppler effect with light: three cases time between ticks is dilated source 1 - observer moving perpendicular to a line between him and light source cp 351 c 1: 7

Doppler effect with light: three cases 2 - observer moving away from source next tick travels farther source time between ticks is dilated cp 351 c 1: 8

Doppler effect with light: three cases 2 - observer moving away towards source next tick travels less distance source time between ticks is dilated cp 351 c 1: 9

Example 1. 2: A driver is caught going through a red light. The driver claims that the color she actually saw was green (n = 5. 60 x 1014 Hz) and not red (n 0= 4. 80 x 1014 Hz) because of the doppler effect. How fast must she have been going? Example 1. 3: A distant galaxy Hydra is receding grom the earth at 6. 12 x 107 m/s. By how much is a green spectral line of wavelength l = 500 nm emitted by this galaxy shifted towards the red end of the spectrum? cp 351 c 1: 10

Length Contraction: determining the extent of a moving object v L 0 =vt v L =vt 0 Example: Muons created in earth’s atmosphere by cosmic rays have speeds of about 2. 994 x 108 m/s(. 998 c) At rest, a muon lifetime is about 2. 2 ms. -How far would these muons travel in 2. 2 ms? -What is the time dialated lifetime? How far do they travel in THIS amount of time? -What is the length contracted distance of the previous answer, as seen in the muons frame of reference? -How far does the earth move relative to the muons during the 2. 2 mslifetime? cp 351 c 1: 11

The Twin “Paradox”: a counter-intuitive result Two twins on earth are separated temporarily when one twin takes a trip at. 80 c to a star twenty light years away. Earth bound twin: round trip takes 2 x(20 ly /. 8 ly/y) = 50 years Traveling twin: Earth-Star distance is length contracted to so round trip takes 2 x(12 ly /. 8 ly/y) = 30 years Twins age differently! Travelling twin does not stay in one inertial reference frame, situation is not symetric. Example 1. 4: Each twin emits a radio signal once a year for the duration of the voyage. How many signals does each twin recieve? cp 351 c 1: 12

Electricity and Magnetism Integral Form Maxwell’s Equations Differential Form Lorentz Force Law Mathematically invariant formulation electric charge is a relativistic invariant quantity Maxwell’s equations -> Wave Equation cp 351 c 1: 13

Relativistic Inertia (“relativistic mass”) consider elastic collision between two identical (when at rest) masses y V’B y’ Y VA S x S’ z z’ x’ Ball A moves vertically only in frame S with speed VA , Ball B moves vertically only in frame S’ with speed VB ’= VA. Ball A rebounds in S with speed VA , Ball B rebounds in S’ with speed VB’. v Y/2 Collision in S’ cp 351 c 1: 14

Example 1. 5: Find the mass of an electron (m 0 = 9. 1 E-31 kg)whose speed is. 99 c. cp 351 c 1: 15

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Example 1. 6: �A stationary body explodes into two fragments each of mass 1 move apart at speeds 0. 6 c relative to the original body. Find the rest mass of the original body. 2. Given Example 1. 7: �Solar energy reaches the earth at a rate of about that 1. 4 k. W/m the average radius of the earth’s orbit is 1. 5 E 11 m, how much mass does the sun lose per second? cp 351 c 1: 19

General momentum-energy relations cp 351 c 1: 20

Electronvolts: typical energy scale for atomic physics 1 e. V = 1. 6 E-19 Coulomb x 1 Volt = 1. 6 E-19 J X-Ray energies ~ ke. V Nuclear Physics ~ Me. V Particle Physics ~ Ge. V Example : �What are momenta the and speeds of a 5 Me. V -electron? (m 0 =. 511 Me. V/c 2, -proton? (m 0 = 938 Me. V/c 2, -photon? (m 0 = 0 Me. V/c 2). cp 351 c 1: 21

Chapter 1 problems: 1, 2, 4, 5, 7, 9, 10, 11, 12, 16, 19, 20, 22, 33, 34, 35, 36, 41, 47, 49 cp 351 c 1: 22