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Time Dilation A clock moves from event A to event B at constant velocity. How much time  passes on the clock? If we were moving with the clock, we would say the clock doesn’t move, and the time was  Note that the ordinary 3D distance is given by Moving clocks run slow: TIME DILATION


Sample problem A neutron at rest lasts an average of 886 s. How fast must it move to make it here from the Sun, a distance of d = 1.501011 m?


Lorentz Contraction A ruler of length Lp moves past us at velocity v. How long does it look to us? x ct x’ ct’ The length Lp is the length measured by an observer moving with the ruler The two ends of the ruler are at x’ = 0 and x’ = Lp Solve for xL and xR. The difference is the length Moving objects look short Contraction only in direction of motion


Galileo almost got it right For ordinary objects, the velocity v is much smaller than the speed of light c Time dilation: Lorentz contraction:


Binomial Expansion It is often helpful to have approximate formulas when is small Often, your calculator will fail where approximations will succeed


Sample Problem Professor Carlson is sprinting at 10 m/s. He is 1.95 m tall, and his nose is 3 cm long. How does his height and nose length change? Contraction only in direction of motion No change in height .0167 fm shorter


Reverse Lorentz Boosts Solve these for (x,y,z,t) in terms of (x’,y’,z’,t’) Multiply by v add add yields substitute Similar computation for t Same formula except: Primed  Unprimed Velocity v  - v If I observe someone moving with velocity v, she observes me moving with velocity –v


Subtraction of velocity Suppose an object is moving with velocity: What does a moving observer see?


Relativistic Doppler Shift Imagine source moving at velocity v at angle  Sending out pulses at time interval  in its own frame What is time interval between signals received by us?  v To observer Frequency of pulses is reciprocal of time gap between pulses


Relativistic Doppler Shift


Paradox Problems Because relativity is so counter-intuitive, it often leads to apparent contradictions These apparent contradictions lead to apparent paradoxes These paradoxes can always be resolved by careful analysis Most common resolution – simultaneity is implicitly assumed Sometimes their resolution leads to important insights


The Twin Pair’o’Docs A pair of identical twin PhD’s both start at Earth. David stays on Earth, while Daniel travels to the star Arcturus ( d = 37 light-yr) at 0.9c. When he gets there, he returns to Earth at the same speed. How much has each of them aged when he returns? One way: Round Trip: David = 82.2 years Daniel = 35.8 years


Twin Pair’o’Docs = Paradox? Why is this an apparent paradox? According to David, Daniel is moving According to Daniel, David is moving How can we say who is really moving? The two cases are not interchangeable Daniel changed speeds, David did not If you want to compute using Daniel’s frame of reference, you must also include a Lorentz transformation at the turnaround point x ct David Arcturus Daniel


Pole Vaulter Paradox A pole vaulter runs towards a barn with a pole of length L. The barn is also length L. Because the man is running, his pole is shortened It looks like the pole will fit in the barn From the viewpoint of the man, the barn is shortened, and it will hit the back before he gets inside Farmer says, “I’ll shut the door when you are inside” Vaulter says, “I’ll stop when the pole hits the back”


Pole Vaulter Resolution What are these coordinates in the unprimed frame? What does the farmer see? At t = -Lv/c2, the runner stops Far end of pole continues on until t = 0 Pole stretches and hits back wall Consider the moment when the pole hits the back of the barn from the vaulter’s perspective


Paradox resolved? What if vaulter uses a stiff pole? What does stiff mean? When you stop one end, other end stops instantaneously Far end can’t “know” that this end stopped instantan-eously, because influences can’t travel faster than light Real poles: information travels along pole at speed of sound Speed of sound must be slower than speed of light In relativity, perfectly rigid objects cannot exist