Einstein Einstein s Postulates Postulate 1 The laws of physics are the same in each reference frame independent of the motion of this reference frame Physics for Scientists Engineers 2 Postulate 2 Spring Semester 2005 The speed of light c is the same in every reference frame Lecture 48 April 21 2005 Physics for Scientists Engineers 2 1 April 21 2005 Beta and Gamma Conventionally we think of time as flowing at a uniform rate in one direction past to the future Introduce a variable beta as the ratio of the speed to the speed of light But if you accept that the speed of light is the same in any reference frame then our old understanding of time runs into conceptual problems v c Another useful variable that depends on the speed and the speed of light April 21 2005 1 1 2 The time interval it takes for an event to happen as observed in a moving rest frame is dilated made bigger as compared to the time interval for the event to occur in the frame where it is at rest 1 1 v c 2 Physics for Scientists Engineers 2 2 Time Dilation It is convenient to express speeds as a fraction of the speed of light Physics for Scientists Engineers 2 3 April 21 2005 Physics for Scientists Engineers 2 4 1 Time Dilation 2 Let s start with two parallel mirrors distance h and send a light beam from one to the other This takes a time Time Dilation 3 Now the mirrors are moving a distance x v t in horizontal direction h h t 0 c Now observe the same event while moving with speed v Light beam has covered a distance Pythagoras L h2 x 2 v h x x v Physics for Scientists Engineers 2 5 April 21 2005 Time Dilation 4 L h x c t c t 0 v t 2 2 Physics for Scientists Engineers 2 6 Once more in words The rate at which time flows is dependent on how fast the observer moves 2 Solve for the time interval in the moving frame t 0 t t 0 1 v c 2 v Time Dilation 5 Put this together 2 v Light moves with c L c t h April 21 2005 h h Note this is not just the subjective perception of time but the objective measurable length of time v t t 0 1 v c 2 t 0 h x April 21 2005 Physics for Scientists Engineers 2 Simply the most astonishing formula derived in the entire course up to now v 7 April 21 2005 Physics for Scientists Engineers 2 8 2 Example Time Dilation Length Contraction Muon Decay If you want to measure the length of a spaceship at rest you can use a laser and measure the time it takes to move the length of the spaceship Muons have a lifetime of 2 2 micro seconds L0 c t 0 CERN experiment muons produced with 0 9994 1 1 28 87 2 1 1 0 9994 2 Now we redo the same experiment with the shuttle and the clock to measure the time inside it are moving by at speed v Lifetime of these moving muons should be 28 87 times longer 63 5 micro secs than those at rest This time the measurement then yields During this time the muons can move a distance L x v v 0 0 9994 c 28 87 2 200 s 19 km Without time dilation this distance would only be 660 m Length contraction by a factor Direct observation of time dilation April 21 2005 c t 0 L0 Note length only changes in the direction of motion not perpendicular Physics for Scientists Engineers 2 9 April 21 2005 Twin Paradox Physics for Scientists Engineers 2 Twin Paradox Solution Alice is not really in an inertial frame because she has to accelerate and slow down Twin sisters Alice astronaut and Betty stays home both age 20 Alice takes a space trip with 10 Alice flies to a nearby star and back True but also not relevant Note the direction of the velocity vector does not matter We can restrict the speeding up and slowing down phases to a very small fraction of the trip so they will not change the answer Alice flies for 5 years and is 25 when she returns home to Earth But in the meantime Betty has aged 5 50 years and is now 70 when her sister returns But wait a minute We can also put ourselves into Alice s rest frame and in this frame Betty and the entire Earth has moved with 10 and so Alice would expect to be 25 on her return but her sister should only be 20 5 years old Big difference Physics for Scientists Engineers 2 Consider that the two endpoints of the trip are given in the coordinate frame of Earth and that they are fixed Earth on one end and a star 25 light years away at the other end In Alice s traveling frame that distance appears length contracted by a factor of 10 which is why she is able to make the roundtrip in 5 years Consequently Betty is 70 and Alice 25 years old when they meet again in particular for Betty Which answer is correct April 21 2005 10 11 April 21 2005 Physics for Scientists Engineers 2 12 3 Frequency Shift Wavelength Shift Time dilation modifies all other observables that are related to time Since c f a shift in the frequency is also a shift in wavelength Example frequency Observed wavelength Relativistic frequency shift acts similarly to Doppler Effect but has origin in time dilation not in movement of the wave medium relative to observer or source f f0 0 again upper signs for moving away from each other lower for towards each other c v c v Red shifted object moves away from us Upper signs for observer and emitted moving away from each other lower for moving toward each other April 21 2005 Physics for Scientists Engineers 2 c v c v Blue shift object moves towards us 13 April 21 2005 Red Shift Physics for Scientists Engineers 2 Lorentz Transformation Note Red shift parameter z is defined as the ratio of the wave length shift of light divided by the wave length of the same light when the source is at rest Lorentz transformation contains both time dilation and length contraction Red shift z 0 0 0 c v 1 c v Observation Practically all galaxies in the universe are red shifted and thus move away from us Physics for Scientists Engineers 2 15 Inverse Galilei x x vt y y z z Lorentz x x vt y y z z x x vt y y z z t t t t vx c 2 t t vx c 2 Small v Galilei special case of Lorentz The further away they are the more red shifted April 21 2005 14 April 21 2005 Physics for Scientists Engineers 2 16 4 Relativistic Velocity Addition Relativistic Momentum Classical Galilean v1 …
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