Lect24 - 11/29/11 Relativity Chapter 27 (1 of 2) • ...

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Unformatted text preview: 11/29/11 Relativity Chapter 27 (1 of 2) •  •  •  •  •  •  •  •  Galilean, classical Relativity Time dilatation Simultaneity Length contraction Addition of velocities Momentum Mass and energy General relativity 11/29/11 Physics 104 Lecture 23 Nov. 28, 2011 Physics 104, Fall 2011 1 Galilean Relativity •  Choose a frame of reference –  Necessary to describe a physical event •  According to Galilean Relativity, the laws of mechanics are the same in all inertial frames of reference –  An inertial frame of reference is one in which Newton’s Laws are valid –  Inertial references •  in a state of constant, rectilinear motion with respect to each other •  they are not accelerated (carousel cabin, accelerating elevator) –  Objects subjected to no forces will move in straight lines November 11 Physics 104, Fall 2011 2 1 11/29/11 Relativity •  The term relativity arises when a situation is described from two different points of view •  When the railroad car moves with a constant velocity, Ted and Alice see different motions of the ball Relativity •  Ted observes the ball’s motion purely along the vertical direction •  Ted would think the ball’s horizontal velocity is zero, but Alice would disagree •  Alice sees the ball undergo projectile motion with a nonzero displacement in both the x- and y-directions 2 11/29/11 Galilean Relativity – Example •  The two observers disagree on the shape of the ball’s path •  Both agree that the motion obeys the law of gravity and Newton’s laws of motion •  Both agree on how long the ball was in the air •  Conclusion: There is no preferred frame of reference for describing the laws of mechanics November 11 Physics 104, Fall 2011 5 Addition of Velocities Non-relativistic •  Velocities can simply be added. •  If in more than one dimension, use vector a ddition. November 11 Physics 104, Fall 2011 6 3 11/29/11 Relative Velocity (Ball) •  Your friend throws a ball at 20 mph towards you. How fast do you think it moves when you are: –  Standing still –  Running 10 mph towards –  Running 10 mph away November 11 20 mph 30 mph 10 mph Physics 104, Fall 2011 7 Addition of velocities Your friend fires a laser at you while you're standing still. You measure the photons to be coming towards you at the speed of light (c = 3.0 x 108 m/s). You start running away from your friend at half the speed of light (c/2 = 1.5 x 108 m/s). Now how fast do you measure the photons to be moving? 1) 0.5 c 2) c 3) 1.5 c Galilean addition of velocities is NOT valid. Speed of light is the same in all inertial reference frames. November 11 Physics 104, Fall 2011 8 4 11/29/11 Galilean Relativity and Light, cont. •  Galilean Rela)vity: Photons go faster (slower) than c when emi:ed from a moving light source. •  Maxwell’s equa)on on electromagne)sm require the the speed of light to be the same in any reference frame. •  Experiments showed that Maxwell’s theory was correct •  The speed of light in a vacuum is always c Einstein’s Postulates •  First Postulate Special Rela)vity, 1905 –  Laws of physics are the same in any inertial frame of reference (principle of relativity) •  No preferred frame of reference, no frame is more “correct”- cannot tell if you are moving at constant velocity •  Second Postulate –  Speed of light in vacuum is the same in all inertial frames •  Independent of the motion of the source •  at rest I measure speed of light c (~3x108 ms-1) •  If it is the headlight of a train moving at u Newton: speed =c+u c u Einstein: speed still = c !!! Newton’s laws are the low velocity limit of Einstein’s special relativity November 11 Physics 104, Fall 2011 10 5 11/29/11 Space travel You discover that you have a long-lost twin who's been on a high-speed spaceship for the last 10 years. When your twin returns to Earth, he or she will be 1)  Younger than you 2)  Older than you 3)  The same age as you Time differences are not the same in two inertial reference frames moving at differing (large) speeds with respect to each other. November 11 Physics 104, Fall 2011 11 Light Clock •  The two postulates lead to a surprising result concerning the nature of time •  A light clock keeps time by using a pulse of light that travels back and forth between two mirrors •  The time for the clock to “tick” once is the time needed for one round trip: 2ℓ / c 6 11/29/11 Moving Light Clock The clock moves with a constant velocity v relative to the ground Ted’s reference frame: •  The light pulse travels up and down between the two mirrors •  Total distance: 2l •  •  •  Alice sees the light pulse travel a longer distance (marked in red According to Alice the light will take longer to travel between the mirrors The relation between observed times is Δt = 2 Δto = c Δto 1−v 2 c2 see simple deriva)on in textbook 27.3 Moving Light Clock The clock moves with a constant velocity v relative to the ground Ted’s reference frame: •  The light pulse travels up and down between the two mirrors •  Total distance: 2l 2 Δto = c •  •  •  Alice sees the light pulse travel a longer distance (marked in red According to Alice the light will take longer to travel between the mirrors The relation between observed times is Δt = Δto 1−v 2 c2 see simple deriva)on in textbook 27.3 7 11/29/11 Moving clocks run slow Time dilatation with respect to “proper time” •  Special relativity predicts that moving clocks run slow •  This effect is called time dilation •  The time interval Δto is measured by the observer at rest relative to the clock: proper time •  The time interval measured by a moving observer is always longer than the proper time Δt = Δto 1−v 2 c2 Experimental evidence Speed of light Historically relevant (1887) •  The Earth travels around sun (as you know) •  u = 30,000 m/s speed of Earth in orbit •  Measure the speed of light in spring and fall – see a difference? •  Michelson-Morley interferometer Result: •  No difference of speed of light for different directions observed. •  (Experiment in 2007 with 100million times better sensitivity: same result) 11/29/11 Physics 104, Fall 2011 16 8 11/29/11 Michelson-Morley Experiment •  •  The Michelson interferometer is based on the interference of reflected waves Historically it has been designed to measure the speed of light in 2 directions –  direction of motion of Earth –  perpendicular to that •  Should get a different result at different times of the year Link to a nice anima)on Muons from Cosmic Rays Fast muons (v=0.99c) travel only 600m before they decay, in their reference frame, in which their average lifetime is about 2.2 µs. But, they reach all the way to the surface of the Earth (4800m). The reason for this is from the point of view of an Earth bound observer, there is time dilation and muon’s can travel much longer: Δt = Δto 1−v November 11 = 2 c 2 2.2 µs 1 − 0.992 = 7 × 2.2 µs = 15.4 µs Physics 104, Fall 2011 18 9 11/29/11 Twin Paradox •  An astronaut, Ted, visits a nearby star, Sirius, and returns to Earth –  Sirius is 8.6 light-years from Earth –  Ted is traveling at 0.90 c •  Alice, Ted’s twin, stays on Earth and monitors Ted’s trip Twin Paradox, Times •  Alice measures the trip as taking 19 years 17.2 ly = 19 years 0.90 c •  Ted’s body measures the proper time of 8.3 years Δt = ( Δto = Δt 1 − v 2 c 2 = 19 years ) ( 1 − 0.90c ) 2 / c 2 = 8.3 years •  Alice knows special relativity and concludes correctly that Ted will be almost 10 years younger than she. Ted: •  Ted calculates the Earth (and Alice) move away from him at 0.90 c •  Ted concludes Alice will age 8.3 years while he ages 19 years •  Ted concludes that Alice will be younger than he is 10 11/29/11 Twin Paradox, Resolution •  Time dilation appears to lead to contradictory results •  Alice’s analysis is correct –  She remains in an inertial frame and so can apply the results of special relativity •  Ted is incorrect –  He accelerates when he leaves and when he turns around at Sirius –  Special relativity cannot be applied during this time spent in an accelerating frame GPS - must take into account special relativity •  The global positioning system (GPS) allows to determine time and location using a system of satellites. •  The satellites travel at a speed of 4km/s in orbit. •  Only with the use of special relativity, the correction for time dilatation can teh GPS work accurately. •  Time dilatation slows down the clocks on the satellites by 7 µsec/day. 11 11/29/11 Simultaneity   Two events are simultaneous if they occur at the same time   Ted is standing the middle of his railroad car   He moves at a speed v relative to Alice   Two lightning bolts strike the ends of the car and leave burn marks on the ground which indicate the location of the two ends of the car where the bolts strike Simultaneity According to Alice •  She is midway between the burn marks •  The light pulses reach her at the same time •  She sees the bolts as simultaneous 12 11/29/11 Simultaneity According to Alice •  She is midway between the burn marks •  The light pulses reach her at the same time •  She sees the bolts as simultaneous According to Ted •  •  The light pulse from at B struck before the bolt at A –  Since he is moving toward B The two bolts are not simultaneous in Ted’s view Simultaneity is relative and can be different in different reference frames This is different from Newton’s theory, in which time is an absolute, objective quantity: It is the same for all observers. All observers agree on the order of the events Length Contraction •  Alice marks two points on the ground and measures length Lo between them •  Ted travels in the x-direction at constant velocity v and reads his clock as he passes point A and point B –  This is the proper time interval of the motion •  •  •  •  •  Δt 2 L Distance measured by Alice = Lo = v Δt = o = 1−v 2 c L0 Δt Distance measured by Ted = L = v Δto Δt Since Δt ≠ Δto, L ≠ Lo Δt = 1−v c Length contraction is a result of time dilation The length measured by Ted is smaller than Alice’s length L = L 1 − v2 / c2 o 2 2 o 13 11/29/11 Proper Length •  Ted is at rest •  Alice moves on the meterstick with speed v relative to Ted •  Ted measures a length shorter than Alice •  Moving metersticks are shortened •  The proper length, Lo, is the length measured by the observer at rest relative to the meterstick L v2 = 1− 2 Lo c Length Contraction Example v=0.1 c v=0.8 c v=0.95 c November 11 Physics 104, Fall 2009 28 14 ...
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This note was uploaded on 11/30/2011 for the course PHYSICS 104 taught by Professor Dasu/karle during the Fall '11 term at Wisconsin.

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