Principle of Relativity

Transcription

1 Principle of Relativity Physical laws are the same in all inertial frames. 1) The same processes occur. But 2) the description of some instance depends on frame of reference.

2 Inertial Frames An inertial frame is a frame in uniform motion relative to the fixed stars. In an accelerated (noninertial) frame such as a merrygo-round, frame dependent (pseudo) forces are felt. Is the floor of Chamberlin Hall an inertial frame?

3 Approximate inertial frame The Earth rotates and the speed at the equator is V = 2 pi r/t = 25,000 miles/24 hours ~ 1000 mph The acceleration is a = v 2 /r <<g = 10 m/s/s For times t<<t, we can often neglect the acceleration. =>the floor of Chamberlin Hall is approximately inertial Note: the Earth orbits the Sun, the Sun goes around the galaxy, our galaxy falls towards Andromeda, clusters of galaxies swirl. We are in motion.

4 Relativity of force free motion Galilean velocity addition rule u = u + v Force free motion is straight line motion in any inertial frame but the position versus time and velocity depend on frame of reference.

5 Relativity in electromagnetism Move a wire loop with speed v towards a magnetic pole. A current flows around the loop due to Lorentz force F=q vxb on moving conduction electrons. Move a magnetic pole with speed v towards a wire loop. A current flows around the loop due to a circulating electric field (E given by Faraday s Law). The same effect has an entirely different description (magnetic vs electric) in the two frames. Relativity is respected - a current flows in either case!

6 Transformations A moving charge carries both E and B fields. A stationary charge carries only an electric field. E and B depend on reference frame! Maxwell s equations and the Lorentz force law describe the physics of E,B fields coupled to matter particles (rho,j). How do we relate E,B,rho,j to their values E,B,rho,j in another inertial frame? Lorentz and Fitzgerald: Consistency requires charge densities contract along their direction of motion and times dilate and E and B are mixed together!!

7 Light speed First measured by Romer: Jupiter moon eclipse time varies by the difference in light travel time (15 minutes) as the Earth moves around the Sun. You can measure c with a fast electronic light sensor and an oscilloscope sensitive to 1 ns/cm

8 Light speed and EM Theory Maxwell s equations predict electromagnetic waves (free E,B fields) which move with speed constructed from fundamental constants If Maxwell s equations are valid in any frame, light speed has the same value c=3e8 m/s in any frame. Crazy but true, the constancy of light speed is inconsistent with Galilean velocity addition, but consistent with relativity - no inertial frame is preferred.

9 Albert A. Michelson Albert A. Michelson ( ). invented the interferometer and spent much of his life making accurate measurements of the speed of light. He was the first American to be awarded the Nobel prize (1907), which he received for his work in optics.

10 Michelson-Morley experiment Observed wave interference for split then recombined beams. The fringe pattern is ensitive to a path length difference a fraction of a wavelength of light or equivalent small differences in light speed along the two orthogonal directions.

11 Interferometer Stretch version View of interference pattern

12 Aether Maybe light is an undulation in frozen absolute space.

13 Michelson-Morley result If light waves move at fixed speed relative to some fixed frame (ether, weakly coupled medium), then the round trip travel time along an arm along Earth s direction of motion will be decreased. The earth s speed is only c but the shift is (in principle) observable. Result: No effect at any time of year!!

14 Analysis Time to travel to M2 and back t(x) = L/(c-v) +L/(c+v) = 2L/(c 2 -v 2 ) Time to travel to M1 and back t(y) = 2L/[c 2 -v 2 ] 1/2 Time difference over light speed for one orientation versus rotated by 90 degrees D = 2L(v/c) 2 = 2e-7 m For L=11 m, wavelength of 5e7 m and v/c=1e-4, the fringe shift is: Shift = D/wavelength=0.4

15 Albert Einstein Relativity:The Special and General Theory By A. Einstein Under $10 at ***.com

16 Postulates 1) The laws of physics have the same form in all inertial frames. 2) The speed of light has the same value in all inertial frames. From these postulates and thought experiments, Einstein deduced the Lorentz transformation rules and many other consequences.

17 Events and measurements You see me as I was L/c ~30 ft/(1 ft/ns)=30 ns ago. The Sun is 8 light minutes away. If it just exploded, we won t know for 8 minutes. We must consider carefully how we make observations when light speed, the maximum speed of any influence, is important. An event is a process in a sufficiently small volume and time interval such that such time delays may be neglected.

18 Relativity of simultaneity Two lightning bolts strike the ends of a moving box car. (a) The events appear to be simultaneous to the stationary observer at O, who is midway between A and B. Light arrives simultaneously at O. (b) Observer O has speed v so travels towards the light from B, away from A and observes light from the front of the train before light from the rear.

19 Both are correct Observers in relative motion disagree as to whether events are simultaneous. Like space coordinates, time depends on frame of reference.

20 A moving light clock O has a light clock that ticks every dt = 2d/c. O sees the light clock tick every dt = 2L/c where L is the hypotenuse shown so dt = 2L/c=(2/c)[d 2 +(v dt/2) 2 ] 1/2 =[(2d/c) 2 +(v dt/c) 2 ] 1/2 = dt 2 = dt 2 + (v dt/c) 2 dt = dt /[1- (v/c) 2 ] 1/2

21 Time dilation factor Gamma is larger than one so dt >dt: the moving clock runs slow. All moving clocks (physical processes) must runs slow relative to stationary clocks in any given reference frame.

22 Time dilation example A jet flies around all day at jet speed (500 mph) carrying a precise atomic clock that was initially synchronized with an identical clock on Earth. At the end of the day, how do the two clocks compare? c = approx 186,290 miles per SECOND, x 3600 seconds/hour = 670,644,000 miles/hour =>v/c~1e-6 be careful with your calculator!

23 Length contraction In a given time, an object moves a distance L p. In the object s frame, the time interval is longer by a factor gamma so the distance (all lengths) must appear shorter to the moving observer by the same factor.

24 A contraction contradiction? A 10 meter long bus drives at high speed with gamma = 2 into a garage of depth 5 meters and the door immediately closed. Possible? Yes! But to a bus driver, the garage is 2.5 m deep! How can be reconcile the length contractions?

25 A contraction contradiction? A 10 meter long bus drives at high speed with gamma = 2 into a garage of depth 5 meters and the door immediately closed. Possible? Yes! But to a bus driver, the garage is 2.5 m deep! Resolution: the events 1) front of bus hits back wall and 2) door closed on rear of bus are not simultaneous to the bus driver. He is crushed long before the the rear of the bus enters the garage.

26 Time dilation in muon decay Muons traveling at 0.99c travel only about 650 m as measured in the muons reference frame, where their lifetime is about 2.2 s. (b) The muons travel about 4700 m as measured by an observer on Earth. Because of time dilation, the muons lifetime is longer as measured by the Earth observer.

27 Space Travel The nearest star is L=4 light years away. How fast must your space ship fly to get there in what to you seems 1 year? To you the star moves a distance L/gamma at speed v so the time is L/(v gamma) = (1/4)L/c. Since v is close to c so we need gamma=4. We will see to achieve this v requires conversion of gamma times the spaceship rest mass entirely to energy.

28 Lorentz transformations Lorentz transformations relate coordinates x,t in on frame to coordinates x,t in aframe with relative velocity v along the x direction.

29 Reverse transformations The reverse Lorentz transformations follow by reversing the relative velocity.

30 Example Consider two events which, according to a moving observer, are separated by distance L'= x' 2 -x' 1 and a time interval T'= t' 2 -t' 1. Relative to the stationary observer, space and time coordinates are

31 Example continued If the two events are simultaneous in the primed frame, the time interval in the primed frame is T =t 2 -t 1 =0 and the distance is the proper length L = x 2 -x 1. What are distance and time intervals in the unprimed frame?

32 Example continued Now suppose the two events are at the same position (x 2 -x 1 =0) but not simultaneous in the primed frame. The time interval in the primed frame is T =t 2 -t 1. What are distance and time intervals in the unprimed frame? The object is stationary in the primed framed but in the unprimed frame moves with velocity L/T = v.

33 A longer space trip How fast must you travel relative to the Earth (x=0) to reach x=l=3e20 m (across the galaxy) in T=10 years=3e8 s of your life? Consider a moving frame with you sitting always at its origin. Starting with x = t = x'=t'=0, the space-time coordinates relative to the moving frame when you arrive are x'=0 and t'=t. The coordinates in the Earth frame are x=l, t=unknown. We want the value of v such that the Lorentz transformation relations x = g(x'+vt') and t = g(t'+v x'/c 2 ) are valid.

34 A space flight continued How fast must you travel relative to the Earth (x=0) to reach x=l=3e20 m (across the galaxy) in T=10 years=3e8 s of your life? Solve the first equation for v Here ct/l = 3e8 (m/s)* 3e8 s/3e20 m=9e-4 so v deviates from c by a tiny amount. The gamma factor is Meanwhile on earth evolves a time t=gamma T = 30,000 yr

35 Relative velocity From the Lorentz transformations, we derive the relativistic velocity addition relation. Note: If v/c and u/c <<1, we recover the Galilean result u =u-v. Change the sign of v to find the inverse rule u = (u +v)/(1+u v/c 2 )

36 Example velocity addition u = (u +v)/(1+u v/c 2 ) Example: A train moves with speed v along the x direction. A passenger sends light at speed u =c in the x direction. With what speed u is the light observed by someone at the station? For u =c we have u = (c+v)/(1+cv/c 2 ) = c The light still moves at speed c relative to the ground.

37 Another example u = (u +v)/(1+u v/c 2 ) Example: A train moves with speed v=>c along the x direction. A passenger runs along x at speed u relative to the train. With what speed u is the passenger observed to move by someone at the station? For v=>c we have u = (u +c)/(1+u /c) = c The passenger still moves at speed c relative to the ground.

38 u = (u +v)/(1+u v/c 2 ) Final example Example: A train moves with speed v=c/2 along the x direction. A passenger runs along x at speed u =c/2 relative to the train. With what speed u is the passenger observed to move by someone at the station? For v=c/2 and u =c/2 we have u = (c/2+c/2)/(1+(c/2)(c/2)/c 2 ) = c/(1+1/4) = (4/5)c The passenger moves at speed u<c relative to the ground.

39 Transformation of E,B fields The transformation of electric and magnetic fields between frames with relative velocity v is: Reverse transformation has v=> - v. Here B stands for cb(mksc) to convert to similar units.

40 Nonrelativistic limit Low velocity (nonrelativistic): limit In particular from the last equation, a single stationary charge in its rest frame has E=E (Coulomb), B=0 and E =E, B = (1/c) v xe

41 Biot-Savart law E field of a stationary line of charge B field of a line current Biot-Savart Law

42 Connection with relativity The B field of a current is the sum of the B fields of the moving charges. Note: Biot Savart is actually more general than the nonrelativistic connection suggests.

43 Relativistic Doppler effect A source of period T=1/f gives wavelength ct with c the velocity of the wave. The wavelength of a source moving towards/away is to (c -/+ v)t. For light c is not sound speed but light speed and T = T/gamma.

44 Doppler shift of light For source approaching with speed v For source receding, v=> -v Light from an atom/galaxy moving away from you is shifted to longer wavelengths (red shifted). An atom/ galaxy moving towards you appears blue shifted.

45 Simulated relativistic flight throughcube_big.html