Chapter 37 Relativity is an important subject that

Chapter 37 Relativity is an important subject that looks at the measurement of where and when events take place, and how these events are measured in reference frames that are moving relative to one another. In this Chapter we will explore with the special theory of relativity (which we will refer to simply as "relativity"), which only deals with inertial reference frames (where Newton's laws are valid). The general theory of relativity looks at the more challenging situation where reference frames undergo gravitational acceleration. In 1905, Albert Einstein stunned the scientific world by introducing two "simple" postulates with which he showed that the old, common-sense ideas about relativity are wrong. Although Einstein's ideas seem strange and counter-intuitive, e. g. , rate at which time passes depends on the speed of reference frame, these ideas have not only been validated by experiment, they are being used in modern technology, e. g. , global positioning 37 - 1 satellites.

Some references on relativity • The original papers of Einstein • N. David Mermin – American Journal of Physics 65, 476 -486 (1997) and 66, 1077 -1080 (1998) – http: //people. ccmr. cornell. edu/~mermin/homepage/mi nkowski. pdf – It's About Time: Understanding Einstein's Relativity N. David Mermin http: //press. princeton. edu/titles/8112. html • Others 2

About that speed • conventional foot (ft) = 0. 3048 m. 1 foot (f) = 0. 299792458 m. 1 f/ns = 299, 792, 458 m/s = c, speed of light. (ns = nanosecond = 10 -9 sec) 3

The Postulates 1. The Relativity Postulate: The laws of physics are the same for observers in all inertial reference frames. No frame is preferred over any other. 2. The Speed of Light Postulate: The speed of light in vacuum has the same value c in all directions and in all inertial reference frames. Both postulates tested exhaustively, no exceptions found! 37 - 4

The Ultimate Speed Experiment by Bertozzi in 1964 accelerated electrons and measured their speed and kinetic energy independently. Kinetic energy →∞ as speed → c Fig. 37 -2 Ultimate Speed→Speed of Light: 37 - 5

Testing the Speed of Light Postulate If speed of light is same for all inertial reference frames, then speed of light emitted by a source (pion, p 0) moving relative to a given frame (for example, a laboratory) should be the same as the speed light that is emitted by a source that is at rest in the laboratory). 1964 experiment at CERN (European particle physics lab): Pions moving at 0. 9975 c with respect to the laboratory decay, emitting two photons (g). The speed of the light waves (g-rays) emitted by the pions was measured always to be c in the lab frame (not up to 2 c!)→same as if pions were at rest in the lab frame! 37 - 6

Measuring an Event: something that happens, can be assigned three space coordinates and one time coordinate Where something happens is straightforward, when something happens is trickier (for example the sound of an explosion will reach a closer observer sooner than a farther observer. Space-Time Coordinates All clocks read exactly the same time if 1. Space Coordinates: three you were able to look at them all at once! dimensional array of measuring rods 2. Time coordinate: Synchronized clocks at each measuring rod intersection How do we synchronize the clocks? Fig. 37 -3 Event A: x=3. 6 rod lenghts, y=1. 3 rod lengths, z=0, time=reading on nearest clock 37 - 7

The Relativity of Simultaneity Sam observes two independent events (event Red and event Blue) occurring at the same time, Sally, who is running at a constant speed with respect to Sam also observes these two events. Does Sally also find that the events occurred at the same time? If two observers are in relative motion, they will not, in general, agree as whether the two events are simultaneous. If one observer finds them to be simultaneous, the other generally will not. Simultaneity is not an absolute concept but a rather relative one, depending on the motion of the observer. WARNING: When we speak of observers like Sam and Sally, we are referring to the entire space-time coordinate system (frame of reference) in which each is at rest. The observer's location within their frame of reference does not affect 37 - 8 the relativistic physics that we discuss here.

A Closer Look at Simultaneity Fig. 37 -4 • Events Blue and Red same distance from Sam and Sally, • Sam at rest→the light from two events reaches him at same time →he concludes that the two events occurred at the same time (in his frame). • Sally is moving to right→sees the light from Red event before the light from Blue event. Distance from Sally to B' and R' same and light travels at c from both events towards Sally →Event Red must have occurred at an earlier time (in her frame)! 37 - 9 • What would a third stationary observer, Bill, standing to the right of Sam observe?

The Relativity of Time The time interval between two events depends on how far apart they occur in both space and time; that is, their spatial and temporal separations are entangled. Fig. 37 -5 37 -10

The Relativity of Time, cont'd When two events occur at the same location in an inertial reference frame, the time interval between them, measured in that frame, is called the proper time interval or the proper time. Measurements of the same time interval from any other inertial reference frame are always greater. In previous example, who measures the proper time? Speed Parameter: Lorentz factor: 37 -11

The Relativity of Time, cont'd Lorentz factor g as a function of the speed parameter b Fig. 37 -6 37 -12

Two Tests of Time Dilation 1. Microscopic Clocks. Subatomic particles called muons are unstable and decay (transform into other particles). The average time from when a muon is produced to when it decays (Dt) depends on how fast the muon is moving. Muon stationary in lab (production and decay in same place, at muon itself) Dt 0=2. 200 ms If muon is moving at speed 0. 9994 c with respect to the lab (production and decay in different places in the lab frame) the lifetime measured by laboratory clocks will be dilated 37 -13

Two Tests of Time Dilation, cont'd 1. Macroscopic Clocks. Super precision atomic clocks (large systems) flown in airplanes b~7 x 10 -7 (Hafele and Keating in 1977 within 10%, and U. Maryland a few years later within 1% of predictions) repeated the muon lifetime experiment on a macroscopic scale If the clock on the U. Maryland flight registered 15. 0000000 hours as the flight duration, how much would a clock that stayed on earth (lab frame) have measured for the duration? More or less? Does it matter whether airplane returns to same place? 37 -14

The Relativity of Length The length L 0 of an object in the rest frame of the object is its proper length or rest length. Measurement of the length from any other reference frame that is in motion parallel to the length are always less than the proper length. 37 -15

Does a moving object really shrink? Fig. 37 -7 Must measure front and back of moving penguin simultaneously to get its length in your frame. Let's do this by having two lights flash simultaneously in the rest frame when the front and back of the penguin align with them. In penguin's frame, your measurements did not occur simultaneously. You first measured the front end (light from front flash reaches moving observer first as in slide 37 -7) and then the back (after the back has moved forward), so the length that you measure will appear to be shorter than in the penguin's rest frame. 37 -16

Proof of Eq. 37 -13 Sam is sitting on bench at train station. Using a tape measure, Sam determines the length of the station in his frame, which is the proper length L 0. Sally is sitting on a train that passes through the station. What is the length L of the train station that Sally measures? Sally v Train A Sam B According to Sam, Sally moves through the station (time interval between passing point A and then point B, different places in Sam's frame) in time Dt=L 0/v : length of train station For Sally, the platform moves past her. She passes points A and B at the same place in her reference frame (proper time) in time Dt 0: 37 -17

The Lorentz Transformation How are coordinates x, y, z, and t reporting an event in frame S related to the coordinates x', y', z', and t' reporting the same event in moving frame S'? Gallilean Transformation Equations Origins coincide at t = t' =0 Fig. 37 -9 Lorentz Transformation Equations 37 -18

The Lorentz Transformation, cont'd What about S coordinates in terms of S' coordinates? Switch from one frame to the other by letting v→ -v What about position and time intervals for pairs of events? 37 -19

Some Consequences of the Lorentz Equations Simultaneity Time Dilation Length Contraction 37 -20

The Relativity of Velocities Fig. 37 -11 37 -21

Doppler Effect for Light Let f 0 represent the proper frequency (frequency in the source's rest frame) If source and detector moving towards one another b → - b Note: Unlike Doppler shift with sound, only relative motion matters since there is no ether/air to be moving with respect to. Low Speed Doppler Effect For b<<1 Same as for sound waves 37 -22

Doppler Effect for Light, cont'd Astronomical Doppler Effect Proper wavelength l 0 associated with rest frame frequency f 0. Replacing b=v/c and using l-l 0 = |Dl| = Doppler shift 37 -23

Doppler Effect for Light , cont'd Transverse Doppler Effect Classical theory predicts no Doppler shift observed at point D when source S is at point P. Fig. 37 -12 For low speeds (b<<1) Transverse Doppler effect another test of time dilation (T=1/f) Proper period T 0=1/f 0) 37 -24

Doppler Effect for Light , cont'd The NAVSTAR Navigation System v 1 f 03 f 01 v 2 vairplane v 3 f 02 Given v 1, v 2, v 3, f 01, f 02, f 03, and measured f 1, f 2, f 3, can determine vairplane, 37 -25

A New Look at Momentum relativistic expression using Dt=Dt 0 g, where the time Dt 0 to move a distance Dx is measured in the moving observer's frame 37 -26

A New Look at Energy Mass energy or rest energy Table 37 -3 The Energy Equivalents of a Few Objects Object Mass (kg) Energy Equivalent Electron ≈ 9. 11 x 10 -31 ≈ 8. 19 x 10 -14 J (≈ 511 ke. V) Proton ≈ 1. 67 x 10 -27 ≈ 1. 50 x 10 -10 J (≈ 938 Me. V) Uranium atom ≈ 3. 95 x 10 -25 ≈ 3. 55 x 10 -8 J (≈ 225 Ge. V) Dust particle ≈ 1 x 10 -13 ≈ 1 x 104 J (≈ 2 kcal. ) U. S. penny ≈ 3. 1 x 10 -3 ≈ 2. 8 x 1014 J (≈ 78 GWh) 37 -27