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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 satellites.

Relativity

37-

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/minkowski.pdf

– It's About Time:Understanding Einstein's RelativityN. David Mermin

http://press.princeton.edu/titles/8112.html• Others

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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)

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4

The Postulates

37-

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!

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Fig. 37-2

Experiment by Bertozzi in 1964 accelerated electrons and measured their speed and kinetic energy independently. Kinetic energy →∞ as speed → c

The Ultimate Speed

37-Ultimate Speed→Speed of Light: 299 792 458 m/sc

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If speed of light is same for all inertial reference frames, then speed of light emitted by a source (pion, 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 0.9975c with respect to the laboratory decay, emitting two photons ().

The speed of the light waves (-rays) emitted by the pions was measured always to be c in the lab frame (not up to 2c!)→same as if pions were at rest in the lab frame!

Testing the Speed of Light Postulate

37-

0

7

Measuring an Event

37-Fig. 37-3

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 Coordinates1. Space Coordinates: three dimensional array of measuring rods

2. Time coordinate: Synchronized clocks at each measuring rod intersection

How do we synchronize the clocks?

Event A: x=3.6 rod lenghts, y=1.3 rod lengths, z=0, time=reading on nearest clock

All clocks read exactly the same time if you were able to look at them all at once!

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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?

The Relativity of Simultaneity

37-

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 the relativistic physics that we discuss here.

937-

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)!

•What would a third stationary observer, Bill, standing to the right of Sam observe?

A Closer Look at Simultaneity

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The Relativity of Time

37-Fig. 37-5

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.

0

2 Sally

Dt

c

2 212

2 21 102 2

2 Sam

Lt

c

L v t D

L v t c t

0

2 (37-7)

1

tt

v c

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The Relativity of Time, cont'd

37-

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.

2 2

1 1 (37-8)

1 1 v c

In previous example, who measures the proper time?

Lorentz factor:

0 (time dilation) (37-8)t t

v c Speed Parameter:

12

Lorentz factor as a function of the speed parameter

The Relativity of Time, cont'd

37-

Fig. 37-6

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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 (t) depends on how fast the muon is moving.

Muon stationary in lab (production and decay in same place, at muon itself) t0=2.200 s

If muon is moving at speed 0.9994c with respect to the lab (production and decay in different places in the lab frame) the lifetime measured by laboratory clocks will be dilated

Two Tests of Time Dilation

37-

2 2

1 1if 0.9994 28.87

1 1 0.9994

0 28.87 2.200 s 63.51 st t

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1. Macroscopic Clocks. Super precision atomic clocks (large systems) flown in airplanes ~7x10-7 (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 15.00000000000000 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?

Two Tests of Time Dilation, cont'd

37-

7

2

0

80

1if 7 10 1.000000000000245

1

1.000000000000245 15.00000000000000 hr

15.00000000000368 hr

1 10 s!

t t

t t

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The Relativity of Length

37-

The length L0 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.

2 00 1 (37-13)

LL L

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Does a moving object really shrink?

37-

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.

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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 L0. Sally is sitting on a train that passes through the station. What is the length L of the train station that Sally measures?

Proof of Eq. 37-13

37-

length of train station

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 t=L0/v :

Train

Sam

Sallyv

A B

0 (Sam)L v t

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 t0:

0 (Sally)L v t

0 0

0

1= or

v t LLL

L v t

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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'?

The Lorentz Transformation

37-

Fig. 37-9

Gallilean Transformation Equations

' (approximately valid

' at low speeds) (37-20)

x x vt

t t

Lorentz Transformation Equations

2

'

' (valid at all

' physically possible speeds) (37-21)

'

x x vt

y y

z z

t t vx c

Origins coincide at t = t' =0

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What about S coordinates in terms of S' coordinates?

The Lorentz Transformation, cont'd

37-

2' and ' (37-21) x x vt t t vx c

Switch from one frame to the other by letting v→ -v

What about position and time intervals for pairs of events?

2 1 2 1- and - x x x t t t 2 1 2 1' '- ' and '- ' x x x t t t

2 2

2 2

1. ' ' 1.' '

2. ' ' 2.' '

1 1

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Frame ' moves with a velocity relative to frame

x x v t x x v t

t t v x c t t v x c

v c

S v S

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Simultaneity

Some Consequences of the Lorentz Equations

37-

2

'' (37-23) v x

t tc

2

'0 (simultaneous events in ')

v xt S

c

Time Dilation ' (events in the same place in ') (37-24)t t S

0 (time dilation)t t

Length Contraction ' (37-25)x x v t

0 (length contraction)L

L

ends measured simultaneously in 0 , , ' ' S t L x L x

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The Relativity of Velocities

37-

Fig. 37-11

2

' '

''

x x v t

v xt t

c

2

' '

' '

x x v t

t t x c

2

' '

1 ' '

x x t v

t v x t c

2

' (relativistic velocity transformation) (37-29)

1 '

u vu

u v c

' (classical velocity transformation) (37-30)u u v

' and '

'

x xu u

t t

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Let f0 represent the proper frequency (frequency in the source's rest frame)

Doppler Effect for Light

37-

0

1- (source and detector separating) (37-31)

1+f f

If source and detector moving towards one another → - 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 EffectFor <<1

210 21- + (source and detector separating, 1) (37-32)f f

Same as for sound waves

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Doppler Effect for Light, cont'd

37-

0

c (radial speed of light source, ) (37-36)v v c

Proper wavelength 0 associated with rest frame frequency f0.

Astronomical Doppler Effect

Replacing =v/c and using -0 = || = Doppler shift

0 1- (37-33)f f

0

-1

0

1-

1- (37-34)

c c

0

0

0

1-

(37-35)

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Doppler Effect for Light , cont'd

37-

Classical theory predicts no Doppler shift observed at point D when source S is at point P.

Transverse Doppler Effect

For low speeds (<<1)

Fig. 37-12

20 1- (transverse Doppler Effect) (37-37)f f

210 21- (low speeds) (37-38)f f

Transverse Doppler effect another test of time dilation (T=1/f)

002

= (37-39)1-

TT T

Proper period T0=1/f0)

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Doppler Effect for Light , cont'd

37-

The NAVSTAR Navigation System

v1

v2

v3vairplane

Given v1, v2, v3, f01, f02, f03, and measured f1, f2, f3, can determine vairplane,

f01

f02

f03

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A New Look at Momentum

37-

(momentum) (37-41)p mv

relativistic expression using t=t0 , where the time t0 to move a distance x is measured in the moving observer's frame

(classical momentum) (37-40)x

p mv mt

0 0

x x t xp m m m

t t t t

(momentum) (37-42)p mv

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A New Look at Energy

37-

20 (37-43)E mcMass energy or rest energy

Object Mass (kg) Energy Equivalent

Electron ≈ 9.11x10-31 ≈ 8.19x10-14J (≈ 511 keV)

Proton ≈ 1.67x10-27 ≈ 1.50x10-10J (≈ 938 MeV)

Uranium atom ≈ 3.95x10-25 ≈ 3.55x10-8J (≈ 225 GeV)

Dust particle ≈ 1x10-13 ≈ 1x104J (≈ 2 kcal.)

U.S. penny ≈ 3.1x10-3 ≈ 2.8x1014J (≈ 78 GWh)

Table 37-3 The Energy Equivalents of a Few Objects

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A New Look at Energy , cont'd

37-

20 (37-47)E E K mc K Total energy

2 (37-48)E mc

The total energy E of an isolated system cannot change

0 0

system's initial system's final

total mass energy total mass energy

or (37-49)i i

Q

E E Q

2 2 2 (37-50)i fQ M c M c Mc

2 2i fM c M c Q

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A New Look at Energy, cont'd

37-

212 (37-51)K mvKinetic energy

2 2 2

2 2

2

= -

1 -1 = 1 (kinetic energy) (37-52)

1-

K E mc mc mc

mc mcv c

Fig. 37-14

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A New Look at Energy, cont'd

37-

2 2 (classical) (37-51)p KmMomentum and kinetic energy

2 2 22 (37-54)pc K Kmc

Fig. 37-15

222 2 (37-55)E pc mc

sin and cos 1 (37-56)