Chapter 37
Relativity
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:
c  299 792 458 m/s
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, p0) 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 (g).
p0 g 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 2c!)→same as if pions were at rest
in the lab frame!
37- 6
Measuring an Event
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.
2D
t0 
c
2L
t 
c
L

1
2
L

1
2
t 
Fig. 37-5
 Sally 
 Sam 
vt   D 2
vt    ct0 
2
2
1
2
t 0
1 v c
2
2
(37-7)
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:
t  gt0
 v c
g
1
1 
2

1
1 v c
2
(37-8)
(time dilation) (37-8)
37-11
The Relativity of Time, cont'd
Lorentz factor g as a function of the speed parameter 
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 (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 ms
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
if   0.9994  g 
1
1 
2

1
1   0.9994 
2
 28.87
t  gt0   28.87  2.200 ms   63.51 ms
37-13
Two Tests of Time Dilation, cont'd
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?
if   7 10  g 
7
1
1  2
 1.000000000000245
t  gt0  1.000000000000245 15.00000000000000 hr 
 15.00000000000368 hr
t  t0  1 108 s!
37-14
The Relativity of Length
L  L0 1   
2
L0
g
(37-13)
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.
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 L0. Sally is
sitting on a train that passes through the station. What is the length L of the
train station that Sally measures?
v
Sally
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 t=L0/v :
L0  vt
length of train station
(Sam)
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:
L  vt0
L vt0 1

=
L0 vt
g
or
L
(Sally)
L0
g
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
x '  x  vt
t't
(approximately valid
at low speeds)
(37-20)
Origins coincide at t = t' =0
Fig. 37-9
Lorentz Transformation Equations
x '  g  x  vt 
y'  y
z' z
t '  g  t  vx c 2 
(valid at all
physically possible speeds)
(37-21)
37-18
The Lorentz Transformation, cont'd
What about S coordinates in terms of S' coordinates?
x  g  x ' vt  and t  g  t ' vx c 2 
(37-21)
Switch from one frame to the other by letting v→ -v
What about position and time intervals for pairs of events?
x '  x2 '- x1 ' and t  t2 '- t1'
x  x2 - x1 and t  t2 - t1
1. x  g  x ' vt '
2. t  g  t ' vx ' c 2 
g
1
1 v c
2

1.' x '  g  x  vt 
2.' t  g  t ' vx c 2 
1
1  2
Frame S ' moves with a velocity v relative to frame S
37-19
Some Consequences of the Lorentz Equations
Simultaneity
Time Dilation
vx ' 

t  g  t ' 2 
(37-23)
c 

vx '
t  g 2  0 (simultaneous events in S ')
c
t  gt '
(events in the same place in S ') (37-24)
t  gt0
(time dilation)
Length Contraction
x '  g  x  vt 
(37-25)
ends measured simultaneously in S  t  0 , L  x , L '  x '
L
L0
g
(length contraction)
37-20
The Relativity of Velocities
x  g  x ' vt '
Fig. 37-11
x
u
t
x '
and u ' 
t '
u ' v
u
1  u ' v c2
u  u ' v
vx ' 

t  g  t ' 2 
c 

x
x ' vt '

t t ' x ' c 2
x
x ' t '  v

t 1  v  x ' t ' c 2
(relativistic velocity transformation)
(classical velocity transformation)
(37-29)
(37-30)
37-21
Doppler Effect for Light
Let f0 represent the proper frequency (frequency in the source's rest frame)
f  f0
1-
1+
(source and detector separating)
(37-31)
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 Effect
For <<1
f  f 0 1- + 12  2 
(source and detector separating, 
1) (37-32)
Same as for sound waves
37-22
Doppler Effect for Light, cont'd
Astronomical Doppler Effect
f  f0 1- 
(37-33)
Proper wavelength 0 associated with rest frame frequency f0.
c


c
0
0   1- 
1- 
  0 1- 
-1
(37-34)
  0

0
(37-35)
Replacing =v/c and using -0 = || = Doppler shift
v

0
c
(radial speed of light source, v
c) (37-36)
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.
f  f 0 1- 2
Fig. 37-12
(transverse Doppler Effect) (37-37)
For low speeds (<<1)
f  f 0 1- 12  2 
(low speeds) (37-38)
Transverse Doppler effect another test of time dilation (T=1/f)
T
T0
1-
2
=g T0
(37-39)
Proper period T0=1/f0)
37-24
Doppler Effect for Light , cont'd
The NAVSTAR Navigation System
v1
f03
f01
v2
vairplane
v3
f02
Given v1, v2, v3, f01, f02, f03, and measured f1, f2, f3, can determine vairplane,
37-25
A New Look at Momentum
x
p  mv  m
t
(classical momentum) (37-40)
relativistic expression using t=t0 g, where the time t0 to move a distance x
is measured in the moving observer's frame
x
x t
x
pm
m
m g
t0
t t0
t
p  g mv
(momentum)
(37-41)
p  g mv
(momentum)
(37-42)
37-26
A New Look at Energy
Mass energy or rest energy
E0  mc 2
(37-43)
Table 37-3 The Energy Equivalents of a Few Objects
Object
Mass (kg)
Energy Equivalent
Electron
≈ 9.11x10-31
≈ 8.19x10-14J
(≈ 511 keV)
Proton
≈ 1.67x10-27
≈ 1.50x10-10J
(≈ 938 MeV)
≈ 3.55x10-8J
(≈ 225 GeV)
Uranium atom ≈ 3.95x10-25
Dust particle
≈ 1x10-13
≈ 1x104J
(≈ 2 kcal.)
U.S. penny
≈ 3.1x10-3
≈ 2.8x1014J
(≈ 78 GWh)
37-27
A New Look at Energy , cont'd
Total energy
E  E0  K  mc 2  K
E  g mc 2
(37-47)
(37-48)
The total energy E of an isolated system cannot change
 system's initial   system's final 


Q
 total mass energy   total mass energy 
or
E0i  E0i  Q
(37-49)
M i c2  M f c2  Q
Q  M i c2  M f c2  Mc2
(37-50)
37-28
A New Look at Energy, cont'd
Kinetic energy
K  12 mv 2
(37-51)
K  E  mc 2 = g mc 2 -mc 2


1
 mc 2  g -1 =mc 2 
 1
 1- v c 2





Fig. 37-14
(kinetic energy)
(37-52)
37-29
A New Look at Energy, cont'd
p 2  2Km
Momentum and kinetic energy
 pc 
2
 K 2  2 Kmc 2
E   pc    mc
2
2

2 2
(classical)
(37-51)
(37-54)
(37-55)
sin    and cos  1 g
(37-56)
Fig. 37-15
37-30