Ben Gurion University of the Negev
www.bgu.ac.il/atomchip, www.bgu.ac.il/nanocenter
Physics 3 for Electrical Engineering
Lecturers: Daniel Rohrlich, Ron Folman
Teaching Assistants: Daniel Ariad, Barukh Dolgin
Week 1. Special relativity – crisis of the aether • equivalence of
inertial reference frames • speed of light as maximum signaling
speed • definition of an event • Lorentz transformations • relativity
of simultaneity • spacetime 4-vectors • timelike, lightlike and
spacelike intervals
Sources: Feynman Lectures I, Chap. 15 Sects. 1-7;
Tipler and Llewellyn, Chap. 1 Sects. 1-4;
4 ‫ פרק‬,1 ‫ יחידה‬,‫פרקים בפיסיקה מודרנית‬
The crisis of the aether
Maxwell’s equations (1865) predict that electromagnetic waves
(including light waves) travel at speed c in vacuum. The
accepted value of c today is 299,792,458 m/s.
B
0  E 
t
Faraday’s law
B 

0      E 
t 

Vector identity


 (  B)
 (  E)   E 
t
Gauss’s law in a vacuum:
2
•E = 0
Ampère’s law, as
modified by Maxwell
2


E
1

E


2
2
  E    0  0
  2 2  E
t 
t  c t
The crisis of the aether
Maxwell’s equations (1865) predict that electromagnetic waves
(including light waves) travel at speed c in vacuum. The
accepted value of c today is 299,792,458 m/s.
BUT…
this speed is relative to…what?
The nineteenth century answer: c is relative to the “aether”.
All wave move in a medium: ripples on water, sounds in the
air, etc. The aether is the medium for light.
We can detect water, air, etc. Can we detect the aether?
The experiment of Michelson and Morley (1887)
direction of earth's motion through the “aether”
tup  D / c  v
2
2
earth
Michelson and Morley
rotated the experiment by
90º to cancel experimental
errors.
D
detector
 tdown
t right  D /( c  vearth )
tleft  D /( c  vearth )
D
The experiment of Michelson and Morley (1887)
direction of earth's motion through the “aether”
tup  D / c  v
2
2
earth
 tdown
Expected path difference:
D
detector
2(tright+ tleft – tup – tdown) c ≈ 2(vearth /c)2 D
= 1 · 10–8 · D
= 1 · 10–8 · (2 · 107 λ)
= 0.2 λ
t
 D /( c  v )
right
earth
tleft  D /( c  vearth )
D
The experiment of Michelson and Morley (1887)
direction of earth's motion through the “aether”
tup  D / c  v
2
2
earth
 tdown
Expected fringe shift (after rotation):
D
detector
2(tright+ tleft – tup – tdown) c ≈ 2(vearth /c)2 D
= 2 · 10–8 · D
= 2 · 10–8 · (2 · 107 λ)
= 0.4 λ
t
 D /( c  v )
right
earth
tleft  D /( c  vearth ) Expected fringe shift:
0.4
Observed fringe shift: < 0.02
D
Result for vearth: vearth < 8 km/s instead of 30 km/s
The experiment of Michelson and Morley (1887)
direction of earth's motion through the “aether”
D
detector
D
If there is no aether, then Maxwell’s
electromagnetism and Newton’s
mechanics are incompatible!
If there is no aether, then Maxwell’s
electromagnetism and Newton’s
mechanics are incompatible!
Einstein’s bold line of attack was to combine
an old principle – relativity, true in Newton’s
mechanics – with a new principle: the speed
of light c is a universal constant.
Einstein’s first postulate:
The equivalence of inertial reference frames
Definition: an inertial reference frame is a system of
coordinates moving with constant velocity (constant speed
and constant direction).
Example: the system of coordinates of a passenger in a train
that moves at constant velocity is an inertial reference frame.
The principle of relativity:
The laws of physics are the same in all inertial reference
frames.
A reference frame made of meter sticks and clocks:
Einstein’s second postulate:
The speed of light is the maximum signaling speed
There is a maximum signaling speed, and it is the speed of
light c. Since this is a law of physics, the speed of light c is
the same in every inertial reference frame. The speed of light
is a universal constant.
Einstein’s second postulate:
The speed of light is the maximum signaling speed
There is a maximum signaling speed, and it is the speed of
light c. Since this is a law of physics, the speed of light c is
the same in every inertial reference frame. The speed of light
is a universal constant.
Einstein’s first and second postulates seem incompatible.
He was the first to understand that they are not!
Events
Definition: an event is any physical process that occurs at a
definite time and at a definite point in space.
Relativity of time for two observers, Alice and Bob:
First event: flashlight emits flash
Second event: flash returns to flashlight
These two events define one clock cycle.
cDt/2
Bob, in his frame, measures time Dt
between the two events. The time Dt
must obey (cDt/2)2 = (vDt/2)2 + L2 so
►(Dt)2= 4L2/(c2–v2) = (4L2/c2 ) g2 ,
where g = (1–v2/c2 )–1/2. On the other
hand in Alice’s frame, where the time
between the events is Dt′, we find: (Dt′)2
= 4L2/c2 (time according to Alice).
Thus
► Dt = g Dt′.
Note Dt′ < Dt since 1 < g < ∞. So Bob
thinks that for Alice, the “light clock” ticks
slower. In fact, Bob will conclude that he
ages faster while Alice lives longer!
L
cDt/2
☺
vDt
Observer A (Alice), on a
train moving with speed
v relative to Bob, shines a
flashlight at a mirror on
the ceiling of the train.
Observer B (Bob) is
outside on the ground.
Relativity of length
☺
ruler
L′
train speed = v
L′ – length of ruler in Alice’s “train frame”. In Alice’s frame, the ruler is at rest.
What is L (the length of the ruler in Bob’s frame)?
“Light measurement” of length in Alice’s frame: L′ = ct1′ and L′ = c(t2′ – t1′ )
where t1′ is when the photon hits the right end of the ruler (starting at time 0 for
both Alice and Bob) and t2 ′ is when it hits the end it started from. So 2L′ = ct2′.
Now, Bob sees Alice and the ruler moving at speed v, so L + vt1 = ct1 and
L – v(t2 – t1) = c(t2 – t1). Eliminating t1 we get 2L = (c2 – v2) t2/c . Combining
with 2L′ = ct2′, we get L′/L = (t2′ /t2) c2/(c2 – v2) = g2(t2′ /t2). But t2 = gt2′ so
► L = L′ /g (again a consequence of the fact that c is the same in all frames).
Let’s find the “dictionary” (transformation of space-time) for going
from one “language” (Alice’s inertial frame) to another (Bob’s
inertial frame) while preserving the meaning of the “sentence” (the
form of the equations of Nature, e.g. Maxwell’s equations).
Criterion: For any light ray,
(Dx)2 + (Dy)2 + (Dz)2 – c2 (Dt)2 = 0 = (Dx′)2 + (Dy′)2 + (Dz′)2 – c2 (Dt′)2
If Alice (primed system) moves along the x-axis with velocity v
relative to Bob (unprimed system), then
Dx′ = g(Dx – vDt)
Dy′ = Dy
Dz′ = Dz
Dt′ = g(Dt – vDx/c2)
Lorentz transformations
And by changing v to –v we obtain the
Dx = g(Dx′ + vDt′)
Dy = Dy′
Dz = Dz′
Dt = g(Dt′ + vDx′ /c2)
Inverse
Lorentz transformations
4-vectors
We can write spacetime coordinates as “4-vectors” i.e.
Dr = (cDt, Dx, Dy, Dz)
where the invariant length of any 4-vector v = (v0, v1 , v2, v3 ) is
v 2  (v1 ) 2  (v2 ) 2  (v3 ) 2  (v0 ) 2
Lorentz transformation of 4-vector:
 v0 '   g
  
 v1 '    g
 v '   0
 2 
 v '  0
 3 
 g
g
0
0
0 0   v0 
 
0 0   v1 
1 0   v2 
 
0 1   v3 
4-vectors
We can write spacetime coordinates as “4-vectors” i.e.
Dr = (cDt, Dx, Dy, Dz)
where the invariant length of any 4-vector v = (v0, v1 , v2, v3 ) is
v 2  (v1 ) 2  (v2 ) 2  (v3 ) 2  (v0 ) 2
Lorentz transformation of 4-vector:
 v0 '   g
  
 v1 '   0
 v '     g
 2 
 v '  0
 3 
0  g
1
0
0
0
g
0
0   v0 
 
0   v1 
0   v2 
 
1   v3 
4-vectors
We can write spacetime coordinates as “4-vectors” i.e.
Dr = (cDt, Dx, Dy, Dz)
where the invariant length of any 4-vector v = (v0, v1 , v2, v3 ) is
v 2  (v1 ) 2  (v2 ) 2  (v3 ) 2  (v0 ) 2
Lorentz transformation of 4-vector:
 v0 '   g
  
 v1 '   0
 v '   0
 2 
 v '    g
 3 
0 0  g   v0 
 
1 0
0   v1 
0 1
0   v2 
 
0 0
g   v3 
Timelike, spacelike and light-like 4-vectors
Light-like vector: u2 = 0
Timelike vector: u2 < 0 (between causally connected events)
Spacelike vector: u2 > 0 (between causally unconnected events)
ct
future light cone
event
x
past light cone
Additional reference:
Experimental tests of time dilation:
CERN (1966):
muons at speed v = 0.9997c
Factor 12 increase in lifetime
(2% accuracy in the prediction)
Daily life: GPS
If we want 30 m accuracy in distance we need 0.1 s
accuracy in time. But a satellite in geosynchronous orbit,
moving at a speed of 11,000 km/h ≈ 3000 m/s ≈ 10–5 c,
loses 4.3 s per day because of time dilation!