TN2612 - introduction to special relativity
Lambert van Eijck
NPM2, dep. Radiation Science and Technology, fac. Appl. Sci, TU Delft
2018-2019
Lambert van Eijck
TN2612 - introduction to special relativity - lecture 0
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~ = m · ~a: classical mechanics is the context, but not the
F
cause of SR
Figure: Classical mechanics is based on standards of length and time
(which we will have to sacrifies for Special Relativity).
~ = m · ~a are so successful that we (you)
Classical mechanics and F
use it still today, on a daily basis. It is based on the notion of
distance and time intervals that seem ’set in stone’.
Lambert van Eijck
TN2612 - introduction to special relativity - lecture 0
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Special Relativity should make you aware!
Figure: The Cave of Plato, where the inhabitants see only the (2D)
projection of the real (3D) world.
Lambert van Eijck
TN2612 - introduction to special relativity - lecture 0
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Special Relativity should make you aware!
Figure: At the end of course you should see the shortcomings of the 3D
’projection’ of the 4D world.
Lambert van Eijck
TN2612 - introduction to special relativity - lecture 0
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A short history on physics at that moment
(a) Isaac Newton
(b) James Clerk
postulates ΣF = m a in
1687
Maxwell postulates
Maxwell equations
in 1873
The theory of force, motion and acceleration, together with the
unifying equations on electro-magnetism were sufficient to describe
nearly everything in the universe.
Lambert van Eijck
TN2612 - introduction to special relativity - lecture 0
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Maxwell Equations
The electro-magnetic plane
wave is the solution for
vacuum:
I
∇ · E = 4πρ
I
∇·B=0
I
∇ × E = − ∂B
∂t
I
∇ × B = µ0 0 ∂E
∂t + µ0 J
For vacuum this means:
I
∇·E=0
I
∇·B=0
I
∇ × E = − ∂B
∂t
I
∇ × B = µ0 0 ∂E
∂t
Lambert van Eijck
Ex (z, t) = Ex sin(k · z − ω t)
propagating
at velocity
q
1
c = µ0 0
TN2612 - introduction to special relativity - lecture 0
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How to measure the speed of light
Date
1676
1726
1849
1862
1879
1907
1926
1947
1958
1973
1983
Author
Olaus Rømer
James Bradley
Armand Fizeau
Leon Foucault
Albert Michelson
Rosa, Dorsay
Albert Michelson
Essen, Gorden-Smith
K. D. Froome
Evanson et al
Lambert van Eijck
Method
Jupiter’s satellites
Stellar Aberration
Toothed Wheel
Rotating Mirror
Rotating Mirror
Electromagnetic constants
Rotating Mirror
Cavity Resonator
Radio Interferometer
Lasers
Adopted Value
Result (km/s)
214,000
301,000
315,000
298,000
299,910
299,788
299,796
299,792
299,792.5
299,792.4574
299,792.458
Error
+-500
+-50
+-30
+-4
+-3
+-0.1
+-0.001
TN2612 - introduction to special relativity - lecture 0
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How to measure the speed of light
h has a smaller distance from the light
emitting body so it takes less to get
there compared to l
(c)
(d)
Figure: Ole Rømer (1700) studied the eclipse of the Io moon of Jupiter
for nautical purposes and noticed unexplained variation in his
measurements. Light returning from eclipsed moons of Jupiter arrive
earlier at earth at L than at K (wikipedia).
Lambert van Eijck
TN2612 - introduction to special relativity - lecture 0
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How to measure the speed of light
Figure: Light coming in from below will hit the detector after 3
reflections at a certain angle 2φ, which depends on the rotation speed
ω = dφ/dt and the traveling distance d (Fizeau-Foucault approx. 1850).
Lambert van Eijck
TN2612 - introduction to special relativity - lecture 0
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Newton and Maxwell, or Newton versus Maxwell?
Maxwell
q discovers that
c = µ010 , in vacuum, a
constant.
Newtons main law is
F = ma = m dv
dt . When a
javelin is thrown, the athlete
adds a speed ∆v to her
running speed V over a time
∆t. This results in an average
acceleration
aavg
∆V
=
.
∆t
Lambert van Eijck
Figure: Why does the athlete run
before throwing the javelin?
TN2612 - introduction to special relativity - lecture 0
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Newton and Maxwell, or Newton versus Maxwell?
The reasoning in the previous slide is based on the assumption that
the speed ∆v can be added to the running speed V. Christian
Huygens noticed in the 17th century that speeds are relative, like
positions, and therefore can be added.
Figure: translation from De Motu Corporum in ”The Discovery of
Dynamics” by J.B. Barbour
Lambert van Eijck
TN2612 - introduction to special relativity - lecture 0
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Throwing a laser pointer
With our common understanding of speed, the laser light has a
speed higher or lower than c when the pointer itself has any
velocity v.
Figure: The speed of a moving pointer should be added to the speed of
light leaving the pointer. Or should it not?
Lambert van Eijck
TN2612 - introduction to special relativity - lecture 0
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Waves and media
Figure: No medium: no waves.
Waves usually occur in some medium as water, air or solid matter.
But light propagates in vacuum. Before Einstein, the assumption
was that an aether exists as a medium for light propagation.
Lambert van Eijck
TN2612 - introduction to special relativity - lecture 0
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An attempt to measure variations in c
The Michelson-Morley experiment (1887) was meant to measure
the speed of light using an interferometer that supposedly moves
through the aether.
Figure: Using the earths rotation speed, one can assume that an
interferometry experiment will take place in some aether wind. If the
speed constant c is tied to the ether, one should measure different c s
depending on the velocity w.r.t. the aether.
Lambert van Eijck
TN2612 - introduction to special relativity - lecture 0
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An attempt to measure variations in c
(a) Orienting one arm along the
aether wind, the other arm is
perpendicular to it.
Lambert van Eijck
(b) The actual interferometer,
placed in a bath of mercury.
TN2612 - introduction to special relativity - lecture 0
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The outcome: c = c
The expected variation in the speed of light was not found. The
conclusion was drawn that the aether that is supposedly at
’absolute rest’ may be partially (and locally) dragged to a certain
speed by the objects it penetrates throughout. Alternatively,
Fitzgerald and Lorentz came up with the proposition that space
was contracted in the direction of the aether wind, explaining the
null result.
At this point Einstein stepped in with a new view, explaining all
previous observations.
Lambert van Eijck
TN2612 - introduction to special relativity - lecture 0
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The postulates of special relativity
1. All laws of physics should hold in any inertial frame (incl.
mechanics and Maxwell).
2. The speed of light is constant in all inertial frames
c = 2.998 · 108 m/s.
Lambert van Eijck
TN2612 - introduction to special relativity - lecture 0
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Definition of two reference frames
S
S’
Y’
Y
V
X
Lambert van Eijck
TN2612 - introduction to special relativity - lecture 0
X’
18
S
The Lorentz transformation in 1 dimension
S’
Y’
Y
V
X
X’
The S 0 systems moves in the x direction with speed V: inspect this
direction first.
0
ct
a00 a01
ct
=
x
a10 a11
x0
Aligning S and S 0 : x = x 0 = 0 when t = t 0 = 0, one can say that
x(t) = Vt for the origin of S 0 (x 0 (t 0 ) = 0). And so:
0
substituting V(t) for x at x' = 0:
ct
a00 a01
ct
x -> Vt
=
x' -> 0
Vt
a10 a11
0
→ a10 2 =
Lambert van Eijck
V2 2
a00
c2
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S
The Lorentz transformation
S’
Y’
Y
V
X
X’
Moreover, the speed of light is c in both frames, so light travels as:
c 2t 2 − x 2 = 0
c 2 t 02 − x 02 = 0
⇒ c 2 t 2 − x 2 = c 2 t 02 − x 02
⇒ (a00 ct 0 + a01 x 0 )2 − (a10 ct 0 + a11 x 0 )2 = c 2 t 02 − x 02
yielding:



Lambert van Eijck
a00 a01 = a10 a11
2
− a11
= −1
2
− a10 = +1
2
a01
2
a00
TN2612 - introduction to special relativity - lecture 0
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S
The Lorentz transformation
S’
Y’
Y
V
X
X’
The resulting Lorentz transformation (LT) matrix is:
0
initial frame, with ct being the y-axis in
other frame
speed of light * t = distance, NOT TIME
ct
1 β
ct
ANYMORE!!
= γ(V )
0
x
β 1
x
x is again position.
x and ct have the same unit of
measurement
with:
β = V /c
s
γ(V ) =
Lambert van Eijck
current speed / speed of light
1
1 − β2
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