Special theory of relativity
Failure of Newtonian mechanics
Newton
Revision: Still remember
Newton's 3 law of motion?
1.
2.
3.
An object at rest will always be in the
wrong place
An object in motion will always be
headed in the wrong direction
For every action, there is an equal and
opposite criticism
=b, just joking
Newtonian view of
space and time
space and time are absolute and do not
mix
Newtonian relativity:
(1) all mechanics laws are the same
(invariant) in all inertial frames
(2) Galilean transformation

Inertial frames
 Definition
 E.g.
of inertial frames: the lab frame and
the constant speed car frame
 All inertial frames are equivalent
Example of inertial frames of reference
Galilean transformation
relates the kinematical quantities, such as
position, velocity, acceleration between two
inertial frames
O: stationary frame (uses x,y,z,t as their
coordinates)
O’: moving wrp to O with constant speed u away
from O (uses x’,y’,z’,t’ as their coordinates)
Galilean transformation for the coordinates (in 1D):
x’ = x – ut, y’ = y, z’=z, t’ = t
Galilean addition law for velocity (in 1-D):
v’x = vx - u
Simply a daily experience
One can derive the invariance of Newtonian law
of mechanics in all inertial frame from Galilean
transformation:
In O: F = m dv/dt = ma
In O’: F’ = m’ dv’/dt’ = m’ a’
In Newtonian view, m’ = m, t’ = t; for inertial
frames: du/dt = du/dt’ = 0
The point is: Newton’s law takes on the same
form in all inertial frames – Galilean
transformation guarantees that Newton’s laws
of mechanics are the same in all inertial frames
Newtonian view fails when applied
to light
 However,
Galilean transformation is going to
fail when u is approaching the speed of light –
it has to be supplanted by Lorentz
transformation
 Galilean transformation is inconsistent with
Maxwell theory of light – a gadanken case: the
EM wave is ``frozen’’ and not waving anymore
in an inertial frame moving at the speed of
light
 Newtonian law of invariance fails for EM
 Galilean
transformation on light is also
rejected based on the law of cause and
effect (causality)
Ether and Michelson-Morley
Experiments

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
In early 19th century
Ether: medium for light to propagate at a speed of
3x108m/s (analogue to sound propagate in the
mechanical medium of air at speed 330m/s)
‘the fifth element’
thought to be the `absolute frame of reference’ that
goes in accordance with Newtonian absolute space
and time
One can imagine that ether frame is fixed wrp to the
Sun
 If
exist the Ether wind will `drift’ with a
relative speed of u wrp to Earth
 The ether drift u can be measured by observing
the effect that arises in speed of light when the
Earth is moving in different direction
 Analogy: speed of sound can be measured by
‘talking’ in different directions in the wind
Talking in the wind in and opposing the
direction of wind blowing
 Two
light beams having a difference in phase,
Df, will interfere to produce interference
pattern
(recall this from optics)
 Df arises because the time of arrival at a
common point in space by the two beams are
different, t1 not equal to t2
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
t1 not equal to t2 due to Galilean transformation
assumed on the the speed of light for the two beams
t1 (parallel to the ether drift) = d /(c - u) +
d /(c + u) = 2dc /(c2 - u2)
t2 (perpendicular to the ether drift) =
2d /(c - u)1/2
Dt approx du2/c2
Df = n 2p c|t1 - t2 | / l (for constructive
interference)
 What
would you expect if the
orientation of the experimental set up is
rotated 90 degree?
 A change in the interference pattern
 The MM apparatus is sensitive to detect
very minute change in the interference
pattern up to a number of fringe shift =
2 (d/l) (u/c) 2 approx 0.4
But MM sees only NULL result – no change in
the interference pattern
 Maxwell theory of EM is inconsistent with
Newtonian notion of an absolute reference
frame
 Einstein said EM is right, the Ether frame notion
is wrong
 Need to revise the Galilean transformation for a
consistent picture of EM and mechanics
