This document discusses Einstein's theory of special relativity and its implications. It begins by describing Galilean relativity and frames of reference. It then discusses Michelson-Morley's famous experiment which found that the speed of light is constant regardless of the motion of the observer, contradicting the theory of the luminiferous aether. This led Einstein to postulate that the laws of physics are the same in all inertial frames and that the speed of light in a vacuum is constant. The document explores the implications of these postulates through various thought experiments, showing that simultaneity is relative, time dilates and lengths contract for moving observers. It concludes by discussing some implications of special relativity like mass-energy equivalence and the twins paradox

1. Topic 9.2
1.5.1 – Einstein and Relativity

2. Galilean Relativity
In the Galilean view point, the laws of
physics should hold true in all
inertial frames of reference.
That is, so long as there is no
acceleration, all experiments should
yield the same result.
An experiment to measure g by using
a pendulum should give the same
result when it is stationary, or when
it is moving at a steady speed on a
train.

3. Frames of Reference
A frame of reference is simply a
“background” against which
measurements can be
measured.
Imagine a fully tiled swimming
pool.
The tiles form a grid on the
walls and floor.
The absolute position of
anything can be measured
relative to this grid.
The absolute displacement of
anything can be defined using
the corner of the pool as the
origin.

4. Galliean Relativity
Differences in observation between different
frames of reference can be explained by
considering the relative motion of the two
frames.
Consider an observer in the red frame of
reference looking at the clock.
They would see the clock ticking but not
moving
Consider an observer in the black frame of
reference looking at the red clock
They would see the clock ticking and
moving with speed v. Its position would
xbe s(x,y,z)=(vt,y,z)
If the black frame were also moving (say at
speed uthen the absolute velocity of the
x red clock would be u- vx
x

5. Newtonian Frames of Reference
A frame of reference is said to be inertial if Newton’s first law is valid
for it.
That is if it is not accelerating in any dimension
A frame of reference is said to be non-inertial if Newton’s second
law is valid for it.
That is it is accelerating in at least one dimension.
Newton realised that no frame of reference would be
more correct than any other. Therefore the concept of
absolute position is meaningless.
This is especially true in space where you have no
background grid to work with.
Different observers can have different frames of
reference, but in Newton’s view, they should agree
about when an event happened even if they both say
it was in a different position.

6. A First Thought Experiment
You are sat on a (very good)
train in a tunnel.
You cannot hear anything.
You cannot feel any
movement.
You are looking out of the
window and see another
train move from left to
right past your window.
What is your train doing?

7. A First Thought Experiment
Are you sat stationary in the red train
and the blue train moves from left to
right at speed v?
0
v

8. A First Thought Experiment
Are you sat in the red train moving
from right to left at speed v and the
blue train is stationary?
v
0

9. A First Thought Experiment
Are you sat in the red train moving
from right to left at speed ½ v and
the blue train is moving at ½ v in the
other direction?
½ v
½ v

10. A First Thought Experiment
Or is there something else going on?
Does it matter to the Physics?
3/2 v
½ v

11. A First Thought Experiment
Imagine that your friend on the red
train now hold 2 charges in her
hands. What do you see if you are
on the red train? On the blue train?
0
v
+
+

12. A First Thought Experiment
On the red train you only see the force of electrostatic repulsion.
On the blue train you see this same electrostatic force but because
the charges are moving, there appears to be a current, therefore
there is a magnetic force in the opposite direction.
The blue train always measures more force than the red train
The laws of physics are different to different observers!!!
FQ
v
+
+
+
FB
+
FQ
FQ FQ

13. Waves and Media
• All waves, such as sound waves and
water waves, had been observed to
require a medium in order to propagate.
• Light waves should also require a
medium.
• This medium was called the Aether (also
ether)
• The Aether was thought to be a transparent,
massless, colourless medium that was
present everywhere including in the vacuum.
• This was the theory in the 1600’s through
until the end of the 1800’s!

14. Michelson and Morley
• Michelson and Morley set out to detect
the Aether in 1887.
• They assumed that the Aether was
moving with constant velocity and that
the Earth moved relative to it.
• They likened their experiment to two
boats moving on a fast flowing river.
• The two boats are capable of exactly the same speed.
• One moves parallel to the bank,
• The other perpendicular to it.
• They both travel the same distance before turning
round and coming back to the start.

15. Michelson and Morley
The speed of the boat is u and
the speed of the water is v.
The boat moving parallel to the
stream has speed u+v and
then u-v which averages to u.
The boat moving across the
stream has speed
in both directions
√ u2−v2
As the speeds are different, but
the distance travelled the
same, the time taken to travel
will be different!
v
v v
√ u2−v2
u u
u+v
u-v

16. The Interferometer
Michelson devised a device known as an
interferometer to test this theory.
Because light is a wave, it can interfere
with other light waves and form an
interference pattern.
Differences in times of travel result in a
“phase difference” which causes a
change in the pattern.
As the Earth rotated around the Sun, it was
expected that changes in the
interference pattern would be observed.

17. The Interferometer

18. The Results
Michelson and Morley detected no
systematic change in the
interference pattern.
Any changes that were observed
were random errors.
More sensitive equipment has since
been built and to this day no
detectable change has been reliably
observed.

19. A Failed Experiment?
The Michelson-Morley experiment is
possibly the most famous “failed
experiment”.
However, whilst the experiment failed to
detect an Aether, it did suggest that the
postulate that “the Aether exists” was
incorrect.
This caused other Hypotheses to be put
forward, including Einstein’s special
relativity.
So the experiment did not truly fail.

20. An Unexpected Result
As shown using vectors, the speed of light
should be different in the two directions.
However, the time of travel is observed to
be the same, as is the distance of travel.
Therefore “the speed of light is the same in
all directions, regardless of motion”.
This is compatible with Maxwell’s equations
which showed that the speed of light is
given by:
c 1
e m
0 0
=

21. The implications of Einstein’s Postulates
1) Two events that are simultaneous
to one observer may not
necessarily appear simultaneous to
another observer in a different
frame of reference.

22. Simultaneity of Events
Consider a train moving at high speed, v, through a station.
As the train passes through 2 small explosions are set off at
either end of the train.
The two explosions are equidistant from a stationary
observer on the platform.
By simple calculation, the time taken for light to travel from
the head of the train (t) is given by t=
d
and from the tail
of the train (t’) is
c
t '=
d '
c
To the observer on the platform the events were
simultaneous.
d' d

23. Simultaneity of Events
Consider an observer now on the train moving with high
speed v.
The two explosions are still set off as before.
However, in the time taken for the light to travel, the observer
has moved forward by vt metres.
By simple calculation, the time taken for light to travel from
the head of the train (t) is now given by t=
d−vt
and
from the tail of the train (t’) is
d ' +vt
c
t '= c
To the observer in the train the events were not
simultaneous.
vt
d' d
d' d

24. Time Dilation
Zoe is travelling in her car at speed v past Jasper on his
verandah.
She has a type of clock that measures the time by reflecting
a light ray between two mirrors.
Jasper can also see the light ray bouncing and use this to
measure his own time.

25. Time Dilation
Before the experiment begins, Zoe parks
her car next to Jasper’s verandah and
they both agree on the timing and both
get the same result.
Zoe then drives from left to right at 0.8c
according to Jasper’s frame of reference.
Of course in Zoe’s frame of reference it is
Jasper’s verandah that moves from right
to left at 0.8c!

26. Time Dilation
The red counters count each time Jasper or Zoe sees a
reflection.

27. Time Dilation
Say the car is “w” wide and the time
Zoe measures between reflections
is t’.
Then the distance is given by
w=ct '
However, if t is the time to tick on
Jasper’s clock, he records the
distance travelled (using
Pythagoras) as
d 2=( ct )2=w2+( vt)2

28. Time Dilation
Combining the previous equations gives:
c2 t2=c2 t '2+v2 t2
This rearranges to:
t '=
t
√(1−v2
This is also written as:
Where tv is the time on the
moving clock as measured
by a stationary observer.
t0 is the time on the moving clock as
measured by an observer in the
same moving frame of reference.
c2 )
tv=
t0
√(1−
v2
c2 )

29. Time Dilation
In summary, a moving clock appears
to run slower according to
tv=
t0
√(1−
v2
c2 )

30. Length Contraction
If the speed of light is constant, and time
gets shorter (dilates) for a moving object,
then the length of the object must get
smaller as measured in the direction on
motion.
lv=l0 √(1−
v2
c2 )
lv is the length of the object when it is moving
l0 is the rest length of the object.

31. Mass Dilation
Experiments in the early 1900s
carried out by Kaufmann showed
that the charge/mass ratio of high
speed electrons decreased with
increasing speed.
However, the charge carried by the
electrons was a constant.
Therefore the mass must have been
increasing.

32. Mass Dilation
Einstein’s theory of special relativity
showed how this could happen.
mv=
m0
√(1−
v2
c2 )

33. Practice.
Consider a pen of mass 100g and
length 15cm. What would be its
length and mass, as measured by a
stationary observer, when it is
moving at 340 ms-1 and 3x107 ms-1
relative to the observer?

34. Practice
NASA have invented a rocket that
can travel at 95% of the speed of
light. How long will it take to get to
the nearest star, 4.5ly away as
measured by the scientists at
Houston and by the astronauts on
board?

35. Practice
A type of subatomic particle called a
muon has a half-life of 2μs. What
will be the half-life of these particles
if they are travelling at 0.99c in the
laboratory?

36. What is a Thought Experiment
• Einstein's understanding of relativity
came about due to his use of thought
experiments.
• These are experiments where the logic of
the situation, and hence the results, is
flawless BUT the situation is usually
impossible (or at least highly improbable)
to replicate in reality.
• However, just because the experiment
cannot really be done does not mean
that the result cannot be significant!

37. Common limitations of thought experiments.
• Trains and spacecraft cannot
travel at relativistic velocities (due
to the consequences of special
relativity!!)
• An observer outside of the train
(travelling at nearly c) would find it
impossible to physically make any
valid or accurate observations of
events inside the train.

38. Common limitations of thought experiments.
• It is impossible to “see” the light
beam travelling from the source to
the mirror and back.
– Either the whole train carriage
lights up (the light spreads out)
– Or you need a laser beam and to
have dust particles in the path to
scatter it.
• Either way the light becomes
spread out.

39. Common limitations of thought experiments.
• It is impossible engineer an
infinitely long train and perfectly
identical fireworks.
• The relative simultaneity
experiment (fireworks at either
end of a long train) requires both
of these in order to register any
noticeable effect.

40. The implications of Special Relativity.
• As an object increases its speed towards c,
its mass increases (or dilates)
• This means that it becomes even more
difficult to accelerate the object ( F=ma=
m( v−u)
)
t
• Even at the maximum theoretical speed
(~0.99c), with the biggest engines possible, it
would take over 4 years to reach just out to
our nearest star. The rest of the galaxy is
just too far away!

41. The implications of Special Relativity
• Consider two identical twins. One remains on Earth
whilst the other takes a long trip to a distant star on a
spacecraft that can travel at 0.99c.
• From the Earth twin's point of view, the space twin is
moving whilst they stand still. Therefore the Space
twin's clock is running more slowly. Therefore the
Space twin will be younger when they return.
• However, from the space twin's point of view, the Earth
twin may just as well be moving whilst they sit still.
Therefore the Earth twin's clock runs slower and
therefore the Earth twin will be younger.
• They can't both be younger can they?
This is the so called twins paradox.

42. The implications of Special Relativity
• The twins paradox is really not a major
problem once you consider the limitation of
Special Relativity.
• Special relativity is ONLY valid in an inertial
frame of reference!
• The space twin must have accelerated to
0.99c, turned around the distant star, and
decelerated again.
• Therefore the space twin has not always
been in an inertial frame of reference,
therefore their conclusion is invalid.
• The space twin would indeed be the younger
of the two.

43. The implications of Special Relativity
• When the spacecraft is moving forward
through space at relativistic speeds it
appears that the space is moving
towards the spacecraft.
• Therefore the distance will appear
shorter to the pilots than the distance
measured by an external observer.
• This is a consequence of a slower
ticking clock (time dilation) and a
constant speed of light.

44. Measuring Time
• The passage of time in an inertial
frame of reference can be very
accurately measured using a
pendulum of a specific length.
• However, this requires knowledge
of the gravitational field strength
to be useful.
• It is also mostly useless in non-inertial
frames of references (i.e.
the deck of a boat that is rocking
up and down)

45. Measuring Time
• Watchmakers developed more and more
precise devices to measure time but the
definition of the second was still as being
1/8600 of the time for the Earth to rotate on
its axis.
• In 1967, following observations that the
Earth's rotation was slowing (very) slightly a
new definition of the second was formulated.
• 1 second is the time taken for an atom of Cs-
133 to oscillate 9129631770 times. In theory
this definition will remain accurate
everywhere in the Universe.

46. Measuring Distance
• The metre was originally defined as being
one ten-millionth of the distance from the
North pole to the Equator passing through
Paris.
• This distance was then scored onto 3
platinum bars kept in Paris.
• This allowed a common scale of distance to
be used by everyone and eventually has
replaced a much more complex English
system of Feet, Yards, Fathoms, Chains,
Furlongs and Miles in most of the world.

47. Measuring Distance
• As the surveyors had actually
made a slight mistake in their
calculations for the metre, the
bars became the standard rather
than the original definition.
• These bars were later replaced
with a more stable platinum-iridium
bar.

48. Measuring Distance
• The need to have an even more
precise standard of length that did
not rely on referencing a metal
bar in a vault somewhere brought
scientists to basing their definition
in the atom.
• The metre is now defined as
being the distance travelled by
light in 1/299792458 of a second.
– Note how this requires that the
speed of light is precisely known!