NATIONAL QUALIFICATIONS CURRICULUM SUPPORT
Physics
Special Relativity
Teacher’s Notes
[HIGHER]
The Scottish Qualifications Authority regularly reviews
the arrangements for National Qualifications. Users of
all NQ support materials, whether published by
Learning and Teaching Scotland or others, are
reminded that it is their responsibility to check that the
support materials correspond to the requirements of the
current arrangements.
Acknowledgement
Learning and Teaching Scotland gratefully acknowledges this contribution to the National
Qualifications support programme for Physics.
© Learning and Teaching Scotland 2010
This resource may be reproduced in whole or in part for educational purposes by educational
establishments in Scotland provided that no profit accrues at any stage.
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Contents
Introduction
4
Relativity – an overview
Relativity before Einstein
5
Special relativity
Time dilation and length contraction
Thought experiment 1
Thought experiment 2
Experimental verification
Relativistic effects regarding mass
7
8
9
13
15
16
Appendix
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INTRODUCTION
Introduction
These Teacher’s Notes cover more than the minimum required for
assessment. Suitable comments regarding assessable material a re included.
For the derivation of the equations, the algebra is kept to a minimum in order
to concentrate on the concepts and correct use of the equations. Teachers can
include additional algebraic steps depending on their students. Confusion and
errors can arise on the part of students by mixing up t, t’, l and l’, often
producing ‘wrong physics’. Students do not need to be able to reproduce the
derivations.
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RELATIVITY – AN OVERVIEW
Relativity – an overview
Einstein’s work on relativity was published in two stages: Special Relativity
was published in 1905 and General Relativity was published in 1916.
However, this work is often perceived as being part of ‘modern physics’. In
the eyes of students this is a curious label. Furthermore, relativity, as a
concept in physics, predates the work of Einstein. It is useful to approach
relativity from a Newtonian perspective.
Relativity before Einstein
Students of Higher Physics will be familiar with Newton’s Laws of Motion.
These laws allow us to describe the motion of objects, re gardless of size or
position. They also allow us to predict subsequent motions. Given starting
data, equations allow us, in theory, to predict subsequent motion. This leads
to a ‘clockwork’-type view of the world. Quantum mechanics, developed in
the 1920s, provides another set of rules for the physics of the very small at
the atomic and subatomic level. This theory and Einstein’s relativity give a
less ‘clockwork’ view of the world, leading to some philosophical
implications. Quantum mechanics is not includ ed in Higher Physics but the
term can usefully be mentioned.
To return to pre-Einstein time, it may be useful to consider the way people
thought about the world and the universe in general. Often time, t, is a time
interval in many physics problems and di scussions. When asked ‘What time
is it?’ there is the implication that if the clocks were accurate enough and set
correctly there would be just one answer to this question. Einstein lived in
Bern where some clocks chimed the hour at different times! Accura te watch
making and clock setting led to thoughts about time.
The Newtonian picture of the universe was built on the idea of absolute space
and time – a rigid framework against which all measurements and
experiments could be carried out.
This implies that there is constancy to the rate at which time passes. In the
Newtonian model, clocks tick at the same rate regardless of their movement.
Furthermore, clocks appear to tick at the same rate relative to observers who
may have a different motion relative to the clock.
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RELATIVITY – AN OVERVIEW
Similarly, absolute space implies a static backdrop against which all
movement can be referenced. It is useful to consider examples of everyday
experiences of movement relative to a stationary background. Consider the
sensation when sitting on a train next to another train, and one train moves.
Sometimes it is difficult to be sure whether you are moving or the other train
is moving. It is only after your train or the other train has moved off that you
can be sure by relating to the background.
One further and hugely important assumption in Newton’s view of the
universe is that the laws of physics remain the same whether one is moving
steadily or at rest. This is also known as Galilean invariance. This
assumption, of the universality of the l aws of physics, remains true regardless
of the introduction of relativity or quantum mechanics or any other theory.
Students may wish to comment on the validity of this assumption. Can we
even do physics and astronomy if we do not make this assumption?
In order to relate measurements taken by observers travelling at different
velocities, relative to absolute space, a number of simple rules were
established. These rules were set out by Galileo and are still known today as
Galilean transformations. Students do not need to be familiar with the formal
statements of the transformations but they should be familiar with their use to
calculate relative velocities. Common examples include passengers walking
towards the front or rear of moving vehicles and the relati ve velocities of two
vehicles moving towards each other. Numerical calculations should be
restricted to movement in one dimension, using simple vector addition.
It is useful to introduce the term ‘frame of reference’, which refers to any
laboratory, vehicle, platform, spaceship, planet etc and often has a relative
velocity to another ‘frame of reference’. These are called inertial frames of
reference when the movement is in a straight line with a constant velocity.
For example, a platform could be one frame of reference and a high-speed
train a different frame of reference, where the platform and the train have a
constant relative velocity.
The Galilean transformations assume that the speed of waves depends on the
motion of the wave source relative to the observer. This is intuitive for a
number of wave motions and it was assumed that it would be true for light,
which was known to be a wave. It is possible to ‘catch up’ a sound wave or a
water wave, and students can consider a number of cases to illustrate how a
wave may be seen to have different velocities relative to moving observers.
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SPECIAL RELATIVITY
Special relativity
Special relativity is ‘special’ because it considers only the case where different
observers (frames of reference) are in relative motion with a constant velocity.
The case of accelerated observers was developed by Einstein 11 years after
special relativity and is described in his general relativity.
Einstein very much supported the Newtonian view that the laws of physics
should apply everywhere and for every observer. However, his knowledge of
the work of James Clerk Maxwell led him to see that there was an apparent
contradiction with this principle.
Maxwell was a Scottish physicist who made an enormous leap forward in
explaining how light is produced. His theory of electromagnetism showed
that light is an electromagnetic wave. Students are not required to consider
Maxwell’s equations. However, it is important that they understand that
Maxwell’s equations allowed the speed of light to be predicte d theoretically.
Experiments undertaken by the American physicist Albert Michelson
obtained excellent agreement with the theoretically predicted value.
Knowing that light is a wave led some physicists to conclude that light must
be passing through something, i.e. ‘something’ must be vibrating, even in the
vacuum of space. This something was called the ether and Michelson and his
colleague Morley set out to detect the motion of the Earth through the ether.
However, their experiment failed to detect any dif ference in the speed of light
when it was measured for different speeds of the Earth in its orbit around the
Sun. The details of the classic experiment to detect the ‘ether drift’ are not
required for assessment but teachers may find it worthwhile to descr ibe this
experiment since it is a very good example of the validity of a ‘null result’
(see Appendix).
Einstein postulated that the speed of light is constant for all observers. The
Michelson-Morley experiment did not detect a difference in the speed of l ight
as the Earth moves in different directions and speeds because the speed of
light does not depend on the motion of the source. All observers measure the
speed of light to be the same, regardless of the motion of the source of light,
or the motion of the observer towards or away from the source of light.
Einstein knew that this tied in with Maxwell’s work in that the theoretical
value of the speed of light made no reference to the motion of the source.
Einstein also abolished the concept of the ether and any ether drift.
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SPECIAL RELATIVITY
There are a number of startling consequences of the postulate that the speed
of light is measured to be the same regardless of the motion of source or
observer. The Newtonian idea of the backdrop of absolute space and time was
firmly rejected. Also, observers moving relative to each other with a constant
velocity, in different reference frames, would disagree about the measured
separation in space and time of events they observed. Two events may be
simultaneous for one observer but not for the other observer.
To summarise
Special relativity (which applies to observers/reference frames in relative
motion with constant velocity) has two postulates:
1.
2.
The laws of physics are the same for all observers in all parts of the
universe.
Light always travels at the same speed in a vacuum, 3.0 × 10 8 m s –1
(299,792,458 m s –1 to be more precise).
(Light does slow down inside transparent material such as glass.)
To give a simple example:
A spaceship travelling at 100 million m s –1 approaches a planet considered to
be at rest. An observer on the planet sends a light signal to the spaceship. The
spaceship will measure the speed as 300 million m s –1 and not as (300 + 100)
million m s –1 . We cannot apply our usual ‘Newtonian rules’ for relative
velocity.
From these two postulates Einstein produced a new theory of motion.
We know that speed = distance/time, and speed = fλ, hence if the speed is to
remain constant ‘something’ must happen to the distance and time! This is
the essence of special relativity.
Time dilation and length contraction
Students at Higher level should be able to follow the derivation of the
equations showing time dilation and length contraction, although the
derivations themselves are not required.
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SPECIAL RELATIVITY
Notation:
– time interval (‘event’) or length of object under discussion in a
frame of reference (eg on a platform)
t’ and l’ – time interval or length of object ‘measured’ by travellers in a
different frame of reference (eg on a train)
v
– relative velocity of the two frames of reference
t and l
(Note: Recall that no one frame of reference is any more ‘stationary’ or
‘moving’ than any other. There is no ‘absolute rest’. We have chosen the
platform to contain the ‘event’.)
Thought experiment 1
Consider a person on a platform who shines a laser pulse upwards, reflecting
the light off a mirror. The time interval for the pulse to travel up and down is
t (no superscript).
Person in same frame as ‘event’ measures a
time t.
Total distance travelled by pulse 2h = ct.
mirror
h
platform
v
h
Travellers in this different frame of reference observe the ‘event’ (eg out of the window
of the train), which takes place in the platform frame of reference and measure a time t’.
A different frame of reference, for example a train moving along the x-axis at
high speed v, passes. From the point of view of travellers on board the train,
the light travels as shown in the diagram above.
The time taken for the light to travel up and back, as measured by travellers
in this frame, is t’ (t dash).
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SPECIAL RELATIVITY
In the time t’ that it takes for the light to travel up and back down the train in
this frame, the train has travelled a distance d.
Both observers measure the same speed for the speed of light.
Platform frame of reference
Train frame of reference
Total distance
travelled by pulse= ct’
2h=ct
Horizontal distance
travelled by train = vt’
d = v t’
A right- angled triangle can be formed
where the vertical side is the height, h, of
the pulse ( 12 ct), the horizontal side is half
of the distance, d, gone by the train
( 12 vt’) and the hypotenuse is half the
distance gone by the pulse as seen by the
travellers on the train ( 12 ct’).
1
2
1
2
ct
1
2
c t’
v t’
Applying Pythagoras to the triangle gives:
( 12 ct’ ) 2 = ( 12 ct) 2 + ( 12 vt’) 2
(ct’ ) 2 = (ct) 2 + (vt’) 2
(c 2 – v 2 ) t’ 2 = c 2 t 2
 v2  2 2
1  2  t’ = t
 c 
t
t' 
1  (v / c ) 2
What assumptions have we made?
(i)
(ii)
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The two frames are moving relative to each other along the x-axis, ie
the train passes the platform. There is no bending or circular motion
involved.
We require two travellers on the train since the start and finish places
are separate. This is fine since two clocks can be synchronised in the
same frame of reference.
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(1)
SPECIAL RELATIVITY
It is useful to mention to students the significance of the term
(The reciprocal 1/
1  (v / c ) 2 .
1  (v / c) 2 is known as the gamma factor.) This term
occurs in relativity equations and its size determines when relativity effects
will be observed. At everyday speeds it is almost unity.
Students should be able to interpret the final equation, stating what each of
the symbols represent, and what the equation means in terms of the time
interval for each observer.
Note: The laser pulse starts and finishes at the same place on the platform.
Thus equation (1) is used to calculate the time interval t’ registered in a
frame of reference, eg the train, for an event which take place in a different
frame of reference from the ‘event’.
For example if v = 0.4c then (v/c) 2 = 0.16 and
1  (v / c ) 2 =
1 0.16 =
0.917.
Let us use this value of v = 0.4c in our thought experiment with a laser pulse
time of 8.0 ns.
Thus, if t = 8.0 ns, we can calculate t’, giving t’ = 8.7 ns. A longer time
interval is ‘measured’ by travellers on the train. Th is effect, known as time
dilation (dilation = expanding), is a direct result of the postulate that the
speed of light is measured to be the same by all observers.
Time dilation leads to observers being unable to agree about simultaneous
events. Two events may appear to be simultaneous to one observer, but may
not be simultaneous for others.
Some comments
Sometimes the pulse is referred to as the ‘tick of a clock’ and t the period (or
twice the period).
(a)
Teachers may find a variety of derivations le ading to different formats
of this equation. The SQA data sheet has equation (1).
(b)
It is important to recall that no one frame of reference is any more
‘stationary’ or ‘moving’ than any other. There is no ‘absolute rest’. It is
the relative motion that matters. We often think of ourselves here on
Earth as ‘stationary’ and solve problems from that point of view. Care
is needed to sort out the frames of reference for different situations or
problems.
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SPECIAL RELATIVITY
Another example and some effects
Consider a space ship passing Earth at a velocity of 0.5c. It emits a pulse (or
on Earth we observe ‘ticks’ of their clock) of duration Δ T = 2.0 ns. We on
Earth can ‘measure’ the duration t’ we observe. Note that we are not in the
same frame of reference as the ‘event’ so our time interval is t’ not t. The
duration of the event, in the frame of the event on the space ship, is t.
Using equation (1) gives us an observed time interval of 1 /0.87 = 2.3 ns.
We consider their clock is running slow, time is passing more quickly for us
so they could end up ‘younger’! Although time dilation can give rise to
interesting discussions on time travel (into the future), the explanation of the
twin paradox requires more consideration since any acceleration may involve
general relativity. Also a returning twin would have to ‘change’ frames of
reference.
When
1  (v / c) 2 is almost unity no effect is noticeable. It is useful to
calculate this term (or the reciprocal) for various speeds, for example:
a supersonic plane 422 m s –1 (900 mph);
0.1 × 10 8 m s –1 ;
0.3 × 10 8 m s –1 (10%c); 1.0 × 10 8 m s –1 ; 2.0 × 10 8 m s –1 ,
2.8 × 10 8 m s –1
and
99% c.
Students can clearly see that effects in everyday life are not noticeable. We
need v > 10%c for any noticeable effects.
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SPECIAL RELATIVITY
Thought experiment 2
For length contraction, a rod of length l is placed in a frame of reference. eg
on the platform. Consider the length l’ measured by travellers in a passing
train or space ship, a different frame of reference. The relative velocity is v.
To measure the length of the rod we
need to determine the start and end
points. A mirror is fixed to one end and
the time taken (t) for a light pulse to be
reflected from the mirror and return is
recorded.
Time taken in this frame is t
Hence distance is 2 l = ct
l
l' + vt1
For the different frame of reference, a
light pulse is started when the left-hand
side of the rod is in line, but the
measured time taken to reach the mirror
(t 1 ) will be greater due to the relative
velocity.
Distance to mirror is c t 1 = l’ + vt t
l’
l' – vt2
As the pulse returns, the frame of
reference is still moving. The measured
time taken for the return distance is t 2 .
Distance back from mirror is
c t 2 = l’ – vt 2
l’
The total time taken for the pulse to measure the rod in the different frame is
t 1 + t 2 = l’/(c – v) + l’/(c + v).
Hence the total time in this different frame is t’ = l’ 2c/(c 2 – v 2 )
and the time in the frame with the rod is t = 2l/c.
The time measurements start and finish at the same place in the platform
frame.
Notice that the rod is on the platform.
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SPECIAL RELATIVITY
Using t’ = t/(1 – v 2 /c 2 ) substitute for t’ and t and simplify, giving:
l '  l 1  (v / c ) 2
We can see that the length l’ is less than l since
(2)
1  (v / c) 2 is less than 1.
This effect is known as length contraction.
For length contraction the equation is expressed using l for the length in the
frame of reference with the actual object and l’ the apparent length measured
in the other frame.
Students are expected to be able to interpret and use the equation.
A specific object, eg a metre rule, will have the same length, L say, when
measured by a person actually in a frame of reference with the object, eg a
person on a spaceship with the actual object would measure length L. (This
length L of a specific object is termed the ‘proper length’ of the object.)
Another example
Suppose we on Earth observe a very fast car moving past at 0.3 c. The length
of the car is 4 m. What length will we observe?
This length l = 4 m (actual length of the car to a person in the car) and l’ is
our observed length in our different frame of reference (we are not in the
same frame of reference as the car).
With v = 0.3c,
1  (v / c) 2 = 0.95.
Using l '  l 1  (v / c) 2 gives l’ = 4 × 0.95 = 3.8 m. We observe a car length
of 3.8 m.
Similarly, a spaceship or passing asteroid would appear shorter to us here on
Earth.
Conversely objects on the Earth, will appear ‘shorter’ to observers o n passing
spaceships. No particular frame of reference is any more ‘stationary’ than any
other.
Special Relativity means that the Newtonian view, that space and time are
absolute and completely separate, has to be rejected. Special relativity
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SPECIAL RELATIVITY
requires us to think of space and time as inextricably linked and our
measurements of distance and time depend on the motion of the observer. The
effects of time dilation and length contraction are only observed at very high
speeds (close to the speed of light). Stude nts should study an experiment in
which the effects are confirmed.
Experimental verification
One verification arises in the study of elementary
Muons
particles moving close to the speed of light. Cosmic rays,
Speed = 99.9%c
believed to be produced in deep space, collide w ith atoms
in the Earth’s upper atmosphere. They produce showers
of muons, which are short-lived elementary particles
about 200 times more massive than electrons. When
muons are produced in laboratories, we find that their
Distance
typical mean lifetime is very short – about two millionths
to us
of a second – before they decay into other particles. The
60 km
cosmic ray muons are moving with a speed of about
Muon distance
99.9% of the speed of light. However, even at this speed,
2
 0.999c 
a muon would travel only about half a kilometre in two
 60 1  

 c 
millionths of a second. Yet substantial numbers of
= 60 × 0.045 = 2.7 km
cosmic ray muons are detected at sea level, about 60 km
below the altitude where the muons are created. This is
Life time to us
because time dilation means that, viewed in our reference
=
2.7
× 103/0.999c
frame, the lifetime of the fast-moving muons is
= 9 s
considerably longer, ie to us time runs more slowly than
for them because they are moving at speeds very close to
the speed of light relative to us. Of course in the
reference frame of the muons, they are not moving at all
– hence their lifetime is still only two millionths of a
second. Also the reason why they are able to reach sea
level is that the distance which they travel – measured to
be 60 km in our reference frame – is considerably less in
their reference frame, an example of length contraction.
(Muon mean lifetime = 2.2 μs, and half-life = 1.56 μs.)
Students are not expected to be able to quote numerical values.
Another example is accurate time measurements on airborne clocks. The time
dilation is small for the relatively low speeds of aircra ft, but clock precision
can be very high. The differences can be measured.
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SPECIAL RELATIVITY
An aside
The Newtonian equations of motion we have already studied give us accurate
results for most of our everyday situations. For high speeds we need to use a
different set of equations, namely the relativistic time dilation and length
contraction above. Special relativity equations reduce to our familiar
equations when the speeds involved are less than 10% of the speed of light.
For situations involving atomic, nuclear and s ub-nuclear ‘particles’ we find
that neither Newtonian or relativity equations give results that agree with
experiment. One of the reasons is that energy is not ‘emitted or absorbed’ as
a continuous stream but is quantised into packets. Also there is a
wave/particle aspect that must be accepted and included. In these situations
we need to use another approach, that of quantum mechanics, which involves
another set of equations and different mathematics but gives excellent
agreement between theory and experiment. Relativity has been included at
this subatomic level to give relativistic quantum mechanics. Details of
quantum mechanics are not required for Higher Physics but the photoelectric
effect and wave/particle duality are in the Higher Physics: Particles an d
Waves unit.
Relativistic effects regarding mass
In Newtonian mechanics mass is considered to be conserved and to remain
constant, but at high speeds this could lead to different effects for different
observers. This could appear to violate our basic assumption that the laws of
physics are the same for all observers. For example, consider a small probe
making an impact on a planet. Momentum is a quantity that is conserved. The
‘damage’ caused on impact will depend on the momentum of the probe. This
momentum should be the same when calculated by a ‘stationary’ person on
the planet or by an observer on a fast-moving spaceship near the rocket. Each
observer will measure a different velocity for the rocket, but there is only one
impact and one lot of damage!
Hence each observer somehow considers the mass to be different, since
momentum p = mv.
The observer on the spaceship could consider the inertial mass of the rocket
to be greater than the observer on the planet.
Einstein showed that there is a mass energy equivalence, leading to his
famous equation E = mc 2 . With relativity there is not a separate conservation
of mass and energy but a single conservation of mass/energy.
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SPECIAL RELATIVITY
The equivalence of mass and energy could be introduced here, but is included
in the Particles and Waves unit in the Nuclear Fusion topic and teachers may
prefer to teach it in that context.
This apparent increase in inertial mass at high speeds means that it is more
and more difficult, requiring a great deal more energy, to increase the speed
of objects near the speed of light and impossible for an object to reach the
speed of light. The muons mentioned above have an extremely small mass
hence they are capable of travelling close to the speed of light.
In the Large Hadron Collider at Cern in Switzerland a great deal of energy is
needed to accelerate protons to speeds near to the speed of light.
To summarise
1.
Defining t, t’, l and l’ as shown below, equations for time dilation and
length contraction can be derived. The equations ( 1) and (2) are
provided on the SQA data sheet.
t and l
t’ and l’
v
time interval (‘event’) or length of object under discussion
in a frame of reference
time interval or length of object ‘measured’ by travellers in
a different frame of reference
relative velocity of the two frames of reference.
t' 
t
(1)
1  (v / c ) 2
l '  l 1  (v / c ) 2
(2)
2.
No object can travel faster than the speed of light.
3.
Relativistic effects are negligible when relative velocity is less than10%
of the speed of light.
4.
Experimental verification is provided by observing the life time of fast moving muons.
5
There is not a separate conservation of mass, but a combined
conservation of mass and energy. A greater energy than expected is
required to increase the speed of an object as its speed approaches the
speed of light.
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SPECIAL RELATIVITY
Appendix
Michelson-Morley experiment
mirror
M2
L
beam splitter
mirror
M1
light source
L
detector
The apparatus consists of a light source, a beam splitter (a half -silvered glass
plate) and two mirrors, M 1 and M 2 , each of which is equidistant from the
beam splitter. The beam splitter is at 45 o to the incident beam and the return
beams pass to the detector, a telescope.
The prevailing theory held that the ether formed an absolute reference frame.
The Earth is orbiting the Sun and therefore moving through the ether. Thus,
sometimes one light beam may be travelling in the same direction as the ether
and the other beam at right angles to the ether. They will have slightly
different speeds. The detector should show interference between the two
beams travelling perpendicular to each other. The aim was to measure the
speed of the Earth relative to the ether.
No discernible fringes were found, despite repeating the experiment at
different times in the Earth’s orbit and with different ori entation. On the
prevailing theory and experimental accuracy, the small destructive
interference should have been observed.
This null result had great importance since it could not be explained.
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