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Special Relativity, b. 1905
What went wrong with Physics?
Nothing.
What went wrong with Understanding Physics?
Aha!
Have we got it right now?
Who knows?
1
.... on a Giant’s Shoulders
• Galileo Galilei first used the example of a ship
travelling at constant speed, without rocking, on
a smooth sea; any observer doing experiments
below the deck would not be able to tell
whether the ship was moving or stationary.
• Or, today, while travelling level in an aeroplane
with constant velocity.
• Or even while on the earth on which we stand
orbiting the sun at approximately 30 km/s.
2
Relative Motion
• Sitting in a car, day-dreaming, looking out of
the window – hey, we are moving backwards!
• No we aren’t.
• We cannot tell if we are moving, only if
something else is moving relative to us.
3
Absolute motion?
• Is there a way of knowing if you are actually still?
How fast you are actually going?
– Is there a stationary point in the universe? Motion
relative to that point would be absolute motion.
• Galileo and Newton based everything on
assuming that you can always detect motion.
– They thought of space and time as absolute quantities
that don’t depend on how an observer is moving.
– so they are the same throughout the universe.
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Defining terms ...
• A frame of reference is a set of fixed markers
relative to an observer’s position.
– markers clearly not moving, stationary, “at rest”
• Inertial frames of reference are those moving
at constant velocity relative to each other.
– Inertia refers to the fact that objects don’t change
velocity without a resultant force acting on them
– Relative velocities are a matter of simple vector
addition, since Galileo’s time
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Some assumptions ...
• Infinitely many inertial frames exist. Any two
frames are in relative uniform motion.
• The inertial frames move in all possible forms of
relative uniform motion.
• There is a universal, or absolute, time and space.
• Two inertial frames are related by (a Galilean
transformation of) relative velocities.
• In all inertial frames, Newton’s laws of motion
and gravity hold.
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Relative velocity scenarios (1)
• Woman sitting in train, man sitting seat on platform. Train
moves forward at 200 km h-1.
– woman’s velocity relative to man: 200 km h-1
• Woman walking forward in train at 5 km h-1.
– woman’s velocity relative to man: 205 km h-1.
• What’s the problem?
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Relative velocity scenarios (2)
• Imagine a very long ship with a hooter in the middle that sends out
a short blast of sound on a wind-free day.
• If the ship is moving forwards, a person in the stern will hear the
sound before a person in the bow, because the ship’s forward
motion moves the person in the stern towards the sound whereas it
moves the person in the bow away from the sound.
• By comparing when the sound is heard at each end, it is possible to
tell if the ship is moving forwards or backwards or not at all.
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Relative velocity scenarios (3)
• Shouldn’t we get the same happening with
light?
– Does a light beam from a torch facing forward in
the train travel faster than a beam facing
backwards?
• Sound actually travels at a fixed speed in air,
whatever the speed of the source (hence
shock waves and sound barriers)
– Does light travel at a fixed speed in a “medium”
called the “æther”? Cue Maxwell.
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Just one of Maxwell’s Equations
• One set of solutions to these equations is in the form
– of travelling sinusoidal plane waves
– with the electric and magnetic field directions perpendicular to one another
and the direction of travel
– with the two fields in phase.
• The changing magnetic field creates a changing electric field through
Faraday's law.
• That electric field, in turn, creates a changing magnetic field.
• This perpetual cycle allows these waves, known as electromagnetic
radiation, to move through space, always at velocity c, with
– ε0 = 8.85 x 10-12 Fm-1 and μ 0 = 4π x 10-7 TmA-1 and :
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Mathematical note: don’t go here
• Vector operator expressed
as determinants and/or
tensors
• Replace vector “ν” by “B”
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Conclusions ...
• Many physicists thought these waves were
oscillations in some hitherto undetected medium
(the æther) permeating the whole of space.
• Light was thought to travel at a fixed speed
relative to the Aether.
• The Earth must be travelling through the æther at
some speed.
• A way of detecting this was proposed by Maxwell
himself.
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Maxwell’s hypothesis
• Split a beam of light (or
microwaves) into two
beams travelling in
perpendicular directions
• Bring them together and
look at the interference
pattern.
• Rotating the apparatus
through 90 about a vertical
axis should produce a
difference in the position of
the fringes, because of
different velocities of the
beams relative to the æther.
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Sums: pulse parallel to Earth’s velocity
• If Earth moves at 0.1c
(in reality 0.0001c)
• Light pulse parallel to
Earth’s velocity:
 ct1 = 300 + 0.1ct1
 0.9ct1 = 300
 t1 = 1.111 μs
• Return:
 ct2 = 300 – 0.1ct2
 1.1ct2 = 300
 t2 = 0.909 μs
• Total t1 + t2 = 2.02 μs
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Sums: pulse orthogonal to Earth’s velocity
• Pulse has to follow diagonal to reach mirror in time “t”,
while Earth travels “vt”. Apply Pythagorus:
• (ct)2 = (vt) 2 + (300) 2
 t = 300/(c2 –v2)½
 t = 1.005 µs
• Total = 2t
= 2.01 µs
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Michelson and Morley have a go
• Light beam splits at back of
semi-silvered block
Their interferometer:
• One goes straight on and is
reflected (twice) into the
eyepiece.
• One is reflected at 900 then
reflected back into the
eyepiece.
• Both beams reflected twice.
Extra block so both beams go
through the same 3
thicknesses of glass
• Rotate through 900 then check
again.
• Accurate to 0.05 of a fringe
width, expecting 0.4, got 0.
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Conclusions
The Aether theory predicted that absolute motion could
be detected by a difference in the time taken by light to
travel the same distance parallel and perpendicular to the
Earth’s motion. A null result meant:
• No ether
• No “absolute zero” of speed
Some wanted to suggest the earth dragged the ether
along with it, so “locally” the ether moved at the earth’s
speed and a null result was to be expected. No evidence
for this. More seriously:
• the laws of Physics didn’t seem to work for light.
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Where do we go from here?
• All motion is relative, absolute rest does not exist.
– We pick up from other factors that we might have
accelerated (push of seat forwards, creaks rattles and
rolls of moving train, ...). The station could as well be
rushing at us as our train rushing at it.
– Nothing can let us detect that we have an absolute
speed as opposed to a relative speed.
• The speed of light in free space is invariant,
independent of the motion of source or observer)
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Key ideas you should have grasped
• Absolute motion can not be detected = no
evidence for the ether.
• They expected light travelling parallel to the
Earth’s motion would take longer than that at
right angles. It did not. Therefore light always
travels at the same speed for all observers.
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Inertial frames key idea
Frame = set of coordinates
Inertial frame one in which Newtons’ first law is
obeyed.
Train not moving
Train moving steady speed
Train accelerating to right.
(not inertial frame)
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Review
• SQs
• Relativity in a nutshell watch this
• http://www.onestick.com/relativity/
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Special Relativity
• Einstein published in 1905, starting from these
key ideas: Postulates
– The speed of light in free space is invariant
– Physical laws have the same form in all inertial
frames of reference.
• For example, for light “s = ct” was no longer valid
in all frames. He worked out how to transform
the distance “s” and time “t” coordinates from
one inertial frame to another so the form of the
“equation” would be the same in every frame to
give consistent results.
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Help!
• What the law is trying to say is that not
everything will have the same value in every
frame, but the physics laws that relate
everything together will still be correct.
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Examples of testing
• In 1964 Bertozzi accelerated electrons and measured
their KE. He found that as he accelerated them their KE
increased but not their speed! He got to
0.999 999 999 95c
• At CERN they tested neutral pions
One of these decays into two gammas travelling at c in
two directions
However when the pion was
accelerated to 0.9997c
γ
γ
the gamma pulse still travelled
away at c
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What does that mean?
• Astronaut in high speed craft carries out a simple experiment.
• A light source on one side emits a pulse of light towards a
bullseye opposite, 4 m away. She times the pulse’s journey.
• The laws of physics have to hold the same if she is moving as
if she is at rest, so:
• t0 = distance/time = 4/c = 13ns
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Mission control’s view ...
• Mission control sees the craft move forward 3m while
the pulse travels, and he has an identical clock.
• From his perspective the pulse travels 5 m (3:4:5
triangle) so the time the pulse takes to hit the target is:
• t = 5/c = 17 ns
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Debrief ..
• When they compare times, the controller says her time
(4/c) is shorter than his timing (5/c), so her clock was
running slow.
• In fact not just the clock, but everything runs slow, half
life of Uranium in the craft’s nuclear power source,
time to boil a kettle, so while moving she cannot tell
that the clock is running slow.
• Time itself is running slower than the controller’s time.
• She ages less than the controller.
• They each have their own “times”: they are not the
same
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And in general ...
•
•
•
•
•
AC2 = BC2 + AB2
AB2 = AC2 - BC2
(ct0) 2 = (ct)2 - (vt) 2
c2t02 = (c2 – v2)t2
t02 = (1-v2/c2)t2
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Worked example for time dilation
• c = 3.00 × 108 m s-1
• The half-life of charged π mesons at rest is 18
ns. Calculate the half-life of charged mesons
moving at a speed of 0.95c.
• t0 = 18 ns, v/c = 0.95
• Therefore t = t0/(1-v2/c2)½
• = 3.2 x 18 = 58 ns
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Time dilation ...
• A moving clock runs more slowly than a stationary
clock
• If the time duration of an event measured by an
observer at rest relative to the event is t0
• then an observer moving at speed “v” relative to the
event will record a longer time interval:
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... is real
• Muons decay with a half-life of 2.2μs
• In 1977 at CERN some were made to travel in a 14m
diameter ring at 0.9994c. Their half life then was ..
More or less?
• t = t0(1 - 0.99942)-½
• 63.5 μs. (actually 64.5 but I rounded the figures slightly)
• Also checked with
– atomic clocks in aeroplanes
– satnav satellites that have to be corrected for relativistic
clock speeds, etc .
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Proper time
If two events occur at the same location in an
interial frame, the time between them, in that
frame is called the ‘proper time’ t0.
Measurement of the same time from any other
location are always greater.
The factor on the
bottom is called
The Lorentz factor.
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Time dilation
• Time dilation is the difference between the
two times measured.
• Phew!
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Examples
Your starship passes Earth with a relative speed
of 0.9990c. After travelling 10 years (your time)
you turn around and come back to Earth. The
trip takes another 10 years (your time). How
long does the round trip take according to
measurements made on Earth?
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answer
• Away journey takes 224years so the answer is
448 years.
• You will have aged 20 years and the Earth will
have aged 448 years.
35
Particle example
Kaon has a lifetime of 0.1237µs when stationary.
If it has a speed of 0.990c relative to the lab,
how far can it travel according to classical
physics and according to special relativity?
36.7m
260m
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Hang on ..
• If the controller sees the astronaut’s clock
running slow, surely the astronaut sees the
controller’s clock running slow?
• Yes!
• But … they can only synchronise at the start,
maybe, but then they get further and further
apart and any further comparison involves the
exchange of signals – flashes of light for e.g. and
can only be calculated … or a return journey.
• And there’s more ..
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The Twins ...
• An astronaut aged 25 says goodbye to his twin and travels at 0.95c
in a spaceship to a distant planet, arriving there 5 years later.
• After spending a few weeks there, the astronaut returns to Earth at
the same speed on a journey that takes another 5 years.
• The astronaut is 35 years old on his return.
• How much time has elapsed on earth? t = ...
• 32 years.
• How old is his son who was 4 when daddy left? ...
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.. paradox
• It could be argued that the Earth travels away from the
spaceship at 0.95c and therefore the ‘Earth’ twin is the
one who ages.
• However, the twin who travels to the distant planet has
to accelerate and decelerate and therefore the twins’
‘journeys’ are not equivalent.
• The twin who has travelled to the distant planet did
not remain in the same inertial frame, broke the
symmetry, and so came back younger than his ‘stay at
home’ brother.
• A fuller answer lies in General Relativity Theory that
includes gravity and acceleration.
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Length Contraction
• Our spacecraft is travelling to a planet at a distance L0
away, at the speed v that controller and astronaut agree
on.
• Astronaut say time to arrive t0 = L0/v
• Controller he say time to get there t = L/v
• L0 = vt0 = vt (1 - v2/c2)-½ = L (1 - v2/c2)-½
• When C tells A that time is up, she should have arrived,
– her clock is slow and she would not have had time to cover the
full distance.
– She has arrived, because the distance has contracted in the
same proportion so time and distance are compatible with the
agreed speed!
– A rod moving in the same direction as its length appear
shorter than when it is stationary.
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... through the looking glass?
• According to mission control, not only is the spacecraft
length contracted, but so are all its contents.
• Not a painful compressing, but space contracting and
carrying contents with it ...
41
To think about
• If you could take a picture of a space ship
coming towards you at a speed close to that of
light, what would the photo look like?
– Hint: at those speeds the extra time light takes to
reach you from the rear of the ship is significant.
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Length Contraction .. a la Jim
• A rod moving in the same direction as its
length (L) appears shorter than when it is
stationary (L0).
• L0 is the proper length (measured by the
observer travelling with the rod)
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Worked example
• A spaceship moving at a speed of 0.99c takes 2.3 seconds
to fly past a planet and reach one of its moons. Calculate:
a.
b.
the distance travelled by the spaceship in its own frame of
reference in this time
the distance from the planet to the moon in their frame of
reference.
• Solution:
a.
b.
Distance = speed × time = 0.99c × 2.3 = 0.99 × 3.0 × 108 × 2.3 =
6.8 × 108 m
The distance from the planet to the moon in their own frame
of reference is the proper distance (L0).
i.
ii.
iii.
The distance travelled by the spaceship is the contracted distance
(L).
Rearranging the length contraction formula to find L0 therefore gives
L0 = 6.8x108/(1-0.992)½ = 7.1 x 6.8x108 = 4.8x109 m
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Some assumptions ... modified
Galileo and Newton
Special Relativity
Infinitely many inertial frames exist. Any two frames
are in relative uniform motion.
O.K.
The inertial frames move in all possible forms of
relative uniform motion.
Relative velocity “bounded”
by speed of light
There is a universal, or absolute, time and space.
Each frame has its own
measurement of time and
space
Two inertial frames are related by (a Galilean
transformation of) relative velocities.
More complex (Lorentz)
transformations between
frames
In all inertial frames, Newton’s laws of motion and
gravity hold.
All laws of Physics are the
same.
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Relativistic mass
• Einstein also showed that the mass of an object
depends on it’s speed v as:
• m0 is its rest mass or proper mass
• So mass increases with speed , getting ever larger
• Checked by measuring e/m for electrons at
different speeds.
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Worked example
• Given that e=(-)1.6x10-19 C and m0=9.1x10-31 kg
1. Calculate e/m for an electron at rest.
2. Calculate e/m at a speed of 0.69c
•
Solution
1. e/m0= 1.76x1011Ckg-1
2. m at 0.69c = m0=9.1x10-31 /(1-0.692)½ = 1.25x10-30 so
e/m = 1.6x10-19/ 1.25x10-30 = 1.28x1011Ckg-1 and this
agreed with the experimental result
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The Speed Limit
•
•
•
•
•
m/m0=1/(1 - v2/c2)½
When v=0, m/m0=1
When v=0.9c, m/m0 ≈2.3
As v→c, m/m0 →∞
It is impossible to take the
kind of matter we are
made of and accelerate it
past the speed of light
– That does not mean there
isn’t another sort of matter
that travels in the speed
range from c to infinity ..
Particles speculatively called
tachyons but no evidence!
48
Mass and Energy
• In special relativity, transferring energy to an
object increases its mass with an equivalence of
E = mc2.
• If m is the totality of mass of an object then:
E = mc2 = m0c2/(1-v2/c2)½
• When v=0, E =m0c2 = its rest energy and anything
above this, when it is moving, is kinetic energy:
K.E. = mc2 - m0c2
49
Kinetic Energy
•
•
•
•
K.E. = mc2 - m0c2
which can come from an accelerating field,
where:
KE = W = QV, for example.
E.g. At 0.9c, m = 2.3m0 so KE = 1.3 m0c2.
If v<<c, (1-v2/c2)-½ becomes (1-v2/2c2) and
mc2 - m0c2 becomes ½mv2.
Remember: energy in joules/1.6x10-19 = energy in
eV
50
Worked example
(e = 1.6 × 10−19 C, electron rest mass m0 = 9.1 × 10 −31kg)
• An electron is accelerated from rest through a
pd of 6.0 MV. Calculate:
a. its gain of kinetic energy
b. its mass in terms of its rest mass
c. the ratio of its speed to the speed of light c.
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Solution:
52
• End of chapter SQs
• If you like maths, read how he worked out the
rules for relativity
• The How Science Works topic suggests a
future for people who ask the right sort of
awkward questions.
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