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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?

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.... 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.

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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.

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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

• 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.

Their interferometer:

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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 coordinatesInertial 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 pionsOne of these decays into two gammas travelling at c in two directionsHowever when the pion was accelerated to 0.9997cthe gamma pulse still travelledaway at c

γ γ

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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.

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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?

36.7m260m

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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 ...

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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. the distance travelled by the spaceship in its own frame of reference

in this timeb. the distance from the planet to the moon in their frame of

reference.

• Solution:a. Distance = speed × time = 0.99c × 2.3 = 0.99 × 3.0 × 108 × 2.3 = 6.8 ×

108 m b. The distance from the planet to the moon in their own frame of

reference is the proper distance (L0). i. The distance travelled by the spaceship is the contracted distance (L). ii. Rearranging the length contraction formula to find L0 therefore gives

iii. L0 = 6.8x108/(1-0.992)½ = 7.1 x 6.8x108 = 4.8x109 m

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Some assumptions ... modifiedGalileo and Newton Special RelativityInfinitely 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 kg1. Calculate e/m for an electron at rest.2. Calculate e/m at a speed of 0.69c

• Solution1. 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!

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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

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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

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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:

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• 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.