M

H

Maps of space

Space:

N & E or M & H

Electromagnetism:

E & B or E & B

time axis

space axis

x = ct

x = –ct

home

uni

PAST

Future

Elsewhere Elsewhere

The Light Cone: Maps of space-time

• Events that happen “elsewhere” can have no effect on here and now

Supernova!

How fast are we going anyway?

Earth’s Orbit around Sun:30 km/s

Milky Way Galaxy trajectory towards Great Attractor:7000 km/s

Sun’s Orbit around galaxy:250 km/s

E=mc² did not appear in the original 1905 paper, but later that year

• How did it arise out of theories of relative motion?– Consequence of the conservation of momentum and the

Lorentz transforms

• Energy becomes relative too - how can that be?– We are used to this concept already! What is my kinetic

energy now? Relative to Earth? Sun? Galaxy?

• Where does all the energy go when the spaceship is not actually getting (much) faster?– Into mass!

The Lorentz transforms

)( vtxx )/( 2cvEpp xx yy yy pp zz zz pp

)/( 2cvxtt )( xvpEE

Space-Time Energy Momentum

22 /1/1 cv

Mass

The Victorian Synchrotron

Data:• 30 GeV electrons• 99.9999% speed

of light• Mass 6000 times

ordinary electrons!

“Synchronise” the ring magnets for the effects of

relativity

person

6000/1/1 22 cv

The Lorentz Transformations

Catch

x

y

v

y

x),( yx

Catch ),( yx

)( vtxx yy zz

)/( 2cvxtt

22 /1/1 cv

The Lorentz Transformations: Special Case 1

x

y

22 /1/1 cv

0x0t

v

y

x

1t12 ttt

Time interval

v

y

x

2t

0x

tt Time Dilation formula. Only if

The Lorentz Transformations: Special Case 2

22 /1/1 cv

v

y

x

Measure Length:1t 1x 2x

The Lorentz Transformations: Special Case 2

x

y

22 /1/1 cv

0x0t

v

y

x

1t 1x12 xxx

Space interval

v

y

x

1t/xx

0t

Lorentz contraction formula. Only if

The Lorentz Energy Momentum Transformations

Energy&

Momentum

x

y

v

y

x),( Ep

Energy&

Momentum ),( Ep

)/( 2cvEpp xx

yy pp

zz pp

)( xvpEE

22 /1/1 cv

The Lorentz Velocity Transformations

Velocity

x

y

v

y

xu

Velocity

u

22 /1/1 cv

)/1( 2cvu

vuu

x

xx

)/1( 2cvu

uu

x

yy

)/1( 2cvu

uu

x

zz

),,( zyx uuuu

vuu xx yy uu zz uu Galilean

How are energy and mass related?

• Does mass have energy, does energy have mass?

– Yes – loosely speaking

• Why did Einstein say we will never get energy from mass?

– In 1905 nuclear reactions had not been discovered

– Chemistry is very feeble

– 1908 Rutherford Nobel prize (for Chemistry)

• Did he realise that was what was powering the Sun?

– Nobody understood the Sun until Hans Betheformulated the basic reactions of thermonuclear fusion in 1939! (Nobel Prize 1967)

How are energy and mass related?

• Does mass have energy, does energy have mass?

– Yes – loosely speaking

• Why did Einstein say we will never get energy from mass?

– In 1905 nuclear reactions had not been discovered

– Chemistry is very feeble

– 1908 Rutherford Nobel prize (for Chemistry)

• Did he realise that was what was powering the Sun?

– Nobody understood the Sun until Hans Betheformulated the basic reactions of thermonuclear fusion in 1939! (Nobel Prize 1967)

Were others on the verge of special relativity? Poincare, Lorentz? Was it just waiting to happen?

– An object once set in motion remains in motion until acted upon by an external force

– The universe is governed by a majestic clockwork where all clocks everywhere at all times tick in perfect synchronisation.

• Galileo 1634

The laws of Physics do not depend on absolute motion

Does this include electromagnetism?

• Newton 1687

• Maxwell 1873

The great treatise of electromagnetism

Electromagnetic fields and waves propagate through the Aether

• Poincare 1898

– Relativity of simultaneity: telegraph

– Concept of “local time”– Stuck on the Aether

1889: Maxwell equations Field of a moving

charge

• Heaviside 1897

BvF

qM

• Lorentz 1904

Maxwell equations invariant under L. transformations

The transition from Special to General Relativity

• What are the differences?– Special Relativity = all reference frames are equivalent

– … and speed of light the same in each

– General Relativity = everything falls at the same rate

– …and can cancel gravity by falling!

• Why is it said that although someone else would have soon come up with special, general was much more of an intellectual triumph?– Equivalence of gravity and acceleration a great mystery to

Newton

– Everyone before Einstein thought this was an amazing coincidence!

– Why is gravity so different to the other forces of nature?

– Idea of curved space arises from Special Relativity and the equivalence principle

Curved space

When we say that an accelerated field is equivalent to a gravitational field …

• What does that really mean?

– Everything falls at the same rate

– Can cancel gravity by falling

• The experiments inside an enclosed rocket either sitting on Earth or out in space accelerating may look equivalent, but if one looks out the window the scene is very different!

– True, but so does looking at the stars from Earth!

– Until Copernicus, humanity thought it was living on a stationary Earth and the Stars orbited the Earth

What’s the difference?

W

N

W

N

AcceleratingGravitating

Equivalence Principle

Equivalence Principle

What’s the difference?

Falling Floating