Why Does E=mc^{2}? Brian Cox & Jeff Forshaw

1. Space & Time

2. The Speed of Light

3. Special Relativity

4. Spacetime

5. Why Does E=mc^{2}?

6. And Why Should We Care?

7. The Origin of Mass

8. Warping Spacetime

The objective is to describe Einstein’s theory of space & time

Energy & mass are interchangeable & the speed of light is the exchange rate

1. Space & Time

a. Hermann Minkowski (Einstein’s teacher) thought of space & time being blended

b. E=mc^{2} Belongs to special relativity

c. Speed of light (c) = 299,792,458 metres per second

i. Sun 8 mins

ii. Width of Milky Way = 100,000 light years

iii. Andromeda is 2 million light years away

iv. Edge of the observable universe is > 10 billion light years away

v. Earth moves at about 67,000 mph around the sun

vi. Sun travels at 486,000 mph around the Milky Way (226m years to orbit once)

d. C is the cosmic speed limit

e. Galileo’s principle of relativity – there is no such thing as absolute motion

2. The Speed of Light

a. Faraday showed that magnetism & electricity are connected

b. Maxwell’s waves showed the relationship between electric & magnetic fields travel at c – there is a link btw electromagnetic fields & c; c is a constant of nature

c. 1881 Michelson & Morley experiment showed there to be no difference in c in any direction at any time of the year

d. That we can never catch up with a beam of light no matter how fast we travel infers that there is no absolute time

3. Special Relativity

a. Light is a symbiosis of electric & magnetic fields

b. 2 axioms:

i. Light always travels through empty space at the same speed regardless of the motion of the source or the observer

ii. No experiment can ever be performed that is capable of identifying absolute motion

c. Thought experiment of the light clock (two mirrors bouncing a beam of light; the mirrors are 1m apart so the light takes 6.67 nanoseconds for a round trip)

i. If light clock on a train & observed from a station the light has to travel further for the station observer. The Newtonian world says the light speeds up due to the motion of the train, so the same time for an observer on the station compared to one on a train. But if light cannot speed up it will take longer from the perspective of the station observer. This implies time passes at different rates depending upon how we are moving.

ii. The clock slows by an amount known as γ (gamma) = 1/(1-(v^{2}/c^{2}))

γ is always >1 because v/c will be <1 (i.e. the speed of the clock will be <c)

When v is very small relative to c, γ is close to 1

At speeds of 90% of c, the time stretching factor is >2, i.e. time would be halved

iii. Experimental proof comes from muons. At rest they live for 2.2 micro-seconds before turning into an electron & a pair of neutrinos

At a speed of 99.94% of c their life extended as Einstein’s theory predicted to 60 micro-seconds

iv. Time is malleable & so is space as objects shrink when they move

4. Spacetime

a. Einstein’s theory can be constructed almost entirely using the language of geometry

b. 3 Concepts:

i. Invariance

1. Rotational invariance: The laws of nature do not change if we spin around. If they are unchanged irrespective of the direction in which we face then there exists a quantity that is conserved called angular momentum [The moon moves 4cm farther away from the earth every year due to friction from the sea on the surface slowing the earth’s spin; & so the earth day lengthens by two-thousandths of a second every century. The conservation of angular momentum is transferred to the moon which speeds up in its orbit around the earth]

2. Translational invariance: The laws of nature are universal

Spacetime is the merging of space & time into a single entity & is invariant. It is an absolute

ii. Causality: the order of cause & effect cannot be reversed

1. Time can be measured in metres if we take any time interval & multiply it by a calibrating speed, such as the speed of light.

2. If S is the distance between points A & B where space is on the x-axis & time on the y-axis, we get S^{2} = (ct)^{2} + x^{2} or S^{2} = (ct)^{2} – x^{2}

3. The latter is a hyperbola & always in an area of a spacetime diagram that is above lines of 45° (the lightcone), & so respects the causality concept

4. C is the cosmic speed limit. Nothing can travel faster than c because if it did it could be used to transmit information that could violate the principle of cause & effect

iii. Distance

1. Distances in spacetime are invariant

2. Everything moves over spacetime at the same speed, c

3. Something moving in space uses up some of its fixed quota of spacetime & so leaves less for its motion through time

5. Why Does E=mc^{2}?

a. It seems that light itself is important in the structure of the universe

b. Momentum = Mass x Speed (p=mv)

c. More mass takes a greater force to move it (F=ma)

d. Mass is assumed to be an invariant quantity

e. Law of conservation of energy (unlike momentum, energy has no direction)

f. Kinectic energy = ½mv^{2}

g. Mass & energy are potentially interchangeable

h. The spacetime momentum vector shows us that in the space direction we have the old law of momentum conservation with a tweak for things moving close to the c

i. Along the time direction of the vector we get a new version of the law of conservation of energy, where ½mv^{2} gets replaced with mc^{2} so that even an object standing still has energy associated with it, E= mc^{2}

j. Energy & mass are different manifestations of the same thing

k. The spacetime momentum vector combines energy, mass & momentum

l. Matter can pop in & out of existence

m. If mass is zero then its speed is c

n. Light has no mass as far as we know

o. The energy locked away inside even quite small masses is mind-bogglingly large. A city of 100k people requires just one-trillionth of a gram of matter to be converted into energy every second, i.e. 3kg will last 100 years.

6. And Why Should We Care?

a. Mass is a measure of the latent energy stored up within matter

b. The more latent energy something has, the more massive it is

c. For anything to happen at all in the universe, energy & mass must be continually sloshing back & forth

d. Mass can be measured in electron volts. One eV is the amount of energy an electron gets when it is accelerated through a potential difference of 1 volt. It is divided by c^{2} to turn it into a measure of mass.

i. Proton = 938,272,013 eV/c^{2}

ii. Electron 510,998 eV/c^{2}

iii. Yet a Hydrogen atom has a mass that is 13.6 eV/c^{2} less than the sum of a proton & electron

iv. Note that the electron has mass

v. This ‘negative energy’ is the amount of effort to dismantle the atom; its binding energy

vi. There are smaller mass differences between the hydrogen & the sum of the proton & electron, as the electron can be in a different orbit

1. The next difference is 10.2 eV/c^{2}

2. The differences are discrete

3. The electron with the smallest mass has the electron closer to the proton (‘ground state’)

e. Left alone, a heavier thing will turn into a lighter thing if at all possible

f. Heavier versions of the hydrogen atom shed mass by emitting single particles of light (but isn’t light supposed to massless?) The excess energy is carried away by a photon

i. Our eyes are photon detectors

g. All of the atoms in nature come in a tower of energies (or masses), depending on where the electrons are, & since there is more than a single electron in every atom except hydrogen, the light emitted from them spans all the colours of the rainbow & beyond, which is why the world is colourful.

h. Molecules are less massive than their comprising atoms as it takes energy to break them apart

i. Uranium has 92 protons & 146 neutrons (in its most stable, naturally occurring form). Its half-life is 4.5 billion years, i.e. in this time half of the atoms in a lump of uranium will have spontaneously split up into lighter things, the heaviest of which is lead, & energy liberated as a result (fission)

i. There exists a mineral known as zircon. It naturally incorporates uranium into its crystalline structure, but not lead. So any lead comes from the radioactive decay of uranium & permits high precision dating.

j. When we bring two protons together one of the protons will spontaneously turn itself into a neutron & the excess positive electric charge is shed as a particle called a positron. They are +ve charged electrons. There is also a neutrino emitted. The proton & neutron come together under the influence of the strong nuclear force as a deuteron & the emission of a positron is called radioactive beta decay. (Fussion).

k. The energy released in a nuclear reaction is typically a million times the energy released in a chemical reaction

l. Temperature is essentially nothing more than a measure of the average speed of things

i. At 10,000 °C electrons are ripped from their orbits around nuclei, leaving behind a gas of protons & electrons known as plasma

ii. At 10 million °C the plasma gets transformed into a star as nuclear fusion takes place. The deuterons fuse with protons to produce helium & the energy released in the star shinning.

iii. The sun loses 4m tons of mass every second as it converts 600m tons of hydrogen into helium every second

iv. When the hydrogen starts to run out the star begins to collapse. At 100 million °C the helium begins to fuse. They form beryllium (4 protons & 4 neutrons)

v. Neutrinos flow out from the sun. About 100 billion pass through every square cm of the earth every second.

vi. Once the helium is exhausted the collapse begins again. At 500 million the °C carbon fuses. It produces heavier elements all the way up to iron.

vii. Neutron stars are more massive than the sun, but the size of a city. They emit beams of radiation known as pulsars. They can become black holes.

7. The Origin of Mass

a. The sun is several thousand times less efficient than the human body at converting mass to energy. 1kg of the sun generates 1/5,000 watts, whereas the human body generates more than 1 watt per kg.

b. The master equation of the Standard Model explains how every particle interacts with every other particle – except gravity

i. The first part explains the kinetic energy carried by W & Z particles, the photon & the gluon

ii. The second part how every particle in the universe interacts with each other & the kinetic energies of all matter particles

iii. The third part is a fix to the problem of gauge symmetry demanding no mass

c. The standard model assumes the existence of:

i. 6 types of quark

ii. 3 types of charged lepton (of which the electron is one)

iii. 3 types of neutrino

iv. For every particle there is a corresponding anti-particle

d. Protons are made mainly of 2 up quarks & one down quark

e. Neutrons of two up & one down quark

f. Up & down quarks & electrons are the predominant particles of everyday matter

g. The W & Z particles, the photon & the gluon are responsible for interactions between particles. They carry the force of interaction.

h. The photon carries the force between electrically charged particles, such as electrons & quarks. It mediates the electromagnetic field.

i. The gluon mediates the force that ‘glues’ protons together inside the atomic nucleus

j. The W particle is responsible for the interaction that turns a proton into a neutron during the formation of a deutron in stars. The force carried by the W & Z particles is extremely weak.

k. As it seems necessary to have only 4 particles to build a universe, the existence of the other 8 is a mystery

l. Particles have wavelike characteristics while remaining particles

m. Electrodynamics: the interaction of light & matter

n. The standard model unifies electromagnetism & the weak force. They are different manifestations of the same thing.

o. The masses of the W & Z particles were predicted before they were discovered

p. Nature is symmetrical – Gauge symmetry

q. An electron & anti-electron (positron) bang into each other & annihilate to produce a single photon (so we start with some mass & end up with no mass?)

r. For every observed matter particle there are around 100 billion photons

s. The stuff of the universe that makes up stars, planets & people is only a tiny residue left over after the grand annihilation of mass that took place early on in the universe’s history

t. We don’t know why the universe is not just filled with light

u. The reason that protons, electrons & photons dominate the stuff of the veryday world is they have nothing to decay into. Up & down quarks are the lightest quarks. The electron is the lightest lepton, & the photon has no mass.

v. The Higgs mechanism solves the problem of Gauge symmetry demanding no mass & the fact that there is mass

i. It is what slows things from moving at c & is responsible for the origin of mass

ii. The Higgs field holds back quarks & leptons, but allows photons t pass unimpeded

8. Warping Spacetime

a. Is spacetime warped & curved differently from place to place in the universe?

b. Einstein’s journey to special relativity was triggered by the simple question: what would it mean if c were the same for all observers?

His journey to general relativity begins with another simple question: why do all things fall to the ground with the same acceleration?

c. In a falling lift there is no experiment to distinguish between the possibility you are plummeting to the earth or floating in space. This universality of freefall is called the principle of equivalence

d. Gravity is stronger the closer to the centre of the earth you are; the gravity at your feet is stronger than at your head

e. The force of gravity is nothing more than a signal that spacetime itself is curved

f. Since gravity is found in the vicinity of matter, we might conclude that spacetime is warped in the vicinity of matter, & since E=mc^{2}, energy

g. The sun is a massive object that distorts spacetime. The earth is moving freely through spacetime but the warping of spacetime makes the earth go in circles.

h. Straight-lines in spacetime are called Geodesics

i. Gravity is geometry & all things move along straight-lines unless they are knocked off course

j. Pulsars (spinning neutron stars) were discovered in 1967. There is only one known instance of 2 pulsars circling each other (discovered in 2004). They cause gravitational waves.

k. Clocks run faster in weak gravitational fields

l. GPS: 24 satellites 20,000km up. Their spacetime is warped so that their clocks speed up at a rate of 45 microseconds per day. But because they travel at 14,000 kmph, time dilation means their clocks slow by 7 microseconds per day. The net effect of would mean the system would be out by more than 10km per day. Hence the satellite clocks are made to run 38 microseconds slower.

m. General relativity is not compatible with quantum theory