The numbers behind Hawkings’ Brief(er) History of Time, and other cosmology & quantum physics etc

Here we explore the maths – the numbers and the algebra – behind Stephen Hawkings’ books A Brief History of Time and A Briefer History of Time. We take a series of topics, which we then do some GCSE / A level maths on. Each one will give you a feel for the numbers behind the amazing physics of the Universe. They go from galaxies and further at the largest scale to the minutest secrets of atoms and their nuclei. We’ll be looking at relativity, gravity, and quantum mechanics. And, yes, Black Holes too.

These explorations arose in discussions between members of the Odiham U3A Briefer History of Time study group in Odiham, Hampshire. The input of the members was hugely valuable both in identifying questions to answer and in discussing them and writing them down in forms that non-specialist people will understand.

We will be simplifying the physics enormously, because this is tricky stuff – Rocket Science is child’s-play by comparison ! But we will still get down to ‘brass tacks’: we’ll get the numbers and the formulae. And we’ll show that they aren’t ‘black magic’ we’ll show how they arise from simple principles, many of which you already know.

We’ll often the answer wrong by a factor of 2 or π or something, by the way. Here we are concerned with understanding rather than detailed, accurate answers. Having said that, please do let me know about errors, especially if they are larger than 2 or π, or if there is something conceptually wrong or misleading.

• E = m c² , Mass Energy Equivalence
• Gravity or Not: the equivalence of gravity and centrifuges
• The Red Shift of Light (photons) by Gravity
• Gravitational Lensing: the deflection of light by Gravity
• Black Hole size: the Schwartzchild Radius
• The Planck Mass
• Virtual Particles: making force fields into particle exchanges: Heisenberg’s Uncertainty Principle as a Creator
• Relativistic Time Dilation with no more that Pythagoras’s triangles
• Planets and Radar: The Slowing of Light by Gravity

Gravity or not: the equivalence of gravity and centrifuges – and another Twin Paradox !

One of the key axioms of modern physics is that you can’t tell the difference between acceleration and gravity. One of the favourite ideas for future space stations is to use this to keep astronauts healthy. In space, in zero gravity, we humans get health problems of various sorts. One serious problem is bone density loss. After a few months in space, astronauts are weak and could easily break bones carrying out normal acitivities. The famous movie 2001, A Space Odyssey, had the solution: a rotating ring for the astronauts to live in. No one has done this yet, but basically, the ring which continuously turns relative to ‘distant stars’ (that is a tricky concept, by the way !), so that they are living in a g-acceleration field with the same or similar value to that of the Earth’s gravity. Here is how to do the design calculations for your space station:

The centripetal acceleration a needed to keep a mass going round in a circle R in radius at a an angular velocity of ω is given by a =  R ω².  So ω = √(a/R).  Noting that revs per second is given by ω/2π and the RPM, revs per minute is just 60x this, i.e. 30ω/π , we have:

RPM = 30 √(a/R) / π

Let’s put some real world numbers in there to see what comes out.  If R was 50m (that’s already BIG –  a Football field sized ring, 100m in diameter !) and if we want a comfy 0.5g, 5 m s-2, then we have to have the ring rotating at 3 RPM.  That might feel rather weird if you looked out of the windows.  But the Acceleration-Gravity Equivalence Principle means that if you don’t look outside, you’ll feel just fine !

The Centrifugal Twin Paradox

So what happens if Castor and Pollux, the spacecrew twins, decide to do an experiment on board the Starship Clarke ? Castor climbs down to the rim of the Clarke’s rotating gravity simulator ring, while his friend Pollux stays in the hub, in zero gravity.

Now in General Relativity, in the presence of a gravity field, the time t’ perceived is shifted from the time of the local observer t, by, in not-too-powerful fields by the formula

t’ = t (1+gh/ c² )

where g is the local gravity, and h is the height difference between the observers.

With the centrifuge, the gh factor is given by an integral from zero radius to the rim radius R of the expression ω² r dr, where r is radius and w the rotation angular speed of the centrifuge i.e.

“gh” = ʃ₀^R ω² r dr , which is equal to ½ ω² r²

so t’ = t (1 + ½ ω² 2R² / c² )

So Castor, the merrily rotating astronaut, will be a little younger than his stick-in-the-hub twin Pollux when they meet up compare notes – and clocks – in the Clarke’s coffee bar.

But… is this the same as the Special Relativity Twin Paradox ? Well maybe. Special Relativity would tell us that t’ = t /√(1 – v²/ c² )

but the value of v here is given by ωR, and this then gives, for smaller values of v, t’ = t (1+ ½ ω² 2R² / c² ) exactly the same as the GR formula !

BTW, The Centrifugal Twin Paradox has (sort of) been done experimentally by a guy called Walter Kündig using the redshift of X-rays. There was an earlier experiment in Harwell, Oxfordshire too. And the experiment works! See W. Kündig. Phys. Rev. 129, 2371 (1963) did it, using a 40,000 rpm centrifuge.

The Red Shift of Light (photons) by Gravity

Gravitational Lensing: the deflection of light by gravity

The deflection of light by the gravity of heavy objects like stars can be understood using classical concepts, once you accept that photons, the particle of light, have mass, by virtue of the mass given them by their energy and E = mc², and momentum too.

Lets start with a large mass M, which the photon is going to zip past at distance R, and lets look for a (very small) deflection θ using Newton’s Law : F = ma where F is force, moving mass m, acceleration. Newton’s Law can be re-expressed as F = m dp/dt, i.e. force is equal to mass times the rate of change of momentum p with time t.

We can see by contemplating the geometry of the situation of a near-miss photon that it will be mostly deflected when it is around R from closest approach, and that it will be in that region for a short time Δt, which is given by R / c where c is the speed of light.

Now the force F applied during that brief time will be on the order of :

F = GmM/ R²

and we can think of a small change of momentum Δp taking place in Δt. G is the gravitational constant and m is the mass accelerated, which we can think of as E /c² , or hf /c² , since the energy E of a photon of frequency f is hf. So we then have:

F = Δp /Δt = c Δp /R = GEM /(c² R² )

So Δp = GEM /(c³ R ) and, noting that Δp /p will give the angular deflection of the photon, and that p for a photon is of course E/c, we have:

θr = G M /(c² R) in radians which is
θ = 180 G M /(π c² R) in degrees

Now lets put some real-world numbers in this. For a star of mass 2 x 10^30 kg mass, a photon passing at radius 7 x 10^9 m, we have, with G = 7 x 10-11, we have θ = 2 x 10-4 degrees for the deflection angle. Not much !

The Schwartzchild Radius: the edge of a Black Hole

“The undiscovered country from whose bourn no traveller returns“, Shakespeare (Hamlet’s soliloquy)

The radius of the ‘Event Horizon’, the edge, of a Black Hole is the distance from the centre of mass of the Blacl Hole from which no traveller – or anyone or anything else – returns. It is simply the radius R at which, for a sufficiently large mass, the escape velocity Vesc is that of the speed of light.

The gravitational potential energy of a small mass m around a mass M is given by

G m M /R

This follows from integration of Newton’s Gravity Law from radius R to infinity. And the energy needed to eject a particle of mass m from a distance R from that mass is simply given by equating the particles Kinetic Energy with the gravitational potential energy:

½ m Vesc²  = G m M /R; So Vesc = √ (2 G M /R)

Now if Vesc = c, then we have what we need to get the event horizon Rs:

Rs = 2 G M / c²

Let’s put some real-world numbers in. What is Rs for our Sun ? Msun =
2 x 10^30 kg, so Rs turns out to be 3km. So unless our sun shrinks to a tiny little dot (as viewed from Earth), a shrink factor of a billion or so, it isn’t going to become a Black Hole any time soon. A really big star can become a black hole, of course, especially because really big stars have a density which reflects nuclear size, 3×10^14 denser than our sun, 3×10^17kg/m3. Take a star of that 3×10^17 density and mass 10x that of the sun, and you can get a Schwartzchild radius of 30km, which would then be bigger than the star: it would be a Black Hole. (Our sun has around the density of water, atomic density, normal matter density, 1000 kg/m3 or so).

This leads on to questions, of course, on what we can know about a black hole, given that nothing comes back. From a distance, we can detect its gravitational field, which reveals its mass, of course. Presumably you can tell the electric charge on a black hole, or whether it has magnetic field, again by the field at a distance. (But surely these fields are mediated by photons – which can’t escape ? Help – ! Does someone know what is going on here ?)

The Planck Mass: largest fundamental particle there can be ?

You might think that the largest fundamental particle is going to really small – like an atom or maybe just a shade bigger. Surely its going to be made of lots of other things if its bigger ? Well, maybe, but there have been discoveries of some quite big fundamental particles in recent years. The top quark, which was once sought in data on the TASSO experiment I worked on (it wasn’t there), has a rest-mass of around 173 GeV, which is about 173 proton masses, or the same as a pretty heavy atom. But a top quark isn’t, like an atom or Rhenium, made of 75 protons and 100ish neutrons and a cloud of electrons. A top quark just a point-like thing, with charge and mass and spin and strong interacting with stuff around it, but all these properties at that point, no extended structure.

Now one of the magic distances in physics is the Schwartzchild Radius Rs (see above).

Rs = 2 G M /c²

where G is the gravitational constant and M the mass of the star or black hole.  Now photons are a fundamental particle of zero rest mass, they are particles of light, the quantized form of waves in the electromagnetic field.  The energy (and mass) of a photon can vary continuously from so close to zero as to be undetectable through radio waves, radar waves, infra-red, optical, ultraviolet, X-ray and gamma ray radiation as the energy in the wave, and its frequency increases. The energy of a photon is simply equal to hf, where h is Planck’s constant, and f is the frequency.  The wavelength λ of a photon is simply found by noting that all photons travel at the speed of light, c, and hence λ = c /f or λ = c h /E or h /(m c)

You will notice that the wavelength of a photon goes down as the mass goes up.  But the Schwarzchild radius goes up as the mass goes up. 

So, what happens when the Schwartzchild radius of a photon is equal to the wavelength of the radiation ?  Surely, because at a certain energy a photon we won’t be able to peek behind the ‘curtain’ – the event horizon – of the object, it doesn’t make any sense to talk about fundamental particle.  It might be fundamental, it might not.  The mass of photon at which this happens is called the Planck mass Mp, and it is given by equating the two lengths

Rs = 2 G M /c²  = h /(M c) = λ

That leads to Mp = √(h c / 2G)

Mp has a value of around 10-8 kg . (Depending upon what convention you adopt in the equations above, you can get numbers which differ by a factor of π or so.)  This is Super Big in terms fundamental particles.  It is after all, 1/100^th of a milligram.  It is of the order of 10¹⁵ or more atoms.   It is so big that you could maybe detect the weight of just a few Planck masses if they were dropped on your finger.  Flea eggs weigh about this much.  A lightly written full stop could weigh more !  (Take 10 km length of a biro track, from 1cm³ of biro ink, mass 1g.  A full stop is just 0.2mm long, i.e. 2×10^-8 g.)

Virtual Particles: making force fields into particle exchanges: Heisenberg’s Uncertainty Principle as a Creator of particles

Force fields can be created by the spontaneous creation of temporary particles, which pop into existence, as allowed for by the Uncertainty Principle, and then pop out of existence, having pushed a more permanent particle a little bit. A photon can create an electric field in this way. Let’s do some figuring on that…

From Newton, we have force F = dp/dt, where p is momentum. 

From Maxwell + photon pressure, we have E  = pc, or P  = E/c for a photon

From Heisenberg, we have Δt ΔE = h connecting uncertainty in energy & time. And  Δp Δx = h, connecting uncertainty in momentum and position

F ~ Δp/Δt ~ h/(Δx Δt)

Further substitute Δt = h/ΔE = h/(Δp c) = Δx/c,   and you have

F ~ h/(Δx (Δx/c)), so F ~ hc/ Δx²

Identifying Δx with the distance R between interacting masses, we have:

F ~ hc / R²     which is a 1/R² field  QED #1   !

Now starting from F ~ hc / R² we can relate this to the Coulomb force by noting that we should probably be reducing the force by the fine structure constant α to get electric field.  A strong field would use the force as we derived above, perhaps.  But in the electromagnetic world, we need to multiply the strength of the interaction by α = 1/137 = e²/(4π ε₀ h c).  So then we have:

F ~ α hc / R²  = [e²/(4π ε₀ h c)] . h c / R², which leads us to …

F ~ e²/(4π ε₀ R²), which is Coulomb’s law between 2 electrons !  QED #2

Particles other than photons can be created. Pairs of particles anti-particles such as electrons and positrons can plop into existence in this way. The spontaneous creation of particles also happens around the event horizon of Black Holes, particularly small ones. Stephen Hawking recognised this process as a possibility and figured out that pairs of particles could plop into existence and then one of the pair goes towards the Black Hole and disappears while the other whizzes onwards and can be seen by us. This remarkable process seems to actually happen.

Virtual Massive Particles: Quarks, Gluons and the Yukawa Potential

The spontaneous creation of strongly interacting particles which have a rest mass can account for the strong force, just as the zero rest mass photons can account for the electric force. Hideki Yukawa thought this up in the 1960s, and he was basically right, although the particle that people picked on at the time (the muon) which they thought was the massive particle he described was not right. He came up with a force field due to massive particles mass m of the form:

V = A exp(-B m r ) /r

where r is range, A and B are constants, V is the potential (the force F of the field is the differential with respect to r of this, i.e. F = dV/dr. That means that the force law goes as exp(-B m r)/ r² .

Based on this, Yukawa estimated that a particle of mass around 200 MeV might account for strong nuclear forces. We now know that the pions, a family of particles made up of quark and an anti-quark do indeed provide forces between neutron and protons and strongly ‘glue’ them together. There are strong field forces between the quarks too, but these are mediated by gluons, which are, along with the quarks more fundamental particles than the pions and nucleons.

Just look at this equation for a moment: what it is saying is that for long distances, the potential is an exponentially small – far smaller, and getting smaller far quicker than 1/r or 1/ r² . And at short distances, the exponential factor is just 1, so the whole thing is just the same as the Coulomb law – its just 1/ r² for the variation of force with distance r.

Just suppose, for the moment that you can create particles of any mass you like, but that this will only happen at a very low rate if you choose a large mass. This a bit like in chemistry, when you need an activation energy to cause a reaction to take place. The reaction will take place, but only at a low rate, governed by an expression like exp(-Ea/kT), where Ea is the activation energy you are need, k is the Boltzmann constant, and T is the absolute temperature of the tube of chemical molecules you are thinking about. kT has the same energy dimension as Ea, so the the exponential is correctly dimensionless.

Now change gear and switch from chemistry and its swirling molecules to the world of quantum mechanical particles and Heisenberg’s Uncertainty Principle. By analogy, maybe we can have heavy particles formed, but only at a rate given by exp(-ΔE Δt / h), because we can have any energy ΔE we like, as long as the time Δt it ‘plops’ into existence for is limited to around the Planck constant h. Now we can get a range r into the equation simply by noting that the speed of the particle will be v, and that r = v Δt. We can further assume that for reasonably energetic particles, the v can be roughly said to be c – if you pick energies which are some reasonable fraction or multiple of the rest mass this is true. So r = c Δt .

So we now have an exponential factor of exp(-ΔE Δr / hc). Noting further that E = mc² we can then make this factor into exp(-m Δt c /h). Using this probability factor as a multiplier for the 1 / r² range equation we had for massless photons above, we now have:

Force F = = exp(-m Δr c v / h) / r²

we now have for the Potential V, something like V = exp(-m Δr c / h) / r which is just like the Yukawa Potential. QED !

Relativistic Time Dilation with no more that Pythagoras’s triangles

Train width L, moving at v.  Inside the train, a light beam pulse hits the sides at intervals of just L/c = t.   Viewed from the platform, light pulse has to go further, from side to side L and along by vt’, along a diagonal.  Note that light always goes at speed c, no matter what speed the train goes. 

We now have a triangle, and that means… we can Pythagoras !. Pythagorus says that the diagonal length d is given by

d² = a² + b²

Here a = L, and b = vt’

So  d² = L² + vt’²

t’ = d/c

where time intervals seen from platform between pulse impacts is t’

ct’ = d, so c² t’² = d²

c² t’² = L² + vt’²

(c² – v²)t’ ² = L²

t’ ² = L²/(c² – v²)

t’ ² = L²/ (c² (1 – v²/c²)) , So t’ = L / (c √(1 – v² / c²))

But L / c = t, so

t’ = t /√(1 – v² / c²)

So the time t’ is increased relative to time t by a factor which is close to 1, unless v is near the speed of light.  With ordinary everyday speeds of at most a few hundred metres per second, the increase in t’ can only be measured with super-accurate atomic clocks.  These can measure time within one part in 10¹⁵ or 10¹⁶, and hence measure time to the nanosecond accuracy over a hundred hours, needed to see this effect when flying around the world in a jet plane.

Planets and Radar: The Slowing of Light by Gravity

Radar works by transmitting a pulse of radio waves in a beam and measuring the time that echoes from objects in its beam come back. It isn’t easy on the face of it: the strength of the outgoing beam decreases at 1/R².  And echo strength also has a 1/R² law.  So the echo signal decreases as 1/R⁴ with distance.  Surprisingly, however, you can make radar work over large distances, by using powerful pulses, narrow beams, looking for strong echoes, and amplifying those echoes like crazy. 

 A radar set is surprisingly bad at measuring the direction of something.  However, it is exceedingly good at measuring the time that an echo comes back and hence good at measuring the distance to that something.  It turns out that this ability to time pulses very accurately can be made into remarkable experiment to confirm the time dilatation due to gravity near the Sun.

The experiment can be done in several ways, but the easiest one to think about is to send a pulse toward Venus or Mars when they are on the opposite side of the sun (but still with line of sight).  Time the pulse coming back, and you see a slowing of the return pulse just as the line of sight goes close in to the surface of the sun.

Now consider a radar beam path from Earth to Venus and that grazes the sun, and is little deflected from a straight line (see above ‘Gravitational Lensing’ for why that is a good assumption). 

Now while they in that high gravity field region, the radar pulse waves will go blue shifted (see above).  Or equivalently, you could say that the clock ran slower while the pulse went through there and made the frequency seem higher.  The speed of the radar pulses, for a local observer in the gravity field, will be the same as normal: c the speed of light.  But the clocks run slower in that region, so the speed of radar pulses must be slowed down as they went through there, so as not to upset that sacrosanct axiom, the constancy of the speed of light.  So the radar pulse time delay is the same effect as blue shift and red shift.

Now let’s talk red shift in numbers (see above again).  Red shift in a gravity potential such as we have on the surface of the Earth is just given by:

Δf/ f = g ΔH /c²

where ΔH is the height between transmitter and receiver. This shift is thus given simply by the the change in gravitational potential divided by c².  The gravitational potential between the two points is just the potential energy difference of a test mass m between the two points, which is m g ΔH, divided by m, giving just m g ΔH.  Now the gravitational potential difference GPD  from the sun’s surface to the Earth is simply given by integrating the Newtonian gravity force Fg with respect to the distance r and dividing by the test mass m:

Fg = G m M / r²

GPD = ʃ_r^∞  G M / r² dr  = G  M /r   

This compares to the g ΔH above.  So the redshift is given by dividing by c² to give

Δf/ f = G M / r c²

For the redshift at any point along a track, where f is the frequency, and Δf the shift in frequency.

We now need to calculate the total time delay, which can be estimated by simply integrating the redshift at any point, which gives the change in the clock and hence the (from an external viewpoint) change in the speed of light.  So the effective speed of light any point will be c (1 – Δf/ f ).  We then need to calculate the transit time T’ for a pulse, and then subtract it from the gravity-free transit time T. The Earth’s orbital radius is Re, and we assume that this is pretty much the same as Venus. Also, take the radius of the Sun id d.

T = ʃ₀^Re 1/c  dr  =  [ r/c ] ₀^Re  =  Re/c

T’ = ʃ₀^Re  (1/c(1 – Δf/ f) dr ,  which approximately = ʃ₀^Re  ((1 + Δf/ f)/c) dr.  Now substituting G M / r c² for Δf/ f we get to

T’ =  ʃ₀^Re  ((1 + G M/r c² )/c) dr = Re/c  + ʃ₀^Re  (G M/c³) dr/r  = Re/c + G M/c³ [ ln(Re/d] substituting the rather more reasonable d (sun radius) for 0 in the integral.  We now need to note that the radar covers approximately 4x this much distance, going to the sun, then to Venus, then back to the Sun, then back to Earth.  Which gives us, for T’ – T, the extra time taken by the radar pulse, the delay ΔT:

ΔT   ~  4 G M/c³ [ ln(Re/d]

Now put some real numbers into this, and you get answers like 200 microseconds, which is roughly what is seen when the experiment is done.

Random Word Pairs: my results…

You have found them ! This is where I put the ‘answers’ that I came up with on the random guide word pairs exercise in the Brainstorming and Inventing page of Saturdayscience.org . I did come up with other concepts too, but I think this is enough to give you a flavour of the rich soup of ideas that can come out of random guide words.

roll test: Could you check the roundness of cylindrical or spherical objects by rolling them along, perhaps down a very gentle slope, video imaging them, and measuring how much their centres accelerate and decelerate as they roll ?

gold test: You can test whether something is made of gold by rubbing it against the right grade of abrasive, often an unglazed ceramic tile. Real gold makes a characteristic yellow.

contract coach: A large passenger-carrying coach can be difficult to park. So what about a vehicle which can telescope to a shorter length, so occupying less space. With the vehicle systems and wheels in the front and rear, maybe these can be jacked together by large leadscrews, with the passenger seats pivotting and stacking together inside ?

vacuum coach: What about a vacuum railroad (as seen in my books on a small scale !

gold beat: This reminded me that gold is the metal which was first converted into ultra-thin sheets, just a few microns thick, or even below one micron, by beating it with hammers, using ‘gold-beaters skin’ (ox-intestine) interleaved with thin layers of gold.

light beat: Everyone has been to discos or other dance venues where the lights switch on and off to the beat of the music.

roll finger: What about a higher-security variant of the well-known fingerprint ID system for accessing computer and other equipment ? You would just roll your finger over the scanner, so giving much more biometric data for the security check.

gas finger / air ringer: how about handling of delicate objects in a non-touch manner by means of directed jets of air or gas, as is done with laser for microscopic objects (see below) ?

light finger: the academic world, at least, now has systems using lasers, often dubbed ‘optical tweezers’ to handle very small objects in mid-air, not touching them with any solid object.

Δ½

.