Category: Physics

The Lorentz Transformation – a minimal proof

Lorentz
Einstein

You can find many ways in the textbooks to derive the Lorentz Transformation, starting from Einstein’s famous two postulates: that the laws of physics are the same in all inertial frames, and that the speed of light is a constant. You can do it in one big chunk, or by starting with length contraction and time dilation.

What I want to do here is show a proof which requires only one, surprisingly minimal, assumption, and  which relegates ‘light’ to its proper place as a subsidiary phenomenon. This is the opposite of the order which is usually taught, so this is not the sort of proof  you get in Relativity101, but after you’ve learnt and are happy with the standard proofs, I think you’ll appreciate this one.

We make some basic assumptions – as indeed we do in a conventional proof, though they’re not usually spelt out.  Events occur in continuous time t and continuous space r, though for simplicity we’ll just consider one space dimension x. Space and time are isotropic and homogeneous – there are no special times or places. We can plot events in space-time diagrams, where the t axis is calibrated using repeated identical processes like the swing of a pendulum or the vibrations of a crystal, and the x axis is calibrated using stationary identical rods.

Events cause, and are caused by, other events. For a pair of events A and B it could be that A→B, A has a (possible) effect on B, or that B→A, B has a (possible) effect on A. In the first case we say that A lies in the past of B, and B is in the future of A. In the second case it’s the other way round. We dismiss the possibility that both A→B and B→A, as that leads to paradoxes of the killing-your-grandfather variety. But what about the possibility that neither A→B nor B→A: that there can be pairs of events for which neither can influence the other?

There’s not an obvious answer. If you were designing a universe you could insist that any pair of events must have a causal connection one way or the other, or you could allow the equivalent of the ‘Don’t know’ box. The choice is not forced on us by logic. But let’s suppose that we do live in a universe where this directed link between events is optional rather than compulsory:

There are pairs of events which are not causally connected.

I promised you a single assumption: there it is.  Now let’s build on it.

For any event there must be some events which are not causally connected. The assumption says this is for true for some events, but all events must be similar (as space and time are homogeneous) , so this is true in general.  So we can drawa  space-time diagram showing the events  that are  past, future, and elsewhere for an event at the origin.

Events lie in definite regions

Causality is transitive: if A→B and B→ C then A→ C, as A can influence C through B. That means that  at any particular point x, events that are in A‘ s past must be followed by elsewhere events and then future events. They can’t be mixed up.  The events occur in defined regions

 

 

The Elsewhere region extends to the origin

Even at small distances there must be elsewhere events – if there were some minimum distance from A, Δ, within which all events were either past or future,  and B is the event at Δ on the division between past and future, then all events within 2Δ of A must be in the past and future, and so on for 3,4,5….

 

 

The Elsewhere region is a simple wedge

The lines separating the past, elsewhere and future regions must be straight lines going through the origin. For any point B on the future light cone of A, the gradient of the line separating B‘ s elsewhere and future must have the same gradient as the light cone for A at x=0. But the future light cone of B defines the future light cone of A. So the gradient must be constant all the way. (The same applies for the past light cone, and symmetry requires that the gradient have the same magnitude.)

So to re-cap: first we establish that there are elsewhere events, then that they lie in regions, then that these regions go all the way to the origin, and finally that the shape of the elsewhere region is a simple double wedge. (It’s called a ‘light cone’ as you can imagine extending the picture to two space dimensions by rotating these 2D pictures about the vertical axis, but you probably knew that already.)

Out of this picture a number emerges: the gradient of the line dividing the elsewhere region from the future (or the past). We have no way of knowing what its value is – only that it is finite. It describes the speed of the fastest possible causal signal and we will, of course, denote it by c. It can be viewed as a fundamental property of the universe, or as a way of relating time measurement units to space ones.

Now we’re on more familiar ground. If an event that we denote by (x,t) is observed by someone in a different inertial frame moving at some constant speed relative to the first, they will ascribe different numbers (x’,t’). What is the transformation (x,t)→(x’,t’)?

  1. Let’s assume that zeros are adjusted so that (0,0) is just (0,0). That’s trivial.
  2. We require that vector equations remain true: if (xA,tA)=(xB,tB)+(xC,tC) then  (x’A,t’A)=(x’B,t’B)+(x’C,t’C). That limits us to linear transformations x’=Ax+Bt; t’=Cx+Dt. So the transformation is completely described by 4 parameters A,B,C and D.
  3. The inverse transform  (x’,t’) to (x,t) must be the same, except that the direction of the speed has changed. That’s the equivalent of changing the sign of x or t. So x=Ax’-Bt’; t=-Cx’+Dt’.   The transformation to the new frame and back again must take us exactly back to what we started with, i.e.  A(Ax+Bt)-B(Cx+Dt)=x.  From which we must have A=D and A2-BC=1. The four parameters are reduced to two.
  4. Finally we impose the requirement that the new co-ordinates (x’,t’) must lie in the same sector (past, present, or elsewhere) as the old. In particular, if x=ct then x’=ct’. That means Act+Bt=c(Cct+Dt) and using A=D from the previous paragraph, this shows B=c2C. The two parameters are reduced to one. This is most neatly expressed by introducing v=-B/A, as then A2-BC=1 gives our old friend A=1/√(1-v2/c2) and substituting A, B, C and D gives the familiar form of the Lorentz transformations.

    The Lorentz Transformation

Inspecting these shows that v, which we introduced as a parameter, describes the motion of the point x’=0, the origin of the primed frame, in the original frame, i.e. the speed of one frame with respect to the other.

A bit of algebra shows that the ‘interval’ of an event is the same: x2-c2t2=x’2-c2t’2. Which is neat, showing that the points lie on a hyperbola of which the light-cone crossed-lines is the limiting case, so they cannot move between sectors . But we didn’t have to assume that the interval is unchanged, only that an interval of zero remains zero.

So the Lorentz Transformation springs from the basic causal structure of space-time, assuming that not all events are causally connected one way or the other, with c the speed of the fastest causal signal, whatever that happens to be. Length contraction and time dilation follow from this. Then you discover that if you have Coulomb’s Law type electrostatics the Lorentz Transformations give you magnetism and Maxwell’s Equations emerge. These have wavelike solutions with wave velocity  c. 

In terms of logical argument, the causal structure of the universe just happens to include the possibility that 2 events cannot affect one another in either way. This fundamental property leads to relativity and the Lorentz Transformation,  which leads to electromagnetism, which then leads to EM waves and light, even though historically and pedagogically the sequence is presented the other way round.

The collapse of the wave function, and two sorts of time

Suppose a star emits a photon. Its wave function spreads over space, perhaps over light years.

On a far planet, a poet is looking at the night sky: if they  see that photon they will be inspired to write a great poem. Meanwhile at the other edge of the galaxy there is a pig, also, for its own porcine reasons, gazing skywards, and the photon wave function also passes through its retina. The pig may see the photon or the poet may see the photon. (Or, of course, it may be seen by neither.)  But it absolutely cannot be seen by both of them. If the photon materialises in an eyeball of one, it cannot do so in an eyeball of the other.  The wave function collapses, instantaneously and simultaneously across all space.

This rings alarm bells. The Theory of Relativity tells us loud and clear that ‘simultaneity’ is a dirty word, but it really does apply here. The arrival of the photon wave function at the pig and the poet may be simultaneous in the picture above, but there will be reference frames in which it arrives at the pig before the poet, and frames where the poet comes before the pig. Whatever frame you’re working in, the wave function collapse is simultaneous everywhere in that frame.

It’s the truly spooky bit of quantum mechanics that really nobody understands. An object – be it a photon, an electron, an atom, or a cat – has a wave function, and we can study the behaviour of that wave function by solving complicated differential equations. Then a measurement is made of some property, and the wave function changes randomly but instantly into a state where that property is defined. If the wave function is ‘real’ it defies relativity. If it is not ‘real’ then what is?

Most physicists solve this puzzle by ignoring it – the so-called ‘shut up and calculate’ school. I’m not going to explain the puzzle here – nobody can. What I do want to do is show that it is not really new, but linked to an older one.

Let’s introduce a logical formalism for discussing the collapse of the wave function in an ordered and well-defined way.  This was done by John von Neumann in his “Mathematical Foundations of Quantum Mechanics”, (Springer 1932, English translation by R T Beyer, Princeton, 1955) and what follows is his development, with modernised notation.

We need to introduce a neat concept called the density matrix.  This combines the two sorts of uncertainty that we have to deal with: quantum uncertainty and the established uncertainty of statistics and thermodynamics.

First, from quantum mechanics we take the idea of a basis set of states |i>, which are eigenstates of some measurement operator  (so Â |i> = ai|i>). In what follows I’ll use as an example the simple case where there are just two states describing the spin of an electron as up |↑> or down |↓>;  they could also be the states |x> of delta functions at particular positions, or the pure sine wave states |p> that have definite wavelength and thus definite momentum.

Secondly, from Statistical Mechanics we take the idea of a large ensemble of N states |ν>, which are the actual states of many systems in equilibrium with one another.  The  |ν> states are (in general) not the same as the |i> states but they can be written in terms of them: |ν> = Σi <i|ν>  |i>, because the |i> states are a basis set. <i|ν>  is a number, the i-component of the ν state in the ensemble.

Right, that’s the apparatus in place. The density matrix is just the average over the ensemble of the Cartesian product of the components

ρij=(1/N) Σν <i|ν> <ν|j>

What can you do with it? Well the diagonal elements ρii correspond to the average over the ensemble of | <ν|i> |2, which, according to the Born interpretation, is the probability of finding state |ν> in state |i> if you measure it with  . It tells us the probability of getting the result ai where the probability includes both the quantum uncertainty and the statistical uncertainty from the ensemble.

Density matrix for an up/down mixture

Let’s take an example.  If we take an ensemble of many electrons of which half are spin up and half are spin down then the matrix has 1/2  on each diagonal element and the off-diagonals are zero, because each    |ν> is either up or down, so the product of  <ν|↑> and <ν|↓> is always zero.  The diagonal tells us that if you pick an electron at random from the ensemble, there’s a 50% chance each for it being spin up or spin down.

Now let’s get a bit less trivial. We’ll take a sample of spin-down electrons, all the same this time,  and then rotate them 90 degrees about the y axis, so they point in the +x direction.

Rotating a spinor from down to sideways
Density matrix for electrons pointing sideways, along the x axis

These states give the density matrix which has the same diagonal elements as the last one, but non-zero off-diagonal elements. The diagonal elements tell us that there is, again, a 50:50 chance of detecting the electron in an up or down state, though this time it’s because of quantum uncertainty.

If the diagonal elements give the probabilities, you might wonder whether there’s any point to the off-diagonal elements. But they do play a part. Suppose that, for both examples, we rotate the spins buy another 90 degrees before we measure them.  Under a rotation R the density matrix becomes  ρ’=RρR†. A bit of  matrix arithmetic shows the matrix for the first example is unchanged, on the second example, the sideways states, is

which is obvious, with hindsight. The second rotation converts the spin direction from the x to the z axis,  so they will always be in the spin up state if you measure them, whereas for the first example the result is still 50:50 unpredictable.

So the density matrix encompasses quantum uncertainties, which may become certain if you ask a different question, and statistical uncertainties which cannot.  Diagonal elements give probabilities and off-diagonal elements contain information on the degree of quantum coherence. If you want to know more about it, try Feynman’s textbook  “Statistical Mechanics: a set of lectures” (CRC press, 1998).

As time goes by the states will evolve, and the evolution of the density matrix has the apparently simple form ρ’=e-iHt/ℏ ρ eiHt/ℏ. I say ‘apparently simple’ because H, which is being exponentiated, is a matrix. But this is standard quantum mechanics and the techniques exist to handle it. The |i> wave functions  oscillate at their characteristic frequencies.

But the matrix can also describe the effect of a measurement of the quantity corresponding to the operator Â. The measurement asks of each state |ν > which of the |i> basis states it belongs to: if it is in one of those states it stays in it, if it is in a superposition then it will drop into one of the|i>, the probability for each being   |<i|ν >|2.  So the density matrix becomes ρ’ij= δij ρij. The diagonal elements are preserved, and all the off-diagonal elements vanish, as each member of the ensemble is now in a definite basis state.

To say ‘a measurement zeroes all the off-diagonal terms of the density matrix’ expresses the ‘collapse of the wave function’ correctly and completely, but in a smooth and non-sensational way.

These two equations,  ρ’=e-iHt/ℏ ρ eiHt/ℏ and ρ’ij= δij ρij, both describe how a system changes with time. They are different because the ‘time’ is different.

It is often (unkindly) said that there are as many philosophies of time as there are philosophers of time, but if you ignore the weirdest ones and brush over the details there are basically two.

In first concept, the time of Parmenides, Plato, Newton, Einstein and Hawking,  sometimes called ‘Being Time‘,  is the 4th dimension, analogous to the dimensions of space.  Events take place in a 4 dimensional space and relativity describes, in great and successful detail, the metric of that space.  Which is fine. But the ‘block universe’ this describes has no sense of direction, no sense of time passing. There is no difference between ‘earlier’ and ‘later’. A space-time diagram completely describes events and the world-lines of objects, including ourselves, but contains nothing to say that you and I are at a particular ‘now’ point on our world-lines, and are making our way along them.  This was encapsulated in the very moving letter that Einstein wrote to the widow of his great friend Michele Besso:

“For those of us who believe in physics, the distinction between past, present and future is only a stubbornly persistent illusion.”

The second concept of time, due to Heraclitus, Aristotle, Leibniz and Heidegger, sometimes called ‘Becoming Time‘, is an ordering relation between events. If event A causes event B, then A comes before B, and B comes after A. We write A→B.  (Actually A→B is shorthand for ‘if a choice is made as to what happens at A, that can affect what happens at B’. If I shoot an arrow (A) it hits the target (B); if I refrain then it does not. If I shoot an arrow A and, due to my incompetence, it misses the target B, we can still say A→B because it might have done.) This is a transitive relation, so if A→B and B→C then A→C , and we can establish an order for all possible events. (Relativity says there are some events in which neither A→B nor B→A, but this can be handled.)  Which is fine. the sequence has a sense of direction, and the past and future are clearly different.  But there is no metric. Events are ordered like competitors in a race where only the final places are given – we know that A, B and C came first, second, and third, but not their individual timings.

Being-time is like a clock with a continuous movement. The hands sweep round smoothly – but the ‘clockwise’ direction is an arbitrary convention. Becoming-time is like a tear-off calendar: the present event is visible on the top, future events are in the stack beneath, past events are in the wastebasket.

So the dual nature of time is a longstanding and unsolved puzzle. We’re not going to solve it here. But we can note that the two ways in which the density matrix changes,  ρ’=e-iHt/ℏ ρ eiHt/ℏ and ρ’ij= δij ρij, correspond to the two different sorts of time. The wave function develops in being-time; measurements are made (and the wave function ‘collapses’)  in becoming-time.  The collapse of the wave function is not a new puzzle produced by quantum mechanics, just a new form of an old puzzle that philosophers have argued about since the time of the ancient Greeks.

 

The Legacy of Stephen Hawking

Few great scientists are famous to the general public, but Stephen Hawking achieved rock star status, thanks to his best selling ‘Brief History of Time’, and to the epic saga of his struggle not merely to survive but to master the disease that struck him – his inspirational refusal to give in to the unfair twists of fate.   But the man or woman in the street would claim no knowledge of his scientific achievements: what it was that made his reputation in the scientific community, enabling him to write and speak to the public with such authority.

Although there was a lot of solid first rate work that gave him a reputation in the specialised community of cosmologists, the big item was undoubtedly his development of the theory of Hawking Radiation (as we now call it).  This was a complete paradigm shift, revolutionising our ideas of black holes, but based on a simple and understandable concept. 

Black holes – at the time they were just an oddity predicted by equations, today they are matter of fact items of galactic astronomy – occur when a body is so massive that its escape velocity is more than the velocity of light, so nothing can escape its gravitational pull.  Such black hole will accumulate matter as it falls into it, never to re-emerge: it will just get more and more massive forever, or until the end of the universe.  

Hawking took this description from General Relativity, and linked it to the picture of the vacuum that comes from particle physics. At the quantum level the vacuum is not empty but abuzz with virtual particle-antiparticle pairs (predominantly electrons and positrons) being created out of nothing and then re-merging into nothing. Hawking pointed out that in a very strong gravitational field, such as you get just outside black holes, one of these particles could fall down into the gravity well, giving the other enough energy to become real. The surface of a black hole will radiate electrons and positrons (“Hawking Radiation”), losing energy and mass as it does so. It will not exist forever, but will evaporate.  Our whole concept of black holes changed: they are not just gobblers of everything, but dynamic objects that can be formed and also destroyed.

As well as its own intrinsic importance, his proposal showed how cosmology and particle physics could be fruitfully combined. Previously the former, which involves physics at the largest scale, had been very separate from the smallest-scale physics of the latter.  Theoretical physicists might specialise in General Relativity or Quantum Field Theory, but very few worked on both, and never at the same time.  Today that’s all changed, and our attempts to understand the earliest stages of the big bang, and the most fundamental laws of nature, are framed using combinations of microscopic and megascopic physics.

So he wasn’t just the author of ‘that book by that wheelchair guy’, as Homer Simpson put it, or the character that enabled Eddie Redmayne to win his Oscar. He revolutionised our ideas about black holes and their role in the universe, and he showed the way to a combination of the physics of the very small and the very large which is being carried forward today. That is his gift to us and future generations.