# Excitement about learning complicated things

**Posted:**29 March 2014

**Filed under:**Childhood, Culture, Education, Language, Mathematics, Philosophy, Physics, Science |

**Tags:**culture, Education, Jargon, language, Linear Algebra, Mid-life crisis, Modules, Rings, Sheldon Cooper, Symbols, The Big Bang Theory 1 Comment

Do we all have things we look forward to learning? In early 1983 I was very excited about learning the meaning of ‘*Direct decomposition of a finitely generated module over a principal ideal domain*’. That was the name of the central section of the main text for my second year uni algebra course. The text was ‘Rings, Modules and Linear Algebra’ by Hartley and Hawkes.

I understood from the course summary and the blurb on the textbook that learning how to do the activity described by the above italicised phrase was one of the main goals of this course.

What I found particularly appealing about the goal was that it referred to three different things, *none* of which I knew what they were. What is direct decomposition? Dunno! What is a finitely generated module? Dunno! What is a principal ideal domain? Dunno!

To add to the titillating obscurity of the subject, each of the three things was qualified by an adjective or adverb. The first two things only had one qualification each: *direct* decomposition rather than just any old ordinary decomposition, and *finitely-generated* module rather than just a commonorgarden module. But the third thing actually had *two* qualifications. This was not just an *ideal* domain or a *principal* domain but it was a *principal, ideal* domain. How exciting is that?

(Mathematicians may wish to object that the comma does not belong, and that the word ‘principal’ actually qualifies the word ‘ideal’ rather than the word ‘domain’, so that ‘a principal ideal is a thing’, whereas a ‘principal domain’ and an ‘ideal domain’ are ‘not a thing’, to borrow the ‘thing’ terminology that seems to be so popular amongst today’s young people. But let’s not allow this minor technical point to spoil a good story).

Why, you might wonder, was I so fascinated by a topic with so much jargon in it? What, you might ask, and perhaps not entirely without reason, is my *problem*?

The answer, I think, is that I have a fascination with jargon, and more generally with weird, obscure and bizarre things. The jargon has to be *justified* though. I have no interest in the jargon invented by some professions (merchant bankers and stock brokers in particular come to mind) to describe perfectly ordinary concepts in obscure ways in order to make them appear clever to others and justify their exorbitant fees. No, what excites me is jargon that people have no choice but to invent because the concepts it is describing are so abstract and complex that ordinary words are useless.

The jargon of mathematics and many of the sciences is of this justified type. When a physicist tells you that the steps necessary for predicting the perihelion of Mercury include performing a contraction of the Riemann tensor and another contraction of the Ricci tensor in a Swarzschild spacetime, or that the possible states of a carbon atom form an exterior algebra generated by the Hilbert spaces of electrons, neutrons and protons, she is describing things that cannot be described in plain language. And yet they are real things, not just insubstantial ideas. They are things that enable humans to perform wonders.

More to the point though, when Doctor Who announces that he has ‘*Reversed the polarity of the neutron flow’*, we learn that as a consequence of this linguistic peculiarity the universe, which was about to have its space time continuum rent asunder (ouch!), has been saved. This is no postmodernist proliferating syllables for the sake of mystery and pomposity, as in:

“We can clearly see that there is no bi-univocal correspondence between linear signifying links or archi-writing, depending on the author, and this multireferential, multi-dimensional machinic catalysis.” [Felix Guattari]

At heart, I think my fascination with genuine jargon is just part of an insatiable curiosity. The world is so full of intricate patterns and amazing phenomena, yet we will only ever get to see a tiny part of what is there. I want to find out what I can while I have the chance.

I can’t remember much about that 1983 algebra course, but I can picture that topic title going around and around in my brain like an obsession. I particularly remember running around the uni oval in athletics training, thinking as I went that within a few months I would actually be one of the privileged few that understood ‘*Direct decomposition of a finitely generated module over a principal ideal domain*’, even though at that time I had no idea of what it even meant. It was like counting down the days to Christmas.

I can’t remember the point at which we finally learned how to do it. Perhaps it was a bit of an anti-climax. Perhaps it turned out not to be as exotic as it sounded, or familiarity had bred contempt (or at least dissipated some of the awe) by the time we neared the end of semester.

As well as spoken jargon I also have a taste for unusual symbols. In primary school, when everything we do in maths is a number, the idea of doing algebra in high school, where we would use *letters* rather than numbers, seemed very grown-up. In junior high school we could look forward to trigonometry with those funny *sin*, *cos* and *tan* words, then logarithms and exponentials with *log* x and superscripts *e ^{x}* and then, even more alluring,

*calculus*in senior high school with those loopy integral signs ∫.

At uni I couldn’t wait to be able to use the ‘plus’ and ‘times’ signs with circles around them – ⊕ and ⊗, the special curly ‘d’s that are used for partial differentials ∂. And the upside-down triangle ∇. I imagined it would be like learning a secret language, into which only specially selected people would be initiated (Yes I was vain! So sue me. What privileged, talented 20-year old isn’t?) As it turned out I didn’t get to use ∂ much, and didn’t get to learn about ⊗ or ∇ at all, because of my subject choices.

I very much wanted to learn about ‘tensors’. I had heard that they were like matrices (rectangular tables of numbers that all maths students have to study in first year uni) only more complex, and that you needed them to do relativity theory. But again I missed out because of my subject choices. Not that I regret that. If I hadn’t made the choices I did, I’d have missed out on learning about *finite-state automata* and *NP-complete* problems, and would never have had the opportunity to design a computer chip that converted binary to decimal or to write parallel-processing computer programs to simulate populations of aliens (that’s beings from other planets, not the human immigrants the Tea Party are so worried about).

Over the last few years I have rectified my lamentable ignorance of tensors, ⊗ and ∇, as a consequence of my mid-life crisis. Some people buy red sports cars and get plastic surgery. I decided I couldn’t live another year without understanding General Relativity and Quantum Mechanics. It takes all sorts, I suppose.

An unexpected bonus of this flurry of crisis-induced self-study was two new hieroglyphics 〈*x*| and |*y*〉 – complete with funny names: ‘bra’ and ‘ket’ (because when you put them together to make an ‘inner product’, written 〈*x*|*y*〉, you make a ‘bra[c]ket’ – get it?).

An unexpected bonus of this flurry of crisis-induced self-study was two new hieroglyphics 〈*x*| and |*y*〉 – complete with funny names: ‘bra’ and ‘ket’ (because when you put them together to make an ‘inner product’, written 〈*x*|*y*〉, you make a ‘bra[c]ket’ – get it?). That affords me the smug satisfaction of being able to understand – if not necessarily able to *follow* – most of what is written on the sitting-room white board that’s often in the background in *The Big Bang Theory*. And yes, it usually is *real* physics or maths, not just made-up jumbles of unrelated symbols. Sometimes it’s even relevant to the story-line, like when Sheldon had Permutations and Combinations of a set of 52 elements written on the board, because he was trying to figure out a magic card trick one of the others had done (not that they ever referred to the board). That’s in contrast to Doctor Who, where they just sling any old combination of fancy words together (‘Reverse the polarity of the neutron flow’ is not a ‘thing’. Or at least, it’s not a ‘thing’ you can do.).

I’m not expecting this essay to resonate with many people. It is a rare perversion to be intrigued by arcane language and symbols. But perhaps it’s not unusual for people to long to learn *something or other* that is currently far beyond their knowledge or abilities. It might be how to crochet an intricate doily, to speak a foreign language fluently, to recite The Rime of the Ancient Mariner from memory or, when a bit younger, yearning to be able to ride a bicycle without training wheels, swim the Australian Crawl or play a piano *with both hands at the same time*.

Andrew Kirk

Bondi Junction, March 2014

# More weird and wonderful observations about the expansion of the universe

**Posted:**21 September 2013

**Filed under:**Physics, Science |

**Tags:**Acceleration, Cosmic Inflation, cosmology, Hubble, physics, Universe Leave a comment

In high school physics we are taught the Galilean principle of relativity (named after Galileo Galilei, not the biblical land of Galilee). That is that there is no such thing as *absolute velocity*. All velocity is relative. Consequently we cannot ‘feel’ velocity. If we are inside a closed compartment with no windows, we cannot tell whether it is ‘moving’. We might think we can, and compare the feeling of being in a moving railway carriage or aeroplane to being in that same vehicle when it is ‘stopped’. However, as our high school science teachers explained to us, the feeling of motion we experience there is not the general forward (horizontal) motion but the minor vertical or horizontal *accelerations *that happen when the vehicle goes over bumps or gaps in the rails (train) or air (plane). If the rails or air were perfectly smooth, and we were travelling straight – not turning – we would not be able to tell whether we were ‘moving’.

If our science teachers were especially good, they might have summed this up for us as ‘*you can feel acceleration, but you can’t feel motion*’.

In my earlier essay ‘Expansion of the Universe’ I pointed out that it is quite possible for two items to be moving apart at a rate faster than the speed of light, and that that does not contradict Einstein’s theory of relativity, as long as those two items are sufficiently far apart. The ‘acceleration’ that arises from the cosmological constant (or ‘dark energy’, if you prefer spooky mystical terms) is so mild that we cannot not feel it.

It has since occurred to me though, that, because that rate of relative acceleration increases with the distance between two objects, there must be objects immensely far from us, relative to which our acceleration will be very large – greater than the several ‘g’s needed to make a person black out. This observation follows from Hubble’s equation.

Those that don’t like maths had better close their eyes for a few seconds. Don’t worry, it won’t last long.

Hubble’s equation tells us that *dr/dt = H r*, where r is the ‘comoving’ distance from us to a distant galaxy, *dr/dt* is the rate of change of that distance (using ‘cosmological time’ for t) and H is the Hubble parameter (2.2 x 10^{-21} s^{-1}). The solution to this differential equation is *r = r*_{0}* e*^{Ht}. where *r*_{0} is the *current* distance from us to the distant galaxy. Hence the relative acceleration between us and the distant galaxy is *d*^{2}*r/dt*^{2}* = r*_{0}*H*^{2}*e*^{Ht}* = H*^{2}*r*.

Almost any human will lose consciousness under an acceleration (‘g force’) of 10 ‘g’s, (‘g’ is the acceleration due to gravity at the Earth’s surface). That is 98ms^{-2}. To find the distance away of objects that are accelerating relative to us at that rate we set 98ms^{-2} = *H*^{2}*r*. So *r* = 98ms^{-2}*/H*^{2} = 98ms^{-2}*/ *(2.2 x 10^{-21} s^{-1})^{2} = 2 x 10^{43}m = 2 x 10^{18} billion light years. That is almost 5 x 10^{16} (50 quadrillion) times the radius of the observable universe.

You can open your eyes again now.

We have just worked out that the relative acceleration between ourselves and galaxies that are 2 x 10^{18} billion light years away is around 10 ‘g’s. One or both of us and the distant galaxy must be accelerating away from the other, and the sum of those two accelerations must be 10g. The Cosmological Principle tells us that the universe is essentially the same all over (‘homogeneity’ and ‘isotropy’), so we should have the same acceleration as that galaxy, which means we should each be accelerating at 5g, in opposite directions. That would be enough to pull the Earth out of its orbit around the Sun, and to pull us off the Earth’s surface. Also, we should be able to feel it – in fact it should make us feel very sick, like a super-extreme roller coaster. Remember that science teacher that told us you can’t feel motion but you can feel acceleration? So why can’t we feel it?

The answer is actually at the heart of Einstein’s General Theory of Relativity and, fortunately, it’s not highly technical. What we can feel is acceleration, but in this context it is an oversimplification to say that acceleration is the rate of change of our speed. In General Relativity the acceleration of a body means the rate of divergence of the body’s path through spacetime (its ‘*worldline*’, in relativistic jargon), from the *tangent geodesic*. Let’s unpack this. A geodesic is the equivalent of a straight line on a curvy surface. For instance the ‘great circles’ that are lines of longitude on the Earth are geodesics, and so is the Equator (but *not *lines of latitude like the two tropics or the Arctic and Antarctic circles). At any point in your life, the tangent geodesic to your body is the path it would travel through four-dimensional spacetime if there were no non-gravitational forces acting on it. Fortunately for you, there *are *non-gravitational forces acting on you (principally from the ground, pushing up on the soles of your feet), otherwise you would follow your tangent geodesic which is a headlong plummet towards the centre of the Earth. The floor is constantly pushing you away from your tangent geodesic in four dimensions, just as a turned steering wheel pushes a car away from the line of the beam of its headlights (which is the tangent in *three-*dimensional space to the car’s curved 3D path).

The acceleration you feel, which is just your feeling of weight, arises because your body, by not plummeting, is constantly diverging from its tangent geodesic. You might think you don’t feel any acceleration when you’re just standing still on the floor, or lying in bed, but that’s because you have felt it ever since you were first snuggled up in your mother’s womb, so you don’t notice it. What you *do* notice is if the acceleration suddenly *stops*. That’s what that ‘stomach in your mouth’ feeling is that you get when you jump off a pier into the sea and are briefly ‘weightless’. For a second or two you are following your tangent geodesic, and it feels odd because you’re not used to it.

The science teacher’s advice, that you can feel acceleration, needs to be refined to specify that you only feel ‘*General Relativistic acceleration*’, which is not rate of change of speed (as it is described in Year 11 Physics) but divergence from our tangent geodesic*. The two are only the same in non-accelerated (‘inertial’) frames of reference. In high school Physics we usually only deal with inertial frames of reference so the distinction doesn’t matter. But it does when we are considering our position relative to very distant galaxies.

Now how does this explain our inability to feel our acceleration relative to that distant galaxy? The answer is that the Earth is following its tangent geodesic which, as well as orbiting the Sun, is heading increasingly rapidly away from the distant galaxy. So we can’t feel that change in motion because it is not diverging from the tangent geodesic. The only thing we can feel is the divergence that arises from the floor pushing us up in opposition to the Earth’s gravity.

In summary, the only ‘acceleration’ you can feel is that which pushes you away from your tangent geodesic. It doesn’t matter how wildly that geodesic may be curving through spacetime – you will never be able to feel it.

* Note: Technically speaking, the strength of acceleration that you feel – the ’g force’ – is the magnitude of the *Covariant Derivative* of your *four-velocity* in its current direction.

# Cosmic Inflation

A much more extreme example of this is in the ‘inflationary era’ of the universe, which is believed to have occupied the first unimaginably tiny fraction of a second after the Big Bang (10^{-32} seconds). In that sliver of time every part of the universe is thought to have expanded by a linear factor of around 10^{26}. Imagine if you had just put your pencil down for an instant, then looked to find it and noticed that it was now a billion light years away. Wouldn’t that be annoying? Fortunately, there were no humans around at the time to be annoyed by such nuisances, and not even any pencils either. I do sometimes wonder though whether a brief, local, reappearance of cosmological inflation might be the reason for the occasional mysterious disappearance of my socks.

Describing that period of inflation in terms of acceleration, the g forces are mind-boggling. If we take a brutally crude and almost certainly horribly wrong (but reasonable enough to make the point) assumption that the acceleration was constant, we can use the high school formula to calculate the average acceleration during this split second.

Mathophobes look away again briefly:

The formula is distance (r) = ½ *a **t*^{2} where *a* is the acceleration, *t* is the time period and *r* is how far away the object is at the end of the period. This gives *a* = 2*r / t*^{2}. If I put my pencil down one metre away and 10^{-32} seconds it is 10^{26} metres away we have *a* = 2 x 10^{26}m / (10^{-32}s)^{2} = 2 x 10^{90}ms^{-2}.

The average acceleration during the inflationary split-second was 2 x 1089 ‘g’s! That would give you a bit of a tummy ache.

Only it *wouldn’t*, for the same reason as why we can’t feel our acceleration relative to the distant galaxy. We would just be following our tangent geodesic, and although that geodesic was horrendously curved, we wouldn’t feel a thing, because we only feel *divergence *from that path.

Only it would be rather uncomfortable because our bodies take up space, and so have many different geodesics going through them. Those geodesics, although initially very close, would diverge from one another so rapidly that a moment later they would be separated by squillions of light years. So your body would be ripped apart by the expansion of space, and the molecules that were neatly collected together to make your kidney one instant would be spread over an unimaginably vast expanse of space an instant later. Fortunately, it would happen so fast that you’d never notice. Even more fortunately, we can be fairly sure that nobody reading this was born before 1879, and hence they probably weren’t around to experience such a cosmic evisceration.

Another odd thing is that none of the molecules in your kidney would break the cosmic speed limit of *c* (the speed of light), even though they become separated by many light years in a tiny fraction of a second. This relies on the (somewhat technical) fact that the molecules not only follow their geodesics, but that those geodesics are *timelike*. It is only non-timelike geodesics that are forbidden. The concept of timelike things is partially explained in my essay ‘Expansion of the Universe’ . For a full explanation, see any good relativity textbook such as Bernard Schutz’s ‘A First Course in General Relativity’.

# Summary

We can only feel acceleration, not constant motion. But the acceleration we can feel is only ‘*General Relativistic acceleration*’ that is divergence of our path from our tangent geodesic. Acceleration that is just an increasing rate of separation between us and something else will not be felt if it is solely because of our following geodesics of spacetime. Such non-General Relativistic Acceleration exists between us and immensely distant galaxies. For sufficiently distant galaxies, that acceleration is greater than that of a ferocious roller-coaster, yet we cannot feel it at all.

In the first split second of our universe, non-General Relativistic acceleration is believed to have occurred – called ‘*cosmic inflation*’ – that is unimaginably huge in terms of ‘g forces’. The acceleration could not have been felt if we had been there at the time, but it would have instantly ripped apart any object that was there, because of the extreme divergence of temporarily nearby geodesics.

Andrew Kirk

Bondi Junction, Australia

September 2013

# Expansion of the universe

**Posted:**15 August 2012

**Filed under:**Physics |

**Tags:**cosmology, physics 1 Comment

I wonder if I am the only person who has found the various explanations available in the public domain, of the expansion of the universe, confusing. For about two years after learning the essentials of general relativity I struggled with the apparent contradictions in the idea of cosmological expansion. I asked several questions on internet forums but never got an answer that helped me resolve the confusion. That was probably because I was asking the wrong questions, but it is so hard to know what are the *right *questions to ask.

I am fortunate to have finally been able to discover the right questions and, together with answers to those, and some useful papers by cosmologists, to piece together a coherent understanding of what is actually going on. This essay is an attempt to explain that for anybody else who finds the concepts puzzling.

I’ll start by describing why the concept of an expanding universe is confusing. The usual explanation involves an analogy with a balloon that is being inflated, on which dots have been marked with a felt pen. As the balloon inflates, the dots – which correspond to galaxies – become further apart. There are three key problems with this analogy, and with the idea of expansion in general:

- The balloon analogy fails because it seems to require the existence of a physical object that is space, corresponding to the rubber of the balloon. But the Michelson Morley experiments showed that there is no ‘absolute space’, which was referred to at the time as the ‘luminiferous ether’. The balloon analogy implies the existence of a ‘privileged reference frame’, and they are not supposed to exist.
- We are told that faraway galaxies are receding from us faster than the speed of light. This appears to contradict the rule that nothing can travel faster than light.
- If the universe’s expansion is accelerating then that seems to imply that energy is being created, as the kinetic energy of the galaxies will be ever increasing, without any compensatory reduction in potential energy.

## The Cosmological Principle – a Privileged Reference Frame

To start on trying to resolve these objections, we need to first state the fundamental principle on which this sort of analysis is based – the Cosmological Principle. This principle effectively says that there exists a system of coordinates for spacetime under which a snapshot of the universe at any ‘cosmic’ time coordinate is isotropic and homogeneous at the large scale.

A *system of coordinates* (also known as a ‘*coordinate system*‘ or ‘*reference frame*‘) is a scheme that assigns a unique set of four numbers to every point in spacetime. This enables any point in spacetime to be referred to by those numbers – its ‘coordinates’. One of those coordinates is a *time* coordinate and the other three are *spatial *– denoting a location in space.

*Homogeneous *means it is the same everywhere. This rules out possibilities such as there being a ‘centre of the universe’ where stars occur most frequently, with the distribution of stars getting ever sparser as you travel further from that centre.

*Isotropic *means it is the same in every direction. So an observer that is stationary with respect to this coordinate system would not for instance observe stars on her left moving towards her on average and stars on her right moving away from her.

*At the large scale *means we ignore local variations. Of course stars are more frequent within a galaxy than outside one, but once we zoom our perspective out enough to incorporate many clusters of galaxies, the density of matter should appear roughly constant. Similarly, we may have more objects within our galaxy moving towards us from the left than from the right – meaning local isotropy does not hold – but if we zoom out enough and take enough distant galaxies into our view, their motion relative to us will be the same in all directions.

The homogeneity assumption is easy enough to grasp. It simply says there is nothing special about our part of the universe, that what we can see is a fair sample of the whole thing.

The isotropy assumption is the really powerful one, because it establishes the privileged reference frame that we can associate with the rubber in the balloon analogy. Given any point P in spacetime, we can define a location in space as being the unique worldline through P for which the universe appears isotropic from every point in that worldline. In the analogy, that means that the worldline follows the path of a dot marked on the balloon’s surface, as it is being blown up.

By the way, an isotropic universe must be homogeneous, but the reverse is not true. A homogeneous but non-isotropic universe could be one in which the speed of light is different depending on which direction the light is travelling.

We can now satisfy the first objection. There is indeed a privileged reference frame, being the one that generates the required conditions of homogeneity and isotropy. It does not require the existence of a substance like the luminiferous ether, whose existence was disproven by the Michelson Morley experiments. All it requires is the existence of matter throughout the universe, and the application of the Cosmological Principle. The existence of this frame does not contradict the principles of Galilean and Special Relativity. Those principles state that there is no privileged ‘*inertial* reference frame’, which means a frame that is not accelerated or subject to gravitational forces. The frame identified by the Cosmological Principle is not inertial and hence does not contradict those theories. In one sense it is similar to the ‘laboratory frame’ on Earth, which is the frame in which the laboratory in which experiments are being conducted is stationary. That frame is privileged in a sense, but like the cosmological frame it is not inertial, because of the effect of the Earth’s gravity.

The spatial coordinates of a point that is stationary with respect to the cosmic reference frame are frequently called *‘co-moving coordinates’*. In the balloon analogy this means that a point with those coordinates is moving in the same way as the rubber, i.e. it is stationary with respect to the rubber.

Given this special reference frame, we can now make sense of another commonly used aspect of the balloon analogy, that of ants crawling on the balloon. In the balloon analogy, there is a clear difference between the relative motion of dots marked on the balloon’s surface and the relative motion of ants crawling on that surface. In the former case, the motion is caused solely by the expansion, deflation or other deformation of the balloon. In the latter, the motion is a combination of that with the motion of the ant relative to the rubber.

We can use the same concept in our universe model. The coordinates of a stationary section of the ‘rubber’ are determined by application of the cosmological principle. Objects that are stationary in those coordinates may still be moving relative to one another due to the expansion of the universe. On top of that, an object may be moving locally, relative to that coordinate system. For instance, the Earth is orbiting the sun, which involves constant changes of direction, incorporating a full 180 degree change of direction of motion every six months. This motion is relative to the cosmic coordinates, and we call it ‘peculiar motion’. It is analogous to an ant walking around in circles on the rubber of the balloon.

The Andromeda galaxy is moving towards ours under gravitational attraction, and they seem destined to collide. Those relative motions are also *peculiar motion*, like two ants rushing towards one another along the surface of an expanding balloon.

One final note on this special cosmic reference frame. The standard way to determine it is by reference to the Cosmic Microwave Background Radiation (CMBR), which is the radiation left over from the Big Bang, that fills the sky in every direction. We can identify the co-moving coordinates of our location by identifying the velocity we would need to have for the wavelength of the CMBR to be the same in every direction. An observer that is not ‘co-moving’ with the cosmic reference frame will see longer wavelengths in one direction than in its opposite, because of the Doppler effect.

## But how can galaxies travel faster than light?

The resolution to the second objection comes from the realisation that the prohibition on faster than light (‘superluminal’) travel, precisely stated, is not the same as how it is often popularly described. The popular description is either:

- an object cannot travel faster than light, or
- two objects cannot have a relative speed that exceeds the speed of light

Neither of these is strictly correct. They should be really stated as:

1a. There is no inertial reference frame in which an object’s velocity is faster than light, and

2a. Two objects within the same inertial reference frame cannot have a relative speed that exceeds the speed of light

We can immediately see that 2a is not breached by the motion of distant galaxies, because there is no inertial reference frame that contains both us and such a distant galaxy. A reference frame is inertial if spacetime is very close to flat within that frame, and there is too much spacetime curvature between us and a distant galaxy for any reference frame containing both to be flat. This can be understood by comparison to the Earth. It is reasonable to assume the Earth is flat in measuring travelling distances between my house and the local shops. However, if I am planning a trip between from my house in Sydney to London, I have to take the curvature of the Earth into account.

1a also is not breached by the distant galaxy, because it is not travelling faster than light in any inertial frame containing it. Nor is it travelling faster than light relative to any object close enough to it to be in the same inertial frame.

This may seem all a little unsatisfactory, as it leaves a big grey area in between the local shops and the distant galaxy, in which we are uncertain how far we can extend an inertial reference frame. Fortunately, we can resolve this by expressing the prohibition on superluminal travel in a more precise way, as follows:

- No object can have a spacelike four-velocity.

A four-velocity is a vector that can be used to represent an object’s motion, which is independent of any reference frame (‘co-ordinate independent’). Like all vectors, it has a magnitude (‘size’) and a direction, but the direction is in *spacetime*, not space. In any given reference frame, a four-velocity can be denoted by four numbers, called *components*, of which one will be a ‘time’ component and the other three will be ‘spatial’ components. The values of the four components will differ between reference frames, but they will all refer to the same physical phenomenon, whose magnitude and direction does *not* differ between reference frames. There is a mathematical formula, involving something called the* metric tensor*, for determining the magnitude of any vector. In relativity, magnitudes can be positive, negative or zero. A vector with negative magnitude can be the velocity of an object with mass, and such vectors are called ‘*timelike*‘. Light rays, which have no mass, must have velocity vectors with zero magnitudes, and those vectors are called ‘*lightlike*‘*. Vectors with positive magnitudes are called ‘*spacelike*‘.

The law 3 is the most general and precise statement of the prohibition on superluminal travel. Prohibitions 1a and 2a are consequences of this rule and, given a local, inertial reference frame, prohibitions 1 and 2 follow.

In an expanding universe, no object, including no galaxy, can have a spacelike velocity, so the prohibition is respected. It will be the case that the distance between two galaxies far away from one another is increasing at more than 3 x 10^{8} ms^{-1}, but that does not breach the prohibition.

One last point on this objection. When we refer to distance in that last paragraph, we mean what is called the ‘proper distance’. That means the shortest distance between the two galaxies in the snapshot of the universe taken at an instant of cosmic time. In an expanding universe that shortest distance will be increasing with cosmic time.

## Does the expansion violate conservation of energy?

If galaxies are rushing apart at ever increasing rates that appears to increase the net energy of the universe on two counts. Firstly, their kinetic energy is increasing with their increasing relative velocities. Secondly, their gravitational potential energy is also increasing as they get farther apart. This appears to violate the principle of conservation of energy.

The answer to this objection is that conservation of energy only applies locally, within an inertial reference frame. There is no coordinate-dependent global equivalent to that local principle. In fact there is not even any global, coordinate-independent definition of total energy. There are approaches using coordinate-dependent *pseudo-tensors*, but these are controversial, and seem unsatisfying given their coordinate-dependence.

Even so, it appears that these pseudo-tensor approaches can be used to derive a conclusion that the net energy of the universe is zero, and will remain zero. Hence, under such an analysis, there does not appear to be any violation of conservation of energy.

* Note: If you are familiar with vectors in two and three-dimensional Euclidean space, you might assume that any vector with zero magnitude must be a trivial vector whose components are all zero in any reference frame, and hence which has no direction. While that conclusion is true for Euclidean space, it is not true for spacetime, which is non-Euclidean. Lightlike vectors have zero magnitudes but they also have a well-defined direction. They will have non-zero components, which cancel one another out when we calculate the magnitude.

# Some random thoughts on whether the world is random

**Posted:**13 July 2012

**Filed under:**Mathematics, Philosophy, Physics |

**Tags:**mathematics, philosophy, physics 1 Comment

In discussions about free will, consciousness or interpretations of quantum mechanics people talk a great deal about whether the world is deterministic, random or something else. Well I don’t know what the ‘something else’ bit could be but I’m also starting to wonder whether the idea of randomness makes any sense.

The general idea of the distinction between a deterministic and a random universe seems to be that, in the former, events are somehow ‘fixed’ before they occur, whereas in the latter they are not – there are multiple different possible events. That sounds clear enough if you don’t think too hard about it, but if we do then we come up against the question of what do we mean by fixed? Fixed how, and by whom?

The apparent answer is that they are fixed by the ‘laws of nature’. But what are the laws of nature? Isn’t this some fairly heavy duty *reifying* to posit that there exist some actual laws, perhaps dwelling in some Platonic kingdom of Forms or written on magical stone tablets? Sure we have useful laws like Schrodinger’s and Einstein’s equations, but all we know for sure is that these are ideas we use to make sense of what we see and make predictions about the future. Whether they have some mysterious metaphysical existence independent of human minds is an entirely separate matter. It seems quite unlikely to me, given that every century or so we have to tweak the laws we use when further experiments and theorising show they are not completely accurate.

A Platonist might argue that there really *are* laws of nature out there, which determine how the universe behaves. OK if that’s the case then what about the law that is simply a description of where every single particle in the universe will be at every instant in the history of time? This is essentially a very, very long shopping list, but it also happens to perfectly describe the behaviour of the universe. Let’s call it the *M-law*, as it is the ‘mother of all laws’. Is that a law of nature? If it is then the universe cannot be random because everything that ever happens is described by the M-law.

So what are our choices about universe types? If we deny that laws of nature have any existence independent of human brains then *everything* in the universe is random because there is nothing that fixes it. If we assert the separate existence of laws of nature then *nothing* is random because it is all predicted by the M-law. Or, we could try and be picky and assert the existence of laws but only count them if they meet certain criteria. But what would such criteria be?

One option is to say that a law only counts if it can be known prior to the event we want to use it to predict. That would certainly disqualify the M-law. The trouble is, it would also disqualify everything else, because we cannot prove any of the laws. All we can do is build up supportive evidence for them, and there is never enough to be certain.

Sure, says the law-enthusiast, you can’t ever be *completely* certain, but in practice, if we are 99% certain of something, we would consider that good enough. Alright then. That does seem a bit arbitrary, but let’s go with it and see where it leads. What can we say about the motions of the planets prior to the discoveries of Kepler, Galileo and Newton? Back then we knew nothing of the ‘laws’ we now use to describe those motions, so under this criterion those laws didn’t apply, which apparently makes the motions of planets in 1347 CE random. Is that what we want?

‘Ah yes’ replies the enthusiast, ‘but you are limiting yourself to what we managed to work out based on our imperfect interpretation of the available information and our limited ability to make observations. The validity of a law should be based on all the information that was available up to that time, to an omniscient – but not future-seeing – observer, who was able to develop the best possible theories with the available information.’

Well I have to say this is getting even weirder and more implausible! We now have an ideal scientist-observer that is our yardstick for what constitutes a law of nature. If we go with that, and accept some arbitrary threshold of confidence – say 99% – on the validity of a law (leaving aside the very difficult question of how we would try to implement that threshold and whether it would be possible to validly calculate probabilities against it) then maybe we have arrived at a definition that could be pressed into use for laws of nature while excluding the M-law.

But – haven’t we ended up with a definition of randomness that is entirely *epistemological*? We have effectively defined a random event as one for which our ideal observer could not have 99% confidence beforehand of what the outcome would be. And the trouble with that is that we can no longer make the distinction that metaphysicists like to make between epistemological uncertainty over deterministic but chaotic events such as a coin toss, and ‘genuine’ randomness such as the decay of a radioactive isotope. With our new definition both of these types of uncertainty are ‘merely’ epistemological, and there is no such thing as metaphysical or ontological randomness. If we take this path we have to conclude that all randomness is epistemological, and there is not distinction from ‘metaphysical randomness’.

One last point to wrap up with. I went searching for a mathematical definition of randomness and came up with a complete blank. There are definitions of random variable, random (stochastic) process, probability space and various other related objects. But none of them have anything in them that captures the idea of ‘metaphysical undeterminedness’ that lurks under the popular conception of randomness. In fact, rather oddly, of the various interpretations of quantum physics, the only one that has close parallels to any of those mathematical objects in the field of probability theory is the ‘many worlds interpretation’, which looks very like the peculiar object that is a ‘stochastic process in continuous time’. That is ironic, as the many-worlds interpretation is regarded as a ‘deterministic’ interpretation, standing in contrast to the most popular ‘Copenhagen interpretation’ which is regarded as indeterministic, ie random.

Andrew Kirk Bondi Junction July 2012