Pointing the Camera at the Monitor – A Puzzle







Question 1: what does the monitor show?










Andrew Kirk

Bondi Junction, February 2017




Acoustic ‘beats’ from mismatched frequencies




Andrew Kirk

Bondi Junction, August 2016

Too many words!
























Andrew Kirk

Bondi Junction, November 2015

Helping People on the Internet

I like helping people with their homework on the internet. I do this when I can on physicsforums.com, which is a forum for discussing physics, and philosophyforums.com, which is for discussing philosophy. Both have sections where people can ask questions about problems they are having trouble understanding – usually for homework in a degree course. The physics forum has Help sections covering physics, mathematics and chemistry. The philosophy forum’s Help section is focused on symbolic logic, which is a branch of maths closely related to set theory.

There’s an art to both asking and answering questions on such forums. It is not acceptable to just post a question straight out of an assignment and hope somebody solves it for you. That would be lazy, and possibly also cheating. Equally, it is not the done thing to just post a solution to somebody’s problem, even though that would often be the easiest thing to do. The paradigm of quality advice on the internet seems to be to give carefully chosen hints that will lead the student towards being able to solve the problem by themself. It is quite challenging to choose one’s hints to be not so obvious that they give the game away, and not so obscure that the student remains helplessly lost.

Now you might expect there to be an imbalance between people needing advice and people offering it, so that many people remain unhelped. Well there is an imbalance, but it’s the other way around. There are far more willing advisers than there are people wanting to be advised! How does such an odd situation arise? I can only guess at it based on my own attitude to internet advice – which is that I love giving it. I think there is a deep-seated need in most humans to feel useful. In some it is stronger than others, and in me it is quite strong. In some professions, like being a doctor, one has frequent opportunities to provide advice to one’s acquaintances. People seem to have an inexhaustible appetite for free medical advice. But the knowledge of art critics, journalists, electricians and accountants is drawn upon far less frequently in private life. Of course people would be very happy to draw upon the skills of an electrician without cost, for instance to get them to fix some circuits in one’s house. But that is very different from asking for advice. Spending an hour alone in a dusty roof cavity fiddling with dangerous wires is far less fun than dispensing advice to an agog and appreciative audience. A notable example of this difference is the poor old professional photographer. Goodness knows how many times they get asked by friends, relatives and acquaintances to come to a wedding and take some photos – for no fee, of course. And like the electrician in the attic, it is solitary work, with little appreciation except at the time when one actually delivers the photos.

The few professional photographers I have known have had policies of not doing photography for anybody’s wedding except as a fully professional, paid gig. I thought that policy a bit harsh when I first came across it. But after reflection I realised it was the only way to end up avoiding spending countless hours doing work for free, as well as missing out on being able to enjoy one’s friends’ weddings. Life is hard enough for photographers these days, with news organisations slashing their headcounts of journalists and photographers – preferring to just get stock clips from Getty images. It adds to the woe to be be constantly asked to work for nothing.

On the other hand, if people were to gather around them in adoring circles at parties, asking their advice on shutter speeds, apertures, lenses and filters, that would be a different thing altogether. So far as I can tell, that doesn’t happen very often.

My areas of expertise are not under such hard times as photographers, but neither are they as socially sought-after as those of doctors. They include finance, music, bicycles and fitness, philosophy, physics and mathematics. Once or twice a year I get asked for advice on bicycles, fitness or investment. I don’t think I’ve ever been asked about physics, philosophy, music or maths, unless we count my children asking for homework help when they were little.

So, to fulfil my need to feel useful, I go to the internet to see who needs help integrating x sin x, working out whether a pendulum has enough energy to swing right over the top of its arc, working out whether ‘Dark Energy’ would break a wire stretched between two galaxies, or understanding the significance of the Bell Inequality or Godel’s First Incompleteness Theorem.

And what do I find when I go there? Any question that is expressed even moderately clearly has received three or more responses within half an hour of being posted. It’s almost as though people are falling over one another to be the first to respond. You might think that one response is all that’s needed. But it’s not. There is often room to contribute even when there have already been two or three responses to the request. Remember that answers are supposed to be hints, not answers to the homework problem. The best answers are often in the form of questions to the original poster, like ‘what is the component of the gravitational force that points diagonally down the banked track?‘ They serve just to get the student thinking in the right direction, or even in just a more productive manner. Often responders need to ask questions to clarify the problem because the student has not explained it well enough. In logic, the standard question is ‘what tools are you allowed to use?‘ because the typical problem is to prove a statement, given a number of premises and a set of deduction rules, and there are many different logical systems, with different ‘toolboxes’ of deduction rules. Most students just post ‘how do I prove such-and-such, given the following premises’, without specifying what tools are available to them.

So I still sometimes find questions to which I can helpfully respond. Some questions are expressed so unclearly that nobody wants to touch them. I quite enjoy the forensic work of trying to work out what the communicationally-challenged student was trying to say. In other cases, the question may be clear enough but the responders may have gone off on a false trail, or been insufficiently clear, leaving room for me to contribute. But it’s only the occasional question that I find has received little enough helpful attention for me to respond without giving too much of the answer away.

I have said that I get enjoyment from answering questions. And maybe I am a little more than averagely susceptible to enjoying the fulfilment that comes from that. But there are many people that seem to have a much stronger desire for that fulfilment than I. Many of the responders have made tens of thousands of posts to the forums, and their presence can be seen in most of the questions ever asked. Those that have responded to very large numbers of questions have little electronic trophy icons below their on-line avatar, saying things like ‘Homework Helper’. It reminds me of the Fat Controller, who would occasionally, when Thomas or one of his friends had done especially well, reward them by proclaiming that they were a Really Useful Engine – the ultimate accolade!

Why do people like helping so much on the internet? This wish to be useful is so widespread, and so strong. And it isn’t mirrored in off-line life – or is it? Perhaps it is. Most doctors appear to enjoy being consulted by family and friends on the selection of a surgeon for Tristram’s grommets, and I’m sure the electrician would revel in the opportunity to dispense wisdom on the difference between the connections of a Neutral and an Earth wire, as long as they weren’t then dispatched to the attic to do some free clipping and splicing. I once consulted an electrician on just this topic and, as far as I could tell, they enjoyed providing the information.

Is it possible to harness this human tendency? There are so many people in the world that need help and don’t get it. Maybe it is. But on the other hand maybe the seeming disconnect between the large number of willing advisers on the internet and the large number of people in real life desperately needing help can be explained by the above-mentioned difference between advice and work. Giving advice is often easy, takes little time, costs nothing and is a social activity, as one does it in conversation with the advisee. Work, on the other hand, whether it be fixing wires, taking photos, preparing a pathology slide, or constructing a Monte Carlo model to measure financial risk is time-consuming, usually solitary and often frustrating.

There are exceptions. I’ve always liked the idea of raising a barn, like the Amish do in Pennsylvania, because it seems such an enjoyable, social, neighbourly form of work. Sadly, not many barn raisings happen in my neighbourhood.

Andrew Kirk

Bondi Junction, August 2015

Metaphysics as a creative craft

















Andrew Kirk

Bondi Junction, January 2015


Excitement about learning complicated things





















Andrew Kirk

Bondi Junction, March 2014

More weird and wonderful observations about the expansion of the universe

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 = r0 eHt. where r0 is the current distance from us to the distant galaxy. Hence the relative acceleration between us and the distant galaxy is d2r/dt2 = r0H2eHt = H2r.

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 = H2r. So r = 98ms-2/H2 = 98ms-2/ (2.2 x 10-21 s-1)2 = 2 x 1043m = 2 x 1018 billion light years. That is almost 5 x 1016 (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 1018 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 1026. 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 t2 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 = 2r / t2. If I put my pencil down one metre away and 10-32 seconds it is 1026 metres away we have a = 2 x 1026m / (10-32s)2 = 2 x 1090ms-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’.


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