I was listening to a talk by Alan Watts about some aspect of Eastern mysticism. I can’t remember the exact context. I think he was describing the impossibility of truly understanding the nature of one’s own mind. He said that trying to use one’s mind to understand one’s own mind was ‘like pointing the camera at the monitor’.
I was immediately struck by this. Partly I was surprised at his using such a simile, which involves common enough concepts in 2017, in a talk that he gave in the sixties, when computers only existed in large research establishments and occupied enormous rooms. There was certainly no such thing as a webcam back then. I realised later that he probably had in mind a closed-circuit television arrangement, which they did have in the sixties.
But beyond that, I was struck by the fact that it’s actually a very interesting question – what does happen when one points the camera at the monitor? It’s a classically self-referential problem. But unlike some self-referential problems, like the question of the truth of the statement ‘This sentence is false’, it must have a precise answer, because we can point a camera at a monitor, and when we do that the monitor must show something. But what will it show?
There are a number of practical considerations that can lead us towards different types of answers. While each of those considerations leads to an interesting problem in its own right, I tried to remove as many of them as possible to make the problem as close to ‘ideal’ as I could. So here it is.
Imagine we have a computer connected to a monitor and a digital video camera. A webcam is a digital video camera but, since the camera we are imagining here needs to be extremely accurate, a high-quality professional video camera would be more suitable. The monitor uses a rectangular array of display pixels to display an image and the camera uses a sensor that is a rectangular array of light-sensitive pixels, and the dimensions of the display and the sensor, in pixels (not in millimetres) are identical.i
On the computer we run a program that shows the image recorded by the camera. The telecommunication program Skype is a well-known such program that can do that, amongst other things. There are also dedicated camera-only programs, which webcam manufacturers typically include on a CD bundled with the webcams they sell. Let’s call our program CamView (not a real program name). We start up CamView on the computer in a non-maximised window, which we’ll call the ‘CamView window’. Then we turn the camera on and point it at the monitor. We aim and focus the camera so precisely that an image of the display area of the monitor fills the image-display area of the CamView window. Ideally this would mean that each pixel on the camera’s sensor is recording an image of the corresponding pixel on the monitor screen. In practice there will be some distortion, but we’ll ignore that for now.
Question 1: what does the monitor show?
Question 2: Next we maximise the CamView window. What does the monitor show now?
Those questions are easy enough to answer, when we remember that the window for any computer program, in default mode, typically has an upper border with tool icons on it, a lower border with status info on it, and sometimes left or right borders as well.
These questions are fairly similar to the question of what one sees when one stands between two parallel, opposing mirrors, as is the case in some lifts (elevators).
Now comes the hard one. In most video-viewing computer programs there is an icon that, upon clicking, maximises the window and removes all borders so that the image-display area occupies the entire display area of the monitor. Call it the ‘full screen icon’ and say that we are in ‘full screen mode’ after it is clicked – until a command is given that terminates that mode and returns to the default mode – ie restores the borders etc. In full screen mode the display area of the monitor corresponds exactly to the images recorded by the camera’s sensor.
Question 3: We now click the full screen icon. Describe what appears on the monitor, and how it changes, from the instant before the icon is clicked, until ten minutes after clicking it – assuming the program remains in full screen mode for that entire time.
That is the difficult one. It took me a while to figure it out, and I was surprised by the answer. It is possible that what I worked out was wrong. If so, I hope that someone will point that out to me.
I have one more question, and it has an even more peculiar answer – one that I found quite charming.
Question 4: Assume the camera is mounted on a very stable tripod. Still in full-screen mode, we pan the camera to the right until it no longer shows any of the monitor. Then we pan the camera back at a constant speed until it again sees only the display area of the monitor, and we stop the panning at that point. What is visible on the monitor after the camera has panned back to the original position? Does that change subsequently? What does it look like ten minutes later? Does the monitor image depend on the panning speed, or on the number of frames per second the camera shoots? If so, how?
In order to avoid spoiling anybody’s fun in trying to work out the answers to these puzzles for themself, I will not post answers now. I will post them a little later on. It will also take me a little while to make some nice pictures to help explain what I am talking about.
Bondi Junction, February 2017
i Although most camera sensors have a 3:2 aspect ratio, which is different from the 16:9 aspect ratio of most modern computer monitors, it is possible on a sophisticated camera to alter the aspect ratio to 16:9, which is achieved by deactivating the sensor pixels in an upper and lower band of the sensor, so that the area used to record an image has the required aspect ratio. We’ll assume that is done and that the number of pixels in the active sensor area equals that on the monitor.
Here’s a piece I wrote explaining the mathematics behind the peculiar phenomenon of acoustic ‘beats’.
It’s a bit maths-y. But for those that don’t love maths quite as much as I do, it also has some interesting graphics and a few rather strange sound clips.
Bondi Junction, August 2016
Books are too long. People talk for too long. Academic papers are too long. Almost everything is too long.
Why? Partly, because to be concise is very difficult. Urban legend has it that Blaise Pascal once wrote at the end of a letter to a friend: ‘I’m sorry this letter is so long. I didn’t have time to write a short one’.
I struggle with conciseness. Part of the problem is that, when I am trying to explain something, I worry about whether what I have said is clear enough, so I keep on saying it over, in a slightly different way each time, in the vague hope that one of the attempts will make the connection.
I think a better strategy might be to make one brief attempt at an explanation and then wait for a response. If more is needed, I imagine my interlocutor will tell me. If they do, the particular nature of their response will better enable me to tailor my next statement to fill in the information that was missing in my first.
But that requires discipline, and nerves of steel. It is like being silent in an interview after giving a short reply to a question – forcing the interviewer (or interrogator) to make the next move. Few people can carry that off, and I suspect I am not one of them.
Academic papers can be particularly irritating, droning on about all the references and who has written what, so that by the time one gets to the bit about what the authors have done that’s actually new, one is exhausted and wants to retire for a tea break. It’s not clear to me whether this is a stylistic practice, imposed by the producers and reviewers of journals, or whether it reflects insecurity on the part of the authors, who may feel that they need to mention some minimum number of other papers in order to be taken seriously.
Arthur Schopenhauer railed against this sort of writing in a series of essays collected under the title ‘The Art of Literature’. He opens with an unrestrained broadside ‘There are, first of all, two kinds of authors: those who write for the subject’s sake, and those who write for writing’s sake.‘ Schopenhauer loved the first (and of course considered himself to be one of them) and loathed the second.
If someone really has something important to say, it usually doesn’t take very long. When Neville Chamberlain announced the grim news to the British people in 1939 that Britain had declared war on Germany, the message had been delivered by the end of the 67th word. I did a test reading just now and it took about 26 seconds, including pauses for effect.
Einstein’s legendary 1905 paper that presented his special theory of relativity to the world, ending decades of confusion amongst physicists, is only 24 pages, and the key part that resolves the paradoxes by which physics was previously beset is complete by the end of page 12! John Bell’s paper that turned the world of Quantum Mechanics upside down in 1964 is only six pages. Bell cited only five references. Einstein cited none.
In general communication, most people use too many words. I do too, but I am trying to correct that. I feel that, where possible, I would like to conduct a post-mortem on every sentence I utter and work out whether that sentence has added any new information. If it hasn’t, then it was probably a waste of everybody’s time.
Politicians exploit this deliberately. They are trained to, when asked a difficult question by a journalist, give a long-winded, emphatic speech about something only tangentially related, thereby avoiding the issue and (they hope) making the journalist despair of persisting with the question because of the pressure of time. Even better, if the politician sounds confident in their ‘answer’, the less analytic watchers will form the impression that the politician is competent and frank. The more analytic types just shrug their shoulders in disgust and turn the telly off.
A sentence can be very long and yet not reveal what information it contains until late in the sentence. Sometimes there is a key word that makes it all fall into place, The words before that one stack up like the numbers in a long calculation on a Reverse Polish calculator, impotent while they wait for release. Then the key word comes and it all falls into place. It attains a meaning. The wait for that word can sometimes be prolonged, like in this:
Though they all came from different social strata, sub-cultures and occupations, crammed together against their will in the prison cell from which they wondered if there would ever be any release, though none of them had known each other – or even known of each other – in their previous lives, though they squabbled and quarrelled over the tiniest of things, the one thing that bound them together despite the rivalries and petty jealousies, the perceived slights and reconciliations, the development, disintegration and reformation of cliques, was a single shared emotion, an emotion so powerful that they could feel it oozing out of one anothers’ pores, smell it on their breath and discern it in the tones of voice – the emotion of fear.
In some cases, the key word never comes. Perhaps the writer or speaker confuses themselves by their excessive verbiage and ends the sentence with an admission of defeat.
Books are too long as well! Novels are generally OK, as it takes time to get to know and care about the characters. But I have a strong sense that non-fiction books are often padded to reach whatever is considered a minimum page count for a book – usually at least 200. There isn’t really a strong market for writings that are halfway between essay and book length. In many cases a book really only has one idea, which could make a decent essay, but doesn’t justify a book. But essays don’t get to be put on a prominent shelf that catches your eye as you enter the bookshop, nor do they get listed on the New York Times best sellers’ list.
Nassim Taleb’s famous book ‘The Black Swan’ is like that. It really only contains one idea, which is that investors, bankers and other financiers have for decades been making crucial financial decisions based on theories in which they assume that the future will be like the past, and that all occurrences of randomness must follow the Normal Distribution (the nice friendly old ‘Bell Curve’). Decisions based on that erroneous, oversimplified assumption have repeatedly led to disasters, because events tend to be more extreme than is predicted by the Bell Curve. Taleb’s is a good insight, and definitely worth saying, but probably not worth stringing out to book length.
And then, if the book sells well, they write it again, ever so slightly differently, and pretend it’s a new book, with new ideas. Taleb did that. Self-help authors do it all the time – which raises the question ‘If your first book about how to live a better life was so incomplete that it needs to be supplemented by a second, why did I waste my time reading it?‘ I suspect Richard Dawkins may do it too. As far as I can tell he has written at least four popular explanations of evolution. I read The Blind Watchmaker and thought it was great (but too long, of course!). But I didn’t read The Selfish Gene, The Ancestors’ Tale or The Greatest Show on Earth because I couldn’t see any indicators that they would contain much substance that hadn’t already been covered in the one I had read. I imagine there is some new material in each of them, but I would guess it’s more likely to be a dozen pages’ worth rather than 200+.
Fiction authors and other creative artists do this too. Stravinsky acidly observed that Vivaldi wrote the same marvellous concerto five hundred times. Bach shamelessly reused his work (goodness knows he was paid little enough for it!) and Enid Blyton invented maybe a dozen adventure and fantasy stories, which she recycled into what seems like hundreds of similar tales (surely I’m not the only one that’s noticed the remarkable similarity between Dame Slap’s School for Bad Pixies and Mr Grim’s School for Mischievous Brownies?). And let’s not even mention Mills and Boon. But somehow I don’t mind that so much. We humans are story-telling animals, and telling the same story repeatedly, changing it just a little every time, is what we have always done. I find myself able to smile indulgently on the prolixity of Enid and Antonio and Mills (?), but alas not on that of Nassim or Richard, or Deepak Chopra.
I think I’ve ranted for long enough now about how We All (including me) need to work on being more concise with our communication. It’s time to relent a little.
Not all language is just about conveying information, so the efficiency with which the information is conveyed is not always the best test. In comforting a frightened child, information communication is not the purpose of our speech. I will restrain myself from objecting that the second half of the soothing phrase ‘There, there‘ is informationally redundant. In fact, I think I could even stretch to approving of its repetition, if its first invocation was insufficient to assuage the poor mite’s distress.
Declarations of love, expressions of support, telling jokes, goodbyes, hellos and well-wishes are all ‘speech acts’ that have important non-informational components. It seems appropriate to apply different expectations to those speech acts from those we apply to informational speech. Even there, there are limits though. Many’s the operatic love aria I’ve sat through where after a while I just feel like screaming ‘OK, you love him, we get it, can we move on with the plot now please?’ And waiting for Mimi to die in La Boheme (of consumption, what else?) in between faint protestations of her love for Rodolfo, can become a little trying on one’s patience after the first ten minutes of the death scene.
But communication of information is the purpose of much of the language we use, especially in our work lives. It is a pity that so much of it is ill-considered.
Hmmm. 1,742 words. I wonder if I could turn this into a book.
Bondi Junction, November 2015
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.
Bondi Junction, August 2015
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 ex 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.
Bondi Junction, March 2014
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.
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.
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