The physics of driving

This is quite a departure, I know, but one of my hobby-horses is how little people know about the physics of driving. Unlike our man-made road rules, the laws of nature are unbreakable, so a rudimentary knowledge can be useful.

But what prompted me to write this post was a road test I read of the new, upmarket Infiniti Q50 in EVO Australia (March 2014). The big-selling feature of the Infiniti Q50 is its so-called ‘direct adaptive steering’; a world first, apparently for a production car (as opposed to a prototype or research vehicle). It’s a totally ‘fly-by-wire’ steering system, so there is no mechanical connection between the helm and the front wheels. Personally, I think this is a dangerous idea, and I was originally going to title this post, rather provocatively, “A dangerous idea”. Not surprisingly, at least to me, the road-tester found the system more than a little disconcerting when he used it in the real world. It was okay until he wanted to push the car a little, when the lack of feedback through the wheel made him feel somewhat insecure.

There are gyroscopic forces on the front wheels, which naturally increase with cornering force and can be felt through the  steering wheel. The wheel weights up in direct proportion to this force (and not the amount of lock applied as some might think). In other words, it’s a linear relationship, and it’s one of the major sources of determining the cornering force being generated.

I should point out that the main source of determining cornering force is your inner ear, which we all use subconsciously, and is why we all lean our heads when cornering, even though we are unaware of it. It’s the muscle strain on our necks, arising from maintaining our inner ear balance, that tells us how much lateral g-force we have generated. On a motorcycle, we do the opposite, keeping our heads straight while we lean our bodies, so the muscle strain is reversed, but the effect is exactly the same.

Therefore, you may think, we don’t need the steering wheel’s feedback, but there is more. The turn-in to a corner is the most critical part of cornering. This was pointed out to me decades ago, when I was a novice, but experience has confirmed it many times over. Yes, the corner can change radius or camber or both and you might strike something mid-corner, like loose gravel, but, generally, if the front wheels grip on entry then you know they will grip throughout the rest of the corner. This is the case whether you’re under brakes or not, wet or dry surface. It’s possible to loosen traction with a heavy right foot, but most cars have traction control these days, so even that is not an issue for most of us. The point is that, if the front wheels grip on turn-in, we ‘feel’ it through the steering wheel, because of the gyroscopic relationship between cornering force and the weight of the wheel. And cornering force is directly proportional to the amount of grip. The point is that without this critical feedback at turn-in, drivers will be dependent on visual cues to work out if the car is gripping or not. What’s more, the transition from grip to non-grip and back won’t be felt through the wheel. If this system becomes the engineering norm it will make bad drivers out of all of us.

While I’m on the topic, did you know that at twice the speed it takes four times the distance to pull up to a stop? Perhaps, you did, but I bet no one told you when you were learning to drive. The relationship between speed and braking distance is not linear – braking distance is proportional to the speed squared, so 3 times as fast takes 9 times the distance to stop. This is independent of road surface, tyres and make of car – it’s a natural law.

Another one to appreciate is that at twice the speed, changing direction is twice as slow. There is an inverse relationship between speed and rate of change of direction. This is important in the context of driving on multi-lane highways. A car travelling at half the speed of another – that’s overtaking it, say – can change direction twice as fast as the faster car. This is also a law of nature, so even allowing for superior tyres and dynamics of the faster car, the physics is overwhelmingly against it. This is why the safest speed to travel on multi-lane highways is the same speed as everyone else. An atypically slow car, in these circumstances, is just as dangerous (to other motorists and itself) as an atypically fast car.

Addendum: I also wrote a post on the physics of riding a motorcycle.

Afterlife belief – a unique human condition

Recently I’ve been dreaming about having philosophical discussions, which is very strange, to say the least. And one of these was on the topic of the afterlife. My particular insight, from my dream, was that humans have the unique capacity to imagine a life and a world beyond death. It’s hard to imagine that any other creature, no matter its cognitive capacity, would be able to make the same leap. This is not a new insight for me; it’s one my dream reminded me of rather than initiated. Nevertheless, it’s a good starting point for a discussion on the afterlife.

It’s also, I believe, the reason humans came up with religion – it’s hard to dissociate one from the other.  Humans are more than capable of imagining fictional worlds – I’ve created a few myself as a sometime sci-fi writer. But imagining a life after death is to project oneself into an eternal continuity, a form of immortality. Someone once pointed out that death is the ultimate letting go of the ego, and I believe this is a major reason we find it so difficult to confront. The Buddhists talk about the ‘no-self’ and ‘no attachments’, and I believe this is what they’re referring to. We all form attachments during life, be it material or ideological or aspirational or through personal relationships, and I think that this is natural, even psychologically necessary for the most part. But death requires us to give all these up. In some cases people make sacrifices, where an ideal or another’s life takes precedent over one’s own ego. In effect, we may substitute someone else’s ego for our own.

Do I believe in an afterlife? Actually, I’m agnostic on that point, but I have no expectation, and, from what we know, it seems unlikely. I have no problem with people believing in an afterlife – as I say, it’s part of the human condition – but I have a problem when people place more emphasis on it than the current life they’re actually living. There are numerous stories of people ostracizing their children, on religious grounds, because seeking eternal paradise is more important than familial relationships. I find this perverse, as I do the idea of killing people with the promise of reaching heaven as a reward.

Personally, I think it’s more healthy to have no expectation when one dies. It’s no different to going to sleep or any other form of losing consciousness, only one never regains it. No one knows when they fall asleep or when they lose consciousness, and the same applies when one dies. It leaves no memory, so we don’t know when it happens. There is an oft asked question: why is there something rather than nothing? Well, consciousness plays a big role in that question, because, without consciousness, there might as well be nothing. ‘Something’ only exists for ‘you’ while you are alive.

Consciousness exists in a continuous present, and, in fact, without consciousness, the concepts of past present and future would have no meaning. But more than that, without memory, you would not even know you have consciousness. In fact, it is possible to be conscious or act conscious, whilst believing, in retrospect, that you were unconscious. It can happen when you come out of anaesthetic (it’s happened to me) or when you’re extremely intoxicated with alcohol or when you’ve been knocked unconscious by a blow. In these admittedly rare and unusual circumstances, one can be conscious and behave consciously, yet create no memories, so effectively be unconscious. In other words, without memory (short term memory) we would all be subjectively unconscious.

So, even if there is the possibility that one’s consciousness can leave behind the body that created it, after corporeal death, it would also leave behind all the memories that give us our sense of self. It’s only our memories that give us our sense of continuity, and hence our sense of self.

Then there is the issue of afterlife and infinity. Only mathematicians and cosmologists truly appreciate what infinity means. The point is that if you have an infinite amount of time and space than anything that can happen once can happen an infinite number of times. This means that, with infinity, in this world or any other, there would be an infinite number of you and me. But, not only am I not interested in an infinite number of me, I don’t believe anyone would want to live for infinity if they really thought about it.

At the start, I mentioned that I believe religion arose from a belief in the afterlife. Having said that, I think religion comes from a natural tendency to look for meaning beyond the life we live. I’ve made the point before, that if there is a purpose beyond the here and now, it’s not ours to know. And, if there is a purpose, we find it in the lives we live and not in an imagined reward beyond the grave.

Quantum mechanics without complex algebra

 

In my last post I made reference to a comment Noson Yanofsky made in his book, The Outer limits of Reason, whereby he responded to a student’s question on quantum mechanics: specifically, why does quantum mechanics require complex algebra (√-1) to render it mathematically meaningful?

Complex numbers always take the form a + ib, which I explain in detail elsewhere, but it is best understood graphically, whereby a exists on the Real number line and b lies on the ‘imaginary’ axis orthogonal to the Real axis. (i = √-1, in case you’re wondering.)

In last week’s New Scientist (25 January 2014, pp.32-5), freelance science journalist, Matthew Chalmers, discusses the work of theoretical physicist, Bill Wootters of Williams College, Williamstown, Massachusetts, who has attempted to rid quantum mechanics of complex numbers.

Chalmers introduces his topic by explaining how i (√-1) is not a number as we normally understand it – a point I’ve made myself in previous posts. You can’t count an i quantity of anything, and, in fact, I’ve argued that i is best understood as a dimension not a number per se, which is how it is represented graphically. Chalmers also alludes to the idea that i can be perceived as a dimension, though he doesn’t belabour the point. Chalmers also gives a very brief history lesson, explaining how i has been around since the 16^th Century at least, where it allowed adventurous mathematicians to solve certain equations. In fact, in its early manifestation it tended to be a temporary device that disappeared before the final solution was reached. But later it became as ‘respectable’ as negative numbers and now it makes regular appearances in electrical engineering and analyses involving polar co-ordinates, as well as quantum mechanics where it seems to be a necessary mathematical ingredient. You must realise that there was a time when negative numbers and even zero were treated with suspicion by ancient scholars.

As I’ve explained in detail in another post, quantum mechanics has been rendered mathematically as a wave function, known as Schrodinger’s equation. Schrodinger’s equation would have been stillborn, as it explained nothing in the real world, were it not for Max Born’s ingenious insight to square the modulus (amplitude) of the wave function and use it to give a probability of finding a particle (including photons) in the real world. The point is that once someone takes a measurement or makes an observation of the particle, Schrodinger’s wave function becomes irrelevant. It’s only useful for making probabilistic predictions, albeit very accurate ones. But what’s mathematically significant, as pointed out by Chalmers, is that Born’s Rule (as it’s called) gets rid of the imaginary component of the complex number, and makes it relevant to the real world with Real numbers, albeit as a probability.

Wootters ambition to rid quantum mechanics of imaginary numbers started when he was a PhD student, but later became a definitive goal. Not surprisingly, Chalmers doesn’t go into the mathematical details, but he does explain the ramifications. Wootters has come up with something he calls the ‘u-bit’ and what it tells us is that if we want to give up complex algebra, everything is connected to everything else.

Wootters expertise is in quantum information theory, so he’s well placed to explore alternative methodologies. If the u-bit is a real entity, it must rotate very fast, though this is left unexplained. Needless to say, there is some scepticism as to its existence apart from a mathematical one. I’m not a theoretical physicist, more of an interested bystander, but my own view is that quantum mechanics is another level of reality – a substrate, if you like, to the world we interact with. According to Richard Ewles (MATHS 1001, pp.383-4): ‘…the wave function Ψ permeates all of space… [and when a measurement or observation is made] the original wave function Ψ is no longer a valid description of the state of the particle.’

Many physicists also believe that Schrodinger’s equation is merely a convenient mathematical device, and therefore the wave function doesn’t represent anything physical. Whether this is true or not, its practical usefulness suggests it can tells us something important about the quantum world. The fact that it ‘disappears’ or becomes irrelevant, once the particle becomes manifest in the physical world, suggests to me that there is a disjunct between the 2 physical realms. And the fact that the quantum world can only be understood with complex numbers simply underlines this disjunction.

The Introspective Cosmos

I haven’t written anything meaty for a while, and I’m worried I might lose my touch. Besides, I feel the need to stimulate my brain and, hopefully, yours in the process.

Just before Christmas, I read an excellent book by Noson S. Yanofsky, titled: The Outer Limits of Reason; What Science, Mathematics, and Logic CANNOT Tell Us. Yanofsky is Professor in the Department of Computer and Information Science at Brooklyn College and The Graduate Center of the City of University of New York. He is also co-author of Quantum Computing for Computer Scientists (which I haven’t read).

Yanofsky’s book (the one I read) covers a range of topics, including classical and quantum physics, chaos theory, determinism, free will, Godel’s Incompleteness Theorem, the P-NP problem, the anthropic principle and a whole lot more. The point is that he is well versed in all these areas, yet he’s very easy to read. His fundamental point, delineated in the title, is that it is impossible for us to know everything. And there will always be more that we don’t know compared to what we do know. Anyone with a remote interest in epistemology should read this book. He really does explain the limits of our knowledge, both theoretically and practically. At the end of each section he gives a synopsis of ‘further reading’, not just a list. I found the book so compelling, I even read all the ‘Notes’ in the appendix (something I rarely do).

Along the way, he explains things like countable infinities and uncountable infinities and why it is important to make the distinction. He also explains the difference between computing problems that are simply incomputable and computing problems that are computable but would take more time than the Universe allows, even if the computer was a quantum computer.

He discusses, in depth, philosophical issues like the limitations of mathematical Platonism, and provides compelling arguments that the mathematics we use to describe physical phenomena invariably have limitations that the physical phenomena don’t. In other words, no mathematical equation, no matter its efficacy, can cover all physical contingencies. The physical world is invariably more complex than the mathematics we use to interpret it, and a lot of the mathematical tools we use deal with large scale averages rather than individual entities – like the universal gas equation versus individual molecules.

He points out that there is no ‘fire in the equations’ (as does Lee Smolin in Time Reborn, which I’ve also read recently) meaning mathematics can describe physical phenomena but can’t create them. My own view is that mathematics is a code that only an intelligence like ours can uncover. As anyone who reads my blog knows, I believe mathematics is largely discovered, not invented. Marcus du Sautoy presented a TV programme called The Code, which exemplifies this view. But this code is somehow intrinsic in nature in that the Universe obeys laws and the laws not only require mathematics to quantify them but, without mathematics, we would not know their existence except, possibly, at a very rudimentary and uninformed level.

Yanofsky discusses Eugene Wigner’s famous declaration concerning ‘The Unreasonable Effectivenessof Mathematics’ and concludes that it arises from the fact that we use mathematics to probe the physical world, and that, in fact, leaving physics aside, there is a ‘paucity of mathematics in general science’. But in the next paragraph, Yanofsky says this:

The answers to Wigner’s unreasonable effectiveness leads to much deeper questions. Rather than asking why the laws of physics follow mathematics, ask why there are any laws at all.

In the same vein, Yanofsky gives a personal anecdote of a student asking him why complex numbers work for quantum mechanics. He answers that ‘…the universe does not function using complex numbers, Newton’s formula, or any other law of nature. Rather, the universe works the way it does. It is humans who use the tools they have to understand the world.’ And this is completely true as far as it goes, yet I would say that complex numbers are part of ‘the code’ required to understand one of the deepest and fundamental mysteries of the Universe.

Yanofsky’s fundamental question, quoted above, ‘why are there any laws at all?’ leads him to discuss the very structure of the universe, the emergence of life and, finally, our place in it. In fact he lists this as 3 questions:

1: Why is there any structure at all in the universe?

2: Why is the structure that exists capable of sustaining life?

3: Why did this life-sustaining structure generate a creature with enough intelligence to understand the structure?

I’ve long maintained that the last question represents the universe’s greatest enigma. There is something analogous here between us as individuals and the cosmos itself. We are each an organism with a brain that creates something we call consciousness that allows us to reflect on ourselves, individually. And the Universe created, via an extraordinary convoluted process, the ability to reflect on itself, its origins and its possible meaning.

Not surprisingly, Yanofsky doesn’t give any religious answers to this but, instead, seems to draw heavily on Paul Davies (whom he acknowledges generously at the end of the chapter) in providing various possible answers to these questions, including John Wheeler’s controversial thesis that the universe, via a cosmic scale quantum loop, has this particular life and intelligence generating structure simply because we’re in it. I’ve discussed these issues before, without coming to any definitive conclusion, so I won’t pursue them any further here.

In his notes on this chapter, Yanofsky makes this point:

Perhaps we can say that the universe is against having intelligent life and that the chances of having intelligent life are, say, 0.0000001 percent. We, therefore, only see intelligent life in 0.0000001 percent of the universe.

This reminds me of John Barrow’s point, in one of his many books, that the reason the universe is so old, and so big, is because that’s how long it takes to create complex life, and, because the universe is uniformly expanding, age and size are commensurate.

So Yanofsky’s is a deep and informative book on many levels, putting in perspective not only our place in the universe but the infinite knowledge we will never know. Towards the end he provides a table that summarises the points he delineates throughout the book in detail:

Solvable computer problems                             Unsolvable computer problems

Describable phenomena                                    Indescribable phenomena

Algebraic numbers                                            Transcendent numbers

(Provable) mathematical statements                 Mathematical facts

Finally, he makes the point that, in our everyday lives, we make decisions based primarily on emotions not reason. We seemed to have transcended our biological and evolutionary requirements when we turned to mathematics and logic to comprehend phenomena hidden from our senses and attempted to understand the origin and structure of the universe itself.

Dr Who 50th Anniversary Special

A bit late, I know, as it was 2 weeks ago, but worthy of a post. Despite my advanced years, I didn’t see Dr Who in my teenage years when it first came to air. I really only became a fan with the resurrection or second coming in 2005, when Russell T Davies rebooted it with Christopher Eccleston as the Doctor. But, personally, I liked David Tennant and then Matt Smith’s renditions and it was a pleasure to see them together in the 50^th Anniversary special, The Day of the Doctor, alongside John Hurt, who was an inspirational casting choice. One should also mention Steven Moffat, who, as chief writer, deserves credit for making the show a monumental success. Writers rarely get the credit they deserve.

I recently re-watched episodes involving David Tennant and Matt Smith, and I particularly liked the narrative involving Martha Jones, played by Freema Agyeman, who, as far as I know, is the first non-white ‘companion’. Arguably, as significant as Halle Berry’s appearance as a ‘Bond girl’. My favourite episode was the ‘Weeping Angels’ because it was so cleverly structured from a time-travel perspective.

I saw the 50^th Anniversary Special in a cinema in 3D (good 3D as opposed to bad 3D) and I’ve since watched it again on ABC’s iview (expires today). It was also great to see Billie Piper recreate her role as Rose Tyler or Bad Wolf, albeit in a subtly different guise. It was one of many clever elements in this special. At its heart it contains a moral dilemma – a la John Stuart Mill – which was mirrored in one of the subplots. The interaction between John Hurt’s Doctor and Billie Piper’s sentient AI conscience is one of the highlights of the entire story, which was reinforced when I watched it for the second time. I know that some people had trouble following the time jumps and plot machinations, but that wasn’t an issue for me. To create a doomsday device to end all doomsday devices and give it a sentient conscience is a stroke of narrative genius. At 1hr 16 mins it’s not quite movie-length, yet it shows that length is not a criterion for quality. I found it witty, clever and highly entertaining, both in story context and execution; suitably engaging for a 50^th Anniversary celebration.

Postscript: I should confess that the Daleks had an influence on ELVENE, which is readily spotted by any fan of popular Sci-Fi culture.

Probability and Causality – can they be reconciled in our understanding of the universe?

In last month’s Philosophy Now (July/August 2013) Raymond Tallis wrote an interesting and provocative article (as he often does) on the subject of probability and its relationship to quantum mechanics and causality (or not). He started off by referencing a talk he gave at the Hay Festival in Wales titled, ‘Has Physics Killed Philosophy?’ According to Tallis, no, but that’s neither the subject of his article nor this post.

Afterwards, he had a conversation with Raja Panjwani, who apparently is both a philosopher and a physicist as well as ‘an international chess champion’. They got to talking about how, in quantum mechanics, ‘causation has been replaced by probability’ unless one follows the ‘many-worlds’ interpretation of quantum mechanics, whereby every causal effect is realised in some world somewhere. One of the problems with the many-worlds view (not discussed by Tallis) is that it doesn’t account for the probability of an event occurring in ‘our world’ as dictated by Schrodinger’s equation and Born’s rule. (I’ve written an entire post on that subject if the reader is interested.)

David Deutsch, the best known advocate of the many-worlds interpretation, claims that the probabilities are a consequence of how many worlds there are for each quantum event, but if there are infinite possibilities, as the theory seems to dictate according to Feynman’s integral path method, then every probability is one, which would be the case if there were an infinite number of worlds. It has to be said that Deutsch is much cleverer than me, so he probably has an answer to that, which I haven’t seen.

Tallis’s discussion quickly turns to coin-tossing, as did his conversation with Panjwani apparently, to demonstrate to ordinary people (i.e. non-physicists) how probabilities, despite appearances to the contrary, are non-causal. In particular, Tallis makes the point, often lost on gamblers, that a long sequence of ‘Heads’ (for example) has no consequence for the next coin toss, which could still be equal probability ‘Head’ or ‘Tail’. But, assuming that the coin is ‘fair’ (not biased), we know that the probability of a long sequence of ‘Heads’ (or ‘Tails’) becomes exponentially less as the sequence gets longer. So what is the connection? I believe it’s entropy.

Erwin Schrodinger in his book (series of lectures, actually), What is Life? gives the example of shuffling cards to demonstrate entropy, which also involves probabilities, as every poker player knows. In other words, entropy, which is one of the fundamental laws of the universe, is directly related to probability. To take the classic example of perfume diffusing from a bottle into an entire room, what is the probability of all the molecules of the perfume ending up back in the bottle? Infinitesimal. In other words, there is a much, much higher probability of the perfume being evenly distributed throughout the entire room, gusts of wind and air-conditioning notwithstanding. Entropy is also linked to the arrow of time, but that’s another not entirely unrelated topic, which I may return to.

Tallis then goes on to discuss how each coin toss is finely dependent on the initial conditions, which is chaos theory. It seems that Tallis was unaware that he was discussing entropy and chaos theory, or, if he did, he didn’t want to muddy the waters. I’ve discussed this elsewhere in more detail, but chaos is deterministic yet unpredictable and seems to be entailed in everything from galactic formation to biological evolution. In other words, like entropy and quantum mechanics, it seems to be a fundamental factor in the universe’s evolvement.

Towards the end of his article, Tallis starts to talk about time and references physicist, Carlo Rovelli, whom he quotes as saying that there is ‘a possibility that quantum mechanics will become “a theory of the relations between variables, rather than a theory of the evolution of variables in time.”’ Now, I’ve quoted Rovelli previously (albeit second-hand from New Scientist) as claiming that at the basic level of physics, time disappears. The relevance of that assertion to this discussion is that causality doesn’t exist without time. Schrodinger’s time dependent equation is dependent on an ‘external clock’ and can only relate to ‘reality’ through probabilities. These probabilities are found by multiplying components of the complex equation with their conjugates, and, as Schrodinger himself pointed out, that is equivalent to solving the equation both forwards and backwards in time (ref: John Gribbin, Erwin Schrodinger and the Quantum Revolution, 2012).

So it is ‘time’ that is intrinsic to causality as we observe and experience it in everyday life, and time is a factor, both in entropy and chaos theory. But what about quantum mechanics? I think the jury is still out on that to be honest. The many-worlds interpretation says it’s not an issue, but John Wheeler’s ‘backwards in time’ thought experiment for the double-slit experiment (since been confirmed according to Paul Davies) says it is.

When I first read Schrodinger’s provocative and insightful book, What is Life? one of the things that struck me (and still does) is how everything in the universe seems to be dependent on probabilities, especially on a macro scale. Einstein famously said “God does not play with dice” in apparent frustration at the non-determinism inherent in quantum mechanics, yet I’d say that ‘God’ plays dice at all levels of nature and evolution. And causality seems to be a consequence, an emergent property at a macro level, without which we would not be able to make sense of the world at all.