One such tangent would be developing a rigorous and robust understanding of why dividing by zero does not work. I am sure many people know this – but a surprising number will not. A lot of people will tell you it’s does not exist – true, but why? Others will say it’s just infinity, which is fine but can I define it using a limit and infinity? No, of course you can’t.

Division is first explained to us as sharing some quantity of objects among another quantity of objects (or people). So 15 shared among 3 gives five. The question of how many items are received “when” no items are shared among one, two, three of four hundred people has no real meaning. When are no items shared? Always? Never? There is no real meaning in this elementary description of division – so we may conclude that it is just undefined. Loosely this is right, but does not really contain a complete description of division.

So why can’t we just attack the problem with limits? This argument would see something like this set up;

Where a and b are real numbers, and b approaches zero from the right. when we examine this limit from the left, we would get the following expression;

So when we combine these two, to calculate the limit as b tends to zero?

So it is wrong to define a/0 as infinity, approaching infinity etc. The limit does not exist. So you are only correct if you say that division by zero is undefined; which above we have shown using calculus. You can also show this using algebraic rings, or inverse multiplication. The most basic, and amusing demonstration as to why this monstrosity cannot exist, is the fallacy we would create.

Doing some really basic maths;

0 x 20 = 0

and;

0 x 5 = 0

In this case, the following must also be true;

0 x 20 = 0 x 5.

In a world where we accept division by zero, we can write;

0/0 x 20 = 0/0 x 5,

20 = 5.

This is no world I want to live in.

Let me know if you want to look into other ways we can demonstrate that a/0, and indeed 0/0 is undefined – there are so many of them and it is fundamental to mathematics as we know it. There are areas of mathematics (such as matrices) where such operations are defined or pseudo defined, but these are special cases which in no way violate the discussion above.

]]>*Photo Credit: NASA*

From these numbers, our characterisation of other stars and some nifty extrapolations, assuming the universe is homogenous, (matter is evenly distributed and the same in all directions) we have calculated that in the universe there* could* be as many as 40 billion earth-like planets! By earth-like we usually mean that they exist in what is known as the habitable zone, a region which is sufficiently far away from its host star that it is not frazzled but close enough such that there is enough heat energy to sustain life i.e. conditions where water can exist. These numbers from exoplanetary research have opened our eyes towards our insignificance, having once thought of ourself as a special and rare ball floating at just the right distance from just the right size star, we now realise that this combination is far from unique. Of course, we should remember that these are the conditions for life *we* expect it, we refine our searches for life based on water. It is the only type of life we know and again we are blinded by our sample size and lack of comparison. Perhaps life has somehow evolved from different building blocks, all depends on your definition of life I suppose.. a tangent on which I shall not divert today.

I was sent an article by a true physics guru the other day pulling together exoplanets and black holes by Sean Raymond which sparked my interest. The premise is funky; *theoretically* what is the astrophysical system that could host the most life. (From now on I will speak of life we understand it, water based life on an earth-like planet.) When I say system, I mean similar to ‘solar system’, in our case we have our star, the sun, at the center with a host planets orbiting it at increasing distances. The restriction with having a singular star at the center of the system, is that planetary orbits can only come so close together before their gravitational effects start to pull on each other and dominate over the gravitational pull felt from the star. If this happens the orbits become unstable and the system falls out of its delicate balance. Key takeaway – there are only so many Earth-like planets you could fit into the habitable zone (set distance from the star) for a sun-like star before things get messy.

*Credit: Sean Raymond*

You need something *heavier* at the center of the system, such that more planets can be packed into the habitable zone whilst their gravitational effects on each other will still be overwhelmed by the gravitational mass in the center, meaning they stick to their orbits. In Raymond’s imaginary system, instead of a singular star at the center there is a *super massive black hole. *Taking a supermassive black hole 1 million times the mass of the sun, *550-Earth* mass planets could fit in stable orbits in a set distance away from the center of the system. Of course replacing the sun with a black hole we loose the definition of habitability altogether, we need a sun-like star for heat and warmth. The funky idea theorised by Raymond to create a habitable zone is place to first place a ring of sun-like stars around the black hole. In fact, 9 sun-like stars 0.5 astronomical units from a one-milton mass sun would make 550 Earth-like planets habitable, with each planet circling in its own orbit. If taken a step further and orbits are shared with the planets spread out evenly throughout the ring, Raymond calculated the super-massive black hole could hold as many as 400 rings, each holding *2,500 planets.* The beauty of this set up would be the sky show inhabitants of the worlds would enjoy, with other planets passing through large areas of the sky, day and night. In fact Raymond even goes so far as to suggest that planets would be close enough such that a space elevator could be constructed to connect them!

*Photo Credit: NASA*

Here’s one of my favourite aspects and of course it’s to do with time. Take a planet in one orbital ring and another in the orbit ring one further away from the black hole. The planet closer to the black hole will feel a significantly larger gravitational pull, hence being in a greater gravitational potential well, time will pass slower for them than planets in further orbits. See my previous posts on relativity and time dilation for some more insight in this disturbing phenomenon. As such, if two babies are born at what can be taken as the same moment in an external reference frame on two planets, the one on the inner ring would age slightly more slowly than the other. If taking the inner most and outer most habitable ring and two babies, the effect could be taken to bizarre extremes. The inner most baby could have reached it’s 2nd birthday while the outermost was seeing its grandchild reach their 2nd. Such discrepancy in one shared system, the effects of relativity plainly observable.

Of course this whole system is all theoretical fun and games and could be criticised for its fantasy, but it brings together ideas in theoretical and astrophysics nicely and i’m all for that at the moment. The system could be criticised by being ‘unnatural’ but if by that we mean statistically unlikely then we run into some trouble. Playing statistical games from our vantage point of the universe is a dangerous business. We only have one data point of life-formation, of data point of a life-bearing planet and a very small sample of universe which we can observe from our little corner of space. Although observations in exoplanetary science are continually stacking up, it is only from the spectra of light that we can do our de-coding, we have *much* still to learn about other systems. [See ‘Strange New Worlds‘ for some detail on these observations] If ‘unnatural’ is *not* meant from a statistically unlikely point of view alone then, what even is an unnatural system except something we as humans find inherently displeasing somehow? It is funny what the human mind can easily normalise, which when actually sat down and deeply thought about is mind-bending. I like this quote a lot to remind us of that;

*“The fact that we live at the bottom of a deep gravity well, on the*

*surface of a gas covered planet going around a nuclear fireball 90*

*million miles away and think this to be normal is obviously some*

*indication of how skewed our perspective tends to be.”*

— Douglas Adams, The Salmon of Doubt: Hitchhiking the Galaxy One

Last Time

]]>

First I want to outline two hypotheses that will be central to our discussion. The weak and strong cosmic censorship hypotheses introduced to constrain the nature of black holes. Remember before when we spoke of black holes, it was explained that the point in the very center is a place of such infinite density that all our theories and frameworks for describing space and time breakdown here. The mathematics simply cannot model the extremity that exists at this point. This point is called the *singularity* and it shrouded from our observation due to the fact that black holes have event horizons, this is surface a certain radius from the center of the black hole from which, when information passes through it cannot then escape. If you fall past this radius there is no coming back. This is why we cannot fully understand the inner nature of a black hole and why of course it appears *black. (For a longer explanation of these phenomena please see the posts linked above).*

Now, the weak cosmic censorship hypothesis reiterates this point and firmly states that all singularities must come with this surrounding horizon, meaning they cannot be directly observed and must *stay* hidden to observers who have not crossed the threshold of no return. This is nicely summed up by the phrase that there can be no* naked* singularities. General Relativity is strictly PG-13. The strong cosmic censorship hypothesis is of a different ilk. This hypothesis states that the general relativity is a *deterministic *theory, deterministic meaning that given the initial conditions of a system, its future state is entirely predictable. This is the clear, stable world we know and love. A bit more of a discussion (with some philosophical consequences) of deterministic theories, along with their antithesis, probabilistic theories, can be found here.

Now to the crux of today’s post. In the 1960s mathematicians discovered a solution of Einstein’s field equations which described a system that was no longer deterministic, i.e. contradicting the strong cosmic censorship hypothesis. This came in the form of a *rotating black hole. *Far outside a black hole we can use classical mechanics to describe the universe, which as we’ve said is a deterministic theory, it gives a clear forecast of how the system we’re looking at will evolve, given its current information. Near a black hole we must move from classical mechanics to general relativity yet we can still think of the world as deterministic, GR and Einstein’s equation still manage provide a single forecast for how space-time will evolve. In fact this continues to work clearly and without ambiguity even when we cross the event horizon, there is still one clear future, the snag comes when we reach the *second horizon*.

This second horizon is known as the Cauchy Horizon, predicted by mathematicians as lying beyond the event horizon, deeper inside the black hole. Here when analysed, Einstein’s equations become erratic, spewing out multiple solutions, telling us that many different configurations of space-time can then occur, none of which are more justified than another. The mathematics of general relativity there suggests the universe is inherently unpredictable beyond this point, that there is no single path that will be followed given the initial conditions. The path at the Cauchy horizon branches off from one into many, like the frayed end of a rope.

So how to deal with this unsettling fact? Mathematician Roger Penrose offered the first argument. His angle was that sticking to the strong cosmic censorship conjecture the universe *must *be inherently deterministic and as such there must be some error in our understanding of the nature of the Cauchy horizon and the effect it has on Einstein’s equations. Penrose suggested that the Cauchy horizon is* unstable *and it cannot exist in the physical sense in which it had been postulated. In fact any matter passing through the black hole and hitting the Cauchy horizon would cause it to instantaneously collapse to a *singularity itself. *Penrose claims that Cauchy horizons can only exist in an idealised universe where *nothing else* exists except a single black hole in question. However if you introduce any other matter into the universe, it will eventually fall into the black hole, hit the Cauchy horizon and cause its dramatic collapse. Therefore Penrose attempts to save the strong cosmic censorship conjecture by claiming the Cauchy horizon cannot split the paths in the first place because it has been postulated in error and is only an *idealised* mathematical solution to General Relativity.

However recently a mathematician called Dafermos at Stanford University has offered a different and insightful alternative. He postulated that the Cauchy horizon does form a singularity but not of the extreme kind as suggested by Penrose or of the kind we know exist at the center* *of black holes. The horizon *pulls* on the surrounding spacetime and matter but does *not* cause it to collapse entirely and as such there is a continuation of spacetime past its border. However (and here is the key part) this continuation is not *smooth*. Einstein’s equations evaluate the changes in spacetime over infinitesimal increments but these infinitesimal increments need to be joint up smoothly for the mathematics to work. The Cauchy horizon causes a discrete *break* in the spacetime so much so that Einstein’s equations are not longer satisfied or even applicable..

*So* because they can no longer be applied correctly, they no longer give us meaningful solutions and we are not faced with having to take the multiple conflicting outputs seriously. Determinism preserved. However the theory is rather unsatisfactory in the sense that we’ve managed to retain what the mathematics has told us about the genuine existence of a Cauchy horizon and do away with the conflicting multiple futures *but *the price to pay is that we cannot extend our mathematical analysis beyond the second horizon because the best theory we have, embodied by Einstein’s equations, breaks down. An alternative theory needs to step in whose mathematics can smoothly transition across the Cauchy horizon, deeper into the Black Hole. Border control at the Cauchy horizon requires documentation in the form of mathematics we unfortunately do not yet possess.

]]>

I am registered to complete all of my second year of my second degree, along with third year studies in complex analysis this coming autumn. In preparation for that I have begun studying two beautiful books;

- The Road to Reality: A Complete Guide to the Laws of the Universe, Rodger Penrose, in the hope of resharpening a broad overview of physics; and
- Complex Analysis: A Hitchhiker’s Guide to the Plane, Ian Stewart, a textbook that seems to be horribly underrated on Amazon that saw me good in my first undergraduate.

As such, you can expect random high level physics posts and more technical complex analysis posts until my studies start again, and I hope to post with a little more depth.

Today I thought I would share with you something interesting I read in Penrose’s book – right near the start about three “realms” that live around us and interact, which brings three mysteries. It raises some interesting questions around how the world should be viewed.

**The physical world **

This world is easy to understand, it is the world of things which take a physical existence. You would fit into this category, as would a rock or a table. There is a deeper philosophical question around what a physical object actually is, but leave that for those who like to get tangled up in all of that nonsense. For us a physical object is an object in the sense that immediately springs to mind – the sense which is inherent in our everyday experiences of what physically exists.

**The mental world**

The mental world extends beyond the physical, and includes a whole host of things which are not present in the physical world. Anger is an example – it does not manifest itself in a physical form in the true sense, but it is of course present in the mental world. You know what it is and you know when you feel it – it is within most people’s emotional repertoire.

**Platonic mathematical world**

The Platonic mathematical world is probably the most contentious of the three worlds, but makes reference to the fact that mathematical entities do not belong in space or time, they are eternal and unchanging. A square for example, lives in this world – you cannot construct a perfect square in the physical world. You can however imagine one, which means it also lives in the mental world – we will discuss the links later. There is a sort of assumption that the mathematical world exists and is eternal, we merely borrow items from this realm and return them when we are done.

There are other more subtle considerations as to what the platonic mathematical world actually means – for example It may be considered a realm of everything which is mathematically true, allowing for mathematical entities to exist even if they cannot all be proven from a consistent set of axioms. I don’t want to get my hands dirty with the philosophy – it’s not my bag and does not interest be greatly.

**Penrose’s prejudiced view**

The above title is not be being dismissive of the great mathematician, this is how he himself labels his view, which is presented in a diagram below, with the three mysteries numbered. We will discuss them one by one.

- The physical world is described in its entirety by the Platonic world of mathematics. This is a large claim, but a belief I certainly hold – that there exists timeless mathematics which can be drawn upon to explain the workings of the entire physical world. Ideas like this, or course, destroy notions of free choice and alike but I think I am OK with that. Only a small portion of the Platonic world of mathematics is required to explain the entire physical world – this makes perfect sense. There are a lot of mathematical beings which exist purely in the mathematical world, we need not imply that every single piece of mathematics explains something in the physical world. It is however interesting, how more and more pure mathematics concepts are finding application in the physical world – applied mathematics seems to grow along with our understanding of the universe.
- The mental world is fully and totally contained in a small section of the physical world. This makes perfect sense, but you won’t like it if you believe in certain religious notions. Everything that comprises the mental world, namely brains, products of the physical world – they are made of stuff, just like everything else. This makes sense to me, I don’t view my consciousness to be anything more than the product of physical building blocks. But clearly not everything physical is mental – such as a stone which has , so a small section of the physical world encompasses the mental world.
- Finally, a small section of the mental world encompasses the Platonic mathematical world – that is to say anything which is capable of being expressed in “true” mathematical terms can be held in the mental state, by some being or another, but that there are things in the mental state beyond the Platonic mathematical – earlier we used the example of anger, but generally any emotion works.

This arrives us at an interesting predicament – if all of the platonic world is contained within some of the mental world, all of the mental world is contained is some of the physical world and all of the physical world is contained within some of the Platonic something is broken! This would break a basic transitivity law!

Penrose points out that there are many possible explanations for this – most interestingly perhaps these worlds are not as distinct as we are making out in this representation, which means the question of one being contained within another is not a proper question. If there is more overlap than we have drawn in the above diagram, where is it? Or maybe it has been drawn wrong…

**The non-prejudiced view **

Like any good and open scientific mind, Penrose presents and considers the alternative view which does embody his own personal views on the world.

Here some alternations have been made:

- We have allowed for the mental world containing things which are beyond the physical world – this will appease those who have a more spiritual view on life.
- Not all of the platonic world is capable of being contained in the mental world – this will upset those who like me, think we have the theoretical potential to decipher all.
- Not all of the physical world can be explained by the Platonic mathematical world – again this goes against the grain to any mathematician or scientist, who will like to believe that the world can be depicted through some continually refined mathematical model.

What is the right answer? At this stage nobody knows. You cannot, for example, tell me mathematics which is incapable of entering the mental state, because in order to do so you have negated your point. I favour Penrose’s view over the one just presented, but I do accept the other opinions.

Of course, there could be a problem with the very way this question is constructed – but none the less it’s good fun to play around.

]]>Today I felt like breaking that silence to come to write about a process, that I like to think embodies the phrase ‘making a mountain out of a molehill’ in the truly cosmic sense. How, quantum fluctuations in the dawn of the universe were the seeds that developed into the stars, nebulae and galaxies we see today. The origin of such rich complex structure can be traced back to tiny quantum fluctuations in those first moments of our universe.

The paradigm most accepted by cosmologists today for describing the early universe is inflation. This is the idea that the early universe underwent a period of rapid expansion, small patches of spacetime were vastly stretched out and the universe grew at an accelerated pace. The theory of inflation is proposed for many reasons, including it’s ability to explain why the universe is observed to be flat and why it is observed to be homogenous. By a homogenous universe we mean it is, on large scales, the same in every direction we look. For example when we measure the Cosmic Microwave Background radiation (for our purposes here think of this as the background temperature of the universe) it is the same on every patch of the sky. Inflation allows these patches that are so far flung apart to have been able to be close enough together and ‘talk’ to each other in the early universe that they could have equilibrated their temperatures. I hope to do a more detailed post on the workings of inflation later. The only idea we need today is the rapid expansion of spacetime and the idea that the driver of inflation is a* field*. For a recap on the idea of a field see What is a Field?

Now as we know that the universe was undergoing such a rapid expansion, logically it must have been much smaller in the past. At very early times the universe was* so small* that the theory that ruled the realm was the theory of the small, Quantum Mechanics. If you want a refresher on Quantum Mechanics see the Laws of Quantum. In quantum physics, a quantum fluctuation is the temporary change in the amount of energy in a point in space, as explained in Werner Heisenberg’s uncertainty principle. Now this point is a little tricky but essentially what happens is that the field driving inflation experiences these quantum fluctuations, and everything else at this time is rapidly expanding, these quantum fluctuations are blown up too.

Now every point in spacetime has a region around it that contains events that can effect it. What I like to do is think of this region as a sphere, with the point of interest at the center. This region is called the Hubble radius (see point two in The Horizon and Beyond). These quantum fluctuations get blown up *so *large that they exit their initial Hubble radius and for the moment we put them to one side, worrying about what is inside only.

Now time progresses in the universe’s expansion, inflation ends and we enter the region thought of as the standard big bang cosmology, temperatures heat up, and particles emerge from the primordial soup that existed. For a punchy recount of these stages of the universe see A brief history of the universe. In short, electrons fly around and eventually combine with nuclei to form atoms and small structure begins to come in existence. All the while the universe is still expanding, but at a much slower rate as when compared with during inflation. This means the Hubble radius or sphere that we were imagining earlier is also increasing and* eventually* it gets big enough that those quantum fluctuations outside that we now like to call *cosmological perturbations* (because they certainly aren’t quantum anymore!) get re-engulfed by the sphere.

[Replace the ‘co-moving horizon’ term with the ‘Hubble radius’ we were talking about, there is a subtle difference but for our purpose the concepts are interchangeable]

The perturbations affect (or if you’re well versed in the language of physics *couple to) *the density of the existing matter and then as a result experience the force of gravity, subsequently undergo gravitational collapse and this process over time is what forms the large scale structure in the universe today. It was the quantum mechanical fluctuations in the initial universe that *broke *the initial smoothness if you like of the universe and provided the initial ‘clumpiness’ that when when coupled to the density of matter, over time caused a coalescing of matter into beautiful structures such as stars, planets and galaxies. The complexity that we see all around can be traced all back to simple fluctuations, that acted as the seeds of structure for the universe and in turn, essentially of life.

]]>

* ‘In light of the theory of Special Relativity is a Passage of Time and the argument of the Presentist untenable?’*

The paper elucidates on the ideas i’ve touched on here at RTU in the past, beginning with the philosophical notion of the present moment and then introducing the mathematics of Special Relativity to deconstruct the nature of time. Conclusions are then drawn on whether we must abandon our familiar notion of the present moment due to the implications of theory. In a nutshell Special Relativity postulates observers moving at different speeds can measure the time interval between events to be different – not good for our common belief of a unique ticking of time!

The paper is fairly long but seeing as I won’t be posting for a little while perhaps some of you may be glad of it. I hope the piece is accessible in both the physics and philosophy content but if there are any questions on the content and ideas please drop me a message below and I will find a time to answer!

]]>There are different ways to think about the importance of QFT, firstly we we can think of it as the extension of Quantum Mechanics from a system of few particles to a system of *many *particles. Quantum Mechanics can explain accurately the behaviour of one particle and therefore it can only operate with a limited *number of degrees of freedom. *(A degree of freedom of a physical system is a variable that is necessary to characterise the state of a physical system. For example a system that is confined to move in *one* direction with a* fixed* velocity has 2 degrees of freedom). As such QFT extends QM so that we are able to handle systems of many particles and* infinite* degrees of freedom.

Quantum Field Theory can also be thought of as the reconciliation of Quantum Mechanics and Special Relativity. The Schrodinger Equation – (the fundamental law for the evolution of* Quantum Mechanical *states in time) cannot obey the requirements of *relativistic* theories. Special Relativity, a relativistic theory as the name would suggest, requires that physical laws of nature are* invariant* under certain transformations (namely Lorentz transformations). For example a law of nature in one reference frame must look exactly in the same in a different reference frame that was shifted say shifted in position or boosted by a certain velocity. However the Schrodinger Equation is *not* invariant under such transformations and a quantum mechanical state will *not* evolve in exactly the same manner as one in a different frame. Additionally, a second clash between Quantum Mechanics and Special Relativity occurs, when particles have velocities close to the speed of light, as here Quantum Mechanics breaks down. But QFT allows us to work in relativistic frames which is extends our understanding of the world of the tiny enormously, as often tiny particles are able to move at very high speeds.

*This diagram illustrates the two points above, N stands for the number of degrees of freedom, SRT for Special Theory of Relativity. *

Quantum Field Theory treats particles as excited states of an underlying field (see ‘What is a Field‘ for an introduction to the concept of a field). In QFT, quantum mechanics interactions among particles are then described by interactions among the *underlying quantum fields*. The notation of the theory combines classical field theory, special relativity and quantum mechanics a new overarching manner. QFT was the pivotal rung in the ladder to elevate our understanding of the tiny into the realm of the fast moving whilst also extending our ability to be able to analyse systems with many particles and infinite degrees of freedom.

QFT is a wonderfully successful theory and one of modern physic’s great accomplishments. It is an effectively field theory and is widely believed to be a good low-energy approximation to a more fundamental theory which could take the physics towards the final frontier of incorporating General Relativity with the quantum world.

]]>

2017 marks the 40 year anniversary of the Voyager missions. The missions were originally conceived with the intention to explore Jupiter and Saturn and then onto, if successful, the further reaches of our solar system. The Voyager missions have become *the *success story of space missions, let’s recap their journey.

Way back in 1960s calculations revealed that, due to the alignment of the planets, it would be possible for a spacecraft launched in the late 1970s to visit all four of the outer giants in the solar system with an orbit that manipulated the gravity of each planet to swing the spacecraft round and on to the next. The technical term for this is a gravity assist, or a cooler term, a gravitational slingshot. And so the mission begins, seizing this opportunity which only comes roughly once every 200 years, truly a *once in a lifetime *chance.

The team at NASA decides to create two twin spacecrafts, Voyager 1 & 2 which are designed to take slightly different orbits. Voyager 2 is launched first on August 20 1977, followed 16 days later by Voyager 1 who receives the title of 1 because it will reach Jupiter and Saturn first. The Voyagers spend the next 20+ years travelling through the solar system sending back the most detailed images we have ever seen. On Jupiter for the first time we can see active volcanoes on the moons, understand that the Red Spot is a enormous cyclone-like storm and detect the presence of lightening. On Saturn we discover three more moons , come to learn that the largest moon Titan harbours a thick Earth-like atmosphere and we propel our understanding of the composition of the rings. And these are just the highlights. The main objective of the Voyager missions is a success, but these marathon runners are only just getting started…

After hurtling past Uranus and Neptune, encountering several new moons around each and sending back high resolution images of these icy worlds Voyager 1 gives us one of its most famous pieces in 1990. At a distance of 4 billion miles from the sun Voyager 1 takes the ‘Solar System Family Portrait’ a series of images that capture all the planets in orbit around the sun and the ‘Pale Blue Dot’ image, the image which captures the Earth suspended as a tiny speak in a sunbeam. This never seen before perspective inspires Carl Sagan, a leader behind the mission to write his piece on the humble nature of our planet and of all those that reside on it.

The Voyagers now continue to press outward and conduct studies of interplanetary space. In 1998 Voyager 1 became the farthest human-man object from Earth in space, going further than any craft has gone before. And the marathon journey just keeps on going. In 2004 Voyager crossed the barrier known as the ‘Termination Shock’ here the solar wind slowly down and heats up as it clashes with the interstellar wind. This new area of space in known as the Heliosheath and officially marks the end of the solar system. Then recently in 2012 Voyager 1 entered interstellar space, passing beyond the finally boundary known as the Heliopause – the boundary between our solar bubble and the matter ejected by explosions of other stars. (Soon to be followed by Voyager 2). The spacecraft continue to study ultraviolet sources amongst the stars in interstellar space and are still capable of sending this data back to Earth. In 2013 the first measurement of the density of the interstellar medium was made when an ‘explosion’ from the sun causes waves to ripple outward, creating a ripple in the plasma of interstellar space.

As well as the collection of data and measurements, the Voyagers have another important purpose, a purpose which may well never be fulfilled. They both contain a ‘Golden Record’ a 12-inch gold-plated copper disk, a beautiful artefact which plays the role of a kind of time capsule. The record would, if obtained by extraterrestrials, attempt to communicate the story of our world. It is contains a variety of images, natural sounds of earth, spoken greetings from languages, and music specifically curated to best attempt to display the diversity of life and culture on Earth. The ultimate *message in a bottle. *

At this moment in time Voyager 1 is 12,999,227,000 miles away from Earth, Voyager 2 is 10,728,140,000 miles away and they are both rack up miles every second – 330 million miles each year. It currently takes 19 hours for a light signal to be sent from Voyager 1 to Earth so safe to say communication isn’t easy. It is amazing to think that even with the 1970s hardware, 40 years on scientists can still communicate with the crafts but how much longer this will continue is uncertain. It is predicted communications will drop off between 2022-2025. However, it is not the growing distance that is the main problem. The issue is fuel supply, because the amount of fuel the crafts could be launched with was finite, eventually they will run out of juice and have to wander the galaxy alone, meaning we’ll no longer be able to locate, transmit or receive. The spaceships will go completely *off the radar, *but the distance they’ve got so far since 1977 without running out of fuel is remarkable!

So the Voyagers, the ultimate expectation exceeders, continue into their VIM phase (Voyager Interstellar Mission) 40 years on and still reach for the stars. Although we will soon loose communication with our old friends, they carry with them the human blueprint and the story of our pale blue dot into deep space, a story we hope one day may be found by another.

]]>

Firstly a group, denoted G, is a collection of elements call them g(1), g(2), g(3)…. and there exists a group operation, denote it *, which determines how the elements act on each other. Now the elements of the group must obey the 4 axioms of group theory. I’ll lay them out first and everything may seem rather abstract to begin with – but bear with me, all will become clear after an example.

Axioms of Group Theory

**Identity**

In each group their must exist an element called the identity, it is denoted e. When the identity element acts on any of the other group elements it essentially does nothing, the element remains the same. In group theory language this is written:

e * g = g or g * e = g

The element is unchanged when acted on by the identity.

**Closure**

This principle states that the product of any two group elements will produce an element that is also part of the same group.

For example if g(1) and g(2) belong to G, then g(1)*g(2) must also belong to G.

**Associativity**

This principle states that the order of operation between elements can be fluid. If g(1) acts on the product of g(2) and g(3), this is the same as the product of g(1) and g(2) acting on g(3). In group theory language this is written:

g(1) * (g(2)*g(3)) = (g(1)*g(2)) * g(3)

**Inverse**

Finally there must exist an element which is the inverse for each pre-existing element. The inverse is denoted with a superscript -1 after the element but to save me from introducing math-type I will denote it with a strikethrough.

So the inverse of element g(1) is ~~g(1)~~

~~When each element is acted on by its inverse it gives… the identity!~~

In group theory language this is written:

g(1) * ~~g(1) ~~ = e or more generally g * ~~g~~ = e

Ok this must be seeming extremely abstract without an example so let’s introduce the square – one of the most simple examples we can work with. Group theory is all about respecting and classifying symmetries in nature so the question we want to ask is what transformations exist that preserve the symmetry of the square?

A square has four sides, forming four right angles. What action can we perform on the square that will preserve its shape/symmetry? If we rotate the square by 90 degrees, we will take point a to point b, point b to point c, point c to point d and point d to point a – but the square will still *look *exactly the same. In fact if we rotate the square by 180 degrees we’ll still get a square as well except point a will go to c, point b will go to d, point c will go to a and point d will go to b! Ok very nice. I think we now see if we rotate by 270 degrees a will go to d, b will go to c, c will go to b and d will go to a. And finally if we rotate by 360 everything goes back to its original place and nothing changes!

These transformations/rotations form the elements of the group G – in this case the cyclic group of a square.

Let g(1) = clockwise rotation by 90 degrees

g(2) = clockwise rotation by 180 degrees

g(3) = clockwise rotation by 270 degrees

and what’s rotation by 360 degrees? Of course! It’s e – the identity element.

Let’s now test the axioms to make sure these elements fit our definition for a group.

**1. Identity** – Already checked, we can see that a rotation by 360 degrees leaves all sides as they were to begin with. The e element exists.

**2. Closure** – Is for example g(2)*g(3) a member of the original group? This would be a rotation of 180 degrees followed by 270 degrees, so in total 450 degrees. Perform this on the square, it’s rotating through by 360 then adding an extra 90. So yes it gives back the same operation as you would if you just performed g(1).

g(2)*g(3) = g(1) – Check

You can try this with any combination of elements and check it works!

**3. Associativity** – Does g(1) * (g(2)*g(3)) = (g(1)*g(2)) * g(3) ?

Let’s try it: The left hand side is just 90 degrees then 450 degrees (total 540 degrees). The right hand side is just 270 degrees then 270 degrees (total 540 degrees). All rotations are taken to be clockwise in this case and so it does not matter in which order you perform them, they will produce the same outcome.

**4. Inverse** – This one is very straight forward. What are the inverse of clockwise rotations? Anti-clockwise rotations.

So for g(1), ~~g(1) ~~ is an anticlockwise rotation by 90 degrees. If we perform g(1) then ~~g(1) ~~we undo our first rotation and get back what we started with a.k.a e

g(1) * ~~g(1) ~~ = e

~~g(2),~~ ~~g(3) ~~ are anticlockwise rotations of 180 and 270 degrees respectively.

So there we have it the lay-out of the group G – in this case the cyclic group of a square.

Many other, far more beautiful, groups exist in nature and this was by far the simplest explanation I could give whilst still having enough elements to explain the axioms clearly. At some point in the future I’ll write again on more complex groups and may even dare to venture into the famous Lie Group. If you’re interested in reading more now wikipedia does a decent job or I would recommend ‘Physics from Symmetry’ for an extremely clear approach, which I must say i’ve found hard to come by in this subject. For now, I hope you have found some* closure* on the subject.

]]>

Hello to everyone still reading Rationalising The Universe! Let me apologies for my radio silence, things have been moving around quite quickly for me so let me explain why I have been so quiet, before we go on a brief foray into the Black-Scholes. So first of all, RTU went to Rome.

Following on from that, the opportunity arose to take a qualification in investment analysis, which features a reasonable amount of mathematics and statistics. Whilst this didn’t fall directly into my recent study patterns, it presented me with a very good career opportunity and has some pretty neat applications of mathematics. As such, I have taken it – which means currently I am studying financial mathematics for the year! This site would be impossible to maintain if I wrote about anything other than the work I am doing, so as a result my writings will have a slightly more financial slant to them. They will remain applications of the beautiful language that I find fascinating.

Following on from this I had intended to write on this topic, but then RTU ended up at Glastonbury.

So that brings me to today – back ready to tap out posts a little more regularly, with a marginally different focus.

**The Black-Scholes equation**

The Black-Scholes equation was developed developed by Fischer Black, Myron Scholes and Robert Merton in the early 1970’s which earned them the Nobel Prize (in Economic Sciences). It is actually one of the most important piece of financial mathematics that exists, something young graduates are supposed to understand when they grilled over a shiny Canary Wharf desk.

There are many different applications of the Black-Scholes, because there are many different types of *options* – but for the purposes of this explanation it isn’t all that relevant. So what is an option? Well really it is as the dictionary would suggest – an option to do something. But in financial terms, it usually means an option to buy something such as a stock (but it could be anything; gold, coal, a bond, rice, coffee etc). Now the important thing about the option is you have the *right *to buy, but you don’t have to. Secondly, there will be a maturity – so after this date, you no longer have the option.

Option contracts are very useful – if you know you might have to buy something in the future, and you want to start to working out what it costs it may be very inconvenient if the costs are fluctuating. An option gives you the certainty of what the price could be, without tying you into a cost you may not need to incur. But what is the value to one of the option? Clearly if I can buy the item cheaper than the option price at the time I need it the option has no value; but if the option price is cheaper than I can get the item elsewhere the option has value.

The value of an option comes from two places:

- The intrinsic value: This is the difference between the market price and the option price – or in other words the amount I would gain or lose if I exercised my option right now.
- The time value: The value that comes from having the option open in the future.

This is all discursive – but how is a bank (who sells the option) going to come up with a price – not some discussion about what it should be worth, an actual price that customers can be charged. Coming up with an accurate way of pricing such options allows them to exist; if this could not be achieved no one would ever sell them.

From now on when we talk about options we are talking about stock options – the capital (money) raised by companies by issuing shares. Stock, share and equity are all interchangeable terms. The option we are considering has the following features:

- The option can only be exercised at maturity. This means that the date the option can be used is predetermined so I can’t just use it whenever I want. I might have an option to buy a share in a company on 31 August 2017 for £100.
- The stock pays no dividend over the life of the option (dividends are the division of profits amongst shareholders, which if paid during the life of the option would drastically change the valuation).
- The markets are ordinary, efficient markets (i.e. cannot be easily predicted).
- There are no transaction costs to buy the option.

The maths assumes that the returns on the underlying stock have a normal distribution (see here for further reading). The actual equation that is known as the Black-Scholes is a fairly gnarly second order partial differential equation – no body wants that.If you are familiar with partial differentials and want to know more about these terms do write to me in the comments section and I would be happy to discuss.

However for the option we have been describing (which in financial literature would be a European stock option), we can derive (which we won’t step by step) a far more useful result to price option contracts.

To make sure you are familiar with the terms, the call premium is what we are looking for – what should we charge for the pleasure of this option. This depends on all the terms on the right hand side. The current stock price, the time until the option matures and the option striking price are the most obvious ones (i.e. if I want an option at todays price for tomorrow it will probably be quite cheap, if I want it for 10 years it would be much more expensive). The risk free interest rate is a little more complicated, but can be summarised as something like a theoretical interest rate an investor would expect in the absence of all risk – the rate on a totally risk free rate. N represents the normal distribution (as a function) and e takes its normal meaning.

All of this information is taken from modelling the relationships with partial differential equations and then finding the solution. The result of this is a solution where values can be plugged in, to instantly spit out prices.

Using this information we can calculate stock option prices – here is one I plugged into a calculator earlier, where the model has told me the option I want would carry a premium of $26.30 – this is what I would need to pay the bank in order to enter into this contract. This is the pricing model that in some form is used by all the major banks.

So why is this important? Well in short it’s big money. When I discussed using the option to mitigate your risks, in reality the sexier applications come when we use the option to speculate. If you think you know the market is going to soar, say for example you think Apple are about to soar on the release of a new phone you may enter into an option to buy that stock close to todays price. You then can buy that stock when it matures, and sell it on for more on the market or (because the contracts are standardised) you can cash in and sell your option onto a third party at a higher price – because your option is now in the money. On the flip side, there is a bank somewhere that has to fulfil this option – so if they really mess up, it costs the bank a lot of money.

So the premium represents a number which is the expected amount that needs to be charged to make the contract rational to both parties. How the individuals view the value of this option however, depends on their view on the world. To me this is quite neat – the maths is strong, and when applied allows us to price up something very complex; a choice. This is however strongly based on statistical distributions – much like in quantum physics where we can only give likelihood a particle will be at a range of values, we can only give likelihood that the stock will arrive at a range of values in the future based on the current prices. What happens could make you very rich, or be a real waste of money.

]]>