One of the popular phrases used to describe the nature of space and time through the eyes of general relativity is as follows;

*“Spacetime tells matter how to move,*

*matter tells spacetime how to curve,”*

Imagine the old bowling ball analogy for general relativity and spacetime. Spacetime is depicted as a rubber sheet whilst an object such as a bowling ball, often representing a planet or star (though it can be any object with mass) then deforms this rubber sheet when it is placed on top. This simple analogy is meant to represent the curvature of spacetime as a result of the mass of an object. The two key words here being; mass and curvature. Mass *produces *curvature. The larger the mass of the matter in question, the greater the curvature of the surrounding spacetime. A black hole for example, one of the heaviest objects in our universe, produces a curvature so large, a warping *so great,* of the surrounding spacetime that phenomena like no where else occur – see here and here. In fact, at the centre of the black hole where bending of the spacetime is so great due to the highly concentrated mass, the curvature is mathematically described as *infinite *– and our understanding of the nature of spacetime here breaks down completely. Such an extremity causes us a problem, we do not like it when our maths doesn’t work as a descriptive language for nature. These points of infinite curvature, called singularities in the nomenclature, create problems in the standard theory of general relativity. Take home message 1; general relativity only includes the mass of matter as a property affecting the behaviour of the surrounding spacetime and it does so by causing it to *curve.* If there was no matter in the universe, or if all matter was massless (for example photons) the surrounding spacetime would be flat – imagine a sheet stretched out completely taut. Take home message 2; the theory of general relativity causes us problems, which come in the form of singularities at points of extreme curvature.

An alternative theory, Einstein-Cartan theory, holds a possible key to these problems. Matter in our universe has two fundamental properties; *mass *and *spin*. Spin is a funny little characteristic of matter, it is not related the spinning of a particle in actual space, so don’t imagine a spinning top on a table but instead is the *intrinsic angular momentum *of a particle. If that doesn’t mean much to you don’t fret, but for the purpose of this article humour me and the particle physicists of the world and accept that spin is a fundamental characteristic of matter. See *here* for more reassurance. Einstein-Cartan theory, claims that both these fundamental properties of matter affect the nature of the surrounding spacetime, whereas general relativity only incorporates mass. Whilst mass created a curvature of the spacetime, the spin creates an effect called the torsion of spacetime – unfortunately a similar analogy to the bowling ball does not exist for a visualisation of this torsion (that I know of). The mathematics of Einstein-Cartan theory, with the inclusion of spin and the resulting torsion into the framework, has some very interesting implications. In standard general relativity, the black hole situation described above creates a singularity at the centre due to the infinite curvature of spacetime. *However, *when torsion is also in play at such extreme matter densities, the torsion field creates a repulsive force that pushes outward against this extreme warping. Instead of a singularity at the centre, in Einstein-Cartan theory the interaction between torsion and curvature creates a wormhole or Einstein-Rosen bridge, at the centre of the black hole. The wormhole creates a passage to a new, growing universe on the other-side of the black hole! The same can be said of the situation at the beginning of the universe. In standard general relativity, the big bang represents a singular point, of infinitely dense matter, from which the universe then somehow comes into existence. However in the Einstein-Cartan formulation, the torsion again creates an outward-type repulsion at these such points of extreme curvature and density, forbidding them to occur. The big bang is then replaced by what is known as a big bounce scenario in cosmology.

The nice thing about Einstein-Cartan theory is that, because the extra features only come into play in extremely high matter density regimes, like the centre of a black hole, the tests which probe astronomical phenomena within our experimental reach still agree with the predictions from general relativity. For example, the perihelion of Mercury, a key test of general relativity, would still be true if working within the Einstein-Cartan formulation. Therefore, we can see the theory not as an opponent to general relativity but a slightly revision, extending its validity and in so doing presenting resolutions (and very exciting ones at that) to our previous trouble points. On the other hand, this is also unfortunate because given our nearby surroundings and inability to accurately probe quantum phenomena, we are not in a position to experimentally test the predictions of Einstein-Cartan theory. The experiments that we can conduct are ones where mass is the heavily dominant player affecting the behaviour of spacetime and spin effects remain physically hidden to us.

What we can say, without even commenting on the mathematical elegance of the theory is that Einstein-Cartan simply seems more wholesome ontologically. Why include one fundamental property of matter (mass) and not the other (spin)? The theory seeks to resolve this omission and unlock the missing results even if they cannot be physically probed. A curveball theory, where curvature is no longer the only behaviour of spacetime.

]]>The key idea is that AdS/CFT is an example of a duality between two types of theories. On the left hand side we have theories in Anti-de-Sitter space, this is a special type of geometry that physicists use to model a particular spacetime. There number of dimensions for this spacetime can be chosen freely but the details of what exact *kind *of geometry this represents need not concern us today. When working with spacetimes it is easy to add gravity into your models and physicists try to model the gravitational interactions here in terms of string theory (see How long is a piece of string). Incorporating string theory to AdS space attempts to create a big and bold theory of quantum gravity. The inner workings of such a theory are still very much in the dark but here the assumed pillars of a quantum gravity set-up have been erected. Take-home message; AdS sets up an arena for physicists where the only field (see What is a Field?) is the gravitational field and the constituents are our little friends the strings. It is a proposed theory of quantum gravity, still being explored.

On the right hand side we have what is known as a Conformal Field Theory. Conformal field theories are particular types of quantum field theories (see What is Quantum Field Theory?) which in turn are models describing the interactions of elementary particles. Elementary particles interact heavily with the other three out of four fields in the universe’s playbook; the electromagnetic, strong and weak field. On this side of the correspondence there is no gravity and as such the geometry of the arena is a flat spacetime. The way Maldacena, the father of AdS/CFT, set up the correspondence is that if you choose the number of dimensions in your AdS theory to be D (any integer) the number of dimensions in the CFT theory must be D-1. We’ll explain the consequences of this later. Take home message; CFT sets up an arena for physicists where they can work with the interactions between the very small constituents of the universe and use all the fields *except* gravity.

Ok so both halves have been defined what next? Well we choose a number for our choice of dimensions and play around which each side of the theory. We then begin to see remarkable similarities in the behaviours of the models of quantum gravity formulated in AdS space and those the conformal quantum field theories. Important characteristics of the models, such as emergent symmetries and levels of chaos mirror each other on both sides of the correspondence and the main breakthroughs of this are twofold;

Firstly, the fields at play in quantum field theories on CFT side of things (electromagnetic, strong and weak) are subject to what is known as ‘strong coupling’. When studying theories in theoretical physics there are a bunch of mathematical procedures we very often use however when using maths to study theories with *strong couplings* our calculations essentially blow up. This is because the strong coupling is representing in the mathematics as a larg pre-factor, call it q here for example. As we try to expand the mathematical terms in the theory we get terms with q, q^2, q^3… and because q is a large number already the whole thing blows up in our face! The fancy terminology is that the theory is not ‘mathematically tractable’. *However*, theories on the AdS side of things only contain the gravitational field, which is subject to ‘weak coupling’ The pre-factor here is very small and less than 1, call it g. So when we expand the terms and get g, g^2, g^3 we can essentially ignore the terms which contain g raised to high powers as they would be so much smaller than 1 (check it!). As such we can work with the first few terms alone and the maths makes us happy. Take home message; we can study examples of CFTs in areas in difficult like nuclear and condensed matter physics by translating the projects into *mathematically solvable* problems of string theories on AdS spaces.

Secondly, the string theory on an AdS space set up is, as we said, a proposed theory of quantum gravity. A theory of quantum gravity is the holy grail of the theoretical physics world and this proposed theory is of course only an attempt at a formulation. However due to the similarity in the behaviours of models on both sides of the *duality*, we can probe the proposed quantum gravity with a quantum field theory – a domain much easier for humans to manipulate. Since we know very little about the inner workings of quantum gravity by considering its quantum analogue without gravity (the CFT theory) we can significantly enhance our understanding. Take home message; we explore the quantum gravity AdS set up with what we know about the corresponding CFT.

A final point worth touching on because the idea is pretty funky and Susskind is a wonderful man, is holography. You may have recalled that in the set up the AdS quantum gravity theory has D dimensions but the CFT quantum field theory had D-1 dimensions, one less. The duality between the two theories was proposed by Susskind and ‘T Hooft as being the same kind of duality seen between a real world object in three-dimensions and its hologram on a two-dimensional surface… The idea is that all of the information in a theory of quantum gravity can be encoded within a theory without gravity on a lower-dimensional space. The geometric visualisation is akin to a sphere (3D) holding the full theory and the information being encoded on its surface, the boundary of the sphere (2D). Susskind’s interpretation of the AdS/CFT correspondence has conjured up an image of us living within an analogous hologram, a four-dimensional space, an embodiment of a richer five-dimensional space somewhere in the universe… a post in more detail on this to come.

]]>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.

]]>In this modern day, what I like to call the ‘projection of the self’ is ubiquitous amongst our generation. The dissemination of selfies and social media posts is prolific and it seems we are constantly putting *out *images of ourselves, trying to convey an idea of *who* we are, *what *we like and* what *we are doing to others. It seems as though conveying this image outward, is becoming one the most important activities amongst young people of our generation yet stepping away from the phone, after the post has been posted, what value did this action add to reaffirming our understanding of the self? I worry that all this ‘projection of the self’ leads to far less time being spent on ‘inward reflection of the self’ and it feels as though these activities lead to a dulling of our own consciousness. By occupying all our free time with social media, even when partaking in activities that may be focused on *ourselves *and our identity we are never truly giving ourselves time to be alone with our thoughts. Our identity becomes that which we project on our social media accounts, we constantly share what we like and what we dislike but we never seem to sit down and reflect on what it is that we *truly *like and dislike, what makes us happy and unhappy in life.

As much as I believe this time for self-reflection should increased, by doing this we reach another hurdling point. What *is it *that inherently makes us who we are. A popular answer is that it is our choices that make us who are. However as discussed in a previous post on free will, from a physicists standpoint on life anyway, free will is a very dubious concept. However if you *do *accept that choice is real and it is these decisions that defines you, it is important to try and analyse why you make the choices you do. When I have tried to do this analysis, it has seemed to me that all my choices can be traced back to a previous thought, idea or experience that I have had. However in the tracing back of these choices we find no logical start point, there is always a predecessor in the chain. It seems to me, a combination of upbringing and surroundings heavily influence a persons choices, memories and experiences are continually stored in ones subconsciousness and are then drawn on when making a decision.

I have also experienced of late, your surroundings and the people you choose to surround yourself with can have a large impact on the choices you make and the type of self you choose to project outward in the moment. It is a worrying thought that the idea of the self is so fluid and of course it is simple human psychology that people choose to reflect aspects of the self that are most in line with their company at the time, out of fear of being treated as a social pariah. Yet these effects compound over time and begin to shape the self we independently choose to project. Identity can then begin to become a product of your social company… However, in this new age of online communications social interactions and the way social company is kept is also changing its form. With conversations held more frequently over a whatsapp screen that in person and love shared in the form of a heart emoji as opposed to a real time exchange of emotions we also seem to be at risk of a lost sense of real time relations. It has been shown that too much time on social media can fuel feelings of loneliness and decrease ones self esteem. Compounding these feelings with the idea of a lost sense of self can lead to a worrying state of mind and lack of purpose.

Perhaps now more than ever, in the age where it is all to easy to become influenced not only as those in the same room as you but those a million miles away yet on your phone screen it is important to spend more time alone with your own thoughts in an attempt to try and find out what you independently value. Ideas of mindfulness and meditation are avenues which seem to be useful in trying and get a better sense of this. Although free will may not be real, human intuition compels us to believe it is, in a bid to give purpose to our lives. And given that we will always experience life from the confines of our own brain, in which choice, consciousness and the idea of the self appears tangible and concrete, to spend time focusing on our own identity and values seems a worthwhile exercise indeed. To then surround yourself with people who share these values in real time may be a step in the right direction for a generation who at times seem to be more concerned with the ‘projection of the self’ than what they themselves want to achieve with their finite time.

]]>*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

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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.

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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.

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* ‘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.

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