Category Archives: Maths

Afternoon Tea at Tiffeny’s, and other randomness

It’s been a busy couple of months, but as I now have the internet working at home and the rare luxury of  a free evening, I’ve decided that it’s about time my blog came back to life again, so here’s an update on life:

 

  • I think I’m over my mini crisis of faith, and have actually found a degree of settledness and peace. I guess I’ve finally grasped that it’s ok just to be me, and not to try and push myself into other people’s boxes, because it doesn’t work. I can’t pretend I’ve got my prayer life magically sorted, but I feel relaxed with God, which is a huge step forward.
  • I’m now a graduate of he University of Durham. The day was amazing, and it was great to be able to enjoy it with my partner, parents and friends. I still can’t quite believe I finally made it, and how different I was when I started the whole thing compared to where I am now. I’ve changed so much for the better…. God is very good.
  • I’m now a Methodist Local Preacher – don’t ask me how that happened! I had my first service early last the month, which went very well, and I’m off out with my mentor in Stony Stratford in a week and a half. It’s exciting and scary, but mostly it just feels very right and very ‘me’ somehow. The bit that scares me is the way the whole ordination thing seems to be inching its way back onto the agenda at unexpected moments…
  • The banking world is pretty much the same. I’m now at the stage where most of the stuff from my predecessor has been gone through, documented and tidied up, and I can begin working on my own projects, which is  a good thing.
  • I’ve almost (finally) finished two papers based on my PhD thesis, which I hope to send to my supervisor soon, and then to get published. I miss the world of maths, and want to keep the door open for getting a post-doctoral research position. It’s been nice getting back to grips with it all 🙂
  • I’ve joined a gym, and am seriously working on getting fit.
  • Most importantly of all, Tractorgirl and I have now been together for eighteen months, and I love her more than ever.

At the weekend, I went on a retreat with some friends from Bang! and had a very random dream about a friend of mine directing and presenting a daytime television show called ‘Afternoon Tea at Tiffeny’s’. I might float the idea to the BBC one day…  In the meantime, here’s a couple of graduation pictures:

Me with my parents
Tractorgirl and me chilling after the graduation dinner

Living in a black hole?

A scientist on a research project hijacks a spaceship which he prepares to enter a black hole, with the expectation that exiting out the other side will take him into a whole new universe. Sounds like science fiction? Well, it was the basic plot of a rather poor movie from the 1970s with the stunningly original title ‘The Black Hole‘, but if a recent paper in the Physical Review, one of the most prestigious journals,  is right, it might not be so mad after all…  

The gist of the paper is described in an article in this week’s New Scientist. According to Nikodem J. Popławski of Indiana University, it is possible that our universe exists inside a black hole, or that by passing through a black hole in our universe, we could enter a whole new world. This apparently bizarre concept requires some explanation:  

It’s hard to underestimate the impact Einstein had on the way we view the world. Working in mid-seventeenth century Cambridge (ironically at Trinity College when he was a Unitarian), Sir Isaac Newton formulated his theory of gravity that stood as the best explanation for nearly three-hundred years till a lowly clerk in the Swiss Patent Office changed everything. The results of this change of dominant theory ranged from the gradual shift in physics (as new ideas often take time to gain credibility, and this really was a revolution) to the impact on theology and philosophy, in which it arguably contributed to the collapse of the Enlightenment world-view that relied heavily on Newton’s mechanistic, clock-work view of the universe for its origins in the thought of the English Deists such as John Locke.  

Newton’s view of the world was essentially a common-sense one. The universe could be modelled by taking time as an absolute, independent quantity and having the usual three spatial dimensions we experience every day. When Einstein formulated his theories of relativity, he took it to be axiomatic that the speed of light was constant in a vacuum and that the laws of physics are the same in every inertial frame (for which read frame of reference, or point of view if you will). One consequence of this is that time is no longer an absolute, but rather is bound up with space and affected by the motion of particles and the presence of massive objects such as stars. That’s why physicists talk about ‘space-time’.  

Now, one perhaps surprising thing to note is that Newton could never pin down was what gravity actually is. He could model its results (and his theory is still a very useful approximation to Einstein’s) yet couldn’t define it. Einstein, faced with the same problem, conceptualised it as being a result of the shape of space-time. In other words, in general relativity (GR),  gravity is geometry. The classic example of this is nicely illustrated in this video.  

A black hole is the result of the gravitational collapse of a massive star – we’re talking something like thirty times the mass of the Sun. When this occurs, the fabric of space-time is severely distorted. In the heart of  a black hole, there is understood, in classical GR, to be a singularity, which is a point where all the laws of physics break down and of infinite density and space-time curvature (which is very bad, as infinities in equations cause no end of bother!). This singularity is surrounded by an event horizon. This marks the point at which even light, the fastest thing there is, cannot escape the gravitational pull upon it. If you go pass that point, you’re stuck in the inevitable path towards destruction at the singularity. In a black hole, no-one outside can hear (or see) you scream…  

Another key object we need to know about here is properly called an Einstein-Rosen bridge (but is commonly known as a wormhole) which is sort of like a tunnel that connects two different regions of space-time, allowing fast travel between them. The problem is the stability of these ‘tubes’; they are liable to collapse upon being entered by matter. This gets us into the wonderful world of quantum theory and negative energy (which is not supposed to be allowed, but might be after all…), and means that such structures are at most theoretical as yet. However, for the sake of the argument, let’s suppose that somehow or other, they exist.  

Now, there are different regions of space-time with differing properties either side of an Einstein-Rosen bridge or a singularity in a black hole. This means that passing through into the interior of a black hole or going through an Einstein-Rosen bridge (if it were possible) would result in us emerging into a different universe or part of the universe. Popławski’s paper suggests that, with a slight modification of classical GR, it could be that “observed astrophysical black holes may be Einstein–Rosen bridges, each with a new universe inside that formed simultaneously with the black hole. Accordingly, our own Universe may be the interior of a black hole existing inside another universe”.  

Crazy, but the maths seems to make sense (I knew there had to be advantages to doing this PhD stuff!). The problem of how to get through a wormhole still remains, alas, but it could be the substance (pardon the pun) for some new sci-fi…  

In terms of the implications for science, I reckon that if true, this research renders problematic the idea of a ‘theory of everything’ as the limits on our ability to travel between universes are such that we would only have very partial knowledge of the way the network of universes operates. We can only talk about our visible universe.  

Cosmic inflation

 

In that sense, it’s a bit like inflationary theory, which predicts a period of rapid expansion shortly after the Big Bang in which quantum fluctuations result in different parts of the universe having different values for the fundamental constants, such as the speed of light, the charge on the electron and so on. 

In the immediate aftermath of the Big Bang, before the fundamental particles, quarks, electrons and so on, have formed we are dependent on the murky and random world of quantum mechanics. One key rule here is the Uncertainty Principle of Heisenberg, which states that we cannot know the position of a particle and its velocity simultaneously with complete accuracy. The more we know about one, the less we can know about the other. This has implications for the vacuum of energy that would be present at that early stage of the universe, in that it would cause fluctuations in that field (as zero is too precise a value for it to take) that result in areas with different values of fundamental constants. As the universe expands, we end up with discrete regions, our visible universe being just one of many. This limits our ability to speak about the universe as a whole, as we can only know anything about our little portion. 

Now, I’ve explored some of the theological implications in my talk on physics and Christianity of current physics thinking, and think the questions raised by the inflationary model apply here. Moreover, in what sense can we speak of the cosmic implications of the life, death and resurrection of Jesus? Does it only apply to our visible universe, or what? I don’t know the answer to this and will, when I get the time, do some reading around what others think, but it’s a fascinating question.

Synaesthesia: a colourful world

I see small numbers, medium-sized and larger numbers in various places in space. The colours are not as I've done them here because each number has a unique colour

Yesterday, TractorGirl mentioned that I have synaesthesia, and Jack the Lass was interested in how that works. Well…

According to the BBC, about one in two thousand people have the condition and scientists are unsure about whether the neurons in the brain work differently or are somehow ‘crossed’, but either way the result is a sort of mixing up of the senses, which is unique from person to person.

I hadn’t realised that other people don’t tend to see colours until a chance conversation with TractorGirl a few months ago, so I’m still getting used to trying to imagine what the world looks like without it. For me, sounds have colours and colours have sounds. Colours are also linked to numbers, emotions and strong physical sensations. What this means in practice is that I see a sort of flash of colour when I hear a particular note, or feel a particular emotion, and numbers have a colour even if I write them all in the same colour ink, so nine is green, for instance.

With music, the colour I feel depends on how the piece makes me feel. High notes tend to have brighter shades than lower notes, and loud sounds produce more colours. Green, blue and purple are good colours, red is usually either excitement, danger or a strong physical sensation, pink and orange happen sometimes but rarely, and yellow and brown are bad colours. People seem to think I’m mad if I describe my mood as a colour, but that’s why. Yellow is sometimes accompanied by a horrible scrapping sound. Sometimes, all the colours, sounds and sensations get too much and I get sensory overload, but mostly synaesthesia is a good thing 🙂

In terms of numbers, they have their own colour and also their own place. I’ve made a picture to try to illustrate what I mean. I can see patterns in the numbers and sort of see if they look right, which helps with arithmetic and seeing patterns in sequences. People often have colours, and God is a particularly velvet turquiose.

Some science and a complicated night out

Couple of interesting articles in this week’s ‘New Scientist’.

The first is about quasars, which are galaxies that periodically emit pulses of light. The Big Bang theory predicts that the universe is expanding, something confirmed by Edwin Hubble’s observations of the red shift of galaxes back in the 1920s. As a result of this expansion, the light from distant objects is ‘stretched’ and the wavelength becomes longer, moving to the red end of the spectrum (this is what red shift means). Now, objects like supernovae (the last mega explosion of stars substantially larger than the Sun) display this behaviour, but Mike Hawkins from the Royal Observatory in Edinburgh has found that light from quasars does not behave like it should. Some have explained this by postulating black holes between earth and the quasars distorting the light and others have suggested that the dark matter that makes up most of the universe is causing it. It’s an interesting mystery.

The second is about something I nearly did my PhD on: random matrix theory. A matrix is a two-dimensional array of data. A football league table would be an example. Random matrix theory came from Eugene Wigner, a German physicist working on quantum theory in the 1950s, when he suggested just guessing some unknown values to do with quantum energy levels. His randomly chosen numbers bore an uncanny resemblance to reality and this approach has proved useful in a range of physical contexts, though no-one knows why this should be the case. Again for an as-yet-unknown reason, random matrices are connected to one of the great unsolved problems in pure maths, the Riemann hypothesis. I find the ‘strange effectiveness’, to quote Wigner, of maths and the interconnectedness of the world of maths and physical reality fascinating. If I ever did a second PhD, I’d love to look at this from a theological point of view. Sigh, one can dream….

On a more mundane note, last night I attempted to take my housemate, from hereon known as Stewpot, to a restaurant I tried with TractorGirl last week. They were fully booked, so we ended up, via a four-mile walk into Sunderland and a half-hour Metro ride, in La Tasca in Newcastle. They have an ‘all you can eat for a tenner’ menu which was great value. The only downer was the way the waitress insisted on calling us ‘guys’ at the end of every sentence – why? We ended up in one of our favourite cocktail bars and it was lovely to spend some time with a nice, relaxed Stewpot after a lovely day with TractorGirl. All in all, a grand day. Eeee!