This will likely be the geekiest thing I ever post but I don’t care. I’ve spent all of today on one particular problem on a General Relativity problem sheet and I have to talk about it.
The problem sheet’s challenge was to mess around with some geometry and see what comes out and lo and behold, the Big Bang came out. This is cool as hell. The process by which you can use nothing but a sheet of paper and a pen to unlock the secrets of the universe has always felt to me like a magic trick. Except of course it’s better than a magic trick. It’s better because it’s true.
There’s another way I think that theoretical physics might just be better than magic and it’s the reasoning behind this post. I’m betting that in revealing my secrets the trick won’t be spoilt but made all the better. I believe this because for me the beauty of physics isn’t simply in the end result but in the process. The twists and turns of both pen and logic are really what it’s all about. So here’s my attempt at getting across the magic trick, the secret to the physics performance. Here’s how to derive the Big Bang.
I started this morning imagining a sheet of space-time. This means imagining something in four dimensions so the first thing I need to do is explain how to see that extra dimension. I promise it’s not as hard as you might think. When we talk about dimensions we normally mean space- up, down, left, right, forwards and backwards or x,y and z, mathematically speaking. But really, a dimension is just something you can measure. In the case of space you measure it with a ruler but speed can also be thought of as a dimension (and it is, in a thing in thermodynamics called phase space) and so can, say temperature. Imagine running your fingers over the surface of a metal ball with one end of the ball close to a heater. Your fingers will tell not just where each point is but how warm it is. As you run your hand over the ball’s surface your brain is interpreting information in not just the three dimensions of space but in a fourth dimension of temperature. So really, we use multiple dimensions everyday without realising. Imagine dripping multicoloured paint over the ball. Now between your eyes and hands you are taking in five dimensions worth of info, with each point on the ball having colour, temperature and position. If now move the ball closer to the fire so it starts to heat up then BAM we’ve introduced the dimension of time into things and now we can use the time to also describe each point on the sphere, e.g. (light blue, 50 degrees, 5 cm up, 4 cm right, 3 cm forwards, half past 6). Congratulations you can now think in multiple dimensions.
So back to that slice of spacetime. What does it look time? For the most part I just think of it like this-
which is only 3D but if I really need to think of that 4th dimension I imagine it have varying colours on a rainbow scale from red to violet. What I’m really interested in is how curvy the slice is. I can talk about this using maths. Bigger numbers mean very curvy, the number 1 means not curvy at all. Now I’m ready to write an equation. I want to show you the equation because its very pretty but don’t worry if you can’t follow the maths too well, I’ll talk you through what everything means.
I’m going to use the letter g to represent curviness and add two little letters to the bottom to show what dimensions I’m talking about. T means time, i j and k mean the x, y and z axis (yes physicists could just say x, y and, but that would just be too sensible). So curviness is written as gij.
My ultimate goal in playing with this slice is to see if it stays still or if it stretches or squashes all by itself. The next step is to talk a walk on my slice and see what happens. I imagine my slice as a great big field with bumps and dips in it.
Let’s run across that field. As you run uphill, especially if it’s steep, you’ll notice you start to lean forwards to keep your balance. The steeper it is, the more you need to lean. Running downhill you now need to lean back to stop yourself falling over. The amount you need to lean forwards or backwards, let’s call it the wobbliness of the terrain, depends on the curviness of your landscape. The wobbliness is an important part of your landscape too so let’s make that into an equation. The symbol for this is Greek. I can’t remember the name for it but it looks like a crane. This has three little letters on it which means it contains even more than our curviness, gij, which only had two. In fact, it contains a few g’s. Here’s what it looks like-
So now we know how to describe our space-time slice, which is great, but what about actual stuff? Planets and stars and beds and dogs- what about matter? Let’s drop a lump of matter into our space-time and see what happens. The lump will experience what in physics is called “stress”. This just means pushing and pulling due, for instance, to any pressure inside it and is affected by the density of the lump. We’re now going to allow for the possibility that the space bit of our space-time is expanding. We’re going to describe this expansion with the letter a with a=1 meaning we don’t have any expansion and the bigger a is, the faster we’re expanding. Remember, we aren’t assuming space is expanding, just allowing for the possibility. If our slice of space is expanding that means that everything in that slice will be stretched, so the stress on the lump of matter will increase. We denote stress with a T and it looks like this-
The capital P is pressure, the curly p is density and the u is speed, which depends on a, how quickly our space is expanding.
Now stress, like energy, is conserved. That means you can’t create or destroy it, i.e. the total amount of it can’t change. We can express the idea that the change in stress is zero like this-
Adding in all of the letters we’ve already work out gives us
where w is just a constant to do with the temperature of our matter. For cold matter, like the stuff around us, w is zero and we get
where k is just some constant. For hot matter, like radiation, w is 1/3 and we get
Now let’s really look at what is going on in these equations. First of all that mysterious t0. It just represents some unknown time but what happens when t=t0? Well a=k(t0-t0)=0, in other words space has zero size. If a is zero then what about our density? We had
so if a=0 and we’re dividing by a then we’re dividing by zero- which give us infinity. So there was some time in our space-time slice, t0, when our lump of matter was infinitely dense and concentrated in a tiny point of zero size before it started to expand. That, my friends, is the Big Bang.