# How will it all end? We have asked where we came from, how about where we are headed? Simon Williams explains the theories which describe the end of the Universe.

Everyone always asks the question: ‘Where did we come from?’. Well why not look from the other perspective, ‘How will it all end?’

The true fate of the Universe lies, as bizarre as it may seem, in its geometry. Einstein’s theory of General Relativity formulates a particular set of field equations that have within them an unknown parameter. This parameter is known as the curvature of space-time, literally the shape of space and time, the very fabric of reality itself. Einstein proposed three situations of this parameter, known as: the “open universe”, a negative curvature leading to a saddle-like shape; the “flat universe”, where curvature is zero and thus the universe is flat; and the “closed universe”, a ballooned shaped universe with a positive curvature. Each of these situations have different repercussions on the fate of the universe.

Before we delve into the death of the universe, we must first think about what geometry really means. For almost 2000 years people thought of geometry in a purely logical way, the famous works of Euclid: flat planes with no edges go on for infinity, and shapes have specific rules. However in the 1800s, French mathematicians like Gauss came along and suggested another logical formulation of 3D space, in which planes no longer need to be infinite, and geometric laws are no longer true. For example try adding the angles of a triangle drawn on the surface of a balloon; you’ll find that they add up to more than 180 degrees! Moreover, the 2 dimensional surface of a sphere is finite, but without any edges. Clearly, in some circumstances, Euclidean geometry breaks down. The mathematicians didn’t stop there, they suggested that 2D surfaces can be finite and curved in their own right, without being a surface of anything! Einstein extended this to 3D spaces, explaining that 3D spaces can be curved in their own right, leading to the formulation of his ground breaking theory of General Relativity.

So the question arises, how do we find out what sort of geometry we are standing on? Not as easy as it sounds, for example can you tell you are stood on the surface of a sphere as you read this? We are essentially blind to the third dimension that the plane we stand on curves in. However, we will not let this stop us. We can use the idea of triangles, which we can draw on the surface of our plane by connecting two points by the shortest line. Do this three times, and you form a triangle. Then measure the angles: if they are greater than 180 degrees then we are on a positive curved surface like a sphere; less than 180 degrees then we are on a negatively curved surface like a saddle; and finally if they are equal to 180 degrees, then we are on a flat plane.

We are now ready to tackle the different curvatures of space. All the situations are suggestions of what might be possible for the universe today, having expanded from the Big Bang, depending on the true geometry of the Universe. We shall start with the “open universe”, which holds an interesting idea for the fate of the universe. The negative curvature of space-time means that the universe would keep expanding forever. At the moment, the Universe has been expanding for about 13.8 billion years. In this time it has undergone periods of great expansion and thus has a radius of approximately 46.6 billion light years according to data from WMAP, an orbiting telescope. In the open universe, this expansion continues to accelerate at the same rate, leading to a cold and lonely death for the universe. Eventually the expansion of space-time will be reaching the speed of light, and the expansion will be so large that everything will be atomised. Each atom will lie so far from another that the light from the other atoms will not reach it due to the rate of the universe’s expansion. The universe would be dark.

The “closed universe” doesn’t hold out much hope either. The universe in this regime would expand to a maximum size, at which point the density of the universe becomes to large, and the universe collapses in a cataclysmic ‘Big Crunch’ destroying the universe in essentially the reverse of the Big Bang. Luckily, if this does happen it is estimated to happen when the universe is 100 billions years old, 86 billion years from now, so we can sleep easy in our beds tonight!

The final situation is the “flat universe” which expands to a certain point and then just stays there at constant expansion rate. The most stable of the situations, Einstein liked this geometry, and to make the theory work properly, he introduced a cosmological constant into his formulae. Although Einstein said this was his biggest mistake, making the theory work instead of being mathematics, but it turns out that he was right! The cosmological constant is experimentally measured as well as being needed to fulfil the field equations.

So what is the geometry of our universe? Well, using the triangle method explained earlier, and universally large triangles, experimentalists have been able to calculate the curvature of the universe. The angles of the triangle add up to roughly 180 degrees in most circumstances, but if a black hole or quasar is within the triangle, then this is not the case. Consequently it turns out that the space we live in is more complicated than just one simple curvature. Mother Nature has eluded us once more! But what we do know, is that the end of the universe is quite a while away.