The current received wisdom is that the universe began with a big bang some 13.7 billion years ago and that it has been expanding ever since. This idea of an expanding universe begs the question where was the centre from which this big bang emanated, where is the point in the universe away from which everything is moving.

At this point the cognoscenti will sigh and condescendingly explain that the universe does not have a centre, that it is expanding everywhere in all directions from everywhere. They will typically explain that the universe is not expanding into space but that space itself is expanding. If pressed this usually leads to an explanation expressed in terms of an analogy based on a balloon.

Picture a balloon, they will say, and imagine that its surface is covered in dots. The dots represent the stars and galaxies and the surface of the balloon represents a two dimensional analog of a three dimensional space. As the balloon is inflated its surface stretches in all directions and the dots move away from one another. If we consider just the surface of the balloon, there is no centre; all of the dots move way from one another.

Of course the balloon does have a centre, but that only exists if we add a third dimension and consider the balloon in a three dimensional space. Here we are constrained to exist only on the surface of the balloon.

At this point most laymen will give up and concede defeat, accepting that science has found the answer, that the universe must be expanding and desperately trying to comprehend what a balloon would look like in a four dimensional space.

There is however a fundamental problem with the balloon analogy and by implication therefore with the theory that it supports. The problem lies in the fact that the balloon’s surface is curved and not completely flat. The surface of the balloon has what is known as positive curvature.

In a positively curved space the angles of a triangle add up to more than 180° whereas in negatively curved space they add up to less than 180° and in a flat space they add up to exactly 180°. We can see this if we consider a traveler setting off from the North Pole and heading due south along the Greenwich meridian. When our traveler reaches the equator he turns right through 90° heading west. Somewhere off the coast of Ecuador at 90°W he turns right again through 90° and heads north, back to the pole, where he arrives at 90° to his original heading. Our intrepid traveler has completed a triangular course, but in doing so has turned through 270° and not the 180° we would expect if the earth was flat.

The balloon analogy only works because, no matter how big the balloon gets, and how close it approximates a flat space, its surface has positive curvature. The problem is that as far as we can tell the universe is flat. It shows none of the characteristics of a curved space. The angles of triangles all add up to 180°.

We can visualise how this affects the balloon analogy by imagining the balloon pressed against a flat sheet of glass. The region where the balloon’s surface comes into contact with the piece of glass is flat. As before if the balloon is inflated the dots in that region all move apart, but there is one important difference. This flat region of the balloon has a boundary. Indeed it has to have a boundary, the line where the balloon membrane leaves the surface of the glass. Such a boundary has to be a closed contour and therefore it has to have what can be regarded as a centre, that is a centroid or a centre of gravity.

It is simply not possible to create a flat membrane in a flat space that does not have a boundary. Exactly the same holds true in a three dimensional space, if space is flat, that is if the angles of triangles add up to 180°, then if it is expanding it must have a boundary. In this case the boundary is a bounding surface, not a contour. Such a bounded region must also have a centre of some description.

So the balloon analogy is invalid and it is perfectly reasonable to ask the question where the centre in an expanding universe is. If the universe is expanding and has no centre then space viewed on a large scale has to be curved, which it is not. If space is flat and expanding it has to have a centre, which it does not. There is of course a third alternative and that is that the universe is not expanding at all. That it is static and infinite. But I will deal with that in a later post.