In his best selling book, A Brief History of Time, Professor Hawking suggests that in order for the "Big Bang" to work, the mathematics requires that the condition of the Universe at the beginning must have been finite and boundless. There must have been no edges, or points of discontinuity. Without this assumption, the laws of physics could not be used to explain the activity and state of affairs in the first moments of the creation of the Universe. By assuming that the Universe was and is finite, yet boundless, physicists are able to avoid the problems created by discontinuities. While the results of having made this fundamental assumption regarding the Universe is quite clearly presented by Professor Hawking, he devotes only a few sentences to explaining the concept of "finite and boundless" as it applies here. The following is offered as an expansion on this point.
Let us first examine the concept of boundless. To understand how this idea is being used, imagine (as you did when you first read Flatland) that you are a creature whose physical makeup is a mathematical point. This means that you have no mass and no dimensions. The world in which you live consists of a single line, joined at the ends to form a circle. As you travel in your world, you are unable to mark any given point on the circle. You may travel around it in either direction, but, because of your rather severe physical limitations, you cannot determine by any means permissible in this example, quite where you are in your world. From your point perspective, you live in a "boundless" Universe. To an extraterrestrial looking down on your world, it may be seen as boundless, yet quite limited in its size. This is the basis of the concept of finite, yet boundless.
Let's expand on this concept a little. Instead of being a one dimensional creature in a two dimensional world, we may examine a two dimensional creature, say, a line segment, in a three dimensional world, for example, a sphere. The line may move around on the sphere, but again is not able to mark its location. There is no way for it to tell when it the same point on the surface of the sphere is being recrossed. Again, from our privileged vantage point above the sphere, we can see that the Universe of the line segment creature is finite, being limited to the surface area of the sphere, and yet we can see how, in the eyes of the line segment, the sphere Universe is boundless.
To describe our own Universe this way, we may look at ourselves as three dimensional creatures living in a four dimensional Universe. Once again, the Universe we live in is finite, possessing a fixed amount of mass, yet it appears to be boundless. As in the simpler Universes, we are unable to locate an edge, or boundary. This is, of course, the very condition which Professor Hawking suggests is necessary to avoid the problems posed by a Universe which has an edge, or boundary, and therefore, a boundary condition which may not be susceptible to mathematical analysis.
Background Photo: The Orion Nebula. Learn More