## By: Stephen Hawking

In science fiction, space and time warps are commonplace. They are used for rapid journeys around the galaxy or for travel through time. But today’s science fiction is often tomorrow’s science fact. So what are the chances of time travel?

The idea that space and time can be curved or warped is fairly recent. For more than 2,000 years the axioms of Euclidean geometry were considered to be self-evident. As those of you who were forced to learn geometry at school may remember, one of the consequences of these axioms is that the angles of a triangle add up to 180 degrees.

However, in the last century people began to realize that other forms of geometry were possible in which the angles of a triangle need not add up to 180 degrees. Consider, for example, the surface of the Earth. The nearest thing to a straight line on the surface of the Earth is what is called a great circle. These are the shortest paths between two points so they are the routes that airlines use. Consider now the triangle on the surface of the Earth made up of the equator, the line of 0 degrees longitude through London and the line of 90 degrees longtitude east through Bangladesh. The two lines of longitude meet the equator at a right angle or 90 degrees. The two lines of longitude also meet each other at the North Pole at a right angle, or 90 degrees. Thus one has a triangle with three right angles. The angles of this triangle add up to 270 degrees which is obviously greater than the 180 degrees for a triangle on a flat surface. If one drew a triangle on a saddle-shaped surface one would find that the angles added up to less than 180 degrees.

The surface of the Earth is what is called a two-dimensional space. That is, you can move on the surface of the Earth in two directions at right angles to each other: you can move north-south or east-west. But of course, there is a third direction at right angles to these two and that is up or down. In other words the surface of the Earth exists in threedimensional space. The three-dimensional space is flat. That is to say it obeys Euclidean geometry. The angles of a triangle add up to 180 degrees. However, one could imagine a race of two-dimensional creatures who could move about on the surface of the Earth but who couldn’t experience the third direction of up or down. They wouldn’t know about the flat three-dimensional space in which the surface of the Earth lives. For them space would be curved and geometry would be non-Euclidean.

But just as one can think of two-dimensional beings living on the surface of the Earth, so one could imagine that the three-dimensional space in which we live was the surface of a sphere in another dimension that we don’t see. If the sphere were very large, space would be nearly flat and Euclidean geometry would be a very good approximation over small distances. But we would notice that Euclidean geometry broke down over large distances. As an illustration of this imagine a team of painters adding paint to the surface of a large ball.

As the thickness of the paint layer increased, the surface area would go up. If the ball were in a flat three-dimensional space one could go on adding paint indefinitely and the ball would get bigger and bigger. However, if the three-dimensional space were really the surface of a sphere in another dimension its volume would be large but finite. As one added more layers of paint the ball would eventually fill half the space. After that the painters would find that they were trapped in a region of ever-decreasing size, and almost the whole of space would be occupied by the ball and its layers of paint. So they would know that they were living in a curved space and not a flat one.

This example shows that one cannot deduce the geometry of the world from first principles as the ancient Greeks thought. Instead one has to measure the space we live in and find out its geometry by experiment. However, although a way to describe curved spaces was developed by the German Bernhard Riemann in 1854, it remained just a piece of mathematics for sixty years. It could describe curved spaces that existed in the abstract, but there seemed no reason why the physical space we lived in should be curved. This reason came only in 1915 when Einstein put forward the general theory of relativity.

General relativity was a major intellectual revolution that has transformed the way we think about the universe. It is a theory not only of curved space but of curved or warped time as well. Einstein had realized in 1905 that space and time are intimately connected with each other, which is when his theory of special relativity was born, relating space and time to each other. One can describe the location of an event by four numbers. Three numbers describe the position of the event. They could be miles north and east of Oxford Circus and the height above sea level. On a larger scale they could be galactic latitude and longitude and distance from the centre of the galaxy.

The fourth number is the time of the event. Thus one can think of space and time together as a four-dimensional entity called space-time. Each point of space-time is labelled by four numbers that specify its position in space and in time. Combining space and time into space-time in this way would be rather trivial if one could disentangle them in a unique way. That is to say if there was a unique way of defining the time and position of each event. However, in a remarkable paper written in 1905 when he was a clerk in the Swiss patent office, Einstein showed that the time and position at which one thought an event occurred depended on how one was moving. This meant that time and space were inextricably bound up with each other.