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By Root 930 0

with a globe and measure out the distances using a piece of string. Now we would be correctly accounting for the curvature of the earth. An airline pilot could stretch a piece of string between two cities on the globe, measure its length with a ruler, and then simply multiply the answer by the ratio in size of the globe and the earth. But maybe we don’t have a globe on hand, or maybe we need to write the computer software that helps airplanes navigate. In either case, we need to do better than a piece of string and figure out an equation that tells us the distance between any two points on the earth’s surface given only their latitude and longitude, and the shape and size of the earth. Such an equation is not too hard to find and if you know a little mathematics you might even try to find it. We don’t need to write it down here, but the point is that an equation exists and it hasn’t got much to do with the Euclidean geometry of a flat tabletop. It does, however, allow one to calculate the shortest distance between two points on a sphere, in much the same way that Pythagoras’ theorem is a recipe for calculating the shortest distance between two points (the hypotenuse) on a tabletop if we know the distances from one corner as measured along the edges of the table. Since straight lines belong in the domain of Euclid, we shall introduce a new term for the shortest distance between two points that applies whether the space is curved or flat. This line is called a geodesic: A great circle is a geodesic on the surface of the earth and a straight line is a geodesic in flat space. So much for distances in three-dimensional space. Now we must decide how to measure distances in spacetime, so let’s go ahead and complicate matters by adding time into the mix.

We already introduced the concepts we will need when we thought about getting out of bed and finishing breakfast in the kitchen. There is no problem in saying that the distance in space between the bed and the kitchen is 10 meters. We could also say, although it sounds rather strange, that the distance in time between getting out of bed and finishing breakfast is 1 hour. This is not how we naturally think about time, because we are not used to describing it in the language of geometry. We would rather say “one hour passed between my getting out of bed and finishing my breakfast.” In the same way, we would not normally say “10 meters have passed since I got out of bed and sat down in the kitchen.” Space is space, and time is time, and never the twain shall be intermingled. But we have set ourselves the task of trying to merge space and time together, because we suspect that this is the only way to rebuild things in a way that fits with Maxwell and Einstein. So let us proceed and see where it leads us. If you are not a scientist, then this may be the most difficult part of the book so far because we are operating in a purely abstract fashion. The capacity for abstract thought is what gives science its power, but also perhaps gives it a reputation as being difficult because it is not a faculty we generally need too much in everyday life. We have already encountered a difficult abstract concept in the form of the electric and magnetic fields, and in fact the abstraction needed to merge space and time together is probably less challenging than that.

What we are doing implicitly in speaking of “the distance in time” is treating time as an additional dimension. We are used to the phrase “3-D,” as in three-dimensional, referring to the fact that space has three dimensions: up and down; left and right; forward and backward. When we try to add time into the framework, so that we can define distances in spacetime, we are in effect creating a four-dimensional space. To be sure, the time dimension behaves differently than the space dimensions. We have complete freedom of movement in space, whereas we go only one way in time, and time doesn’t feel anything like space. But that need not be an insurmountable hurdle. Thinking of time as “just another dimension” is the abstract leap we have