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By Root 917 0

to take. The trick, if it sounds too confusing, is to imagine how you might feel if you were a creature that could only ever move forward, backward, left and right. You have never experienced up and down—you live in a flat world. If someone asked you to imagine a third dimension, your flat mind would not be able to grasp it. But if you had a mathematical bent, you might be happy to accept the possibility and, in any case, you could still do the maths even if you couldn’t picture the mysterious extra dimension in your mind’s eye. Likewise for human beings and four-dimensional space. It should become more natural to think of time as “just another dimension” as our story unfolds. If there is one thing we try to teach our students when they first arrive at the University of Manchester, ready to learn to be physicists, it is that everyone gets confused and stuck. Very few people understand difficult concepts the first time they encounter them, and the way to a deeper understanding is to move forward with small steps. In the words of Douglas Adams: “Don’t panic!”

Let us continue in a gentler vain for a moment by noticing something very simple: Things happen. We wake up, we make breakfast, we eat breakfast, and so on. We’ll call the occurrence of a thing “an event in spacetime.” We can uniquely describe an event in spacetime by four numbers: three spatial coordinates describing where it happened and a time coordinate describing when it happened. Spatial coordinates can be specified using any old measuring system. For example, longitude, latitude, and altitude will do if the event is occurring in the vicinity of the earth. So your coordinates in bed might be N 53° 28’ 2.28”, W 2° 13’ 50.52”, and 38 meters above sea level. Your time coordinates are specified using a clock (because time is not universal, we’ll have to say whose clock in order to be unambiguous) and might be 7 a.m. GMT when your alarm goes off and you wake up. So we have four numbers that uniquely locate any event in spacetime. Notice that there is nothing special about the particular choice of coordinates. In fact, these particular coordinates are measured relative to a line passing through Greenwich in London, England. This convention was agreed upon in October 1884 by twenty-five nations, with the only dissenting voice being San Domingo (France abstained). It is a very important concept that the choice of coordinates should make absolutely no difference.

Let’s take the moment when I wake up in bed as our first event in spacetime. The second event could be the event that marks the end of breakfast. We have said that the spatial distance between the two events is 10 meters and the distance in time is 1 hour. To be unambiguous we’d need to say something like “I measured the distance between my bed and my breakfast table using a tape measure whose ends were stretched directly from bed to table” and “I measured the time interval using my bedside clock and the clock sitting in my kitchen.” Don’t forget that we already know that these two distances, in space and in time, are not universally agreed upon. Someone flying past your house in an aircraft would say that your clock runs slow and the distance between your bed and your breakfast table shrinks. Our aim is to find a distance in spacetime upon which everyone agrees. The million-dollar question is then “how do we take the 10 meters and the 1 hour to construct an invariant distance in spacetime?” We need to tread carefully and, just like in the case of distances on the earth’s surface, we shall not assume Euclidean geometry.

If we are to compute distances in spacetime, then we have an immediate problem to resolve. If distance in space is measured in meters and distance in time in seconds, how can we even begin to contemplate combining the two? It is like adding apples and oranges, because they are not the same type of quantity. We can, however, convert distances into times and vice versa if we use the equation we met earlier, υ = x/t. With a miniscule bit of algebra we can write time t = x/υ, or distance x =