Why Does E=mc2_ - Brian Cox [43]
All of this talk of objects in spacetime may sound rather abstract but there is a point to it. So far we have met one “mathematical representation of an object that has a universal meaning in spacetime”—the spacetime distance between two events. There are others.
Before we grapple with a new type of spacetime object we shall take one step back and introduce its analogue in the three dimensions of our everyday experience. It should come as no surprise (especially having read this far) that any reasonable attempt to describe the natural world exploits the notion of the distance between two points. Now, a distance is a special type of object—one that is characterized by a single number. For example, the distance from Manchester to London is 184 miles and the distance from the soles of your feet to the top of your head (more usually referred to as your height) is, at a guess, around 175 centimeters. The word following the number (cm or miles) just explains how we’re doing the counting but in both cases a single number suffices. The distance from Manchester to London provides some useful information—enough to know how much fuel to put in your car, for example, but not quite enough to make the journey. Without a map we might well head off in the wrong direction and end up in Norwich.
A slightly surreal and very impractical solution to that problem would be to construct a giant arrow whose length is 184 miles. We could place one end of the arrow in Manchester and the tip could sit in London. Arrows are useful objects when physicists set about the business of describing the world: They capture simultaneously the idea that something can have a size and also a direction. Obviously our giant Manchester-London arrow makes sense only once it is placed in a particular orientation; otherwise we might still end up in Norwich. That is what we mean when we say that the arrow has both size and direction. The arrows used by weather forecasters to illustrate how the wind blows provide another example of how arrows can help us describe the world. The swirling arrows capture the essence of the flow of the wind, telling us in which direction it blows at any particular point on the map as well as the wind speed: The bigger the arrow, the stronger the wind. Physicists call objects that are represented by arrows vectors. The wind speed as demonstrated on the weather map and the giant Manchester-London arrow are vectors in two dimensions, needing only two numbers for their description. For example, we might say that the wind is blowing at 40 miles per hour in a southeasterly direction. By showing us arrows in only two dimensions, the weather forecasters are not giving us the whole story—they are not telling us if the air is moving upward or downward and by what degree, but that isn’t something we are usually very interested in.
FIGURE 9
There can also be vectors in three or more dimensions. If we began our journey from Manchester to London in one of the old villages in the Pennine Hills north of Manchester, we would have to point our arrow slightly downward since London sits on the banks of the River Thames at sea level. Vectors living in the three dimensions of everyday space are described by three numbers. By now, you might have guessed that vectors can also exist in spacetime, and these will be described by four numbers.
We are now about to reveal the two remaining pieces on the road to E = mc2. The first piece should come as no surprise—we are only ever going to be interested in vectors in the four dimensions of spacetime. That is easy to say but a weird