Mathematics is the exquisitely perfect language needed for describing how the theory of relativity applies to the physical Universe and all of spacetime, and that description includes the strange behaviour that occurs near black holes. A mathematical description, while powerful and exact, even so can be something of a foreign and forbidding language for those without the appropriate technical training. Descriptive words, however eloquent, lack the rigour and power of a mathematical equation and can be imprecise and limiting. Pictures however, being(it is said) worth a thousand words, can be not only a useful compromise but a very helpful way to visualize what is going on. For this reason, it is well worth spending a little effort to understand a particular type of picture, called a spacetime diagram. This will help in understanding the nature of spacetime around black holes.
Spacetime diagrams
The cartoon in Figure 3 shows a simple spacetime diagram.Following tradition, the `time-like' axis is the one that is vertical on the page and the `space-like' axis is drawn perpendicular to this.Of course, we really need four axes to describe spacetime because there are three space-like axes (usually denoted x, y, and z) and one time-like axis. However, two axes will suffice for our purpose(and of course four mutually perpendicular axes are impossible to draw!).Where these two axes intersect is called the origin, and this may be regarded as the point of `here and now' for the observer who has constructed their spacetime diagram. An idealized instantaneous event, say the click of a camera shutter,occurs at a particular moment in time and at a particular location in space. Such an instantaneous event is represented by a dot on a spacetime diagram, appropriate to the time and spatial location in question. There are two dots in Figure 3, which are spatially separated (they do not occur at the same point on the space axis)but they are simultaneous (they have the identical coordinate on the time axis). You could imagine these two dots correspond to the simultaneous shutter presses of two photographers who are standing some distance apart from one another, photographing the same spectacle. If points represent events, what do lines in a spacetime diagram represent? A line simply shows a path of an object through spacetime. As we live our lives, we journey through spacetime and the path we leave behind us (somewhat as a snail leaves a glistening trail of slime behind it) is a line in spacetime,and in the jargon this is called a worldline. If you stay at home all day, your worldline is a vertical path through spacetime (with space coordinate = `22 Acacia Avenue', for example). You move forward in time but are fixed in space. If on the other hand you made a long journey, your worldline slants over because your distance changes with time, because you move in space as well as time.