Just as one studies flat (2-dimensional) Euclidean geometry before studying curved surfaces, one studies special relativity before general relativity. But the analogy goes much further.

The basic notion in Euclidean geometry is the distance between two points, which is given by the Pythagorean theorem. The basic notion on a curved surface is still the distance “function”, but this is now a statement about infinitesimal distances. Euclidean geometry is characterized by the line element \begin{equation} ds^2 = dx^2 + dy^2 \end{equation} which can be used to find the distance along any curve. The simplest curved surface, a sphere of radius $r$, can be characterized by the line element \begin{equation} ds^2 = r^2 \, d\theta^2 + r^2 \sin^2\theta \, d\phi^2 \end{equation} Remarkably, it is possible to calculate the curvature of the sphere from this line element alone; it is not necessary to use any 3-dimensional geometry. It's also possible to calculate the “straight lines”, that is, the shortest path between two given points. These are, of course, straight lines in the plane, but great circles (equators) on the sphere.

Similarly, special relativity is characterized by the line element 1) \begin{equation} ds^2 = - dt^2 + dx^2 \end{equation} which is every bit as flat as a piece of paper. You get general relativity simply by considering more general line elements!

Of course, it's not quite that simple. The line element must have a minus sign. And the curvature must correspond to a physical source of gravity — that's where Einstein's field equations come in. But, given a line element, one can again calculate the “straight lines”, which now correspond to freely falling observers! Matter curves the universe, and the curvature tells objects which paths are “straight” — that's gravity!

Here are two examples to whet your appetite further.

The (3-dimensional) line element \begin{equation} ds^2 = - dt^2 + \sin^2(t) \left( d\theta^2 + \sin^2\theta \, d\phi^2 \right) \end{equation} describes the 2-dimensional surface of a spherical balloon, whose radius changes with time. This roughly corresponds to a cosmological model for an expanding universe produced by a Big Bang. By studying the properties of this model more carefully, you will be led to ask good questions about relativistic cosmology.

And finally, the line element \begin{equation} ds^2 = - \left(1-\frac{2m}{r}\right) dt^2 + \frac{dr^2}{\displaystyle 1-\frac{2m}{r}} \end{equation} describes a simplified model of a black hole, with an apparent singularity at $r=2m$, which is, however, just due to a poor choice of coordinates. Trying to understand what actually happens at $r=2m$ will give you some understanding of what a black hole really is.

If you wish to pursue these ideas further, you may wish to take a look at an introductory textbook on general relativity, such as d'Inverno [ 9 ] or Taylor & Wheeler [ 10 ].