Hi Tedman,
Position in a curved spacetime is described just as you would describe the position on a circle or a sphere (curved surfaces): you use coordinates that have a special relation between them.
For example: to locate a point on a sphere, you need to know two coordinates: two angles are sufficient (in a 3D world) or, if you are living on the sphere, you can use x and y coordinates (only two, the surface of a sphere is two-dimensional). When you use x and y, you need to take the curvature into account. The distance d between two points (x1,y1) and (x2,y2) on a sphere is not the Euclidian:
d<sup>2</sup> = (x2 - x1)<sup>2</sup> + (y2 - y1)<sup>2</sup>
since that is the length of a straight line between the two points. What we want to know is the distance between the points on the sphere. I am a bit too lazy to do the math myself, but I am sure you will agree that you need to take a different formula to calculate the length of a line that connects two points on the sphere.
For curved spacetime things work in exactly the same way: we use the coordinates (x,y,z,t) to locate an object in spacetime, but since spacetime is curved, we need to use a special relationship between the coordinates to calculate a length. This special relationship, that defines the distance between two points, is called a metric. I could be wrong on this one (never really got into the maths myself) but I believe the metric used for curved spacetime is called the Riemann-metric.
Hope this more or less answers your question,
Crisp
