The Geometry of Special Relativity

Simple geometric properties of spacetime and free particles underlie the theory of relativity just as Euclidean geometry follows from simple properties of points and straight lines. The vacuum, the empty four-dimensional curved spacetime, determines straight lines and light rays. In the absence of gravity, the vacuum is isotropic and homogeneous and does not allow to distinguish rest from uniform motion. Therefore, contrary to Newton’s opinion, the vacuum cannot contain the information about a universal time which could be attributed to events. Whether two different events are simultaneous depends on the observer – just as in Euclidean geometry it depends on a given direction whether two points lie on an orthogonal line. 1.1 Properties of the Vacuum We can denote a point in space by specifying how far away it is ahead, to the right and to the top of a chosen reference point. These specifications are called coordinates of the point. One needs three coordinates in order to specify any one point. Space is three-dimensional. The coordinates of a point depend of course on the choice of the reference point and on which directions the observer chooses as ahead, right and above. As for appointments in daily life, for physical processes not only the position is important, where an event takes place, but also the time when it occurs. The set of all events, spacetime, is four-dimensional, because to specify a single event one needs four labels, the position where it takes place and the time when it occurs. The position and time specifications which label an event depend – just as the three coordinates of a position – on the observer. The four-dimensional spacetime fascinates and beats our imagination which is trained in everyday life. Nevertheless it is quite simple. We can easily envisage a stack of pictures, as they are stored in a film reel, which show the sequel of threedimensional situations. Thereby one conceives the four-dimensional spacetime the