Special relativity assumes the existence of Minkowski coordinates, such that the line element takes the form (2.1). In that case one says that spacetime is flat, otherwise one says that spacetime is curved. Spacetime is flat only insofar as gravity can be ignored.

In this chapter we first see how to write the equations of special relativity using generic coordinates. Then we consider curved spacetime, the equivalence principle and the Einstein field equation. All of these things taken together are called general relativity. We end the chapter by giving a basic description of the particular curved spacetime that corresponds to a homogeneous and isotropically expanding universe.

Special relativity with generic coordinates: mathematics

To handle curved spacetime we have to learn how to use generic coordinates. It is helpful to do this first in the familiar context of special relativity, where Minkowski coordinates do exist.

Once a coordinate choice xµ has been made, it defines a threading of spacetime into lines (corresponding to fixed xi) and a slicing into hypersurfaces (corresponding to fixed x0), as shown in Figure 3.1. The threads are chosen to be timelike, so that they are the worldlines of possible observers, and the slices are chosen to be spacelike. The coordinate choice uniquely defines the threading and slicing, but the reverse is not true. Given a slicing and threading there is still freedom in choosing the coordinates which label the slices and threads.