Spacetime continuum

Curved manifolds

For physical reasons, a spacetime continuum is mathematically defined as a four-dimensional, smooth, connected Lorentzian manifold${\displaystyle (M,g)}$. This means the smooth Lorentz metric${\displaystyle g}$ has signature${\displaystyle (3,1)}$. The metric determines the geometry of spacetime, as well as determining the geodesics of particles and light beams. About each point (event) on this manifold, coordinate charts are used to represent observers in reference frames. Usually, Cartesian coordinates ${\displaystyle (x,y,z,t)}$ are used. Moreover, for simplicity's sake, units of measurement are usually chosen such that the speed of light ${\displaystyle c}$ is equal to 1.