Geometrical Issues: Space-Time Curvature

In Sections 13.4 and 13.5.2 we discussed the notion of geodesics in a 4-dimensional space-time. The discussion at that point concerned the geometry of a Riemannian manifold at a stage where it was merely endowed with a way of relating different points through a connection: the Christoffel connection. The geodesic in the space time to which we had attached a coordinate system and a metric was given by the important Equation (13.37). The importance of that equation was that it provided a direct link between geodesics in general relativity and particle trajectories in Newtonian theory (Equation 13.47).

The flow of a fluid in 4-dimensional space-time is visualised as the collection of trajectories of the fluid elements in the space-time, the fluid world lines. If the fluid has zero pressure, we call it dust and the fluid elements, or particles in this case, move along trajectories that are geodesics in the underlying space-time manifold. The instantaneous 4-velocity of the particle is a tangent to the trajectory. Locally, at least, there is a hyper-plane perpendicular to the trajectory at each point. In our case of a 4-dimensional space-time, the hyper-plane is a 3-dimensional space, which we refer to as a hyper-plane by analogy with our experience in 3-dimensional Euclidean space.