Introduction to Tensor Calculus for General Relativity

There are three essential ideas underlying general relativity (GR). The first is that spacetime may be described as a curved, four-dimensional mathematical structure called a pseudo-Riemannian manifold. In brief, time and space together comprise a curved fourdimensional non-Euclidean geometry. Consequently, the practitioner of GR must be familiar with the fundamental geometrical properties of curved spacetime. In particular, the laws of physics must be expressed in a form that is valid independently of any coordinate system used to label points in spacetime. The second essential idea underlying GR is that at every spacetime point there exist locally inertial reference frames, corresponding to locally flat coordinates carried by freely falling observers, in which the physics of GR is locally indistinguishable from that of special relativity. This is Einstein’s famous strong equivalence principle and it makes general relativity an extension of special relativity to a curved spacetime. The third key idea is that mass (as well as momentum density and momentum flux) curves spacetime in a manner described by the tensor field equations of Einstein. These three ideas are exemplified by contrasting GR with Newtonian gravity. In the Newtonian view, gravity is a force accelerating particles through Euclidean space, while time is absolute. From the viewpoint of GR as a theory of curved spacetime, there is no gravitational force. Rather, in the absence of electromagnetic and other forces, particles follow the straightest possible paths (geodesics) through a spacetime curved by mass. Freely falling particles define locally inertial reference frames. Time and space are not absolute but are combined into the four-dimensional manifold called spacetime. In special relativity there exist global inertial frames. This is no longer true in the presence of gravity. However, there are local inertial frames in GR, such that within a