Riemannian Geometry

Abstract

In the previous chapters, we studied non-gravitational phenomena in inertial reference frames, and often we limited our discussion to Cartesian coordinate systems. Now we want to include gravity, non-inertial reference frames, and general coordinate systems. The aim of this chapter is to introduce some mathematical tools necessary to achieve this goal. We follow quite a heuristic approach. The term Riemannian geometry is used when we deal with a differentiable manifold equipped with a metric tensor.

Problems

5.1

Write the components of the following tensors:

$$\begin{aligned} \nabla _\mu A_{\alpha \beta } \, , \quad \nabla _\mu A^{\alpha \beta } \, , \quad \nabla _\mu A^\alpha _{\,\,\beta } \, , \quad \nabla _\mu A_\alpha ^{\,\,\beta } \, , \quad \end{aligned}$$

(5.99)

5.2

Write the non-vanishing components of the Riemann tensor, the Ricci tensor, and the scalar curvature for the Minkowski spacetime in spherical coordinates.

5.3

Check that the Ricci tensor is symmetric.