Curvature of Spacetime in Schwarzschild and Kerr Metrics

Ever since Einstein proposed his theory of general relativity, the notion of curved spacetime has become the orthodox view of physics. According to this concept, the gravity would cause the spacetime to warp, or reversely, the warping of spacetime causes gravity. In this article, we will calculate the Riemann curvature scalar R of the Schwarzschild metric and the Kerr metric, the two analytical solutions of Einstein field equation. If the Riemann curvature scalar equals to zero, the spacetime is flat. The procedure of calculation is:



1) Calculate the metric connection from the metric tensor g_mn;

2) Calculate the Riemann curvature tensor elements from the metric tensor elements;

3) Calculate the Ricci tensor by contracting the Riemann curvature tensor;

4) Calculate the Riemann curvature scalar by contracting the Ricci tensor.



The Schwarzschild metric and the Kerr metric are the two known analytical solutions to Einstein field equations. With the metric elements given, we can then calculate the Ricci tensor elements and the Riemann curvature scalar according to the above procedure. The calculations show that the Ricci tensor and the Riemann curvature scalar of the Schwarzschild metric and the Kerr metric are all identically zeros. It proves that the spacetime described by Schwarzschild metric and Kerr metric are flat.