Einstein Equation and its Spherical Solution

Abstract

Abstract The general relativity (GR) field equation relates spacetime curvature to mass/energy distribution. Its solution is the metric, determining spacetime geometry. This chapter introduces curvature in differential geometry, and then shows that in the geometric theory of gravitation, curvature has a physical interpretation as tidal gravity. The nonrelativistic theory of tidal forces is reviewed in terms of the Newtonian deviation equation. The symmetries and contractions of the Riemann curvature are presented in a search for a symmetric rank-2 curvature tensor for the GR field equation. A spherically symmetric metric involves two unknown scalar functions; they can be determined by the Schwarzschild solution of the Einstein equation. Embedding diagrams can be used to visualize such warped spaces. GR predicts solar deflection of light that is twice as large as implied by the equivalence principle alone. GPS and the precession of Mercury’s perihelion are worked out as successful applications of GR.