The Einstein equations
Valeria Ferrari, Leonardo Gualtieri, Paolo Pani · 2020
The chapter starts with a discussion on the information which can be obtained by evaluating the geodesic equation in the weak-field limit, comparing the result with the Newtonian equation of motion of a particle in a gravitational field. It is shown that the 00-component of the metric tensor can be written in terms of the Newtonian potential, solution to Poisson&s;s equation. The constraints that have to be imposed on the tensor $G_{\mu u}$which has to appear on the left-hand side of the sought equations are then discussed, and the form of the tensor is derived which is shown to depend on two constants. These are determined using the Bianchi identities and the constraint that, in the weak-field limit, the field equations must reduce to Poisson&s;s equation. Einstein&s;s equations are finally obtained and their gauge invariance is discussed.