19 - Gravitational Field Equations

Abstract

In this chapter we derive the gravitational field equations from the action integral of the gravitational field. We use the fact that the components of the metric tensor of a given space-time manifold are the potentials of the field of gravity and the gravitational effects are described by the metric itself. We therefore begin with defining the action integral for the gravitational fields and calculating its variation. Thereafter, we study the action integrals for the matter fields as sources of the gravitational fields, and calculate their variation. Combining these results, we finally derive the Einstein field equations for the gravitational fields.