The preceding chapter dealt with the kinematics of the curved spacetime geometry, that is, the description of the geometry and its curvature. We now come to the dynamics of the geometry, that is, the interaction of the geometry and matter. This interaction is the essence of Einstein’s equations for the gravitational field.

There are several routes that lead to Einstein’s equations; they differ in their starting points. One route begins with the equations of the linear approximation of Chapter 3 and adds the assumption that the exact, nonlinear equations are of second differential order and are endowed with general invariance. These assumptions suffice to completely determine the exact, nonlinear equations for the gravitational field.

That the linear equations imply the full nonlinear equations is a quite remarkable feature of Einstein’s theory of gravitation. Given some complicated set of nonlinear equations, it is always easy to derive the corresponding linear approximation; but in general, if we know only the linear approximation, we cannot reconstruct the nonlinear equations. What permits us to perform this feat in gravitational theory is the requirement of general invariance. As we will see later, this requirement states that the form and the content of the equations must remain unchanged under all coordinate transformations.