I am only just starting to realize that there is a correspondence between quantum field theory and Einsteins field equations. In QFT the approach is to write the Lagrangian and then to solve the Euler Lagrange equation to obtain the equations of motion of the field. In GR it seems that the starting point is not the Lagrangian but instead the metric - but that the end result, the stress energy tensor, is again, the equations of motion of the field. Now, in QFT the simplest lagrangian is for a scalar field. Does one take a similar approach in GR. When one solves the metric what do the eqtns of motion correspond to - ie are they for a field, or a particle or both?