ABSTRACT
This chapter shows that Einstein’s equations can be derived from a variational principle, i.e. by defining an action for the gravitational field and by requiring this action to be stationary. It firstly recalls how the variational approach can be applied in Special Relativity to derive Euler-Lagrange’s equations for a given field. Using It derives Einstein’s equations in vacuum, and generalizes the entire procedure to generic fields in the presence of gravity. The non-homogeneous form of Einstein’s equations can be derived using the variational principle. Ostrogradsky’s theorem guarantees that the only viable action constructed solely from the metric tensor is the Einstein-Hilbert one, plus at most a cosmological constant term.
