A derivation of the Einstein Lagrangian density from the conservation of a well-defined global energy-momentum tensor

We present a novel approach to deriving the Einstein Lagrangian density based on energy-momentum conservation. In this framework, starting from a field-theoretic setting with background Minkowski spacetime and Poincaré invariance, we investigate whether conservation of the total energy-momentum tensor, in which the field contribution is given by the symmetrized Belinfante tensor, determines the form of the field Lagrangian density for a symmetric rank-2 tensor field. We find that the requirement of total energy-momentum conservation is satisfied only if the field Lagrangian density is proportional to the Einstein Lagrangian density.