Energy-Momentum Conservation as a Constraint to Restrict the Space of Viable Field Lagrangians: Vector Field versus Spin-Two Field

We investigate whether the Lagrangian density for an interacting vector field or an interacting massless spin-2 field can be determined by imposing Poincaré invariance and the conservation of energy-momentum for the entire system. We adopt the Belinfante-Rosenfeld energy-momentum tensor for systems involving either a vector field or a spin-2 field. For the vector field coupled to a system of point masses, it is not possible to determine the Lagrangian density of the vector field. On the other hand, we show that for the spin-2 field coupled to a material system such as a system of point particles, its Lagrangian density is uniquely given by the Einstein Lagrangian density. Furthermore, the Belinfante-Rosenfeld tensor for the spin-2 field becomes Papapetrou’s gravitational energy-momentum pseudotensor.