The simplest form of the $\kappa$ and $\kappa^2$ order graviton self-interaction Lagrangian density in the weak field approximation

Abstract

We derive the simplest form of the $\kappa(\kappa^2)$ order graviton self-interaction Lagrangian density $\lag{1}(\lag{2})$ in the weak field approximation. With the divergenceless condition, de Donder gauge and some combinatoric techniques, we derive the mathematically independent basis expressions for $\lag{1}(\lag{2})$ composed of 8(24) terms. By using the basis expressions $\lag{1}(\lag{2})$ is reduced to 6(18) terms.