Self-sourcing versus action-level self-coupling in spin-2 self-interaction

Determine whether the interaction of the classical spin-2 field h with an energy-momentum tensor in the self-interaction program should be modeled primarily as self-sourcing at the level of the equations of motion or as self-coupling at the level of the action, noting that for a self-energy-momentum tensor these descriptions are not equivalent.

Background

Within the self-interaction approach, a free classical spin-2 field h on Minkowski spacetime is iteratively coupled to sources, including its own energy-momentum, with the aim of recovering the Einstein field equations. A crucial step requires specifying how h couples to its energy-momentum tensor, but multiple inequivalent formulations exist.

The authors highlight a conceptual uncertainty: for a self-energy-momentum tensor that contains h itself, describing the interaction as equation-of-motion-level self-sourcing (analogous to electrodynamics) is not equivalent to describing it as action-level self-coupling. Establishing which description is physically primary—and under what principles—remains unresolved within the self-interaction program.