Special-relativistic status of the recursive functional derivative as an energy-momentum tensor

Determine whether the functional derivative δS_{n−1}[γ,h]/δg^{μν} evaluated at γ = η, arising in the recursive expansion of the Einstein–Hilbert action, can be interpreted within special relativity as an energy-momentum tensor derived via Noether’s theorem with Poincaré symmetries, thereby justifying its role as a physically meaningful self-energy contribution for the spin-2 field h.

Background

One way to avoid explicit energy-momentum sourcing is to define the gravitational action recursively by expanding the Einstein–Hilbert action around a background metric γ and relating successive terms via functional derivatives. In the flat-background case γ = η, such derivatives are sometimes informally treated as energy-momentum contributions.

The authors point out that it is not established whether these derivatives admit a special-relativistic interpretation as energy-momentum tensors in the Noether sense, which would be needed to justify their physical role as self-energy sources for h.