Upon building up supersymmetric Supergravity in 4D, is it necessary to modify the Einstein's vacuum field equations (apart from adding the Rarita-Schwinger Lagrangian for the gravitino) in order to obtain Supergravity 4D or can it be built up by simply adding the Hilbert-Einstein Lagrangian and the Rarita-Schwinger Lagrangian ? I guess, dealing with Einstein's nonlinear equations a modifiication is necessary. If this is the case, that would suggest that such a modification could be checked by experiments. Are such experiments conceivable ? Can experiments with this aim give an answer on the search of supersymmetry ? I am aware that most physicists are more interested in Supergravity in 11D, so perhaps my question is rather academic. I can also make the assumption that Supergravity in 11D upon compactification reduces to Supergravity in 4D, then my question is possibly more interesting. But may be this assumption is wrong and my idea is for this reason (or others) not viable. I would appreciate if somebody could explain it. Thank you.
The simplest supergravity theory (N=1, D=4) has the gravity part expressed in the so-called vielbein-spin connection formalism. Without vielbeins, you cannot properly put spinorial fields (Weyl, Dirac, Rarita-Schwinger) in curved spacetimes [chapter #13 of Wald's book]. Therefore, the Lagrangian density and the integration measure look different, the field equations of the combined HE-RS look different. The Wikipedia article on this uses the superspace formalism.
At an elementary level, Boulanger and Esole (https://arxiv.org/abs/gr-qc/0110072) have proven the uniqueness of the SUGRA N=1, D=4 in curved bosonic spacetime starting by BRST-deforming spin-2 fields and spin-3/2 fields in flat spacetime. The linearized vielbein is essentially the spin-2 Pauli-Fierz field (i.e. the 1st order perturbation of the metric).