# For making gravity supersymmetric is a modification of Einstein's vacuum field equations necessary ( apart from adding the R-S Lagrangian)?

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.

## 1 Answer

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).

• Thank you for the answer. I already read such kind of articles (above all from Samtleben) in vielbein-formalism. My question actually aims more at the physics than the purely necessary mathematics with all its requirements. Are possible modifications -- even in the necessary mathematical formalism -- of the field equations measurable ? – Frederic Thomas Dec 12 '17 at 14:32
• Field equations have relevance in classical physics. Supersymmetry and supergravity are thought as quantum theories, so any predictions of these two theories must be quantum. Any experiment could first tell us something about the quantum theory first and then, theoretically, also about the (semi)classical theory and its field equations. More precisely, let us say there is no experimental trace of the gravitino, then a quantum theory of the R-S field in the presence of the quantized gravitational field would be incorrect. You could infer that the E-H field equations of classical GR modified by – DanielC Dec 12 '17 at 15:31
• [ctd] the presence of the R-S field are also incorrect, but to what use? – DanielC Dec 12 '17 at 15:36
• I think, it has use. Imagine the large number of physicists desperately searching for physics beyond the SM, in particular for its supersymmetric extension. Any sign in that sense would be a large encouragement. What if dark matter consists mainly of gravitinos ? – Frederic Thomas Jan 3 '18 at 21:22
• You are free to make any speculations you want, as long as you can get them passed the test of peer-review from a journal with a big TR impact factor... – DanielC Jan 4 '18 at 5:47