Why aren't gravitons spin 1?

Expressing the metric as $$g_{\mu \nu} = \eta_{\mu \nu} + h_{\mu \nu}$$, assuming $$h_{mu \nu} \ll 1$$ we can write the Einstein Hilbert action to leading order in $$h_{\mu \nu}$$ and quantize the linearized Einstein Hilbert action to construct the graviton field. Gravitons are spin 2 particles, which is easiest to see by noting that $$h_{\mu \nu}$$ has two indices. These enjoy a "gauge symmetry" corresponding to diffeomorphisms.

However, classically, gravity can be understood to be largely analogous to a gauge theory. The chrisoffel symbol $$\Gamma^{\alpha}_{\beta \mu}$$ takes the place of $$A^a_\mu T^a$$ as the gauge field.

Note that $$\Gamma^{\alpha}_{\beta \mu}$$ has three indices, however the $$\alpha,\beta$$ indices can be understood as a matrix, much like the lie algebra elements $$T^a$$ in Yang Mills theory.

If we quantize this field instead shouldn't we not be able to realize gravity as a theory mediated by spin 1 gauge particles?

(This should be especially true if we take the action to be the Kretschmann scalar, which seems to be equivalent to the Yang Mills Lagrangian.

$$\mathrm{Tr}(F_{\mu \nu} F^{\mu \nu}) \leftrightarrow R^a_{b \mu \nu} {R^b_{a}}^{\mu \nu}$$

However, this would obviously give a different theory than Einstein gravity.)

• – Cosmas Zachos May 12 at 18:24
• Related: physics.stackexchange.com/q/108230/2451 , physics.stackexchange.com/q/263572/2451 and links therein. – Qmechanic May 12 at 18:31
• see my answer here physics.stackexchange.com/questions/11542/… .charges play a role in attraction and repulsion together with the spin. – anna v May 12 at 18:38
• It sure feels like 3 questions: a) Physically, coupling to the energy-momentum tensor dictates spin 2. b) Formally, the weak field expansion of Einstein's equations produce a metric perturbation field which is spin 2. c) Indeed, the intuitive gauge-theory simulacrum for gravity is the gauged tangent space Lorentz group, effected by the spin connection ω, related to Γ, which is not a tensor, and which is utilized in supergravity; however, possibly counterintuitively, it demonstrably leads to spin 2. You apparently want to focus on 3)? – Cosmas Zachos May 12 at 19:32
• ...perhaps (3.3) here might help--unless it doesn't. – Cosmas Zachos May 12 at 20:27