In the context of [mimetic gravity](http://www.google.com/search?as_epq=mimetic+gravity) at some point one try to add to an already modified Einstein-Hilbert action also a term like
Sχ=∫d4x√−g12γχ2,(⋆)
where
χ=gμν∇μ∇νϕ=gμν∇μ∂νϕ. Variation of this lead to the two contributes
δS1χ+δS2χ=∫d4x[δ(√−g)12γχ2+√−gγχδχ].
Looking only at the second term, after integration by parts, it become
δS2χ=∫d4x[√−gγχδgμν∇μ∂νϕ]=−∫d4xγ[∇μ(√−gχ)∂νϕδgμν]
or
δS2χ=−∫d4xγ√−g∂μχ∂νϕδgμν
because the metric is covariantly conserved. So I found that the contribute to the right side of the Einstein equations of the term
δS2χ that originate from (
⋆) is
Gμν=⋯+γ∂μϕ∂νχ.
My question is: the reason that i find in literature that
δS2χ contribute as
γ(∂αϕ∂αχδμν−(∂νϕ∂μχ+∂νχ∂μϕ))
is because the missing step is to decompose the rank two tensor into its antisymmetric, trace and symmetric trace free parts
∂μϕ∂νχ=Kμν=K[μν]+1nδμνδαβKαβ+(K(μν)−1nδμνδαβKαβ)
or I'am missing something?