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  Variation of quadratic term in modified Einstein-Hilbert actions

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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=d4xg12γχ2,()


where χ=gμνμνϕ=gμνμνϕ. 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[gγχδgμνμνϕ]=d4xγ[μ(gχ)νϕδgμν]

or
δS2χ=d4xγgμχνϕδgμν

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=+γμϕνχ.

My question is: the reason that i find in literature that δS2χ contribute as
γ(αϕαχδμν(νϕμχ+νχμϕ))

is because the missing step is to decompose the rank two tensor into its antisymmetric, trace and symmetric trace free parts
μϕνχ=Kμν=K[μν]+1nδμνδαβKαβ+(K(μν)1nδμνδαβKαβ)

or I'am missing something?

asked Aug 6, 2015 in Theoretical Physics by yngabl (10 points) [ no revision ]

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