I have read that the D'Alembertian for a scalar field is ◻=gνμ∇ν∇μ=1√−g∂μ(√−g∂μ).
Exactly when is this correct? Only for ◻ϕ where ϕ is a scalar-field?
How exactly is it shown?
Is it true only on-shell, from the Euler-Lagrange equations of a scalar-field?
Answer
This is based on the observation that, given some vector Vμ,
∇μVμ=1√−g∂μ(√−gVμ)
We can show explicitly that this is true:
∇μVμ=∂μVμ+ΓμμλVλ
Let's examine the last term:
Γμμλ=12gμρ(∂μgλρ+∂λgμρ−∂ρgλμ)
The idea is to show that this equals 1√−g∂λ√−g:
∂λ√−g=−12√−g∂λg=−12√−g|gμν+δgμν|−|gμν|δxλ=−12√−g|gμν||I+(gμν)−1∂λgμνδxλ|−|gμν|δxλ=−12√−g|gμν|(1+Tr((gμν)−1∂λgμνδxλ))−|gμν|δxλ=√−g12(gμν∂λgμν)
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