# D'Alembertian operator in linearized gravity

In linearized gravity, where we take a general background metric $$g$$ with perturbation $$h$$, the linearized Einstein equations become

$$-\square h_{\alpha\beta}+\nabla^{\delta}\nabla_{\alpha}h_{\beta\delta}+\nabla^{\delta}\nabla_{\beta}h_{\alpha\delta}-\nabla_{\alpha}\nabla_{\beta}h-g_{\alpha\beta}\bigg(\nabla^{\gamma}\nabla^{\delta}h_{\gamma\delta}-\square h\bigg)=0$$

where the box operator is $$\square=g^{\alpha\beta}\nabla_{\alpha}\nabla_{\beta}$$. I was wondering, how to interpret the term $$\square h_{\alpha\beta}$$. Do I have to interpret is component-wise, i.e. in the sense of $$\square$$ is acting on each component function, which would yield

$$\square h_{\alpha\beta}=\square (\underbrace{h_{\alpha\beta}}_{\text{function}})=g^{\mu\nu}\nabla_{\mu}(\partial_{\nu}h_{\alpha\beta})=g^{\mu\nu}(\partial_{\mu}\partial_{\nu}h_{\alpha\beta}-\Gamma^{\rho}_{\mu\nu}\partial_{\rho}h_{\alpha\beta})$$

or really as the douple connection acting on a tensor (the so-called "connection Laplacian" in math terminology), which would least a much longer expression like

$$\square h_{\alpha\beta}=g^{\mu\nu}\nabla_{\mu}\nabla_{\nu}h_{\alpha\beta}=g^{\mu\nu}(\partial_{\mu}\nabla_{\nu}h_{\alpha\beta}-\Gamma^{\rho}_{\mu\nu}\nabla_{\rho}h_{\alpha\beta}-\Gamma^{\rho}_{\mu\alpha}\nabla_{\nu}h_{\rho\beta}-\Gamma^{\rho}_{\mu\beta}\nabla_{\nu}h_{\alpha\rho})$$

with

$$\nabla_{\nu}h_{\alpha\beta}=\partial_{\nu}h_{\alpha\beta}-\Gamma^{\rho}_{\nu\alpha}h_{\rho\beta}-\Gamma^{\rho}_{\nu\beta}h_{\alpha\rho}$$ and etc for the others? I am pretty sure its the second, more complicated, option, but I did not found any reference confirming my guess. (Most references are discussing Minkowski background, where there is no difference, since all Christoffel symbols are anyway zero).

Notice that, in your first development of the D'Alembertian operator, you implicitly assume that $$\nabla_\mu h_{\alpha\beta} = \partial_\mu h_{\alpha\beta}.$$ You have rightly identified this only holds in a Minkowski background.