Perturbation theory covariant derivative

Question

Let $\varphi$ be a scalar field and consider the metric $g_{\mu \nu}=\eta_{\mu \nu} + h_{\mu \nu}$. I want to compute $\nabla_\mu \varphi \nabla^\mu \varphi$ to first order.

$$\nabla_\mu \varphi \nabla^\mu \varphi = g^{\mu \nu} \partial_\mu \varphi \partial_\nu \varphi\approx (\eta^{\mu \nu} - h_{\alpha \beta} \eta^{\alpha \mu} \eta^{\beta \nu}) \partial_\mu \varphi \partial_\nu \varphi. \tag{1}$$

Or via:

$$\nabla_\mu \varphi \nabla^\mu \varphi=g_{\mu \nu} \nabla^\mu \varphi \nabla^\nu \varphi \approx (\eta^{\mu \nu} + h_{\alpha \beta} \eta^{\alpha \mu} \eta^{\beta \nu}) \partial_\mu \varphi \partial_\nu \varphi.\tag{2}$$

My question is why do these methods not coincide? Which one is correct?

Answer 1

In OP's last $\approx$ sign, the index on the covariant derivatives $\nabla \varphi$ should be lowered by use of the inverse metric. This yields 2 minus contributions, so that the corrected eq. (2) agrees with OP's eq. (1).

See also this related Phys.SE post.