Given the four-dimensional metric tensor $g_{\mu\nu}$, written in the perturbed form as: $$g_{\mu\nu}=\eta_{\mu\nu}+h_{\mu \nu}, \qquad |h_{\mu \nu}|<<1$$ In deriving the linearized Einstein's equations in Braneworld models, I got terms like:
$$\partial_{\mu}\partial_{\nu}h \ \ \ \ \mathrm{and} \ \ \ \ -\frac{\square}{4}h\eta_{\mu\nu} \ ,$$ with $h$ being the trace of $h_{\mu \nu}$, i. e. $$h=\eta^{\mu\nu}h_{\mu \nu} \ \ \ \ \mathrm{and} \ \ \ \ \square=\partial_{\mu}\partial^{\mu} \ .$$ Knowing that $\eta_{\mu\nu}\eta^{\mu\nu}=4$ in $4$D, I wonder if it is possible to write: $$\partial_{\mu}\partial_{\nu}h= \frac{4}{4}\partial_{\mu}\partial_{\nu}h=\frac{1}{4}\eta_{\mu\nu}\eta^{\mu\nu}\partial_{\mu}\partial_{\nu}h=\frac{1}{4}\eta_{\mu\nu}\partial_{\mu}\partial^{\mu}h \ ,$$ so that the two terms cancel out.
