For linearized gravity $G_{\mu\nu} = 8 \pi G T_{\mu\nu} $, for $$ G_{\mu \nu} = \frac{1}{2} \left[ \partial_\sigma \partial_\nu {h^\sigma}_\mu + \partial_\sigma \partial_\mu {h^\sigma}_\nu -\partial_\mu \partial_\nu h - \square h_{\mu\nu} - \eta_{\mu \nu} \partial_\rho \partial_\lambda h^{\rho \lambda} + \eta_{\mu \nu} \square h \right] $$ We let $T_{\mu \nu}$ be describing dust. By deriving the transformation properties of $h_{\mu \nu}$, we want to show that this describes a Lorentz-invariant theory of a symmetric rank-2 tensor field (h) on flat spacetime.
My attempt: I thought the only straight forward way to do this was by showing that the Lagrangian density of the theory is lorentz invariant. According to Carroll the Lagrangian density is $$ \mathcal{L} = \frac{1}{2} \left[ (\partial_\mu {h^{\mu \nu }} )(\partial_\nu h) - (\partial_\mu {h^{\rho \sigma}} )(\partial_\rho {h^{\mu }}_{ \sigma} ) + \frac{1}{2} \eta^{\mu \nu} (\partial_\mu {h^{\rho \sigma }} )(\partial_\nu {h}_{\rho \sigma } ) - \frac{1}{2} \eta^{\mu \nu}(\partial_\mu h )(\partial_\nu h ) \right] $$ However, every attempt I made failed which makes me wonder if I didn't understand the question properly or if I am plain stupid. Consider the last term for instance: $$ \eta^{\mu \nu}(\partial_\mu h )(\partial_\nu h ) = (\partial_\mu h )(\partial^\mu h ) $$ We must have $(\partial_\mu h )(\partial^\mu h ) \to (\partial_\mu h )(\partial^\mu h )$, otherwise $\mathcal{L}$ is not lorentz invariant. However, when I try to Lorentz transform one of the factors: $$ \partial_\mu h \to \eta^{\rho \sigma} {\Lambda_\mu}^\nu {\Lambda^\alpha}_\rho {\Lambda^\beta}_\sigma \partial_\nu h_{\alpha \beta}, $$ which gets me nowhere. I get a similar mess for all the other terms. Any hints to help me make some progress or spotting mistakes would be appreciated.
