Note on linearized “new massive gravity” in arbitrary dimensions

Abstract

By means of a triple master action we deduce here a linearized version of the "New Massive Gravity" (NMG) in arbitrary dimensions. The theory contains a 4th-order and a 2nd-order term in derivatives. The 4th-order term is invariant under a generalized Weyl symmetry. The action is formulated in terms of a traceless $\eta^{\mu\nu}\Omega_{\mu\nu\rho}=0$ mixed symmetry tensor $\Omega_{\mu\nu\rho}=-\Omega_{\mu\rho\nu}$ and corresponds to the massive Fierz-Pauli action with the replacement $e_{\mu\nu}=\p^{\rho}\Omega_{\mu\nu\rho}$. The linearized 3D and 4D NMG theories are recovered via the invertible maps $\Omega_{\mu\nu\rho} = \epsilon_{\nu\rho}^{\quad\beta}h_{\beta\mu} $ and $\Omega_{\mu\nu\rho} = \epsilon_{\nu\rho}^{\quad \gamma\delta}T_{[\gamma\delta]\mu} $ respectively. The properties $h_{\mu\nu}=h_{\nu\mu}$ and $T_{[[\gamma\delta]\mu]}=0$ follow from the traceless restriction. The equations of motion of the linearized NMG theory can be written as zero "curvature" conditions $\p_{\nu}T_{\rho\mu} - \p_{\rho}T_{\nu\mu}=0$ in arbitrary dimensions.