# Linearized variation of $\sqrt{-g}(\nabla^\mu \xi^\nu - \nabla^\nu \xi^\mu)$

The Noether current associated with invariance of the Einstein-Hilbert action, $$S=\int d^dx \sqrt{-g}\,R[g_{\mu\nu}]\,,$$ under infinitesimal diffeomorphisms $\delta_\xi g_{\mu\nu}=\nabla_\mu \xi_\nu + \nabla_\nu \xi_\mu$ is equal on-shell to $j^\mu = \partial_\nu \kappa_\xi^{\mu\nu}$, where $$\kappa_\xi^{\mu\nu}=\sqrt{-g}(\nabla^\mu \xi^\nu - \nabla^\nu \xi^\mu).$$ On the other hand, the Noether current corresponding to the invariance of the action for a linearized perturbation around Minkowski space $h_{\mu\nu}=g_{\mu\nu}-\eta_{\mu\nu}$, $$S=\int d^dx\, h^{\mu\nu} \big(\Box h_{\mu\nu}-\partial_{(\mu}\partial\cdot h_{\nu)}+\partial_\mu\partial_\nu h-\eta_{\mu\nu}\Box h + \eta_{\mu\nu}\partial\cdot\partial\cdot h\big)\,,$$ ($h$ being the trace of $h_{\mu\nu}$) under linearized diffeomorphisms $\delta_{\xi} h_{\mu\nu}=\partial_\mu\xi_\nu-\partial_\nu\xi_\mu$ is $J^\mu = \partial_\nu K^{\mu\nu}_{\xi}$, where $$K^{\mu\nu}_{\xi}=\partial^\nu h{^\mu_{\ \ \rho}}-h^{\rho\nu}\partial_\rho \xi^\mu+\frac{1}{2}h\,\partial^\nu \xi^\mu+\big\{\xi^\nu(\partial^\mu h-\partial\cdot h^\mu)\big\}-[\mu\leftrightarrow\nu]\,.$$ The claim in the literature (see e.g. this paper, eq. (95) and below eq. (101)) is that $K^{\mu\nu}_\xi$ should also be the linearized perturbation of $\kappa^{\mu\nu}_\xi$ around Minkowski background. However, trying to reproduce this result, I appear to be missing the terms in the curly brackets: letting $\delta g_{\mu\nu}=h_{\mu\nu}$ and calculating to first order in $h_{\mu\nu}$, $$\delta \kappa_\xi^{\mu\nu}=\delta(\sqrt{-g}) \partial^\mu \xi^\nu+ \delta g^{\mu\rho}\partial_\rho \xi^\nu + \eta^{\mu\rho}\delta\Gamma^{\nu}_{\rho\sigma}\xi^\sigma \\ =\left(\frac{1}{2}h\right)\partial^{\mu}\xi^\nu+(-h^{\mu\rho})\partial_\rho \xi^{\nu}+\eta^{\mu\rho}\left(\frac{1}{2}(\partial_\rho h^{\nu}_{\ \ \sigma}+\partial_\sigma h_\rho^{\ \ \nu}-\partial^\nu h_{\rho\sigma})\right)\xi^\sigma-[\mu\leftrightarrow\nu]\\ = \partial^\nu h{^\mu_{\ \ \rho}}-h^{\rho\nu}\partial_\rho \xi^\mu+\frac{1}{2}h\,\partial^\nu \xi^\mu-[\mu\leftrightarrow \nu]\,.$$ First, am I missing something in the calculation? (I hope so) If I didn't make any mistake, is this discrepancy related to the ambiguity present in the definition of $k^{\mu\nu}$, due to the fact that any current of the type $$\partial_{\nu}(\kappa^{\mu\nu}+\lambda^{\mu\nu})$$ would still be conserved, provided $\lambda^{\mu\nu}$ be ansitymmetric?