Riemann curvature tensor in first order perturbation theory as a Lie derivative of Riemann curvature tensor in zero order I am having a difficulty solving my homework so I was hoping
I could get some help, so here it is. 
It is about gravitational waves and first order gravitational perturbation theory,
I have to prove that under the gauge transformation: 
$$h_{ab} \rightarrow h_{ab}+ \nabla{a} \xi_b + \nabla_{b} \xi_a$$
the curvature tensor: 
$${R^{i(1)}}_{klm}=\frac{1}{2} (\nabla_{l} \nabla_{m}{h^i}_k +\nabla_{l} \nabla_{k}{h^i}_m-\nabla_{l} \nabla^{i}h_{km}-\nabla_{m} \nabla_{l}{h^i}_k-\nabla_{m} \nabla_{k}{h^i}_l+\nabla_{m} \nabla^{i}h_{kl})$$
changes by:
$$\delta {R^{(1)}}_{mnrs}=\xi^t \nabla_{t} {R^{(0)}}_{mnrs}+{R^{(0)}}_{tnrs} \nabla_{m} \xi_{t} - {R^{(0)}}_{tmrs} \nabla_{n} \xi_{t}+{R^{(0)}}_{mntr} \nabla_{s} \xi_{t}-{R^{(0)}}_{mnts} \nabla_{r} \xi_{t}$$
and hence is not gauge invariant.
$h_{ab}$ is the metric perturbation of the first order which changes because of infinitesimal coordinate transformation, $x^{a} \rightarrow x^{a}+\xi^{a}$.
This is given as exercise 9.6. in T.Padmanabhan, Gravitation - Foundations and Frontiers.
I have tried all sorts of manipulations with covariant derivatives, but it all resulted in a bunch of asymetrical expressions with no connection with the solution which is the Lie derivative of Riemann curvature tensor in zero order.
 A: In general relativity, a diffeomorphism, i.e. gauge transformation, is infinitesimally represented by a shift on the manifold by a vector field, wherein
$$X_a \to X_a + \xi_a$$
By definition, the metric tensor changes by a Lie derivative,
$$g_{ab}\to g_{ab}+\mathcal{L}_\xi g_{ab} = g_{ab} + 2\nabla_{(a}\xi_{b)}$$
Hence it induces a non-physical linear perturbation, which we denote $h_{ab}$. The connection coefficients, or 'Christoffel' symbols experience a variation,
$$\delta \Gamma^{a}_{bc} = \frac{1}{2}\left(\nabla_c h^{a}_{b}+\nabla_{b}h^{ac}-\nabla^{a}h_{bc} \right)$$
The variation of the Riemann tensor can also be expressed in terms of covariant derivatives with respect to the unperturbed background metric,
$$\delta R^{\rho}_{\sigma \mu \nu} = \nabla_\mu (\delta \Gamma^{\rho}_{\nu \sigma}) - \nabla_\nu (\delta \Gamma^{\rho}_{\mu \sigma})$$
The variation in terms of $h_{ab}$ is straightforwardly computed by inserting the Christoffel variations,
$$\delta R^{\rho}_{\sigma \mu \nu} = \frac{1}{2} \left[\nabla_\mu\nabla_\sigma h^{\rho}_{\nu} + \nabla_\mu \nabla_\nu h^{\rho}_\sigma - \nabla_\mu \nabla^\rho h_{\nu \sigma}   -\nabla_\nu \nabla_\sigma h^{\rho}_\mu - \nabla_\nu \nabla_\mu h^{\rho}_\sigma + \nabla_\nu \nabla^{\rho} h_{\mu\sigma}\right]$$
To combine terms, one can use the Riemann identity to exchange covariant derivatives, but at the price of introducing the original Riemann tensor. Finally, to obtain the variation in terms of the original vector field, simply input $h_{ab}=2\nabla_{(a}\xi_{b)}$.
