Is there a Lorentz invariant approximation to General Relativity? Since General Relativity is the most accurate description of gravity is there any possible way to derive a Lorentz invariant theory from: $$R_{\mu\nu}-\frac{1}{2} g_{\mu\nu}R+\Lambda g_{\mu\nu}=kT_{\mu\nu}$$
Assuming the metric is in the form: $g_{\mu\nu}=\eta_{\mu\nu}+h_{\mu\nu}$ where $\eta_{\mu\nu}$ is the flat space time metric and $h_{\mu\nu}$ is a disturbance in the metric. 
If there is a derivation for a Lorentz invariant theory, what would each component of the metric correspond to?
Thanks
 A: If we take a flat background and perturb around it, $g_{\mu\nu}=\eta_{\mu\nu} + h_{\mu\nu}$, the Riemann tensor takes the form,
$$R_{\alpha\beta\gamma\delta} = \frac12 (h_{\alpha\delta,\beta\gamma}+ h_{\beta\gamma,\alpha\beta} - h_{\alpha\gamma,\beta\delta}-h_{\beta\delta,\alpha\gamma}),$$
where commas denote covariant derivatives, which are reduced to partial derivatives, since they are taken with respect to the background in perturbation theory, and in this case it is just $\eta_{\mu\nu}$.
Noting that $g^{\mu\nu}=\eta^{\mu\nu}-h^{\mu\nu}$ (a good exercise to show this using the properties you know about a metric), the Einstein field equations read,
$$\partial^\alpha\partial_\nu {h}_{\mu\alpha} + \partial_\mu \partial^\alpha h_{\nu\alpha} - \partial^\alpha\partial_\alpha h_{\mu\nu} - \partial_\mu\partial_\nu h^{\alpha\beta}h_{\alpha\beta}-\eta_{\mu\nu}(\partial^\alpha\partial^\beta h_{\alpha\beta} - \partial^\alpha\partial_\alpha h^{\gamma\delta}h_{\gamma\delta}) = 16\pi T_{\mu\nu}.$$
We have yet to choose a gauge, and the choice often depends on the problem at hand. It can at times reduce it to simply the wave equation, which one can solve with Green's functions.
Knowing the form of the curvature tensor, you can compute the action $S \sim \int \mathrm{d}^dx \sqrt{|g|}R$ and find a Lorentz invariant theory, which can in fact be quantised as well, like with other field theories, though it leads to a non-renormalisable field theory.
