# Linearized gravity and perturbation theory

This is a question regarding a calculation in perturbative GR. We have :

$$g_{\mu\nu} = \eta_{\mu\nu}+h_{\mu\nu}$$

where $$h_{\mu\nu}$$ is a small perturbation around the flat spacetime metric. In linearized theory, we ignore terms which grow as $$h^2$$ and higher.

Can you point me to a reference which provides the Curvature tensor, Ricci tensor and Ricci scalar to cubic powers in h, and not just the linear term?

Expansions around an arbitrary background. You can later put background curvatures to zero to get the expansion around flat space.

Ricci scalar up to $$\mathcal{O}(h^3)$$:

$$R + \epsilon (- h^{\alpha \beta } R_{\alpha \beta } + \nabla_{\beta }\nabla_{\alpha }h^{\alpha \beta } - \nabla_{\beta }\nabla^{\beta }h^{\alpha }{}_{\alpha }) + \epsilon^2 (2 h_{\alpha }{}^{\gamma } h^{\alpha \beta } R_{\beta \gamma } - h^{\alpha \beta } h^{\gamma \lambda } R_{\alpha \gamma \beta \lambda } + h^{\alpha \beta } \nabla_{\beta }\nabla_{\alpha }h^{\gamma }{}_{\gamma } - \tfrac{1}{4} \nabla_{\beta }h^{\gamma }{}_{\gamma } \nabla^{\beta }h^{\alpha }{}_{\alpha } - \nabla_{\alpha }h^{\alpha \beta } \nabla_{\gamma }h_{\beta }{}^{\gamma } + \nabla^{\beta }h^{\alpha }{}_{\alpha } \nabla_{\gamma }h_{\beta }{}^{\gamma } - 2 h^{\alpha \beta } \nabla_{\gamma }\nabla_{\beta }h_{\alpha }{}^{\gamma } + h^{\alpha \beta } \nabla_{\gamma }\nabla^{\gamma }h_{\alpha \beta } - \tfrac{1}{2} \nabla_{\beta }h_{\alpha \gamma } \nabla^{\gamma }h^{\alpha \beta } + \tfrac{3}{4} \nabla_{\gamma }h_{\alpha \beta } \nabla^{\gamma }h^{\alpha \beta }) + \epsilon^3 (-2 h_{\alpha }{}^{\gamma } h^{\alpha \beta } h_{\beta }{}^{\lambda } R_{\gamma \lambda } + h_{\alpha }{}^{\gamma } h^{\alpha \beta } h^{\lambda \sigma} R_{\beta \lambda \gamma \sigma} - \tfrac{3}{4} h^{\alpha \beta } \nabla_{\alpha }h^{\gamma \lambda } \nabla_{\beta }h_{\gamma \lambda } + \tfrac{1}{4} h^{\alpha \beta } \nabla_{\alpha }h^{\gamma }{}_{\gamma } \nabla_{\beta }h^{\lambda }{}_{\lambda } - h^{\alpha \beta } \nabla_{\beta }h^{\lambda }{}_{\lambda } \nabla_{\gamma }h_{\alpha }{}^{\gamma } - h^{\alpha \beta } \nabla_{\beta }h_{\alpha }{}^{\gamma } \nabla_{\gamma }h^{\lambda }{}_{\lambda } - h_{\alpha }{}^{\gamma } h^{\alpha \beta } \nabla_{\gamma }\nabla_{\beta }h^{\lambda }{}_{\lambda } + \tfrac{1}{2} h^{\alpha \beta } \nabla_{\gamma }h^{\lambda }{}_{\lambda } \nabla^{\gamma }h_{\alpha \beta } + h^{\alpha \beta } \nabla_{\gamma }h_{\alpha }{}^{\gamma } \nabla_{\lambda }h_{\beta }{}^{\lambda } + 2 h^{\alpha \beta } \nabla_{\beta }h_{\alpha }{}^{\gamma } \nabla_{\lambda }h_{\gamma }{}^{\lambda } - h^{\alpha \beta } \nabla^{\gamma }h_{\alpha \beta } \nabla_{\lambda }h_{\gamma }{}^{\lambda } + h^{\alpha \beta } h^{\gamma \lambda } \nabla_{\lambda }\nabla_{\beta }h_{\alpha \gamma } - h^{\alpha \beta } h^{\gamma \lambda } \nabla_{\lambda }\nabla_{\gamma }h_{\alpha \beta } + 2 h_{\alpha }{}^{\gamma } h^{\alpha \beta } \nabla_{\lambda }\nabla_{\gamma }h_{\beta }{}^{\lambda } - h_{\alpha }{}^{\gamma } h^{\alpha \beta } \nabla_{\lambda }\nabla^{\lambda }h_{\beta \gamma } + h^{\alpha \beta } \nabla_{\beta }h_{\gamma \lambda } \nabla^{\lambda }h_{\alpha }{}^{\gamma } + \tfrac{1}{2} h^{\alpha \beta } \nabla_{\gamma }h_{\beta \lambda } \nabla^{\lambda }h_{\alpha }{}^{\gamma } - \tfrac{3}{2} h^{\alpha \beta } \nabla_{\lambda }h_{\beta \gamma } \nabla^{\lambda }h_{\alpha }{}^{\gamma })$$

Ricci Tensor up to $$\mathcal{O}(h^3)$$:

$$R_{\mu \nu } + \epsilon (\tfrac{1}{2} h_{\nu }{}^{\alpha } R_{\mu \alpha } + \tfrac{1}{2} h_{\mu }{}^{\alpha } R_{\nu \alpha } - h^{\alpha \beta } R_{\mu \alpha \nu \beta } - \tfrac{1}{2} \nabla_{\alpha }\nabla^{\alpha }h_{\mu \nu } + \tfrac{1}{2} \nabla_{\mu }\nabla_{\alpha }h_{\nu }{}^{\alpha } + \tfrac{1}{2} \nabla_{\nu }\nabla_{\alpha }h_{\mu }{}^{\alpha } - \tfrac{1}{2} \nabla_{\nu }\nabla_{\mu }h^{\alpha }{}_{\alpha })+ \epsilon^2 (- \tfrac{1}{2} h^{\beta \gamma } h_{\nu }{}^{\alpha } R_{\mu \beta \alpha \gamma } + h_{\alpha }{}^{\gamma } h^{\alpha \beta } R_{\mu \beta \nu \gamma } - \tfrac{1}{2} h^{\beta \gamma } h_{\mu }{}^{\alpha } R_{\nu \beta \alpha \gamma } - \tfrac{1}{4} \nabla_{\alpha }h^{\beta }{}_{\beta } \nabla^{\alpha }h_{\mu \nu } + \tfrac{1}{2} \nabla^{\alpha }h_{\mu \nu } \nabla_{\beta }h_{\alpha }{}^{\beta } + \tfrac{1}{2} h^{\alpha \beta } \nabla_{\beta }\nabla_{\alpha }h_{\mu \nu } - \tfrac{1}{2} \nabla_{\alpha }h_{\nu \beta } \nabla^{\beta }h_{\mu }{}^{\alpha } + \tfrac{1}{2} \nabla_{\beta }h_{\nu \alpha } \nabla^{\beta }h_{\mu }{}^{\alpha } + \tfrac{1}{4} \nabla_{\alpha }h^{\beta }{}_{\beta } \nabla_{\mu }h_{\nu }{}^{\alpha } - \tfrac{1}{2} \nabla_{\beta }h_{\alpha }{}^{\beta } \nabla_{\mu }h_{\nu }{}^{\alpha } - \tfrac{1}{2} h^{\alpha \beta } \nabla_{\mu }\nabla_{\beta }h_{\nu \alpha } + \tfrac{1}{4} \nabla_{\mu }h^{\alpha \beta } \nabla_{\nu }h_{\alpha \beta } + \tfrac{1}{4} \nabla_{\alpha }h^{\beta }{}_{\beta } \nabla_{\nu }h_{\mu }{}^{\alpha } - \tfrac{1}{2} \nabla_{\beta }h_{\alpha }{}^{\beta } \nabla_{\nu }h_{\mu }{}^{\alpha } - \tfrac{1}{2} h^{\alpha \beta } \nabla_{\nu }\nabla_{\beta }h_{\mu \alpha } + \tfrac{1}{2} h^{\alpha \beta } \nabla_{\nu }\nabla_{\mu }h_{\alpha \beta }) + \epsilon^3 (\tfrac{1}{2} h_{\beta }{}^{\lambda } h^{\beta \gamma } h_{\nu }{}^{\alpha } R_{\mu \gamma \alpha \lambda } - h_{\alpha }{}^{\gamma } h^{\alpha \beta } h_{\beta }{}^{\lambda } R_{\mu \gamma \nu \lambda } + \tfrac{1}{2} h_{\beta }{}^{\lambda } h^{\beta \gamma } h_{\mu }{}^{\alpha } R_{\nu \gamma \alpha \lambda } + \tfrac{1}{4} h^{\alpha \beta } \nabla_{\alpha }h_{\mu \nu } \nabla_{\beta }h^{\gamma }{}_{\gamma } - \tfrac{1}{2} h^{\alpha \beta } \nabla_{\alpha }h_{\mu }{}^{\gamma } \nabla_{\beta }h_{\nu \gamma } - \tfrac{1}{2} h^{\alpha \beta } \nabla_{\alpha }h_{\mu \nu } \nabla_{\gamma }h_{\beta }{}^{\gamma } + \tfrac{1}{2} h^{\alpha \beta } \nabla_{\alpha }h_{\mu }{}^{\gamma } \nabla_{\gamma }h_{\nu \beta } - \tfrac{1}{2} h_{\alpha }{}^{\gamma } h^{\alpha \beta } \nabla_{\gamma }\nabla_{\beta }h_{\mu \nu } + \tfrac{1}{2} h^{\alpha \beta } \nabla_{\beta }h_{\nu \gamma } \nabla^{\gamma }h_{\mu \alpha } - \tfrac{1}{2} h^{\alpha \beta } \nabla_{\gamma }h_{\nu \beta } \nabla^{\gamma }h_{\mu \alpha } - \tfrac{1}{2} h^{\alpha \beta } \nabla_{\beta }h_{\alpha \gamma } \nabla^{\gamma }h_{\mu \nu } + \tfrac{1}{4} h^{\alpha \beta } \nabla_{\gamma }h_{\alpha \beta } \nabla^{\gamma }h_{\mu \nu } - \tfrac{1}{4} h^{\alpha \beta } \nabla_{\beta }h^{\gamma }{}_{\gamma } \nabla_{\mu }h_{\nu \alpha } + \tfrac{1}{2} h^{\alpha \beta } \nabla_{\gamma }h_{\beta }{}^{\gamma } \nabla_{\mu }h_{\nu \alpha } + \tfrac{1}{2} h^{\alpha \beta } \nabla_{\beta }h_{\alpha \gamma } \nabla_{\mu }h_{\nu }{}^{\gamma } - \tfrac{1}{4} h^{\alpha \beta } \nabla_{\gamma }h_{\alpha \beta } \nabla_{\mu }h_{\nu }{}^{\gamma } + \tfrac{1}{2} h_{\alpha }{}^{\gamma } h^{\alpha \beta } \nabla_{\mu }\nabla_{\gamma }h_{\nu \beta } - \tfrac{1}{2} h^{\alpha \beta } \nabla_{\mu }h_{\alpha }{}^{\gamma } \nabla_{\nu }h_{\beta \gamma } - \tfrac{1}{4} h^{\alpha \beta } \nabla_{\beta }h^{\gamma }{}_{\gamma } \nabla_{\nu }h_{\mu \alpha } + \tfrac{1}{2} h^{\alpha \beta } \nabla_{\gamma }h_{\beta }{}^{\gamma } \nabla_{\nu }h_{\mu \alpha } + \tfrac{1}{2} h^{\alpha \beta } \nabla_{\beta }h_{\alpha \gamma } \nabla_{\nu }h_{\mu }{}^{\gamma } - \tfrac{1}{4} h^{\alpha \beta } \nabla_{\gamma }h_{\alpha \beta } \nabla_{\nu }h_{\mu }{}^{\gamma } + \tfrac{1}{2} h_{\alpha }{}^{\gamma } h^{\alpha \beta } \nabla_{\nu }\nabla_{\gamma }h_{\mu \beta } - \tfrac{1}{2} h_{\alpha }{}^{\gamma } h^{\alpha \beta } \nabla_{\nu }\nabla_{\mu }h_{\beta \gamma })$$

Riemann tensor up to $$\mathcal{O}(h^3)$$:

$$R_{\mu \nu \rho \sigma } + \epsilon (\tfrac{1}{2} h_{\sigma }{}^{\alpha } R_{\mu \alpha \nu \rho } - \tfrac{1}{2} h_{\rho }{}^{\alpha } R_{\mu \alpha \nu \sigma } + \tfrac{1}{2} h_{\sigma }{}^{\alpha } R_{\mu \nu \rho \alpha } - \tfrac{1}{2} h_{\rho }{}^{\alpha } R_{\mu \nu \sigma \alpha } - \tfrac{1}{2} h_{\sigma }{}^{\alpha } R_{\mu \rho \nu \alpha } - \tfrac{1}{2} h_{\nu }{}^{\alpha } R_{\mu \rho \sigma \alpha } + \tfrac{1}{2} h_{\rho }{}^{\alpha } R_{\mu \sigma \nu \alpha } + \tfrac{1}{2} h_{\nu }{}^{\alpha } R_{\mu \sigma \rho \alpha } + \tfrac{1}{2} h_{\mu }{}^{\alpha } R_{\nu \rho \sigma \alpha } - \tfrac{1}{2} h_{\mu }{}^{\alpha } R_{\nu \sigma \rho \alpha } - \tfrac{1}{2} \nabla_{\rho }\nabla_{\mu }h_{\nu \sigma } + \tfrac{1}{2} \nabla_{\rho }\nabla_{\nu }h_{\mu \sigma } + \tfrac{1}{2} \nabla_{\sigma }\nabla_{\mu }h_{\nu \rho } - \tfrac{1}{2} \nabla_{\sigma }\nabla_{\nu }h_{\mu \rho }) + \epsilon^2 (- \tfrac{1}{4} \nabla_{\alpha }h_{\nu \sigma } \nabla^{\alpha }h_{\mu \rho } + \tfrac{1}{4} \nabla_{\alpha }h_{\nu \rho } \nabla^{\alpha }h_{\mu \sigma } + \tfrac{1}{4} \nabla^{\alpha }h_{\nu \sigma } \nabla_{\mu }h_{\rho \alpha } - \tfrac{1}{4} \nabla^{\alpha }h_{\nu \rho } \nabla_{\mu }h_{\sigma \alpha } - \tfrac{1}{4} \nabla^{\alpha }h_{\mu \sigma } \nabla_{\nu }h_{\rho \alpha } + \tfrac{1}{4} \nabla_{\mu }h_{\sigma \alpha } \nabla_{\nu }h_{\rho }{}^{\alpha } + \tfrac{1}{4} \nabla^{\alpha }h_{\mu \rho } \nabla_{\nu }h_{\sigma \alpha } - \tfrac{1}{4} \nabla_{\mu }h_{\rho }{}^{\alpha } \nabla_{\nu }h_{\sigma \alpha } + \tfrac{1}{4} \nabla_{\alpha }h_{\nu \sigma } \nabla_{\rho }h_{\mu }{}^{\alpha } - \tfrac{1}{4} \nabla_{\nu }h_{\sigma \alpha } \nabla_{\rho }h_{\mu }{}^{\alpha } - \tfrac{1}{4} \nabla^{\alpha }h_{\mu \sigma } \nabla_{\rho }h_{\nu \alpha } + \tfrac{1}{4} \nabla_{\mu }h_{\sigma \alpha } \nabla_{\rho }h_{\nu }{}^{\alpha } - \tfrac{1}{4} \nabla_{\alpha }h_{\nu \rho } \nabla_{\sigma }h_{\mu }{}^{\alpha } + \tfrac{1}{4} \nabla_{\nu }h_{\rho \alpha } \nabla_{\sigma }h_{\mu }{}^{\alpha } + \tfrac{1}{4} \nabla_{\rho }h_{\nu \alpha } \nabla_{\sigma }h_{\mu }{}^{\alpha } + \tfrac{1}{4} \nabla^{\alpha }h_{\mu \rho } \nabla_{\sigma }h_{\nu \alpha } - \tfrac{1}{4} \nabla_{\rho }h_{\mu }{}^{\alpha } \nabla_{\sigma }h_{\nu \alpha } - \tfrac{1}{4} \nabla_{\mu }h_{\rho \alpha } \nabla_{\sigma }h_{\nu }{}^{\alpha }) + \epsilon^3 (- \tfrac{1}{4} h^{\alpha \beta } \nabla_{\alpha }h_{\mu \sigma } \nabla_{\beta }h_{\nu \rho } + \tfrac{1}{4} h^{\alpha \beta } \nabla_{\alpha }h_{\mu \rho } \nabla_{\beta }h_{\nu \sigma } - \tfrac{1}{4} h^{\alpha \beta } \nabla_{\alpha }h_{\nu \sigma } \nabla_{\mu }h_{\rho \beta } + \tfrac{1}{4} h^{\alpha \beta } \nabla_{\alpha }h_{\nu \rho } \nabla_{\mu }h_{\sigma \beta } - \tfrac{1}{4} h^{\alpha \beta } \nabla_{\mu }h_{\sigma \beta } \nabla_{\nu }h_{\rho \alpha } + \tfrac{1}{4} h^{\alpha \beta } \nabla_{\alpha }h_{\mu \sigma } \nabla_{\nu }h_{\rho \beta } - \tfrac{1}{4} h^{\alpha \beta } \nabla_{\alpha }h_{\mu \rho } \nabla_{\nu }h_{\sigma \beta } + \tfrac{1}{4} h^{\alpha \beta } \nabla_{\mu }h_{\rho \alpha } \nabla_{\nu }h_{\sigma \beta } - \tfrac{1}{4} h^{\alpha \beta } \nabla_{\beta }h_{\nu \sigma } \nabla_{\rho }h_{\mu \alpha } + \tfrac{1}{4} h^{\alpha \beta } \nabla_{\nu }h_{\sigma \beta } \nabla_{\rho }h_{\mu \alpha } - \tfrac{1}{4} h^{\alpha \beta } \nabla_{\mu }h_{\sigma \beta } \nabla_{\rho }h_{\nu \alpha } + \tfrac{1}{4} h^{\alpha \beta } \nabla_{\alpha }h_{\mu \sigma } \nabla_{\rho }h_{\nu \beta } + \tfrac{1}{4} h^{\alpha \beta } \nabla_{\beta }h_{\nu \rho } \nabla_{\sigma }h_{\mu \alpha } - \tfrac{1}{4} h^{\alpha \beta } \nabla_{\nu }h_{\rho \beta } \nabla_{\sigma }h_{\mu \alpha } - \tfrac{1}{4} h^{\alpha \beta } \nabla_{\rho }h_{\nu \beta } \nabla_{\sigma }h_{\mu \alpha } + \tfrac{1}{4} h^{\alpha \beta } \nabla_{\mu }h_{\rho \beta } \nabla_{\sigma }h_{\nu \alpha } - \tfrac{1}{4} h^{\alpha \beta } \nabla_{\alpha }h_{\mu \rho } \nabla_{\sigma }h_{\nu \beta } + \tfrac{1}{4} h^{\alpha \beta } \nabla_{\rho }h_{\mu \alpha } \nabla_{\sigma }h_{\nu \beta })$$

Computations performed using xAct. You might also find perturbative expansions of curvatures in some PhD theses, though I don't remember coming across one that has them up to the cubic level.

• Did the software also write the LaTex code for you? In the linear approximation, one should use the flat spacetime derivative, no longer the covariant/contravariant one. – DanielC Jun 24 '19 at 11:52
• @DanielC There is a command to typeset expressions in TeX format. The expressions above are for an arbitrary background, as mentioned in the beginning. – Avantgarde Jun 24 '19 at 12:27
• When I do the expansion of $R$, I don't get the term $h^{\alpha\beta}h^{\gamma\lambda} R_{\alpha\gamma\beta\lambda}$. Is it easy to see, what I do wrong? DefManifold[M, 4, {\[Alpha], \[Beta], \[Gamma], \[Lambda], \[Sigma], \ \[Rho]}] DefMetric[-1, metric[-\[Alpha], -\[Beta]], cd, {";", "\[Del]"}, PrintAs -> "g"] DefMetricPerturbation[metric, pert, \[Epsilon]] PrintAs[pert] ^= "h" tmp = (Perturbation[RicciScalarcd[], 1] + Perturbation[RicciScalarcd[], 2] + Perturbation[RicciScalarcd[], 3]) // ExpandPerturbation tmp /. pert[LI[n_], __] :> 0 /; n > 1//ToCanonical – p6majo Jun 25 '19 at 19:51
• Thank you for the xAct reference. Very powerful!!! – p6majo Jun 25 '19 at 20:02
• @p6majo You're welcome. I can't read your comment because it's not in mathjax format. Anyway, provided you've coded correctly, you shouldn't be worried. In order to recover 'missing curvatures', you need to commute some covariant derivatives. That's all. – Avantgarde Jun 25 '19 at 20:35