- #1
balaustrada
- 1
- 0
- Homework Statement
- I need to compute the variation of two covariant derivatives applied to a tensor, and I'm not sure how it should be computed.
- Relevant Equations
- $$ \delta ( \nabla_\rho \nabla_\nu \left[ F'(G) R_{\mu}^{\hphantom{\mu} \rho} \right])$$
I'm working with modfied gravity models and I need to consider the perturbation of field equations. I have problems with the term were I have two covariant derivatives, I'm not sure if I'm doing it right.
I have:
$$\delta(\nabla_\rho \nabla_\nu \left[F'(G)R_{\mu}^{\hphantom{\mu} \rho}\right])$$
where F(G) a function of the Gauss-Bonet comination.
What I consider, is the following:
$$ \delta (\nabla_\rho V^\mu) = \delta ( \partial_\rho V^\mu + \Gamma^\mu_{\rho \xi} V^\xi) = \partial_\rho \delta(V^\mu) + \Gamma^\mu_{\rho \xi} \delta(V^\xi) + \delta (\Gamma^{\mu}_{\rho \xi}) V^\xi \\
= \nabla_\rho \delta(V^{\mu}) + \delta(\Gamma^\mu_{\rho \xi} V^\xi) $$
Applied with two covariant derivatives gives:
$$\delta(\nabla_\rho \nabla_\nu \left[ F'(G) R_{\mu}^{\hphantom{\mu} \rho} \right]) = \nabla_\rho \delta(\nabla_\nu \left[ F'(G) R_{\mu}^{\hphantom{\mu} \rho} \right]) - \delta(\Gamma^\xi_{\rho \nu}) \left(\nabla_\xi \left[ F'(G) R_{\mu}^{\hphantom{\mu} \rho} \right] \right) - \delta(\Gamma^\xi_{\rho \mu})\left(\nabla_\nu \left[ F'(G) R_{\xi}^{\hphantom{\mu} \rho} \right] \right)
+ \delta(\Gamma^\rho_{\nu \xi})\left(\nabla_\nu \left[ F'(G) R_{\mu}^{\hphantom{\mu} \xi} \right] \right)$$
and finally:
$$ \delta(\nabla_\rho \nabla_\nu \left[F'(G)R_{\mu}^{\hphantom{\mu} \rho}\right]) = \nabla_\rho \left[ \nabla_\nu \delta( \left[ F'(G) R_{\mu}^{\hphantom{\mu} \rho} \right] ) - \delta(\Gamma^\xi_{\nu \mu}) \left[F'(G)R_{\xi}^{\hphantom{\xi} \rho} \right] + \delta(\Gamma^\rho_{\nu \xi}) \left[F'(G) R_{\mu}^{\hphantom{\xi} \xi} \right] \right] \\
- \delta(\Gamma^\xi_{\rho \nu}) \left(\nabla_\xi \left[ F'(G) R_{\mu}^{\hphantom{\mu} \rho} \right] \right) - \delta(\Gamma^\xi_{\rho \mu})\left(\nabla_\nu \left[ F'(G) R_{\xi}^{\hphantom{\mu} \rho} \right] \right) + \delta(\Gamma^\rho_{\nu \xi})\left(\nabla_\nu \left[ F'(G) R_{\mu}^{\hphantom{\mu} \xi} \right] \right)$$
Is this reasoning true?
I have:
$$\delta(\nabla_\rho \nabla_\nu \left[F'(G)R_{\mu}^{\hphantom{\mu} \rho}\right])$$
where F(G) a function of the Gauss-Bonet comination.
What I consider, is the following:
$$ \delta (\nabla_\rho V^\mu) = \delta ( \partial_\rho V^\mu + \Gamma^\mu_{\rho \xi} V^\xi) = \partial_\rho \delta(V^\mu) + \Gamma^\mu_{\rho \xi} \delta(V^\xi) + \delta (\Gamma^{\mu}_{\rho \xi}) V^\xi \\
= \nabla_\rho \delta(V^{\mu}) + \delta(\Gamma^\mu_{\rho \xi} V^\xi) $$
Applied with two covariant derivatives gives:
$$\delta(\nabla_\rho \nabla_\nu \left[ F'(G) R_{\mu}^{\hphantom{\mu} \rho} \right]) = \nabla_\rho \delta(\nabla_\nu \left[ F'(G) R_{\mu}^{\hphantom{\mu} \rho} \right]) - \delta(\Gamma^\xi_{\rho \nu}) \left(\nabla_\xi \left[ F'(G) R_{\mu}^{\hphantom{\mu} \rho} \right] \right) - \delta(\Gamma^\xi_{\rho \mu})\left(\nabla_\nu \left[ F'(G) R_{\xi}^{\hphantom{\mu} \rho} \right] \right)
+ \delta(\Gamma^\rho_{\nu \xi})\left(\nabla_\nu \left[ F'(G) R_{\mu}^{\hphantom{\mu} \xi} \right] \right)$$
and finally:
$$ \delta(\nabla_\rho \nabla_\nu \left[F'(G)R_{\mu}^{\hphantom{\mu} \rho}\right]) = \nabla_\rho \left[ \nabla_\nu \delta( \left[ F'(G) R_{\mu}^{\hphantom{\mu} \rho} \right] ) - \delta(\Gamma^\xi_{\nu \mu}) \left[F'(G)R_{\xi}^{\hphantom{\xi} \rho} \right] + \delta(\Gamma^\rho_{\nu \xi}) \left[F'(G) R_{\mu}^{\hphantom{\xi} \xi} \right] \right] \\
- \delta(\Gamma^\xi_{\rho \nu}) \left(\nabla_\xi \left[ F'(G) R_{\mu}^{\hphantom{\mu} \rho} \right] \right) - \delta(\Gamma^\xi_{\rho \mu})\left(\nabla_\nu \left[ F'(G) R_{\xi}^{\hphantom{\mu} \rho} \right] \right) + \delta(\Gamma^\rho_{\nu \xi})\left(\nabla_\nu \left[ F'(G) R_{\mu}^{\hphantom{\mu} \xi} \right] \right)$$
Is this reasoning true?