Hi all,(adsbygoogle = window.adsbygoogle || []).push({});

I'm following along the derivation of the field equations for f(R) gravity, and there's one step I don't understand entirely. There's just something in the math that's eluding me. So wiki has a pretty good explanation:

http://en.wikipedia.org/wiki/F(R)_gravity#Derivation_of_field_equations

So there's a step where you have:

[tex]\delta S = \int \frac{1}{2\kappa} \sqrt{-g} \left(\frac{\partial f}{\partial R} (R_{\mu\nu} \delta g^{\mu\nu}+g_{\mu\nu}\Box \delta g^{\mu\nu}-\nabla_\mu \nabla_\nu \delta g^{\mu\nu}) -\frac{1}{2} g_{\mu\nu} \delta g^{\mu\nu} f(R) \right)\, d^4x[/tex]

Now the next important step, the wiki article says, is to integrate the second and third terms by parts to yield:

[tex]\delta S = \int \frac{1}{2\kappa} \sqrt{-g}\delta g^{\mu\nu} \left(\frac{\partial f}{\partial R} R_{\mu\nu}-\frac{1}{2}g_{\mu\nu} f(R)+[g_{\mu\nu}\Box -\nabla_\mu \nabla_\nu] \frac{\partial f}{\partial R} \right)\, \mathrm{d}^4x [/tex]

In other words, integrating by parts should yield:

[tex]\int \sqrt{-g} \left(\frac{\partial f}{\partial R} (g_{\mu\nu}\Box \delta g^{\mu\nu}-\nabla_\mu \nabla_\nu \delta g^{\mu\nu}) \right)\, d^4x = \int \sqrt{-g}\delta g^{\mu\nu} \left([g_{\mu\nu}\Box -\nabla_\mu \nabla_\nu] \frac{\partial f}{\partial R} \right)\, \mathrm{d}^4x[/tex]

And from there getting the usual f(R) field equations is trivial. What I'm confused by is how to integrate by parts to get that. Naïvely I think that the left side should be 0, since the connection has metric compatibility so any covariant derivatives of [tex]g_{\mu \nu}[/tex] should vanish. But apparently they don't, and somehow integrating by parts ends up moving the [tex]\partial f / \partial R[/tex] into the covariant derivatives. Any help here? I'm pretty confused by how the math is supposed to work.

Thanks!

**Physics Forums - The Fusion of Science and Community**

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Deriving f(R) field equations

Loading...

Similar Threads - Deriving field equations | Date |
---|---|

I [itex]f(R)[/itex] gravity field equation derivation mistake? | Jul 4, 2017 |

I Help with derivation of linearized Einstein field equations | Oct 25, 2016 |

Einstein-Cartan Field Equations Derivation | Jul 8, 2015 |

How to derive the field equations from a Gauss-Bonnet Lagran | Dec 5, 2014 |

By what method did Einstein derived his gravitational field equation? | Nov 13, 2014 |

**Physics Forums - The Fusion of Science and Community**