- #1
ramparts
- 45
- 0
Hi all,
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!
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!