Palatini f(R) gravity and the variation

[shadi_s10]

Hi friends,

going through Palatini gravity, I cannot do the variation for palatini f(R) gravity and get to the famous equation (Tsujikawa dark energy book equation 9.6):
$$R_{\mu\nu} - \frac{1}{2}g_{\mu\nu}R =\frac{\kappa^2 T_{\mu\nu}}{F} - \frac{FR-f}{2F}g_{\mu\nu} + \frac{1}{F}(\nabla_\mu \nabla_\nu F - g_{\mu\nu} \Box F)- \frac{3}{2F^2}(\partial_\mu F\partial_\nu F - \frac{1}{2}g_{\mu\nu} (\nabla F)^2)$$

I tried but it does not work!

 

[bcrowell]

If you want your math to show up correctly, you need to surround it with itex tags, like this: $a^2+b^2=c^2$. Click on the QUOTE button in my post to see how I did that.

 

[shadi_s10]

If you want your math to show up correctly, you need to surround it with itex tags, like this: $a^2+b^2=c^2$. Click on the QUOTE button in my post to see how I did that.

Thanks!

 

[bcrowell]

If you edit your #1, we'll be able to read it.

 

[shadi_s10]

Hi friends,

going through Palatini gravity, I cannot do the variation for palatini f(R) gravity and get to the famous equation (Tsujikawa dark energy book equation 9.6):

R$_{\mu\nu}$-$\frac{1}{2}$ g$_{\mu\nu}$ =
$\frac{\kappa^{2} T_{\mu\nu}}{F}$ - $\frac{F R -f}{2F}$ g$_{\mu\nu}$ + $\frac{1}{F}$ ($\nabla$ $_{\mu}$ $\nabla$ $_{\nu}$ F - g$_{\mu\nu}$ d'lambert F) - $\frac{3}{2 F ^{2}}$ [ $\partial$ $_{\mu}$ F $\partial$$_{\nu}$ F - $\frac{1}{2}$ g$_{\mu\nu}$ ( $\nabla$ F)$^{2}$]


I tried but it does not work!

 

[Mentz114]

$$R_{\mu\nu} - \frac{1}{2}g_{\mu\nu}R =\frac{\kappa^2 T_{\mu\nu}}{F} - \frac{FR-f}{2F}g_{\mu\nu} + \frac{1}{F}(\nabla_\mu \nabla_\nu F - g_{\mu\nu} \Box F)- \frac{3}{2F^2}(\partial_\mu F\partial_\nu F - \frac{1}{2}g_{\mu\nu} (\nabla F)^2)$$

 

[shadi_s10]

$$R_{\mu\nu} - \frac{1}{2}g_{\mu\nu}R =\frac{\kappa^2 T_{\mu\nu}}{F} - \frac{FR-f}{2F}g_{\mu\nu} + \frac{1}{F}(\nabla_\mu \nabla_\nu F - g_{\mu\nu} \Box F)- \frac{3}{2F^2}(\partial_\mu F\partial_\nu F - \frac{1}{2}g_{\mu\nu} (\nabla F)^2)$$

Yes this is the exact equation.
But I do not know how they reach to this by combining
$\nabla_{\lambda}$ ( $\sqrt{-g}$ G g$^{\mu\nu}$)=0

and

F R$_{\mu\nu}$ - $\frac{1}{2}$ f g$_{\mu\nu}$ = $\kappa$ $^{2}$ T $_{\mu\nu}$

!

 

[shadi_s10]

Yes this is the exact equation.
But I do not know how they reach to this by combining
$\nabla_{\lambda}$ ( $\sqrt{-g}$ G g$^{\mu\nu}$)=0

and

F R$_{\mu\nu}$ - $\frac{1}{2}$ f g$_{\mu\nu}$ = $\kappa$ $^{2}$ T $_{\mu\nu}$

!

It is interesting to mention that there are actually two different R in this equation.
The first one on the left hand side is R(g) and the second one in the right hand side is R(T)