I need to calculate δR: R is Ricci scalar

sourena

I need to calculate δR: R is Ricci scalar

nicksauce

This should be shown in most GR textbooks...

sourena

Thank you for your reply. I know the answer of calculation but I couldn't derive it.
This is not a homework.

JustinLevy

This should help:
http://en.wikipedia.org/wiki/Einstei...e_Ricci_scalar

sourena

Thank you for your time and care, but I need to obtain this:
δR=Rμσ δg+gμσ δg^μσ -∇μ ∇σ δg^μσ
in f(R) gravity:
http://en.wikipedia.org/wiki/F(R)_gravity

arkajad

Do you have any problem with these equations:

sourena

No, I don't have problem with these equations, but I have problem to calculate this equation from them:
δR=R_ab δg^ab+g_ab δg^ab -∇a ∇b δg^ab

arkajad

First step:
The term [tex]\nabla_\sigma \left( g^{\mu\nu}\delta\Gamma^{\sigma}_{\mu\nu}-g^{\mu\sigma}\delta\Gamma^{\rho}_{\rho_\mu}\right)[/tex]
is of the form [tex]\nabla_\sigma A^\sigma[/tex].
But [tex]\nabla_\sigma A^\sigma=|\det g]^{-\frac12}\partial_\sigma (A^\sigma |\det g|^{\frac12})[/tex]
for any vector field [tex]A^\sigma[/tex]. This is handy.

P.S. For some reason my display is not displaying correctly one character in your first term on the right. So, I do not know what is exactly the formula you would like to have.

JustinLevy

Alright, I tried to work it out, but it looks like I got a sign error somewhere. I wrote everything out in more detail than was probably necessary, so where this occurs can stand out.

Okay, so we at least have a starting point we can agree on:
[tex]\delta R = R_{ab} \delta g^{ab} + g^{ab} \delta R_{ab}[/tex]
[tex]\delta R_{ab} &= [\nabla_c \delta \Gamma^c_{ab} - \nabla_b \delta \Gamma^c_{c a}][/tex]

As in wikipedia noting that [tex]\delta \Gamma^\lambda_{\mu\nu}\,[/tex] is actually the difference of two connections, it should transform as a tensor. Therefore, it can be written as
[tex]\delta \Gamma^\lambda_{\mu\nu}=\frac{1}{2}g^{\lambda d}\left(\nabla_\mu\delta g_{d\nu}+\nabla_\nu\delta g_{d\mu}-\nabla_d\delta g_{\mu\nu} \right)[/tex]

and substituting in the equation, after playing with it in excruciating detail one finds:
[tex]\begin{align*}
\delta R_{ab} &= [\nabla_c \delta \Gamma^c_{ab} - \nabla_b \delta \Gamma^c_{c a}] \\
&= [\nabla_c \frac{1}{2}g^{c d}\left(\nabla_a\delta g_{db}+\nabla_b\delta g_{da}-\nabla_d\delta g_{ab} \right)
-\nabla_b \frac{1}{2}g^{c d}\left(\nabla_c\delta g_{da}+\nabla_a\delta g_{dc}-\nabla_d\delta g_{ca} \right)] \\
&= \frac{1}{2}g^{c d}[\left(\nabla_c\nabla_a\delta g_{db}+\nabla_c\nabla_b\delta g_{da}-\nabla_c\nabla_d\delta g_{ab} \right)
-\left(\nabla_b\nabla_c\delta g_{da}+\nabla_b\nabla_a\delta g_{dc}-\nabla_b\nabla_d\delta g_{ca} \right)]
\end{align*}[/tex]
swap the derivative order on the fourth term
http://en.wikipedia.org/wiki/Covaria...ative#Examples
[tex]\nabla_b\nabla_c\delta g_{da} = \nabla_c\nabla_b\delta g_{da} + R^{e}{}_{dbc} \delta g_{ea} + R^{e}{}_{abc} \delta g_{de}[/tex]
[tex]\begin{align*}
g^{ab}\delta R_{ab} &= g^{ab} \frac{1}{2}g^{c d}[\left(\nabla_c\nabla_a\delta g_{db}-\nabla_c\nabla_d\delta g_{ab} \right)
-\left(R^{e}{}_{dbc} \delta g_{ea} + R^{e}{}_{abc} \delta g_{de} + \nabla_b\nabla_a\delta g_{dc}-\nabla_b\nabla_d\delta g_{ca} \right)] \\
&= \frac{1}{2}g^{c d}[\left(\nabla_c\nabla^b \delta g_{db}-\nabla_c\nabla_d g^{ab} \delta g_{ab} \right)
-\left(R^{e}{}_d{}^a{}_c \delta g_{ea} + R^{eb}{}_{bc} \delta g_{de} + \nabla_b\nabla^b \delta g_{dc}-\nabla^a \nabla_d\delta g_{ca} \right)] \\

&= \frac{1}{2}[\left(\nabla^d\nabla^b \delta g_{db}-\nabla^d\nabla_d g^{ab} \delta g_{ab} \right)
-\left(R^{eca}{}_c \delta g_{ea} + R^{eb}{}_b{}^d \delta g_{de} + \nabla_b\nabla^b g^{c d} \delta g_{cd} -\nabla^a \nabla^c\delta g_{ca} \right)] \\

&= \frac{1}{2}[\left(\nabla^d\nabla^b \delta g_{db}-\nabla^d\nabla_d g^{ab} \delta g_{ab} \right)
-\left(R^{ea}\delta g_{ea} - R^{de} \delta g_{de} + \nabla_b\nabla^b g^{c d} \delta g_{cd} -\nabla^a \nabla^c\delta g_{ca} \right)] \\

&= \nabla^a\nabla^b \delta g_{ab} - g^{ab} \nabla^c\nabla_c \delta g_{ab}
\end{align*}
[/tex]

Hmm...
maybe there isn't an error. Does
[itex]\delta g_{ab} = - \delta g^{ab}[/itex] ?
I'm too tired to check right now.

arkajad

Not exactly.

Use:

[tex]0=\delta ( \delta^a_b) =\delta (g^{ac}g_{cb})= ....[/tex]

Now use the Leibniz rule, calculate what you need.

JustinLevy

Thanks.

Alright, so
[tex]0=\delta ( \delta^a_b) =\delta (g^{ac}g_{cb})= g_{cb} \delta g^{ac} + g^{ac} \delta g_{cb}[/tex]
[tex]g_{cb} \delta g^{ac} = - g^{ac} \delta g_{cb}[/tex]

Thus manipulating the previous posts result gives
[tex]\begin{align*}
g^{ab}\delta R_{ab}
&= \nabla^a\nabla^b \delta g_{ab} - g^{ab} \nabla^c\nabla_c \delta g_{ab} \\
&= \nabla^a\nabla_c g^{bc} \delta g_{ab} - \nabla^c\nabla_c g^{ab} \delta g_{ab} \\
&= - \nabla^a\nabla_c g_{ab} \delta g^{bc} + \nabla^c\nabla_c g_{ab} \delta g^{ab} \\
\end{align*}
[/tex]
which is what sourena wanted to find:
[tex] g^{ab}\delta R_{ab} = g_{ab} \nabla^c\nabla_c \delta g^{ab} - \nabla_a\nabla_b \delta g^{ab}[/tex]

...

arkajad,
I assume your post #8 hint leads to the result faster, but I'm not sure how to apply that. Once we've converted from covariant to ordinary coordinate derivative, how do we recognize the result as terms with two covariant derivatives?

arkajad

The trick with ordinary derivatives is useful when you calculate under the integral. There, in variational calculus, when you assume that variations vanish at the boundary (usually at infinity), you need a true divergence of a vector field, and not some "covariant one". You discard such terms converting volume integral into surface integrals using ordinary rules of the calculus.

sourena

Dear JustinLevy
Sorry for being late to answer your posts. I value your hard work to obtain this expression a great deal. Thank you very much for your time and care.

sourena

Dear arkajad
I wanted to calculate this:
[tex] g^{ab}\delta R_{ab} = g_{ab} \nabla^c\nabla_c \delta g^{ab} - \nabla_a\nabla_b \delta g^{ab}[/tex]
or
[tex]\delta R = R_{ab} \delta g^{ab} + g_{ab} \nabla^c\nabla_c \delta g^{ab} - \nabla_a\nabla_b \delta g^{ab}[/tex]

haushofer

Do you know how to derive

[tex]
\nabla_{\mu}\delta\Gamma^{\lambda}_{\nu\rho} = \frac{1}{2}g^{\lambda\alpha}[\nabla_{\mu}\nabla_{\nu}\delta g_{\rho\alpha} + \nabla_{\mu}\nabla_{\rho}\delta g_{\nu\alpha} - \nabla_{\mu}\nabla_{\alpha}\delta g_{\nu\rho}]
[/tex]
?
This can be shown directly by varying the connection, but you can also guess the form of it; varying the connection gives a tensor, and the partial derivatives in it can only become covariant ones (convince yourself if you're not!). You can plug this in the Palatini equation, which gives

[tex]
\delta R_{\mu\nu\rho}^{\ \ \ \ \lambda} = \frac{1}{2}g^{\lambda\alpha}[\nabla_{\mu}\nabla_{\nu}\delta g_{\rho\alpha} + \nabla_{\mu\rho}\delta g_{\nu\alpha}-\nabla_{\mu}\nabla_{\alpha}\delta g_{\nu\rho}]
- \frac{1}{2}g^{\lambda\alpha}[\nabla_{\nu}\nabla_{\mu}\delta g_{\rho\alpha}+\nabla_{\nu}\nabla_{\rho}\delta g_{\mu\alpha} - \nabla_{\nu}\nabla_{\alpha}\delta g_{\mu\rho}]
[/tex]

Does this help?

haushofer

Do you know how to derive

[tex]
\nabla_{\mu}\delta\Gamma^{\lambda}_{\nu\rho} = \frac{1}{2}g^{\lambda\alpha}[\nabla_{\mu}\na_{\nu}\delta g_{\rho\alpha} + \nabla_{\mu}\na_{\rho}\delta g_{\nu\alpha} - \nabla_{\mu}\na_{\alpha}\delta g_{\nu\rho}]
[/tex]

This can be shown directly, but you can also guess the form of it; varying the connection gives a tensor, and the partial derivatives in it can only become covariant ones (convince yourself if you're not!). You can plug this in the Palatini equation, which gives

[tex]
\delta R_{\mu\nu\rho}^{\ \ \ \ \lambda} &=& \frac{1}{2}g^{\lambda\alpha}[\nabla_{\mu}\nabla_{\nu}\delta g_{\rho\alpha} + \nabla_{\mu}\nabla_{\rho}\delta g_{\nu\alpha}-\nabla_{\mu}\nabla_{\alpha}\delta g_{\nu\rho}]
- \frac{1}{2}g^{\lambda\alpha}[\nabla_{\nu}\nabla_{\mu}\delta g_{\rho\alpha}+\nabla_{\nu}\nabla_{\rho}\delta g_{\mu\alpha} - \nabla_{\nu}\nabla_{\alpha}\delta g_{\mu\rho}]
[/tex]

yuiop

The equation and mathematics posted in this thread are very impressive, but I always wonder how useful they are (or what the point of them is) if these sort of calculations seemingly can not be used to answer "simple" questions like the ones posed here http://www.physicsforums.com/showthread.php?t=431712 and here http://www.physicsforums.com/showthr...38#post2901038 ?

This is one reason I have never made the effort to really try and learn the apparatus of tensors, because of the seeming limited applicability of these formalisms. Is it that tensors are "overkill" to solve the questions posed in those links (which have not yet been solved), like using a sledgehammer to crack a walnut, or is it just that the people that like to play with tensors never look at or try to solve the "simple" questions?

My real question is this. If a person makes an effort to learn tensors would they able to answer those "simple" questions and is it worth the effort to learn these formidable looking formalisms, if all you want to do is solve the sort of simple questions posed in those threads? (which up to now are seemingly unsolvable).