Metric variation of the covariant derivative

[Theo1808]

Homework Statement

Hi all, I currently have a modified Einstein-Hilbert action, with extra terms coming from some vector field $A_\mu = (A_0(t),0,0,0)$, given by

$\mathcal{L}_A = -\frac{1}{2} \nabla _\mu A_\nu \nabla ^\mu A ^\nu +\frac{1}{2} R_{\mu \nu} A^\mu A^\nu$.

The resulting field equation has been given as $R_{\mu \nu} - \frac{1}{2} R g_{\mu \nu} = 8 \pi G (T_{\mu \nu} + T^A _{\mu \nu})$ i.e. the same as Einstein's equation but with an extra energy-momentum term coming from $A_\mu$. I'm trying to find the form of $T^A _{\mu \nu}$ and I only know that the density associated with it (I assume $T^A _{00}$) is $\rho_A = \frac{3}{2}H^2A_0^2+3HA_0 \dot{A}_0 - \frac{1}{2} \dot{A}^2 _0$. (The metric used is FRW $ds^2 = dt^2 - a^2(t) \delta_{ij}dx^idx^j$).

Homework Equations

Searching around has found these (I think) useful relations: $\frac{\delta g_{\mu \nu}}{\delta g^{\rho \sigma}} = -g_{\mu \rho}g_{\nu \sigma}$, $\frac{A^\mu}{\delta g^{\rho \sigma}} = 0$ and $\frac{A_\mu}{\delta g^{\rho \sigma}} = -g_{\mu \rho}A_\sigma$

The Attempt at a Solution

So my problem is getting a analytical equation for variations in the covariant derivative. I can rewrite $R_{\mu \nu} A^\mu A^\nu$ as $\nabla _\mu A^\mu \nabla _\nu A^\nu - \nabla _\mu A^\nu \nabla _\nu A^\mu$, so everything is in terms of covariant derivatives. The problem arises with variations in the Christoffel symbols. This gives me terms that are derivatives of $\delta g_{\mu \nu}$, which I'm not sure how to deal with (ultimately I'd want some expression all multiplied by $\delta g^{\rho \sigma}$. In the original EH action for example the variations in the connections are in a total derivative so they aren't needed to be evaluated. I tried to see what would happen if I ignored variations in the connection and with $T^A _{\mu \nu} = - 2 \frac{\delta \mathcal{L}}{\delta g^{\rho \sigma}} + g_{\rho \sigma}\mathcal{L}$, which gave me some of the terms in the density, but ultimately I believe I'm missing extra terms. Ignoring the variations in the connection, I get

$T^A _{\rho \sigma} = \nabla_\rho A_\nu \nabla_\sigma A^\nu - g^{\mu \beta}\nabla_\mu A_\rho \nabla_\beta A_\sigma -\frac{1}{2} g_{\rho \sigma} g^{\alpha \mu} \nabla _\mu A_\nu \nabla _\alpha A ^\nu +\frac{1}{2} g_{\rho \sigma} R_{\mu \nu} A^\mu A^\nu$.

The $R_{\mu \nu} A^\mu A^\nu$ term gives me (finding the (00) term in T) a $-\frac{3}{2}(\dot{H} + H^2)A_0^2$ term. I also get that the first 2 terms in $T_{00}$, after using the substitution of connections in FRW, all cancel out or go to zero. While the third term gives me the $-\frac{1}{2} \dot{A}^2 _0$ and $+\frac{3}{2}H^2A_0^2$. The problem is then I'm missing a few terms which I think come from me ignoring variations in the connection.

Thanks for any help you might be able to give me.

 

[demon]

If I am understanding you, you arrive at terms like ∂(δg_μν) and you don't know how to factor it in terms of δg_μν, is that right?

If so, I recently arrived to the same problem through a different action. Sadly I haven't found an answer yet, so I'll be watching this space... Also, if I find an answer I'll post it here.

By the way, would you mind letting me know where you found the relation
δg_μν/δg_ρσ=−g_μρg_νσ

Thanks in advance

 

[demon]

By the way, would you mind letting me know where you found the relation
δgμν/δgρσ=−gμρgνσ

Please forget this last point, I got confused with δ and ∂ again...