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Theo1808
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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.