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I'm given the cosmological metric

$$ds^2 = -dt^2 + a(t)^2dx^2 +a(t)^2dy^2 + a(t)^2dz^2$$

and perfect fluid with stress-energy tensor

$$T^{\alpha \beta}=\left(\begin{array}{cccc}{\rho} & {0} & {0} & {0} \\ {0} & {a^{-2} p} & {0} & {0} \\ {0} & {0} & {a^{-2} p} & {0} \\ {0} & {0} & {0} & {a^{-2} p}\end{array}\right)$$

I want to examine $T_{; \nu}^{\mu \nu} = 0$ to determine the conditions imposed on $p$ and $\rho$.

My thought process is this: evaluate the time and spatial components separately. I have also calculated the non-vanishing Christoffel symbols from the given metric: $\Gamma_{x x}^{t}=a(t) \dot{a}(t)$, $\Gamma_{y y}^{t}=a(t) \dot{a}(t)$, $\Gamma_{z z}^{t}=a(t) \dot{a}(t)$, $\Gamma_{t x}^{x}=\dot{a}(t)/a(t)$, $\Gamma_{t y}^{y}=\dot{a}(t)/a(t)$, $\Gamma_{t z}^{z}=\dot{a}(t)/a(t)$. I know this is also related to the Friedmann equations.

I am just unsure of how to calculate $T_{; \nu}^{\mu \nu} = 0$ and then find the constraints on $p$ and $\rho$. Thanks!

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  • $\begingroup$ I've added the homework-and-exercises tag. In the future, please use this tag on this type of question. $\endgroup$ – Ben Crowell Oct 23 '19 at 2:34
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See this link for a list of how to compute tensors of various types. The one you are interested in is the following one: $$ T^{ab}_{\ \ \ \ \ ;c} = \partial_{c} T^{ab} + \Gamma^{a}_{\ cd} T^{db} + \Gamma^{b}_{\ cd} T^{a d} $$

You are contracting your indices in the sense that $$ T^{\mu\nu}_{\ \ \ \ \ ;\nu} = \partial_{\nu} T^{\mu\nu} + \Gamma^{\mu}_{\ \nu d} T^{d\nu} + \Gamma^{\nu}_{\ \nu d} T^{\mu d} $$

You now just need to plug into this your expressions for $T^{\mu\nu}$ as well for $\Gamma^{a}_{\ bc}$.

EDIT: It will probably be useful to note that your $T^{\mu\nu}$ is diagonal. It will simplify the above expression considerably.

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