# Constraints on $p$ and $\rho$ from cosmological metric and $T_{; \nu}^{\mu \nu} = 0$

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!

• I've added the homework-and-exercises tag. In the future, please use this tag on this type of question. – Ben Crowell Oct 23 '19 at 2:34

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.