A spacetime has line element

$$d s^{2}=-d t^{2}+t^{2 p_{1}} d x_{1}^{2}+t^{2 p_{2}} d x_{2}^{2}+t^{2 p_{3}} d x_{3}^{2},$$

where $p_{1}, p_{2}$ and $p_{3}$ are constants. Calculate the Christoffel symbols.

Find the constraints on $p_{1}, p_{2}$ and $p_{3}$ for this spacetime to be a solution of the vacuum Einstein equations with zero cosmological constant. For which values is the spacetime flat?

Show that it is not possible to have all of $p_{1}, p_{2}$ and $p_{3}$ strictly positive, so that if they are all non-zero, the spacetime expands in at least one direction and contracts in at least one direction.

[The Riemann tensor is given in terms of the Christoffel symbols by

$$\left.R_{b c d}^{a}=\Gamma_{d b, c}^{a}-\Gamma_{c b, d}^{a}+\Gamma_{c f}^{a} \Gamma_{d b}^{f}-\Gamma_{d f}^{a} \Gamma_{c b}^{f}\right]$$