The Riemann tensor and the metric

For a maximally symmetric $\gamma_{ij}$ the Riemann tensor of the 3d space satisfies

$$R_{ijkl}=k(\gamma_{ik}\gamma_{jl} - \gamma_{il}\gamma_{jk})$$ (1.2)

where k is some constant. Contracting the tensor using the fact that the trace of the metric is three gives us the Ricci tensor given by

$$R_{jl}=2k\gamma_{jl}$$ (1.3)

A maximally symmetric metric is also spherically symmetric and from our knowledge of the Schwarzschild solution we have

$$d\sigma^{2}=\gamma_{ij}du^{i}du^{j}=e^{2\beta(r)}dr^{2}+r^{2}(d\theta^{2}+sin^{2}\theta d\phi^{2})$$ (1.4)

Using this metric we calculate the diagonal components of the Ricci tensor as

$$R_{11}=\frac{2}{r}\partial_{1}\beta$$ (1.5)

$$R_{22}=e^{-2\beta}(r\partial_{1}\beta -1)+1$$ (1.6)

$$R_{33}=(e^{-2\beta}(r\partial_{1}\beta -1)+1)sin^{2}\theta$$ (1.7)

Using equation $1.3$ we solve for $\beta$ to get

$$\beta=-\frac{1}{2}ln(1-kr^{2})$$ (1.8)

which gives

$$ds^{2}=-dt^{2}+a^{2}(t)\left[dr^{2}\frac{1}{1-kr^{2}}+r^{2}(d\theta^{2}+sin^{2}\theta d\phi^{2})\right]$$ (1.9)

This is called the Robertson Walker metric.
Here k takes the values 1, -1 or 0 depending upon the curvature of space that we are talking about. The connection coefficients can be calculated using

$$?\Gamma\sigma_{\mu\nu}?=\frac{1}{2}g^{\sigma\rho}(\partial_{\mu}g_{\nu\rho}+\partial_{\nu}g_{\rho\mu}-\partial_{\rho}g_{\mu\nu})$$ (1.10)

Calculating the connection coefficients we proceed to calculate the components of the Ricci tensor which come out to be

$$R_{\mu\nu}=0 \hspace{.5cm} \mu\neq\nu$$ (1.11)

$$R_{00}=-3\frac{\ddot{a}}{a}$$ (1.12)

$$R_{11}=\frac{a\ddot{a}+2\dot{a}^{2}+2k}{1-kr^{2}}$$ (1.13)

$$R_{22}=r^{2}(a\ddot{a}+2\dot{a}^{2}+2k)$$ (1.14)

$$R_{33}=R_{22}sin^2\theta$$ (1.15)

On contracting the Ricci tensor we get the Ricci scalar as

$$R=\frac{6}{a^{2}}(a\ddot{a}+\dot{a}^2+k)$$ (1.16)

This forms the numbers for various further calculations which we shall do. In the next chapter we will see the Einstein's equations and derive some special relations connecting the properties of matter in the universe and then we shall use them to extract information about the development of the Universe from the Robertson Walker metric.