# Appendix A

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Published 2023-10-25

For those who are interested...

Introducing particles with mass into space will result in a gravitational field. Einstein wanted to describe this gravitational field using geometry of spacetime. In other words, the presence of the massive particles (with energy and momentum) should influence the curvature of the spacetime.

With that Einstein went in search for an equation that links between energy-momentum and geometry of spacetime. But how do we link these two totally different things together? The hint comes when we consider that there is a law that governs the conservation of energy-momentum. The conservation law for energy momentum is:

\begin{equation}
\nabla_{c}T_{ab}=0
\end{equation}

where \nabla_{c} is some form of differentiation, and T_{ab} is a quantity that describe energy-momentum content. Can we find a similar one for geometry, like

\begin{equation}
\nabla_{c}G_{ab}=0
\end{equation}

where G_{ab} is some geometrical term?

Indeed after many years of searching, Einstein found a "simple" geometrical term that satisfy the above equation. Without doing the derivation, we state the Einstein's field equation as follows:

\begin{equation}
R_{ab}-\frac{1}{2}Rg_{ab}=\frac{8\pi G}{c^{4}}T_{ab}
\end{equation}

The terms on the left hand side, R_{ab}-\frac{1}{2}Rg_{ab} is Einstein's choice for G_{ab} (Yes it satisfies Eq.(\ref). Each of the terms is only dependent on the metric.

\begin{equation}
\text{Ricci tensor: }\;R_{ab}=\sum_{c}\partial_{c}\Gamma_{ab}^{c}-\sum_{c}\partial_{b}\Gamma_{ac}^{c}+\sum_{c,d}\Gamma_{ab}^{c}\Gamma_{cd}^{d}-\sum_{c,d}\Gamma_{ad}^{c}\Gamma_{cb}^{d}
\end{equation}

\begin{equation}
\Gamma_{bc}^{a}=\Gamma_{cb}^{a}=\sum_{d}\frac{1}{2}g^{da}(\partial_{b}g_{cd}+\partial_{c}g_{db}-\partial_{d}g_{bc})
\end{equation}

\begin{equation}
\text{Ricci scalar: }\;R=\sum_{a,b}g^{ab}R_{ab}
\end{equation}

\begin{equation}
\sum_{b}g^{ab}g_{bc}=\delta_{\phantom{a}c}^{a}
\end{equation}

The right hand side of Einstein's field equation is one that describe the matter/energy content in the system.

\begin{equation}
\text{Energy momentum tensor }T_{ab}=\sum_{d}g_{ad}T_{\phantom{c}b}^{d}
\end{equation}

In the case of a homogeneous and isotropic Universe,

\begin{align}
T_{\phantom{0}0}^{0} & =-\rho(t)c^{2}\nonumber \\
T_{\phantom{0}1}^{1} & =T_{\phantom{0}2}^{2}=T_{\phantom{0}3}^{3}=p(t)
\end{align}

where \rho(t) is the energy density and p(t) is the pressure of the matter/energy content in the system (the Universe in this case).

From the FLRW metric

\begin{equation}
ds^{2}=-c^{2}dt^{2}+S(t)^{2}\left(\frac{dr^{2}}{1-kr^{2}}+r^{2}d\theta^{2}+r^{2}\sin^{2}\theta d\phi^{2}\right), \nonumber
\end{equation}

\begin{equation}
g_{00}=-c^{2},\,g_{11}=S(t)^{2}\frac{1}{1-kr^{2}},\,g_{22}=S(t)^{2}r^{2},\,g_{33}=S(t)^{2}r^{2}\sin^{2}\theta
\end{equation}

All other g_{ab} are zero.

\begin{equation}
g^{00}=\frac{-1}{c^{2}},\,g^{11}=\frac{1}{S^{2}}(1-kr^{2}),\,g^{22}=\frac{1}{S^{2}r^{2}},\,g^{33}=\frac{1}{S^{2}r^{2}\sin^{2}\theta}
\end{equation}

All other g^{ab} are zero.

We are not going to torture ourselves with calculating R_{ab} and R here. For now the result will be stated.

The non-zero components of the Ricci tensor are

\begin{align}

R_{00} & = & -\frac{3\ddot{S}}{S} \nonumber \\
R_{11} & = & \frac{1}{1-kr^{2}}\frac{1}{c^{2}}(S\ddot{S}+2\dot{S}^{2}+2kc^{2})\nonumber \\
R_{22} & = & r^{2}\frac{1}{c^{2}}(S\ddot{S}+2\dot{S}^{2}+2kc^{2})\nonumber \\
R_{33} & = & r^{2}\sin^{2}\theta\frac{1}{c^{2}}(S\ddot{S}+2\dot{S}^{2}+2kc^{2})

\end{align}

The Ricci scalar is

\begin{equation}
R=\frac{6}{c^{2}}(\frac{\ddot{S}}{S}+\frac{\dot{S}^{2}+kc^{2}}{S^{2}})
\end{equation}

With the above information, we can derive the first and second Friedmann equations using Einstein's field equation by

(i) setting a=0,b=0,

\frac{\dot{S}(t)^{2}+kc^{2}}{S(t)^{2}}=\frac{8}{3}\pi G\rho(t)

(ii) set a=1,b=1 (or both to 2 or both to 3)

-2\frac{\ddot{S}(t)}{S(t)}-\frac{\dot{S}(t)^{2}+kc^{2}}{S(t)^{2}}=\frac{8\pi G}{c^{2}}p(t)