we simplify the notation by omitting the bar over $F$ in (\[Kasner-like metric in terms of F-bar\]) and the index $^{(4)}$ in $^{(4)}T_{\mu\nu}$)

$$\begin{aligned}
\label{EMT}
8 \pi G T_{0}^{0} &=& \frac{a^2 c \dot{F}}{(a + b)^2 \dot{V} F^2}, \nonumber \\
8\pi G T_{1}^{1} &=& \frac{a \left\{(\gamma + \beta)(a + b) F \left[\frac{\ddot{F}}{\dot{F}} - \frac{\ddot{V}}{\dot{V}}\right] - 2 c(\alpha - \beta - \gamma )\dot{F}\right\}}{2(a + b)^2 \dot{V} F^2}, \nonumber \\
8\pi G T_{2}^{2} &=& \frac{a \left\{(\gamma + \alpha)(a + b) F \left[\frac{\ddot{F}}{\dot{F}} - \frac{\ddot{V}}{\dot{V}}\right] - 2 c(\beta - \alpha - \gamma )\dot{F}\right\}}{2(a + b)^2 \dot{V} F^2}, \nonumber \\
8\pi G T_{3}^{3} &=& \frac{a \left\{(\alpha + \beta)(a + b) F \left[\frac{\ddot{F}}{\dot{F}} - \frac{\ddot{V}}{\dot{V}}\right] - 2 c(\gamma - \alpha - \beta )\dot{F}\right\}}{2(a + b)^2 \dot{V} F^2}.\end{aligned}$$

Certainly the specific shape of the EMT depends on the embedding. However, there are a number of relationships, between the components of the EMT, which are “embedding-independent". These are $$\begin{aligned}
\label{rel between the stresses}
(\gamma - \beta)T_{1}^{1} + (\alpha - \gamma)T_{2}^{2} + (\beta - \alpha)T_{3}^{3} = 0,\end{aligned}$$ and $$\begin{aligned}
\label{rel between T0 and the stresses}
(\alpha + \gamma)T_{1}^{1} -  (\beta + \gamma)T_{2}^{2} &=&  (\beta - \alpha)T_{0}^{0},\nonumber \\
(\alpha + \beta)T_{1}^{1} -  (\beta + \gamma)T_{3}^{3} &=&  (\gamma - \alpha)T_{0}^{0},\nonumber \\
