Dust stress-energy in a boosted inertial frame

Boosted components. The boost matrix has ${\mathrm{\Lambda }}^0{}_0=\gamma$, ${\mathrm{\Lambda }}^0{}_1={\mathrm{\Lambda }}^1{}_0=-\gamma \beta$, ${\mathrm{\Lambda }}^1{}_1=\gamma$ and preserves the other two spatial directions (The passive Lorentz boost with parameter beta). Since $T^{ab}=\rho {\delta }_0^a{\delta }_0^b$, [F1] gives $T^{\mathrm{\prime }ab}=\rho {\mathrm{\Lambda }}^a{}_0{\mathrm{\Lambda }}^b{}_0$; using ${\mathrm{\Lambda }}^0{}_0=\gamma$, ${\mathrm{\Lambda }}^1{}_0=-\gamma \beta$, ${\mathrm{\Lambda }}^2{}_0={\mathrm{\Lambda }}^3{}_0=0$ this is $T^{\mathrm{\prime }00}={\gamma }^2\rho$, $T^{\mathrm{\prime }01}=T^{\mathrm{\prime }10}=-{\gamma }^2\beta \rho$, $T^{\mathrm{\prime }11}={\gamma }^2{\beta }^2\rho$ and $T^{\mathrm{\prime }22}=T^{\mathrm{\prime }33}=T^{\mathrm{\prime }02}=T^{\mathrm{\prime }03}=T^{\mathrm{\prime }12}=T^{\mathrm{\prime }13}=0$.

Factorization and conservation. For the same field $n(p)=e_0$, [F1] gives $n^{\mathrm{\prime }a}={\mathrm{\Lambda }}^a{}_b{n}^b={\mathrm{\Lambda }}^a{}_0$, hence its primed components are $n^{\mathrm{\prime }0}=\gamma$, $n^{\mathrm{\prime }1}=-\gamma \beta$, $n^{\mathrm{\prime }2}=n^{\mathrm{\prime }3}=0$. Step 1.1 therefore reads $T^{\mathrm{\prime }ab}=\rho n^{\mathrm{\prime }a}{n}^{\mathrm{\prime }b}$; the passive change preserves the fields $n$ and $T$. The primed components satisfy ${\eta }_{ab}{n}^{\mathrm{\prime }a}{n}^{\mathrm{\prime }b}=-{\gamma }^2+{\gamma }^2{\beta }^2=-1$ and $n^{\mathrm{\prime }0}>0$, consistent with the future unit normalization of $n=e_0$. All primed components are constant in the primed chart, so ${\mathrm{\partial }}_a^{\mathrm{\prime }}{T}^{\mathrm{\prime }ab}=0$, consistent with [F2] applied to the constant future unit field $n$ and constant $\rho >0$.

The primed coordinate coefficient $T^{\mathrm{\prime }00}={\gamma }^2\rho$ is the numerical factorization ${\gamma }^2\rho =\gamma \cdot (\gamma \rho )$; its source interpretation as one energy factor and one volume factor belongs to the separately adopted dust model reading and is not proved by this component computation. The example is a component computation and asserts no interaction, no dynamics and no measurement.