I just don't understand this tensor and would like to go through an example with you to somehow make sense of it.
I consider two spheres with masses $m_1$ and $m_2$, densities $\rho_1$ and $\rho_2$ and radii $R_1$, and $R_2$, which are located at coordinates $\vec{x}_1$ and $\vec{x}_2$, move on an ellipsoidal orbit around their center of mass with period duration $T_1$ and $T_2$ and rotate around the axes $\vec{\Omega}_1 = \Omega_1 \cdot \frac{1}{\sqrt{3}}\begin{pmatrix} 1 \\ 1 \\ 1 \end{pmatrix}$ and $\vec{\Omega}_2 = \Omega_2 \cdot \begin{pmatrix} 0 \\ 0 \\ 1 \end{pmatrix}$ with the speed $|\vec{\Omega}_1|$ and $|\vec{\Omega}_2|$.
How do I now set up the stress-energy tensor? It is not dust and also not a perfect fluid.
If I use the formula for dust, how do I bring the rotation and the ellipsoidal orbit into the equation? Does this even play a role here?
