I'm working with electrostatic interaction tensors, which are defined as follows:
\begin{align} T &= \frac{1}{r} \\ T^\alpha &\equiv \nabla T = -\frac{r^\alpha}{r^3} \\ T^{\alpha\beta} &\equiv \nabla \otimes \nabla T = \frac{1}{r^3}\left[ \frac{3r^\alpha r^\beta}{r^2} - \delta_{\alpha\beta} \right] \\ T^{\alpha\beta\gamma} &\equiv \nabla\otimes\nabla\otimes\nabla T = \frac{3}{r^5}(r^\alpha \delta_{\beta\gamma} + r^\beta\delta_{\alpha\gamma} + r^\gamma\delta_{\alpha\beta}) - \frac{15}{r^7}r^\alpha r^\beta r^\gamma \\ T^{\alpha\beta\gamma\delta} &= \cdots \end{align}
where $r = |\vec{r}|$ is a range between two particles, and Greek indices are Cartesian components $\{x,y,z\}$, so $r^\alpha$, for example, is some component of a vector $\vec{r}$ between two particles.
These are written in terms of their components. However, I wonder if there's any way to represent them as some kind of formula with other tensors.
For example, the second order tensor $T^{\alpha\beta}$ can (if I'm not mistaken) be conveniently written as $$ T^{\alpha\beta} = \frac{1}{r^3}\left[ \frac{3r^\alpha r^\beta}{r^2} - \delta_{\alpha\beta} \right] = \frac{1}{r^3} \left[\frac{3 \vec{r}\otimes\vec{r}}{r^2} - \mathbb{I}\right] $$ where $\mathbb{I}$ is a 2x2 identity matrix.
But when it comes to higher-order tensors, I'm kinda stuck. The second term in the third-order tensor is, as far as I understand, can be written as just $\frac{15}{r^7}\vec{r}\otimes\vec{r}\otimes\vec{r}$. But what about the first term with many Kronecker deltas?
So far I've came up with something like this monster $$ (r^\alpha \delta_{\beta\gamma} + r^\beta\delta_{\alpha\gamma} + r^\gamma\delta_{\alpha\beta}) = r^\alpha\{1,1,1\}\otimes\mathbb{I} + r^\beta \mathbb{I}\otimes\{1,1,1\} + r^\gamma (\mathbb{I}\otimes\{1,1,1\})^T $$ where $()^T$ means transposing last two dimensions.
So, my questions are
- Any way of representing this tensor as some tensor products sum?
- Is there a way outside of guessing to come up with such representations?