What's the physics origin and meaning of $\delta_{ij}-k_ik_j/k^2$ The projection tensor $P_{ij}\equiv\delta_{ij}-k_ik_j/k^2$ happens frequently in fluid mechanics or field theory. What's the physics origin of this structure and what's the meaning?
 A: Let $\mathbf k$ be a vector of $\mathbb R^3$, and let $k_i$ denote its components in the standard basis. The expression $k_ik_j/k^2$ is a rank-one operator on $\mathbb R^3$, which specifically is also the projection onto the direction of $\mathbf k$. So I will assume in what follows that $\mathbf k$ has norm 1. Some standard notation is the dyadic expression $\mathbf k\otimes\mathbf k$, or the linear operator $\theta_{\mathbf k,\mathbf k}$ which is given by
$$\theta_{\mathbf k,\mathbf k}(\mathbf v) = (\mathbf k,\mathbf v)\mathbf k,\qquad\forall\mathbf v\in\mathbb R^3.$$
where $(\ \cdot\ ,\ \cdot\ )$ denotes the usual inner product $\mathbf k\cdot\mathbf v$. Therefore $P=\operatorname{Id} - \theta_{\mathbf k,\mathbf k}$ represents the projection orthogonal to $\theta_{\mathbf k,\mathbf k}$, or equivalently the projection onto the plane for the origin in $\mathbb R^3$ which is perpendicular to the line (through the origin) generated by $\mathbf k$.
A: The matrix $P$ obeys $P \vec{k} = 0$, as is hopefully easily verified. As for a physical interpretations, often times laws in continuum theories can be broken up into effects that are parallel and perpendicular to some direction. That is, spherical symmetry gets broken when a particular axis is preferred, and gets broken into into rotational symmetry about that axis. The projection operator not only projects out vectors along $\vec{k}$, but leaves alone vectors perpendicular to $\vec{k}$. An example is the derivation of the electromagnetic stress-energy tensor from the dynamics of a system of charges. There, the Poynting vector picks out a preferred direction in space along which energy is transferred.
