Physical interpretation of the following stress-energy tensor : $T^{\mu\nu}=X^\mu X^\nu$ I have come across the following stress-energy tensor and  I was wondering if anyone know of a physical system this could correspond to?
$$
T^{\mu\nu}=\pmatrix{(X^0)^2 & X^0X^1 & X^0X^2&X^0X^3 \\ X^1X^0&(X^1)^2&X^1X^2&X^1X^3\\X^2X^0&X^2X^1&(X^2)^2&X^2X^3\\X^3X^0& X^3X^1&X^3X^2&(X^3)^2}
$$
where $X^0,X^1,X^2$ and $X^3$ are fields, functions of $\mathbb{R}^4\to\mathbb{R}$.
It is obtained by taking a vector of fields, such as
$$
\mathbf{v}:=\pmatrix{X^0[x,y,z,t]\\X^1[x,y,z,t]\\X^2[x,y,z,t]\\X^3[x,y,z,t]}
$$
And then taking product of the vector with its transpose:
$$
\mathbf{v}\mathbf{v}^T=\pmatrix{(X^0)^2 & X^0X^1 & X^0X^2&X^0X^3 \\ X^1X^0&(X^1)^2&X^1X^2&X^1X^3\\X^2X^0&X^2X^1&(X^2)^2&X^2X^3\\X^3X^0& X^3X^1&X^3X^2&(X^3)^2}
$$
 A: If this stress–energy tensor corresponds to some realistic matter then  for every observer (characterized by the timelike 4-velocity $W_\mu$) the local energy density must be non-negative ($T^{νμ} W_ν W_μ\ge 0$) and the local energy flow vector ($F^μ=T^{μν}W_ν$) must be non-spacelike (formally  we would say that stress–energy tensor must satisfy the dominant energy condition). This would mean that the field $X^μ$ is everywhere either timelike or null.  We would discuss those possibilites (timelike and null $X^μ$) separately. We would also assume that $X^μ$ is future pointing (otherwise we could just change its sign).
If $X^μ$ is timelike, then the stress-energy tensor has only one nonzero (and positive) eigenvalue which would be the energy density in the comoving frame. We   can interpret such stress–energy tensor as that of a pressureless  (dust) matter. We could then define 4-velocity field of this dust $\mathbf{u}$ , and energy density in comoving frame $\rho$ through vector field $\mathbf{X}$:
$$u^μ=\frac{1}{\sqrt{X^ν\phantom{|}\! X_ν}} X^μ, \qquad \rho = X^ν X_ν ,$$
assuming “mostly minus” metric signature.
Using these quantities the stress–energy tensor could be written in a possibly more familiar form:
$$
T^{μν}=\rho\, u^μ u^ν.
$$
In the comoving frame it would take particularly simple form: $T^μ_ν=\rho \mathop{\mathrm{diag}}(1,0,0,0)$.
If $X^μ$ is null then the stress–energy tensor would correspond to a null dust matter and may be seen as either a limiting case of dust matter or pure radiation matter and could also arise as a stress energy tensor  of null Maxwell field. The vector $X^μ$ would then be its principal null direction.
