Solving for the metric of a spherically symmetric stationary (but not static) energy distribution that only moves in positive $r$ direction I want to solve for the metric of a spherically symmetric stationary, but not static energy Distribution. Where there is time translation symmetry, but not time reflection symmetry, and the distribution only moves along the positive $r$ direction.
For example, it could be a spherical torus (a ball not just a shell), where energy only moves in the positive $r$ direction, and energy that travels beyond a finite radius $R$ re-emerges from the center of the ball. So at any time, $t$, the total distribution is always the same, but there is not time reflection symmetry, because energy flows from center outward and never inward.
My proposed strategy is analogous to the derivation of the Kerr metric, but where $r$ changes with time instead of $\phi$. I anticipate having to derive the 4 diagonal elements of the metric in addition to two identical d$r$d$t$ and $\mathrm dt\mathrm dr$ terms with the other 10 terms being zero.
Does this approach make sense? Is there any book, publication, or other reference that demonstrates a solution for a similar problem? Any other advice or guidance?
If the proposed $g$ has 6 non-zero terms as described above, my understanding is that there exists a tensor $B$, such that $B^{-1}gB$ is diagonal. Might this correspond to a coordinate transformation from time, $t\rightarrow t'$, $T$, such that the above stationary system becomes static with regards to $T$?
If the velocity of the Energy distribution: depends on $r$ ($\mathrm dV(r)/\mathrm dr$ non-zero) and travels at relativistic velocities where gravitation fields/potential are strong, changing and depend on $r$, there would significant GR and SR time dilation for any r1 relative to a different r2. What I am trying to understand: how does a simple transformation from $t\rightarrow T$ exist? Certainly, there exists a transformation to $T$ that makes the system static at a specific $r_1$. How can that same transformation result in the system being static at a different $r_2$ where there is a different relative clock rate?
Thank you, Samuel, for your helpful comments and references. Right on page 594 of my Gravitation book, there is a transformation, $e^p dt' = a dt + b dr$, that addresses my question.
 A: I'm a bit confused about some parts: For example, you want "time translation symmetry", but say that "energy flows"? Or want a "spherically symmetric" metric but later consider a "torus"? In the following, I will just consider a spherically symmetric and in the interior non-static metric:
Consider the exterior solution for $r\geq R$: According to the Birkhoff theorem, every spherically symmetric solution of the vacuum field equations is static and asympotically flat, hence is the Schwarzschild solution. In particular, a spherically symmetric and radial movement as you have described won't emit any gravitational radiation.
Consider the interior solution for $r\leq R$: The energy just reemerging from the center as it reaches the boundary, basically teleportation as you said the flow is "never inward", would violate special relativity. This would be manifested in the violation of the stress-energy conservation ($\nabla_\mu T^{\mu\nu}\neq 0$), which is a contradiction to the definition of the Einstein tensor (with $\nabla_\nu G^{\mu\nu}=0$) and the field equations ($G_{\mu\nu}=\kappa T_{\mu\nu}$).
Finding a solution in the presence of matter is pretty difficult and Einstein himself guessed (wrongly as Schwarzschild showed just months later), that it would even be impossible. So depending on what flow you want, the problem might be way too difficult to solve. However, since you use the Kerr metric as a reference, an interior solution has been found in 2017. There are also results to the opposite problem, which you could find useful as reference, where matter only moves inward and never outwards, which is of interest, as it describes the gravitational collapse (See this paper by Roger Penrose from 1964) of stars into black holes. Every good book about general relativity covers the description of a gravitational collapse. I would try looking through "Relativistic Stars", the fifth chapter of "Gravitation" by Misner, Throne and Wheeler found here, starting at page 591.
