I had an argument with contributor @safesphere regarding this problem. He insists that an infinitely large plate with a finite surface density of $\sigma$ would eventually collapse into a black hole because the mass tends to infinity as well as the plate's spatial dimensions. He believes that an infinite plate that doesn’t form a black hole produces zero gravity. Taking the Schwarzschild solution for a thin hollow shell and making its radius tend to infinity, we see that no uniform gravity exists.
However, using Newtonian mechanics, I calculated the G-field near an infinitesimally thin cylinder with an infinite radius to be $g=2πGσ$, where $σ$ is the surface density of the cylinder. I am really doubtful that GR predicts something far away from the Newtonian mechanics for this problem.
On the other hand, to defend my claim, I found an article in which the authors try to find solutions similar to those that occur for a plane of charge of constant density $σ$, i.e., $E=σ/2$. In the Conclusion section, the authors claim:
In this paper we have investigated in full detail the most general solution associated with a source localized on a plane with no behavior more singular than a Dirac delta function. For two very special equations of state, we find one-parameter families of solutions very analogous to those of the electrostatic problem of a sheet of charge, including one solution that is reflection-symmetric and others where the two sides are qualitatively the same (both flat or both curved). But in general, a flat solution on one side dictates a curved one on the other and vice versa, and the one-parameter freedom in the plate’s location is lost.
I want to know if this article demonstrates correct calculations and a plausible discussion and if there exists a certain traditional solution to this problem like that mentioned by contributor @safesphere.