# Does an infinitely large thin plate produce a uniform $g$-field near its surface according to GR?

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.

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.

There are some pathologies with static gravitating infinite plate in Newtonian mechanics, that are not obvious if one is only interested in the gravitational field. Namely, in order for the plate to retain its static equilibrium, material of the plate must have elastic stresses. If we consider this infinite plate as a limit ($$L\to \infty$$) of some finite system with characteristic length scale $$L$$ while keeping the surface density constant, then the stresses would be diverging as $$\mathcal{O}(L)$$. Thus for an infinite static plate, mechanical stresses would be infinite, which could be considered unphysical. The main problem is that a closed and consistent system of Newtonian mechanics that includes both gravity and elastic forces does not allow any static solution describing infinite plate with constant surface density.

In general relativity we cannot just ignore elastic forces when calculating gravitational field, because elastic forces enter in the stress–energy–momentum tensor and thus constitute the source of gravity. To overcome this obstacle we can try several approaches:

• Dynamic solutions. That is the route (I believe) advocated by safesphere. In both GR and Newtonian mechanics an infinite plate with constant mass density but without unphysical infinite stresses cannot be in equilibrium. If we require that the planar symmetry is preserved at all times and the equation of state for the plate is realistic then the solution must include singularity either in the future or in the past or both. Note, that I would not call such solution a “black hole” without analyzing its causal structure first.

• A limit of a static finite system in which the surface density goes to zero and the size goes to infinity while stresses remain constant. I believe that the solutions from the quoted paper, Fulling et al., 2015 that have $$\rho_0=0$$, $$p_0\ne 0$$ could be seen as originating from such a limit.

• “Antigravity”: the curved solution (37) of Fulling et al., 2015 (also known as Taub's plane-symmetric spacetime) could be interpreted as a repulsive field of a singularity corresponding to negative mass (see e.g. this paper). The repulsion from such mass (or from two masses on both sides of the plate) could prevent our plate (with a positive surface density and a reasonable equation of state) from collapsing onto itself.

• Non-vacuum spacetimes. Thin plate could be in static equilibrium if other fields are supporting it. For example, a plate could carry a surface electric charge. A nonzero cosmological constant also could allow static equilibrium.

In conclusion: there could be several nontrival solutions of general relativity serving as analogues of Newtonian gravitational field of infinite thin plate. The static solutions found by Fulling et al., 2015 have the drawbacks of either unrealistic equation of state ($$\rho_0=0$$, $$p_0\ne 0$$) for the plate, or require “negative mass” singularity(ies) for support. If we want the plate to have somewhat realistic matter it must either be nonstationary (as suggested by safesphere) or must also include additional matter fields.

would eventually collapse into a black hole because the mass tends to infinity as well as the plate's spatial dimensions.


These seem not good reasons to form a black hole, IMHO. A very large object with low density will not form a black hole. In order to form one you must put a large enough mass into a given distance, in practice you need the Schwarzchild radius $$r_g$$ to exceed the size of the region on which you concentrate the mass. So the Sun does not turn into a black hole because its $$r_g$$ is way smaller than it size.

As a typical counter example you can think about star formation, a large low density cloud of gas makes a gravitationally bound object, a star, as soon as pressure does not hold it from doing it. Not all stars become black holes.

Maybe you are trying to argue that by plate you meant something with special properties?