# How can we quantify gravity within and outside a non-rotating cylinder in outer space?

Initial research

I have found many questions and answers about O'Neill cylinders but they are assumed to be rotating. https://en.wikipedia.org/wiki/O%27Neill_cylinder

I am also aware of the Shell Theorem https://en.wikipedia.org/wiki/Shell_theorem

Background

However I would like to understand the nature of the gravitational field surrounding and inside an open non-rotating hollow cylinder floating freely in space.

Intuition tells me that an object floating freely along the axis of an open-ended cylinder would be attracted towards the cylinder's centre of gravity and that the object would continue to oscillate along that axis indefinitely (possibly slowing because of tidal forces?)

Question

We can draw notional lines of force surrounding a magnet. Can we draw a similar diagram showing lines of gravitational force around and inside an open-ended hollow cylinder? I anticipate that the diagram would show arrows that indicate strength of field and direction.

Specifically (1) Is such a diagram trivial in a mathematical sense (2) If so, how would I calculate it? (3) If not, is it in some way intractable and why?

Note

Here is the sort of diagram I am looking for. It shows tidal flow where the direction and velocity are indicated by arrows of different lengths. In my case the arrows would show force and direction.

Can we draw a ... diagram showing lines of gravitational force around and inside an open-ended hollow cylinder?

Yes.

Is such a diagram trivial in a mathematical sense?

No. The mathematics is complicated.

If so, how would I calculate it?

Suppose the cylindrical shell has radius $$R$$, length $$L$$, and surface mass density (e.g., some number of kilograms per square meter) $$\sigma$$. Take the $$z$$ axis along the cylinder's axis of symmetry, and let the cylinder extend from $$z=0$$ to $$z=L$$. Then the gravitational potential $$\varphi$$ at any point in cylindrical coordinates $$(\rho,\phi,z)$$ can be shown to be

$$\varphi(\rho,\phi,z)=-G\sigma(2\pi R)\int_0^Ldz'\int_0^\infty dk\,J_0(k\rho)\,J_0(kR)\,e^{-k|z-z'|}$$

where $$J_0$$ is a Bessel function of the first kind. I'll skip the derivation since this is a graduate-level homework problem.

One can numerically evaluate this to obtain the potential everywhere and plot the equipotential lines. By taking the gradient, one can obtain the gravitational field. I will leave it to someone else to produce plots if they are interested in doing so.