Thoughts on Gravitational Fields on Shells

Models for gravity on cool shapes!

I was reading Unruh's 'Notes on Black Hole Evaporation' and I came across an analysis on the gravitational fields of shells with mass, along with particles living on this object. I wanted to quickly write some of my thoughts in this post.

General relativity helps us answer questions about the gravitational fields that are spawned by different shapes. What is the metric, and therefore the gravitational field, of a shell? This is quite an interesting question to ask. I think that this problem is quite interesting since shells aren’t the typical natural structures you might think of in gravitation. To start with, let’s consider a shell in one time dimension, and one space dimension. A shell is considered the surface of a sphere, and in one dimension it’s simply a point R along the x-axis.

$$ \text{Shell} =  \{ R  \} \subset \mathbb R
$$

A shell in one space dimension is simply a dot at a given radius, or length, from the origin. The best way to approach a problem like this would be to split the problem into two parts - the region outside the shell, and the region inside the shell. Inside the shell, we expect the physics to look like that of an inertial frame - Minkowski space with one time dimension and one space dimension. In this region, there is no active gravitational field at work. Here, an object at rest will stay at rest. Similarly, an object moving at constant speed will stay that way. Therefore, we expect the metric in this region to look like that of flat space. Suppose that the radius of this shell is R, then we should have that

$$ ds ^ 2 = d \tau ^ 2- dr^ 2 ,\quad r < R $$

In this region, we have two numbers that represent coordinates. The tau represents time, and r represents space. It is also important to note that these coordinates only really make sense, and are defined, within the shell only. Now, outside the shell, we should expect that objects do indeed feel some gravitational force coming from the mass of the shell. Since the shell is spherically symmetric in one spatial dimension, we can expect the metric to take the form of the Schwarzchild metric in one dimension. This metric looks like the following, where M is the mass of the shell.

$$ ds ^ 2 = ( 1 - 2M / r ) dt^ 2 - \frac{ dr ^2 } { ( 1- 2M / r ) } , \quad r > R $$

There is a curious thing to note here. The regions outside and inside the shell have different time coordinates. One reasonable question could be to ask - how are they related? One approach to answering this question might be to impose reasonable continuity conditions on the boundary of the shell itself. Outside the shell - we ask about what the dynamics of objects are. To do this, we compute geodesics, which I’ll talk about in the next section.

References

[1] Notes on black-hole evaporation, W. G. Unruh. Phys. Rev. D 14, 870 – Published 15 August 1976