Discussion of Exercise 3 of Section 4.3

Gerard Venema of Calvin College in Grand Rapids, Michigan, writes:

I have enjoyed very much using your textbook this semester. I have been holding problem sessions each Tuesday at which my students discuss their solutions to problems. As usual, today they had animated discussions of some of them–I can hardly get them to stop, so we have been meeting for longer sessions on Tuesdays.

Today we discussed Exercise 3 in Sectin 4.3: What happens to an astronaut who flies through a point with neighborhoods bounded by the projective plane? The students asked me to contact you to find out what you think the answer is. Can you tell me? Thanks.

Gerard

Robert Messer replies:

Here are some thoughts I have had about this problem.

First, need to agree on what it means to fly through a point. Are we modeling the astronaut as a point moving in this space? Or does the astronaut shrink down to a point as she approaches this singularity? If the astronaut is or ever becomes a point, we lose track of orientation and other aspects of the 3-dimensional object. So let's reject these possibilities.

Instead, let's start with the 3-dimensional astronaut some distance from the singularity. Let's parameterize the distance from the singularity by a ray [0, 100) with the singularity at 0 and the astronaut between 40 and 60. Assume this is far enough from the singularity that any intersection of the astronaut with the projective planes is contained a nice disk D in each projective plane. So, I am thinking that the astronaut starts off in the Euclidean neighborhood D x [40, 60].

Now, let's think of each of the projective planes as a sphere with antipodal points identified. Then there will be another copy of the astronaut in the disk D' (the set of antipodal points of D in the sphere) cross [40, 60]. There are not two astronauts. Nor is it the case that one is the real astronaut and the other is an image. It is more like looking through curved space and seeing yourself at a distance. Your students may enjoy flying around in some of the spaces Jeff Weeks has created.

Well, as the astronaut approaches the singularity, eventually she will no linger fit in such nice neighborhoods. But small enough parts of her body will. So, for example, if she approaches the singularity at waist level, she will see her feet approaching her head. Ouch, I hope she wasn't going too fast when they meet. Don't you hate it when that happens? But that's what you get when you go flying around in pseudo-manifolds.

Phil Straffin adds:

Yes, the astronaut is to have physical extension. As long as the singularity is outside the astronaut's body, she's fine, though she will certainly see strange things ahead of her. But as soon as the singularity enters her body, I think she's in trouble, since opposite points on spheres around the singularity are identified. I doubt she'll come through the experience intact.

I actually don't think there's a completely convincing answer to this question, but my students have always enjoyed thinking about it. Many of them think the astronaut will be turned inside out, but they haven't been able to explain to me exactly what they mean by that!