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## Main Question or Discussion Point

I'm currently teaching a gen ed course called Relativity for Poets. This is the first semester I've taught it, and it's been a ton of fun so far. If anyone is curious, here is the class's web page with links to the syllabus and lecture notes. The required texts are Takeuchi, Stannard, and Ostriker and Mitton. We have currently gone through SR (Takeuchi) and most of GR (Stannard plus my lecture notes). Although this is a class for non-science majors, I have decided that any conceptual apparatus that doesn't require an equation is fair game, so I've taught them some pretty advanced ideas, including parallel transport and Penrose diagrams.

In my office hours this week, one of my students asked what parallel transport was really good for, which seemed like a fair question at this point in the course. So far I've explained it using pictures on a sphere and used it as a qualitative explanation for the geodetic effect as a test of GR. One additional application I had in mind was that one can use it to answer the FAQ about where the universe gets the energy required for accelerating expansion -- but we haven't done cosmology yet, so they aren't really ready for that.

What we are doing right now is black holes, and off the cuff, I proposed parallel transport to my student as a way of explaining how the speed of light near a black hole can appear "wrong" to a distant observer. Now that I think about it some more, I'm not so sure that this explanation makes sense. There are certainly difficulties in GR with defining how an observer describes a velocity vector that exists at a distant point in space. However, parallel transport preserves the norm of a vector, so if a ray of light at the event horizon of a black hole has a lightlike (unnormalized) velocity vector according to a local observer, then no matter where we transport the vector or what path we transport it along, it will still be lightlike (although possibly rescaled).

Is there any way to salvage my idea of applying parallel transport to this topic?

The only book I could find that had a relatively elementary treatment of this in any detail was Taylor and Wheeler, Exploring Black Holes. They refer to Schwarzschild coordinates as "bookkeper coordinates," and say that the bookkeeper's low speed of light is what is observed in the Shapiro time delay.

This seems fine, but doesn't necessarily fit the pedagogy of my course, because I never even define the Schwarzschild coordinates or metric. We have discussed the fact that the event horizon is lightlike and the singularity is spacelike, and they've also seen the Penrose diagram for a black hole.

Does this topic work at all as an application of the concept of parallel transport, or if not, is there some other application to black holes?

In my office hours this week, one of my students asked what parallel transport was really good for, which seemed like a fair question at this point in the course. So far I've explained it using pictures on a sphere and used it as a qualitative explanation for the geodetic effect as a test of GR. One additional application I had in mind was that one can use it to answer the FAQ about where the universe gets the energy required for accelerating expansion -- but we haven't done cosmology yet, so they aren't really ready for that.

What we are doing right now is black holes, and off the cuff, I proposed parallel transport to my student as a way of explaining how the speed of light near a black hole can appear "wrong" to a distant observer. Now that I think about it some more, I'm not so sure that this explanation makes sense. There are certainly difficulties in GR with defining how an observer describes a velocity vector that exists at a distant point in space. However, parallel transport preserves the norm of a vector, so if a ray of light at the event horizon of a black hole has a lightlike (unnormalized) velocity vector according to a local observer, then no matter where we transport the vector or what path we transport it along, it will still be lightlike (although possibly rescaled).

Is there any way to salvage my idea of applying parallel transport to this topic?

The only book I could find that had a relatively elementary treatment of this in any detail was Taylor and Wheeler, Exploring Black Holes. They refer to Schwarzschild coordinates as "bookkeper coordinates," and say that the bookkeeper's low speed of light is what is observed in the Shapiro time delay.

This seems fine, but doesn't necessarily fit the pedagogy of my course, because I never even define the Schwarzschild coordinates or metric. We have discussed the fact that the event horizon is lightlike and the singularity is spacelike, and they've also seen the Penrose diagram for a black hole.

Does this topic work at all as an application of the concept of parallel transport, or if not, is there some other application to black holes?

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