How to explain the curvature of space-time to students

[supernova88]

I teach a class on astronomy and recently tried to explain the curving of space time by massive objects like neutron stars and black holes. I even used a sheet of spandex to represent space-time which we bent using different weights. However my students were very confused how space, which they perceive as "emptiness" or "nothing," could be bent. In their minds you can't change the shape of something that isn't there. Other than that they're perfectly content with the idea that gravity pulls on objects and can even make light bend, but I was hoping to explain "why" this happens. To be honest I don't quite know how to explain this concept any better since we can't directly "see" space curve, and I'd definitely appreciate some insight.

 

[Nugatory]

Our own A.T. has some really excellent short videos. Search this forum and you'll find many references.

The model of a sheet being distorted by different weights is in many ways misleading... It certainly didn't help with my understanding. You'll find much complaining about it here too :)

 

[Bandersnatch]

The thing about A.T.'s videos is that they explain (superbly) the how, not the "why". If your students are stuck on the "why", then maybe it'd be a good idea to spend some time on the philosophy of science. Try and get them to appreciate that science is only about precise mathematical descriptions of how the nature works, and that trying to pursue the why question is ultimately a red herring - an attempt to visualise a phenomenon using an analogy with what you're intuitively more familiar with. At some point the phenomenon you're trying to describe gets so far removed from the common sense gained in everyday life (the "lions and savannah" scale), that it's impossible to further the understanding by draping the thing in imperfect analogies.

Make your students examine their biases and preconceptions. Make them explain why they think only "things" can curve, warp and stretch. Compare space-time(4d space) to 1 dimensional distance - do they find it equally odd that this immaterial thing can grow and shrink? How about straight lines on a sphere (meridians)? It's not like they're made of something, yet they curve.

Have you seen this "rant" of Feynman's? He goes on length about the difficulty of asking the "why" questions. It may help you guide your students.
Also, this:

 

[A.T.]

I even used a sheet of spandex to represent space-time which we bent using different weights.

Shame on you. Read this:
https://www.physicsforums.com/threa...visualization-of-gravity.726837/#post-4597121
http://en.wikipedia.org/wiki/Gravity_well#Gravity_wells_and_general_relativity

Here are some better analogies:
http://www.physics.ucla.edu/demoweb..._and_general_relativity/curved_spacetime.html
http://www.relativitet.se/spacetime1.html
http://www.adamtoons.de/physics/gravitation.swf

However my students were very confused how space, which they perceive as "emptiness" or "nothing," could be bent.

You could equally ask: How could that "nothing" be flat? Why should we assume that empty space has Euclidean geometry, not some other geometry.

We had many discussions on this. The last one is here:
https://www.physicsforums.com/threads/on-space-time.783833/#post-4923002

 

[Matterwave]

What helped me visualize most the concept of spacetime and curvature is probably MTW's introduction to the concept in chapter 1 of their book. The ants crawling around on an apple, being "attracted" to the stem/dimple simply because the apple is curved there. Cutting a slice of the skin of the apple reveals a straight line path that the ants took. The book is slightly advanced, but the concepts in chapter 1 are quite good. You can take a look. :)

 

[pervect]

I liked the parable of the apple as well. I also liked the "Distances determine geometry" lecture from "Exploring Black Holes",the chapter which has this is online at http://www.eftaylor.com/pub/chapter2.pdf.

I would consider introducing an even simpler example before giving Taylors. That would be to compute the lengths of the diagonals of a square on a plane, then do a similar computation on a sphere, using segments of great circles for the sides of the square (a quadrilateral with 4 equal sides) and the diagonals. It might be necessary toe explain first that a great circle is the curve of shortest distance between two points on a sphere.

Depending on the level of the students, this could be given as a homework problem?

[add]Hints if this were a HW problem would be that you can project a square (with diagonals) drawn on a plane onto the surface of a sphere, and the length of the straight line segment lying in the plane is $r \sin \theta$, while the length of the projected great circle segment is $r \theta$, where $\theta$ is the central angle subtended by the line/arc at the center of the sphere.

The point of the exercise, which might need to be spelled out, is that the difference between the geometry of the sphere and the geometry of the plane affects the distances - in particular the lengths of the diagonals of a "square".

I *am* a bit concerned that some students might see Taylors example, and draw a blank at the main point about how you could use the information on the distances between pairs of points to recreate the shape of the boat :(. It's not like Taylor specified an exact algorithm for this process. Perhaps some questions on this topic could determine if the students "got it".

There's also the issue that in GR, the "distances" are the space-time interval of special relativity. So a "review" of this concept , if it's not already part of the course, would be a good idea.

This would all be laying groundwork for the very idea of curvature at the most basic level. AT's ideas are better for explaining why space-time curvature causes effects which we label as gravity.