When analogies are taken too far: Spacetime is bent, but it’s not quite a stretchy membrane

Last week I was asked by some school students about the nature of gravity. What is it?

Isaac Newton, and a whole pile of textbooks following him, treat gravity as an attractive force between two objects. It’s a force that is proportional to the product of the masses of the two objects, and is inversely proportional to the distance between the objects squared. That is, F = G M m / r^2, where M and m are the masses of the two objects and r is the distance between them. The constant G, known as Newton’s Constant of Gravitation (or ‘big G’, for short), is given by  6.6741 x 10-11 m3 kg-1 s-2, with an uncertainty of about 5 in that last digit. ‘Big G’ is really hard to measure well, and its value is far more uncertain than any other fundamental constant in physics.  We used to have apparatus set-up in a student lab here at the University of Waikato, for students to have a go at measuring G. To be honest, anyone who got 6-something x 10-11 m2 kg-1 s-2 was doing well.

This idea of gravity as a force serves physics well in many respects. But it doesn’t when we come to astrophysics. Einstein, as part of General Relativity, developed another picture. In this description, spacetime (one cannot separate space from time) is bent by the presence of massive objects (e.g. stars). Thus a straight line path is no longer a straight line. The physical analogy that’s often used at this point is the ball on the stretchy sheet – something like this Youtube clip. The mass at the centre distorts the sheet, and smaller balls don’t roll in straight lines anymore:

Now, in this stretchy-sheet analogy, the ball follows a curved path because the distorted sheet creates a force on it towards the big mass at the centre. But this isn’t Einstein’s thinking. Einstein would say that the ball follows a curved path because space-time is curved (distorted). A path that looks straight to the ball, isn’t straight to an external observer. Unfortunately the stretchy-sheet analogy doesn’t quite show this.   I think a better analogy is Einstein’s own – illustrated by Jim Al-Khalili in this video – of a beetle moving on a curved tree branch. (It starts at about 3:45 into the movie). We perceive forces, because we are moving in a straight line in  curved space-time.

Note how the stretchy-sheet analogy can create a misunderstanding – that the distorted spacetime (the sheet) creates a pushing force (gravity) on the small particles. It’s not how Einstein sees it.

The ‘force or curvature’ is a subtle distinction perhaps, and not likely to be relevant to most of us. But it does serve as an illustration of how analogies, which are very useful in physics, can be taken too far. Teachers like myself need to be careful when presenting them.

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