General relativity is one of the more challenging ideas in science. That’s hardly surprising given the complexity of the mathematics involved.

And yet one of relativity’s crowning glories is its public appeal. Scientists and laypeople alike are fascinated by the idea that gravity is the natural result of the warping of spacetime.

This insight is almost entirely the result of one the most famous analogies in science. While few people can grasp the subtleties if non-Euclidean geometry or Ricci calculus, almost everyone can visualise the circular motion of a ball rolling on a deformed rubber sheet.

Indeed, it’s straightforward to set up a demonstration in which a marble rolls around a central mass deforming a rubber trampoline. Countless students of relativity have gained the essential insight from experiments just like this

But here’s the problem. Physicists have long suspected that the orbit of such a marble is not at all like a planet orbiting a star. Indeed, some ten years ago, researchers showed that there a rubber sheet deformed by a central mass simply cannot adopt a shape that reproduces the real orbits of objects in spacetime.

Now it turns out the problem is even worse than first thought. Today, Chad Middleton and Michael Langston at Colorado Mesa University in Grand Junction show that the properties of the rubber itself can have a bigger influence on the motion of the marble than the central mass that deforms it. That’s entirely unlike spacetime where the motion is determined by the masses of the objects involved.

In a paper to appear in the American Journal of Physics, These guys even demonstrate their result with the help of, yes, a spandex trampoline and a marble.

Middleton and Langston begin by deriving the equations that govern the motion of a marble rolling on the cone-like surface that forms when a central mass deforms a rubber sheet.

However, the actual shape of such a sheet is different from a cone in significant ways so they also have to calculate the real shape that it forms.

That turns out to be tricky. Middleton and Langston approach the problem by imagining that the elastic sheet is divided into concentric rings around the central mass and that these increase in width as the sheet deforms. They then add up the changes to get the actual shape (a process known as integration).

Finally, they plug this shape into the equations that describe the motion of a marble on a cone-like surface to work out its actual motion.

The results will be a shock for students of general relativity. In ordinary spacetime, the equations of general relativity produce Kepler’s famous relation that the square of the orbital period is proportional to the radius cubed, in other words T^2 is proportional to r^3.

But on a rubber sheet that is deformed by a small amount, this relation turns out to be reversed— T^3 is proportional to r^2.

And this changes as the deformation becomes larger. For a heavily deformed sheet, Middleton and Langston show that T is actually proportional to r.

So the motion of a marble on a deformed rubber sheet is entirely unlike the motion of a plant around star (or any small object moving around a large one under the force of gravity). Indeed, physicists have known for ten years that a rubber sheet deformed by a central mass can never take on a shape that reproduces the gravitational effects of spacetime.

Middleton and Langston go on to test their mathematical model by measuring the motion of a real marble on deformed spandex sheet. This confirms their thinking but throws up some further surprises.

They say that the mass of the sheet itself causes a deformation and that this significantly influences the behaviour of the marble when the overall deformation is small. What’s more, the way that spandex deforms depends on the deformation itself and this further complicates matters.

So the bottom line is that the rubber sheet analogy with spacetime is fundamentally flawed—it simply cannot reproduce the motion that general relativity predicts.

That’s interesting and certainly a useful exercise for those interested in the mathematical validity of the analogy.

But the truth is that this work cannot diminish the extraordinary utility of this analogy. And so the public love affair with general relativity is safe. Long may it continue!

Ref: arxiv.org/abs/1312.3893 : Circular Orbits On A Warped Spandex Fabric