wormy apple [image: pinclipart.com] |

Wait, what? That doesn’t make sense. Certainly a maggot-free apple is not maximally yucky. Where did our math fail us?

It didn’t, really. The beheaded maggot is an example of a discontinuous or “singular” limit and originally due to Michael Berry*. You know you have a discontinuous limit if the function whose limit you are taking (that’s the increasing “yuck factor” of the maggot) does not approach the value of the function at the limit (unyucky).

A less fruity example is taking the y-th power of x and sending y to infinity. If x is any positive number smaller than 1, taking its exponent to infinity will give zero. If x is equal to one, all values of y will give back 1. If x is larger than one, the result of taking y to infinity will return infinity. If you plot the limit y to infinity as a function of x, it’s discontinuous.

Such singular limits are not just mathematical curiosities. We have them in physics too.

For example in thermodynamics, when we take the limit in which the number of constituents of a system becomes infinitely large, we see phase transitions where some quantities, such as the derivative of specific heat, become discontinuous. This is, of course, strictly speaking an unrealistic limit because the number of constituents may become very large, but never actually infinite. However, the limit isn’t always unrealistic.

Take the example of massive gravity. In general relativity, gravitational waves propagate with the speed of light and the particle associated with them – the graviton – is massless. You can modify general relativity so that the graviton has a mass. However, if you then let the graviton mass go to zero, you do not get back general relativity. The reason is that if the graviton mass is not zero, then it has additional polarizations and those are independent of the mass as long as the mass isn’t zero**.

The same issue appears if you have massless fields that can propagate in additional dimensions of space. This too gives rise to additional polarization which don’t necessarily disappear even if you take the size of the extra dimensions to zero.

Discontinuous limits are often a sign that you have forgotten to keep track of global, as opposed to local properties. If you for example take the radius of a sphere to infinity the curvature will go to zero, but the result is not an infinitely extended plane. For this reason, there are certain solutions in general relativity that will not approximate each other as you think they should. In a space with a negative cosmological constant, for example, black hole horizons can be infinitely extended planes. But these solutions no longer exist if the cosmological constant vanishes. In this case, black hole horizons have to be spherical.

Why am I telling you that? Because discontinuous limits should make you skeptical about any supposed insights gained into quantum gravity by using calculations in Anti de Sitter space.

Anti De Sitter (AdS) space, to remind you, is a space with a negative cosmological constant. It is popular among string theorists because they know how to make calculations in this space. Trouble is, the cosmological constant in our universe is positive. And there is no reason to think the limit of taking the cosmological constant from negative values to positive values is continuous. Indeed, it almost certainly is not because the very reason that string theorists prefer calculations in AdS is that this space provides additional structure that exists for any negative value of the cosmological constant, and suddenly vanishes if the value is zero.

String theorists usually justify working with a negative cosmological constant by arguing it can teach us something about quantum gravity in general. That may be so or it may not be so. The case with the negative cosmological constant resembles that of finding a piece of a maggot in your apple. I find it hard to swallow.

* ht Tim Palmer

** there are ways to fix this limiting behavior so that you do get back general relativity.