Geometry and Physics

Luboš Motl has a long, interesting post on what he calls the myths of quantum gravity. I have a slight quibble with one point.

Myth: But a geometric approach is better, isn't it?

In physics, the primary way of dividing theories is into correct theories and wrong theories. A general attempt to divide ideas and tools into geometric ones and non-geometric ones is typically ill-defined - it depends on the definition of "geometry" which is a matter of historical and social coincidences in mathematics rather than a matter of well-defined differences. Our understanding what geometry is has been evolving for centuries. More importantly, the approach that is labeled "more geometric", whether or not the reasons behind this terminology are rational or not, doesn't have to be "more correct".

Physics of string theory can be defined to be the right "generalized geometry". At this level, it is just an empty word.

The basic dynamics of general relativity admits a geometric interpretation whether or not we like to use the word "geometry" more often than "fields" or less often. Arguing how often certain words and dogmas should be repeated doesn't belong to physics.

In some general sense his point is certainly valid. Poincare emphasized that geometry is ultimately a choice. In Newtonian physics space and time are independent - space can be considered to be a three-dimensional fiber bundle over a time manifold. Special relativity seemed to require multiple times and spaces depending on reference frames, but Minkowski showed that they could be considered parts of a unified, flat, psuedo-Riemannian manifold. That discovery not only provided an elegant picture of special relativity but set the stage for the discovery of general relativity. The additional structure provided by the metric provided an essential key.

Couldn't you just make do with tensor fields as Lumo suggests? Well, yes, sort of. By themselves, though, the fields are just a cross-section of a fiber bundle on the manifold. The existence of a metric structure on the manifold has far richer implications - implications that have been important in discovering the nature of general relativity.

So what about such things as torsion in string theories? Are the geometric implications important? I have no idea, but it seems unreasonable to dismiss the possibility out of hand.

The point is that certain choices of geometry are simpler, more elegant, or more fecund for future developments than others. One of the tasks of physics is to discovery those better alternatives.