This is what a crackpot looks like

John Baez famously drew up his Crackpot Index, a sort of ready reckoner in place of a fully technological crank-o-meter, scoring a novel suggestion according to how wacky it is. Here is a fully paid-up member of the Crack Pot Brigade (against my better judgment, link) on curved space:

Curved Space: The concept of a 'curved space', which is essential for present cosmological models, is logically flawed because space can only be defined by the distance between two objects, which is however by definition always given by a straight line. Mathematicians frequently try to illustrate the properties of 'curved space' through the example of a spherical (or otherwise curved) surface and the associated geometrical relationships. However, a surface is only a mathematical abstraction within the actual (3-dimensional) space and one can in fact connect any two points on the surface of a physical object through a straight line by drilling through it.

Ahem. *headdesk*

The whole point is that we can only begin to imagine a space which is curved by immersing a lower-dimensional curved space in the three-dimensional flat space we are used to. (Technically, not entirely flat, but the curvature is so small we never notice it in everyday life.) The surface of a sphere is two-dimensional, as is a flat sheet of paper. However, the two are not equivalent (you cannot flatten out an orange peel onto a piece of paper, as we all remember from geography lessons), and if we write the distance between two points, then the function from the first is different from the one from the second.

This matters because as I suggested, space-time is curved albeit only slightly. Particularly, when electrons accelerate in a curved space-time, they may behave a bit differently from a flat space-time: and that possible effect matters, because I hope to get a PhD on the back of it.