Thinking Straight About Curved Space

Thinking Straight About Curved Space

Raymond Tallis

We are now in a position to understand the origin of the counter-intuitive – indeed unintelligible – notion of space being curved (or indeed, having any kind of topology). It arises from projecting into space our mathematical portrait of the influence of gravity on the trajectories of objects. However, by using an everyday term such as ‘curved’, we are conflating an appearance, as exemplified in the surface of a spherical object or the parabolic flight of a thrown missile, with a mathematical abstraction capturing the change in the position of an accelerated object with time. This is a prime example of what in The Concept of Nature (1920) Alfred North Whitehead called, “the muddle of importing the mere procedures of thought into the facts of nature.” In this case, we are translating handy ways of depicting (and hence predicting and calculating) the influences of gravity on the motion of objects (and indeed of light) through space – bending their trajectories – into the structure of space itself. Nevertheless, the fact that the mathematics of motion in space and the curved shape of graphs of position against time are congruent does not mean that space is itself curved, nor that it makes sense to say so.

It is not idiotic to be puzzled by the very idea of expanding space, given that ‘expansion’ is usually of stuff into relatively empty space. Thinking of emptiness expanding into something that is not even emptiness is more than a little odd.

Physics, and the technology based on it (and indeed our civilisation), has flourished by being prepared to set aside the common sense that tells us that the earth must be flat otherwise people will fall off it, that a small object will always fall slower than a big one, and that the state of rest and motion in a straight line are fundamentally different. But we should not conclude from this that the mathematical portrait of the world is the last word on what is really there, or that everyday experience of lived space is in some profound sense defective or even wrong.

In The Production of Space (1991), Henri Lefebvre, the ‘philosopher of everyday life’ reflected that, “Not many years ago, the word ‘space’ had a strictly geometrical meaning… In scholarly use it was generally accompanied by some such epithet as ‘Euclidean’… and the general feeling was that the concept of space was ultimately a mathematical one. To speak of ‘social space’, therefore, would have sounded strange.” We should recognise that the notion of ‘curved space’ is less legitimate than that of ‘social space’, however useful the former might be for the development of mathematical physics. And we should not be pressured into thinking that the space of daily life, which is neither Euclidean nor non-Euclidean, is somehow not the real thing.