Space, time and space-time

Vigorous debate about the foundations of space and time has to begin with Poincare's arguments for the thesis that one can always choose a Euclidean geometry as the basis for a physical theory, provided one is ready to make enough sacrifices in the structure of the theory itself. This led to the well-known position that "no choice of geometry is a choice of spatial facts"; in fact, "there are no spatial facts but only conventions which made our world-pictures more elegant without any possible risk of trespassing beyond the bounds of truth".

Graham Nerlich is against such a philosophy. His arguments aim to support the standpoint, which he calls realism, that "space-time is a concrete particular and the practice of scientific explanation and invention firmly commits us to its existence". However, the unconventional character of space-time becomes evident if its structure is taken into account.

We can ask "objectual questions": how do objects (such as curves, vector fields, light cones) exist in space-time? This leads to an "objectual ontology". But we can ask "structural questions", ie questions about the kinds of relations in which the objects stand (without bothering whether relations exist over and above the objects themselves). This leads to a "structural ontology". The theory of relativity "could be best interpreted not as the unfolding of physical processes in space and time but as their pattern in space-time". One could say that "a basic object in the ontology of special relativity and general relativity physics is space-time itself".

Many discussions of the theory of relativity resolve themselves if looked upon from the structural perspective. For instance, the problem of the conventional character of simultaneity disappears if special relativity is regarded as primarily a theory of space-time rather than of reference frames in space and time. This is certainly true, but we should not forget that both special and general relativity theories can be expressed in terms of the space of all possible (Lorentz) frames of reference (called the fibre bundle of Lorentz frames).

Such an approach is par excellence structural, but space-time is not regarded in it as a primary object. It can be derived from the space of all reference frames and the fact that they are locally connected by Lorentz transformations (space-time turns out to be the quotient of the space of all reference frames by the action of the Lorentz group). In the truly structuralist approach, I think one can start with various objects, conventionally regarded as primary ones, the only constraint being that structural relations must be preserved.

Nerlich deals extensively with paradoxes which may arise if the conscious observer is put into the world with closed (or semi-closed) time-like curves (eg one could kill one's father to prevent one's birth). However, we should remember that the conscious observer is beyond the scope of both special and general relativity theories - indeed, beyond the scope of existing physics. If the presence of, say, an electromagnetic field alters global properties of space-time, we should expect that taking into account all fields and processes necessary to produce a conscious observer will drastically change the space-time geometry. A conscious observer is simply not part of the structural space-time lattice of our present physics. When making the analysis of temporal loops it would also be useful to take into account Hawking's theorem stating the necessary and sufficient condition for the existence of a global time: global time exists if, and only if, a given space-time is causally stable, ie if small perturbations of its Lorentz metric do not produce temporal loops.

There exists a certain tension between two approaches to the philosophical analysis of physical theories. Either an analysis goes deeply into a problem, and then it must take into account all its mathematical subtleties, or it is intended to be easily accessible to every kind of philosopher, and then it runs the risk of being trivial. By adopting "a more leisured approach", Nerlich has exposed himself to this risk.

Michael Heller is a professor of philosophy, Pontifical Academy of Theology, Krakow.