The Metaphysics of Space-Time
Space and time have been favourite subjects for philosophers since at least the ancient Greeks. The paradoxes of the infinite and the infinitesimal are reinvented each day by children with inquisitive minds. How can space be infinite? If it is not infinite what would lie beyond the end? Can the universe have a beginning and an end? How have modern physicists and philosophers learnt to deal with these questions?
The simplest answer is that they use mathematics to construct models of the universe from basic axioms. Mathematicians can define the system of real numbers from set theory and prove all the necessary theorems of calculus that physicists need. With the system of real numbers they can go on to define many different types of geometry. In this way it was possible to discover non-Euclidean geometries in the nineteenth century which were used to build the theory of general relativity in the twentieth.
The self consistency of general relativity can be proven mathematically from the fundamental axioms. This does not make it correct, but it does make it a viable model who's accuracy can be tested against observation. In this way there are no paradoxes of the infinite or infinitesimal. The universe could be infinite or finite, with or without a boundary. There is no need to answer questions about what happened before the beginning of the universe because we can construct a self-consistent mathematical model of space-time in which time has a beginning with no before. Notice that I say no before not nothing before. There is not even a time before when there was nothing.
So long as we have a consistent mathematical model we know there is no paradox, but nobody yet has an exact model of the whole universe. Newton used a very simple model of space and time described by Euclidean geometry. In that model space and time are separate, continuous, infinite and absolute.
This is consistent with what we observe in ordinary experience. Clocks measure time and normally they can be made to keep the same time within the accuracy of their working mechanisms. It as if there was some universal absolute standard of time which flows constantly. It can be measured approximately with clocks but never directly.
So long as there is no complete theory of physics we know that any model of space-time is likely to be only an approximation to reality which applies in a certain restricted domain. A more accurate model may be found later and although the difference in predicted measurement may be small, the new and old model may be very different in nature. This means that our current models of space and time may be very unrealistic descriptions of what they really are even though they give very accurate predictions in any experiment we can perform.
Philosophers try to go beyond what physicists can do. Using thought alone they consider what space and time might be beyond what can be observed. Even at the time of Newton there was opposition to the notion of absolute space and time from his German rival Leibnitz. He, and many other philosophers who came after, have argued that space and time do not exist in an absolute form as described by Newton. Newton himself appreciated that he was making a big working assumption.
If we start from the point of view of our experiences, we must recognise that our intuitive notions of space and time are just models in our minds which correspond to what our senses find. This is a model which exists like a computer program in our head. It is one which has been created by evolution because it works. In that case there is no assurance that space and time really exist in any absolute sense.
The philosophical point of view developed by Leibnitz, the Bishop Berkeley and Mach is that space and time should be seen as formed from the relationships between objects. Objects themselves are formed from relationships between our experiences. Only our experiences are absolute. The mathematical models used by physicists turn this upside-down. They start with space and time, then they place objects in it, then they predict our experiences as a result of how the objects interact.
Mach believed that space and time do not exist in the absence of matter. The inertia of objects should be seen as being a result of their relation with other objects rather than their relation with space and time. Einstein was greatly influenced by Mach's principle and hoped that it would follow from his own principles of relativity.
In the theory of special relativity he found that space and time do not exist as independent absolute entities but space-time exists as a combination of the two. In General Relativity he found, ironically, that the correct description of his theory must use the mathematics of Riemannian geometry. Instead of confirming Mach's principle he found that space-time can have a dynamic structure in it's own right. Not only could space-time exists independent of matter but it even had interesting behaviour on it's own. His most startling prediction that there should exist gravitational waves, ripples in the fabric of space-time itself, may soon be directly confirmed by detection in gravitational wave observatories.
Einstein's use of geometry was so elegant and compelling that physicists thereafter have always sought to extend the theory to a unified description of matter through geometry. Examples include the Kaluza-Klein models in which space-time is supposed to have more than four dimensions with all but four compacted into an undetectably small geometry. Thus physicists and philosophers have become alienated during the twentieth century.
Recent theories of particle physics have been so successful that it is now very difficult to find an experimental result which can help physicists go beyond their present theories. As a result they have themselves started to sound more philosophical and are slowly reviewing old ideas. The fundamental problem which faces them is the combination of general relativity and quantum theory into a consistent model.
According to quantum theory a vacuum is not empty. It is a sea of virtual particles. This is very different from the way that space and time were envisioned in the days of Mach. In a theory of quantum gravity there would be gravitons, particles of pure geometry. Surely such an idea would have been a complete anathema to Mach. But suppose gravitons could be placed on a par with other matter. Perhaps then Mach would be happy with gravitons after all. The theory could be turned on its head with space-time being a result of the interactions between gravitons.
In string theory, the most promising hope for a complete unified theory of physics, we find that gravitons are indeed on an equal footing with other particles. All particles are believed to be different modes of vibration in loops of string. Even black holes, one of the ultimate manifestations of the geometry of space-time are thought to be examples of single loops of string in a very highly energised mode. There is no qualitative distinction between black holes and particles, or between matter and space-time.
The problem is that there is as yet no mathematical model which makes this identity evident. The equations we do have for strings are somewhat conventional. They describe strings moving in a background space-time. And yet, the mathematics hold strange symmetries which suggest that string theories in different background space-times and even different dimensions are really equivalent. To complete our understanding of string theory we must formulate it independently of space-time. The situation seems to be analogous to the status of electrodynamics at the end of the 19th century. Maxwell's equations were described as vibrations in some ether pervading space. The Michelson-Morley experiments failed to detect the hypothetical ether and signalled the start of a scientific revolution.
Just as Einstein banished the ether as a medium for electromagnetism we must now complete his work by banishing space-time as a medium for string theory. The result will be a model in which space-time is recovered as a result of the relationship between interacting strings. It will be the first step towards a reconciliation of physics and philosophy. Perhaps it will be quickly followed by a change of view, to a point from where all of our universe can be seen as a consequence of our possible experiences just as the old philosophers wanted us to see it. What other ways will we have to modify our understanding to accommodate such a theory? Not all can be foreseen.