As I prepare to head for Cambridge (the Brits' one) for the conference to mark Stephen Hawking's 60th birthday, I know that the suggestion I am just about to make will strike the great and the good who are assembling for the event as my scientific suicide note. Suggesting time does not exist is not half as dangerous for one's reputation as questioning the expansion of the universe. That is currently believed as firmly as terrestrial immobility in the happy pre-Copernican days. Yet the idea that the universe in its totality is expanding is odd to say the least. Surely things like size are relative? With respect to what can one say the universe expands?
When I put this question to the truly great astrophysicists of our day like Martin Rees, the kind of answer I get is that what is actually happening is that the intergalactic separations are increasing compared with the atomic scales. That's relative, so everything is fine. Some theoreticians give a quite different answer and refer to the famous failed attempt of Hermann Weyl in 1917 to create a genuinely scale-invariant theory of gravity and unify it with electromagnetism at the same time. That theory, beautiful though it was, never made it out of its cot. Einstein destroyed it before it was even published with the simple remark that Weyl's theory would make the spectral lines emitted by atoms depend on their prior histories, in flagrant contradiction to observation. Polite in public, Einstein privately called Weyl's theory 'geistreicher Unfug' [inspired nonsense].
Ever since that time it seems to have been agreed that, for some inscrutable reason, the quantum mechanics of atoms and elementary particles puts an absolute scale into physics. Towards the end of his life, still smarting from Einstein's rap, Weyl wrote ruefully "the facts of atomism teach us that length is not relative but absolute" and went one to bury his own cherished ambition with the words "physics can never be reduced to geometry as Descartes had hoped".
I am not sure the Cartesian dream is dead even though the current observational evidence for expansion from a Big Bang is rather impressive. The argument from quantum mechanics, which leads to the identification of the famous Planck length as an absolute unit, seems to me inconclusive. It must be premature to attempt definitive statements in the present absence of a theory of quantum gravity or quantum cosmology. And the argument about the relativity of scale being reflected in the changing ratio of the atomic dimensions to the Hubble scale is vulnerable.
To argue this last point is the purpose of my contribution, which I shall do by a much simpler example, for which, however, the principle is just the same. Consider N point particles in Euclidean space. If N is greater than three, the standard Newtonian description of this system is based on 3N + 1 numbers. The 3N (=3xN) are used to locate the particles in space, and the extra 1 is the time. For an isolated dynamical system, such as we might reasonably conjecture the universe to be, three of the numbers are actually superfluous. This is because no meaning attaches to the three coordinates that specify the position of the centre of mass. This is a consequence of the relativity principle attributed to Galileo, although it was actually first cleanly formulated by Christiaan Huygens (and then, of course, brilliantly generalized by Einstein). The remaining 3N - 2 numbers constitute an oddly heterogeneous lot. One is the time, three describe orientation in space (but how can the complete universe have an orientation?), one describes the overall scale, and the remaining 3N - 7 describe the intrinsic shape of the system. The only numbers that are not suspect are the last: the shape variables.
Developing further ideas first put forward in 1902 in his Science and Hypothesis by the great French mathematician Poincare [ascii does not allow me to put the accent on his e], I have been advocating for a while a dynamics of pure shape. The idea is that the instantaneous intrinsic shape of the universe and the sense in which it is changing should be enough to specify a dynamical history of the universe. Let me spell this out for the celebrated 3 body problem of Newtonian celestial mechanics. In each instant, the instantaneous triangle that they form has a shape that can be specified by two angles, i.e., just two numbers. These numbers are coordinates on the space of possible shapes of the system. By the 'sense' in which the shape is changing I mean the direction of change of the shape in this two-dimensional shape space. That needs only one number to specify it. So a dynamics of pure shape, one that satisfies what I call the Poincare criterion, should need only three essential numbers to set up initial conditions. That's the only ideal that, in Poincare's words, would give the mind satisfaction. It's the ideal that inspired Weyl (though he attacked the problem rather differently).
Now how does Newtonian dynamics fare in the light of the Poincare criterion? Oddly enough, despite centuries of dynamical studies, this question hardly seems to have been addressed by anyone. However, during the last year, working with some N-body specialists, I have established that Newtonian mechanics falls short of the ideal of a dynamics of pure shape by no fewer than five numbers. Seen from the rational perspective of shape, Newtonian dynamics is very complicated. This is why the study of the Moon (which forms part of the archetypal Earth-Moon-Sun three-body problem) gave Newton headaches. Among the five trouble makers (which I won't list in full or discuss here), the most obstreperous is the one that determines the scale or size. The same five trouble makers are present for all systems of N point particles for N equal to or greater than 3. Incidentally, the reason why 3-body dynamics is so utterly different from 2-body dynamics is that shape only enters the picture when N = 3. Most theoretical physicists get their intuition for dynamics from the study of Newtonian 2-body dynamics (the Kepler problem). It's a poor guide to the real world.
The point of adding up the number of the variables that count in the initial value problem is this. The Newtonian three-body problem can be expressed perfectly well in terms of ratios. One can consider how the ratios of the individual sides to the perimeter of the triangle change during the evolution. This is analogous to following the evolution of the ratio of the atomic-radii to the Hubble radius in cosmology. To see if scale truly plays no role, one must go further. One must ask: do the observable ratios change in the simplest way possible as dictated by a dynamics of pure shape, or is the evolution more complicated? That is the acid test. If it is failed, absolute scale is playing its pernicious role. The Poincare criterion is an infallible test of purity.
Both Newtonian dynamics and Einstein's general relativity fail it. The fault is not in quantum mechanics but in the most basic structure of both theories. Scale counts. In fact, seen from this dynamical perspective Einstein's theory is truly odd. As James York, one of John Wheeler's students in Princeton, showed 30 years ago (in a beautiful piece of work that I regard as the highest point achieved to date in dynamical studies), the most illuminating way to characterize Einstein's theory is that it describes the mutual interaction of infinitely many degrees of freedom representing the pure shape of the universe with one single solitary extra variable that describes the instantaneous size of the universe (i.e., its 3-dimensional volume in the case of a closed universe). From Poincare's perspective, this extra variable, to put frankly, stinks, but the whole of modern cosmology hangs on it: it is used to explain the Hubble red shift.
I have stuck my neck out in good Popperian fashion. Current observations
suggest I will have my head chopped off and Einstein will be vindicated.
Certainly all the part of his theory to do with pure shape is philosophically
highly pleasing and is supported by wonderful data. But even if true
dynamical expansion is the correct explanation of the Hubble red shift,
why did nature do something so unaesthetic? As I hope to show very shortly
on the Los Alamos bulletin board, dynamics of pure shape can mimic a
true Hubble expansion. The fact is that Einstein's theory allows red
shifts of two kinds: one is due to stretching (expansion) of space,
while the other is the famous gravitational red shift that makes clocks
on the Earth run at a now observable amount slower than clocks in satellites.
It is possible to eliminate scale from Einstein's theory, as Niall O'Murchadha
and I have shown. This kills the stretching red shift but leaves the
other intact. It is just possible that this could explain the Hubble
challenge to the theoreticians is this: Are you absolutely sure Einstein
got it exactly right? Prove me wrong in my hunch that the universe obeys
a dynamics of pure shape subtly different from Einstein's theory. If
size does count, why should nature do something so puzzling to the rational