Thursday, June 08, 2006

Time Times Two

Philosophy of physics geek day at the Playground!

Richard over at Philosophy, et cetera had an interesting post the other day on time travel. While I'm not so interested here in the questions of causation that are raised, he did touch on an issue that I've been interested in since it was raised by my general relativity prof in grad school, the question of multiple times.

In his 1916 paper, "Foundations of the General Theory of Relativity," Einstein sets out what has come to be called the "point-coincidence argument" wherein he argues that there are certain properties that change with one's reference frame and others that do not. Those which have the same value in the reference frames of all possible observers are to be considered objectively real. (This notion of invariance and its ontological weight finds its way into the epistemology of the Logical Empiricists and the psychology of the Gestalt theorists, but that's another post.) In general relativity, one such real quantity is called the space-time interval between events (in fact, Minkowski first points this out for the Special Theory). It is a combination of spatial distance and the time between two events, say, the snapping of two sets of fingers, such that,

S= x^2 + y^2 + z^2 - ct^2,

where S is the space-time interval, x, y, and z are the distances in three perpendicular directions from some reference frame, c is the speed of light, and t is the time measured between the snaps by the same observer. While the measured quantities x, y, z, and t will all change with observers in different states of motion relative to the snapping fingers, the combination above won't. Everyone who measures x, y, z, and t will get the same value for the space-time interval.

Notice that the only difference between space and time in this formalism is the minus sign (and the c, but this is a constant and if you pick your units right, it goes to 1 and disappears for calculation and really only serves to make the units come out right). It would be a perfectly straightforward mathematical exercise to work through the usual exercises for a space-time in which we just happen to flip the second plus into a minus, giving us as an invariant interval,
S = x^2 + y^2 - cz^2 - ct^2.

Notice what we did in flipping that one sign, we went from a world with three spatial dimensions and one time to a flat two dimensional world with two times. Perfectly trivial in terms of the math.

But the question my grad school prof asked (and he actually asked it in a letter to Richard Feynman!) is what would it be like to live in such a world. (Feynman wrote back that he and his son puzzled over it during a weekend at the beach and they have no clue.) The times would be perpendicular, that is, completely independent. You could be early in one direction and late in another. You would be simultaneously aging in two different ways. Your birthday in one direction could last an infinite amount of time in the other direction. We can think of space having a fourth dimension with little problem, but why is two spatial dimensions so darn strange? Space and time seem to play very similar roles in the formalism, why would it be so hard to think about life in a four dimensional world with a second temporal dimension? (If you want the fifth dimension, of course, you need to wait for the dawning of the age of Aquarius.)

Turns out it isn't only hard to imagine, it also makes the physics really, really weird. Things that are just quantities like energy, suddenly become vectors, that is directional quantities and this makes certain little things like the stability of matter go away. There is an interesting paper, On the Dimensionality of Spacetime, on this by MIT physicist Max Tegmark where he argues that the only stable dimensional set up is one in which there are three spatial and one temporal dimension.

It is a wonderful paper. But, be careful, Tegmark does have a philosophical ax to grind here. He is one who has a soft spot for the Anthropic Principle, the cosmological version of Intelligent Design. The idea is that the universe and its constants are so perfectly picked that the universe as a stable enough place to support matter, much less life, is evidence of Divine Creation. If any of the universal constants had been slightly different, the argument goes, a stable universe with us in it would have been impossible, so its existence stands as good reason to believe that their is a Cosmic Fine Tuner.

If one were to take this work as evidence for the Anthropic Principle (note: Tegmark does not make this move or suggest it anywhere in this paper, although you could see his interest in the work being related), one would have to assume that while the number and arrangement of dimensions may change, the foundational laws governing matter is to be held constant. The argument is that if P are our best current physical laws and Gn is some geometry n of the universe (with G0 being the geometry of our universe with its dimensionality), P + G0 is the only stable combination. (This, of course, is modeled on Einstein's explanation of Poincare's conventionalism from Experience and Geometry) The conclusion, therefore, at best shows that if our current physical laws or ones very close to them are necessarily true, then space must have three spatial and one temporal dimension. Far weaker than the complete anthropic result, but completely fascinating in its own right.