| MadSci Network: Physics |
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i realize its somewhat silly to consider, but a science fiction novel once
referred to changing the value of pi in *a* universe. what explicit
effects would this have on the world as we know it?
thanks for all your help! steve Steve, you have it exactly backwards. p (Pi) is a ratio - the circumference of a circle divided by its diameter - defined by the geometric properties of the universe one inhabits. In a "flat" or Euclidean universe - named for Euclid of Alexandria, who wrote the Element s, the definitive geometry text for thousands of years - parallel lines never meet or diverge. In such a universe the value of p is 3.14159... (or approximately 3 1/7). The universe we live in is Euclidean on most scales, in fact on almost every scale we can measure; at the very largest scales it may be non-Euclidean. Actually, General Relativity says that the presence of matter curves space, so that any local area of space will be non-Euclidean, even if only a little. But if the total curving effect of matter cancels out, the universe as a whole can be Euclidean. This is equivalent to saying that "the density of the universe is one," just enough to stop the universal expansion at infinite time.I'm not enough of a geometer to tell you the effect on the value of p, if a universe is something other than flat. But it can be determined; you might consult a mathematician who is familiar with non-Euclidean geometry. Qualitatively, the value of p in a universe with positive curvature (a 2-D version is the surface of a sphere) should be less than 3 1/7 because a circle would "bulge" in the middle. I suppose that in a universe with negative curvature (like the surface of a saddle) it should be more than 3 1/7, though this doesn't seem right to me; maybe the Euclidean value of p is an upper limit rather than a median. Anyhow, if the curvature is very gradual - as it is in our universe - the difference would not be detectable by measurement. Determinations of p to umpteen decimal places are based on Euclidean geometrical algorithms, not on measurement!The effects of a significant curvature might or might not be nothing. p does come into a great many physical laws, because the mathematics which describe them involve circles, spheres, or ellipsoids. For example, Neils Bohr calculated the radius of allowed orbits for the hydrogen atom assuming circular electron orbits, and found it to be
However, this radius is not an observable. When we go on to calculate the observable energy of the electron for a particular allowed orbit n, we find that the factor of p cancels:
In general I think this is true for most physical laws, and so the exact value of p cannot be sensitively detected by observation of the universe in most cases. (There is the famous quantum mechanical constant "h-bar" (h/2p, but again the value of p probably cancels most of the time for observable quantities.)
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