Einstein, Curved Space, and Pi
Date: 10/09/1999 at 13:35:35 From: Steve Subject: Pi and curved space... I am in 8th grade and recently became interested in Einstein's General and Special Relativity theories. (Just so you know my question is NOT about Einstein's Relativity but how it relates to pi). In Einstein's general relativity, space is described as being curved. If space is curved, then Euclidean geometry doesn't apply. And since one of the parts to Euclidean Geometry is that the relation between a circle's circumference and its diameter (pi) is always 3.1415926535897923... doesn't that mean that the value of pi changes when space is curved? Doesn't that also mean that sometimes the value of pi can actually be rational? Do you know of any methods to calculate pi in curved space? Thanks for your help.
Date: 10/10/1999 at 21:49:27 From: Doctor Rick Subject: Re: Pi and curved space... Hi, Steve. In general relativity, it is space-time (4-dimensional "space"), not just space, that is curved. But that concept is very hard to grasp (I'm not sure I can really picture it), so we get at the idea by considering curved space. Even curved 3-dimensional space is hard to picture. It's easier to use a 2-dimensional example. The surface of a sphere is one good example; it has a constant curvature. But if you examine what happens to circles on the surface of a sphere, you may be surprised. What you'll find is that pi doesn't just have a different value, it actually has no meaning. Pi is defined as the ratio of the circumference of a circle to its diameter. In flat 2-dimensional space (Euclidean geometry), it can be proved that this ratio is the same for every circle. Once you know this, you can calculate the value of the ratio - not exactly, but to any desired precision. You can also prove that pi is also the ratio of the area of a circle to the square of its radius, and so forth. What happens on the surface of a sphere? Instead of straight lines, we have "geodesics" - the shortest distance between two points without leaving the surface. A geodesic on the surface of a sphere turns out to be a portion of a great circle - a circle, like the equator or a line of longitude on a globe, whose center is at the center of the sphere. Look at a globe. It probably has a set of circles of different radii, all centered at the North Pole. These are lines of latitude. The line at 45 degrees of latitude, for instance, has a radius that is the portion of any line of longitude between 45 degrees and the North Pole. How long is this radius? It's half the distance from pole to equator, or 1/8 of the full line of longitude. The length of a line of longitude is the same as the length of the equator. Thus the circle's diameter is 1/4 of the length of the equator. How long is the circumference of the circle? A little geometry will convince you that it is the length of the equator divided by the square root of 2. Now, what is the ratio of the circumference of this circle to its diameter? It's 1/sqrt(2) : 1/4, or 2*sqrt(2) = 2.8284... That's noticeably less than pi. But does this mean that "pi" on the surface of a sphere is 2.8284? No. Consider another circle on the globe, the equator. Its radius is 1/4 the length of the equator. Its circumference is the length of the equator. So the ratio of the circumference of this circle to its diameter is exactly 2. You might say that you were right, it is possible for "pi" to be rational in a curved space. But if "pi" depends on the radius of the circle, then the ratio of circumference to diameter is not fixed, and if the ratio isn't constant, there is no meaning in the definition of pi as THE ratio of circumference to diameter. That's a pretty interesting result, even though it's a negative one. What it says is that we shouldn't take even the existence of pi for granted: there are geometries in which there is no such thing as pi. I went a little fast at a few points. If you can't figure out what I am saying, please write back. Understanding curved space, even in 2 dimensions, is a challenge. - Doctor Rick, The Math Forum http://mathforum.org/dr.math/
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