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### 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/

Associated Topics:
High School Geometry
High School Higher-Dimensional Geometry
Middle School Geometry
Middle School Higher-Dimensional Geometry
Middle School Pi

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