Chronology Current Month Current Thread Current Date
[Year List] [Month List (current year)] [Date Index] [Thread Index] [Thread Prev] [Thread Next] [Date Prev] [Date Next]

Re: General Relativity and Pi



Chuck Britton asked:
...
Does this imply that here, near the surface of the earth, the value of pi
is not exactly equal to the value currently calculated to a gazillion
decimal places?
(These calculations are, of course, based on Euclidian Flat space assumptions.)

This is an excellent question. Since the definition of [pi] *is* for
circles in a 'flat' space, the value of [pi] can't, therefore, change by
going to a curved space. What *can* happen, though, is that the quotient
of a circle's circumference divided by its diameter may no longer *be* [pi]
in a curved space. The amount of discrepancy would increase with the size
of the circle. In the asymptotically small limit of infinitesimal-sized
circles the ratio remains [pi] and all flat-space geometric relationships
hold in the curved space to whatever degree of accuracy you require for
sufficiently small geometric figures.

For instance, consider the curved 2-d space of an idealized perfectly
spherical earth's surface. If we draw a tiny circle around the south pole
on the ground and measure the circumference to diameter ratio (C/D) for
this circle we would get a number very close to [pi]. If we kept making
ever-larger circles concentric to the south pole on the earth's surface we
would begin to notice that the C/D ratio is less than [pi] and this ratio
continues to shrink with increasing circle size. As a matter of fact, in
this example C/D = [pi]*sin(x)/x where we defined x == D/(earth's diameter).
Notice that when the circle is so large that it is the earth's equator that
then x = [pi]/2 and we have C/D = 2. This is the largest sized circle that
can exist on the sphere. If we continue to increase the circle's radius its
circumference begins to shrink. Once the x value reaches [pi] the circle
has shrunk to a point at the north pole and the circle's radius is then
half to the earth's circumference. In this case the entire finite
spherical surface (except for the north pole point) is 'inside' the circle.

For the curved space in the vicinity of the earth due to the GR curving
effects of the earth's mass, the amount of curvature is *very* small. In
fact, if we drew a perfect circle around the earth's equator and then
measured its circumference and divided the result by [pi], then the quotient
wound be about 3 mm less than the actual diameter of the circle measured
by boring a hole through the earth. This discrepancy is about 1 part in
4.3 x 10^9. If much smaller sized circles were drawn in the vicinity of
the earth the effect would be correspondingly smaller. It should be noted
that the effects of the earth's exterior Schwarzschild geometry on various
sized circles also depends on both the orientation of those circles wrt the
earth's center and on their distance from the earth.

How many decimal places out would we have to go to see the difference?

For the equatorial circle given above the effect shows up in the 10th
sig fig. For smaller circles the effect would show up in a greater
sig fig.

Couldn't this '# of decimal places' be a somewhat 'conceptual' measure of
the local curvature of space/time?

The relative rate at which the discrepancy in the circle ratio:
(1 - [pi]*D/C) increases with circle size for small circles *is* a local
measure of the curvature of the space. For each orientation of a circle it
measures a different component of the curvature of the space (I think it
actually measures a component of the Ricci tensor for a spatial section
submanifold of the spacetime, but I could be wrong about this).

This idea just occured to me when a student asked a math-list how pi is
actually determined. (and the expected answer of various infinite
expansions was given)

Using certain recursive modular functions is more efficient than evaluating
infinite series for this purpose.

David Bowman
dbowman@gtc.georgetown.ky.us