From: ted@rosencrantz.stcloudstate.edu
Subject: Re: Straight line
Date: Thu, 3 Feb 2000 04:22:21 GMT
Newsgroups: sci.physics.research
Summary: [missing]

In article <3891BE81.354A91E6@csc.albany.edu>,
Frank Wappler  <fw7984@csc.albany.edu> wrote:
>MMM wrote:
>
>> Could anybody tell me how a straight line is defined
>> in the context of curved space?
>
>Following Heron's definition, a line is "straight"
>if for all triples A, B, C of its elements (points)
>
>(distance( A, B ) + distance( B, C ) - distance( C, A )) *
>(distance( B, C ) + distance( C, A ) - distance( A, B )) *
>(distance( C, A ) + distance( A, B ) - distance( B, C )) = 0;

This is certainly not the definition people typically use nowadays to
define a straight line, either in curved space or flat!  It's a
correct (but needlessly complicated) definition in good old Euclidean
space, but it's not even correct on a general curved manifold.

First, let's unpack what this definition is saying.  I'll use the more
compact notation AB to mean distance between A and B.  Then Heron's
definition says a line is straight if, for any 3 points on the line,
either AB + BC = AC, BC + CA = BA, or CA + AB = CB.  The first
equality holds if B is between A and C, the second if C is between A
and B, and the third if A is between C and B.

Suppose we want to apply this definition to see if a particular path
in some curved space is straight.  First, we have to know what the
definition of "distance" is.  At least for a physicist, "distance" is
typically defined in terms of a Riemannian metric.  If you don't know
what a Riemannian metric is, don't worry about it.  The key point here
is that it lets you define the length of a path.  If you want to
define the distance between two points, you imagine all possible paths
joining those two points and pick the shortest one.

So in order to apply Heron's rule, we already need the notion
of the shortest path between two points.  In that case, how about
just taking that to be the definition of a straight line instead
of Heron's definition!  Indeed, that's what most people who have
responded in this thread have suggested.  That's certainly a better
answer to MMM's question than Heron's definition, although neither
is quite correct as they stand.

To see why, consider a sphere.  The "straight lines" (geodesics) on a
sphere are great circles.  A great circle does not satisfy Heron's
definition.  Let A, B, C be three points equally spaced 120 degrees
apart around the great circle.  Then AB, BC, and AC are all equal to
each other (2 pi / 3 if it's a unit sphere), so no two add up to equal
the third.

Even the shortest-distance definition of a straight line segment isn't
great in this situation.  Imagine a path that starts at A and follows
a great circle almost (but not quite) all the way around the sphere,
ending at a point B that's quite near A.  This segment clearly isn't
the shortest path from A to B, but it is a geodesic.

Both of these definitions can be made OK if you throw some words like
"sufficiently nearby" into them: for a sufficiently small segment of
any geodesic, it is true that the geodesic does give the shortest path
between all pairs of points.  

In fact, though, most mathematicians and physicists adopt a different
definition for a geodesic.  Instead of talking about the distance
between two points, they define it in terms of the notion of "parallel
transport."  Essentially, a path on a curved surface is a geodesic if
a tiny ant walking along that path wouldn't have to turn either left
or right to stay on the path.  This turns out to be equivalent to the
shortest-distance definition.  In general, these days, people take the
parallel-transport property to be the definition of a geodesic and
prove the shortest-distance property as a theorem, although I suppose
there's no reason you couldn't do it the other way around.

For more detail, you might want to look at John Baez's relativity
tutorial, which talks quite a bit about parallel transport
and geodesics.

http://math.ucr.edu/home/baez/gr/gr.html

>and, if the measured pairwise distances are not unidirectional,
>i.e. if distance( A, B ) =not_necessarily= distance( B, A ), etc.

This has no correspondence with any notion of distance I've ever
heard of.

-Ted