From: ted@physics2 (Emory F. Bunn)
Newsgroups: sci.physics
Subject: Re: Curved space?
Followup-To: sci.physics
Date: 21 Dec 1992 05:57:31 GMT
Organization: University of California, Berkeley
Distribution: world
NNTP-Posting-Host: physics2.berkeley.edu

In article <BzL7As.A81@usenet.ucs.indiana.edu> mkohlhaa@silver.ucs.indiana.edu (mike) writes:
>I've heard "the shortest distance between two points is not necessarily
>a straight line."  I'm not as knowledgable about physics as most of you
>probably are, and would like a explanation to this interesting
>statement. 

I'll take a shot at it.  First let me restate the question, and then I'll
give some answers.

First of all, we can surely all agree that in Euclidean space (that is,
the space in which the ordinary rules of geometry, including the ones
about parallel lines, hold), a straight line IS the shortest path between
two points.  However, people have known for a while that there are geometrical
systems one can design that do not follow the Euclidean rules, and 
people have known since the dawn of general relativity that our own
Universe may have such a non-Euclidean geometry.  So it's reasonable to
ask the question, "If our Universe has a non-Euclidean geometry, is
it possible that a straight line isn't always the shortest path between
two points?"

Well, first of all, we need to define some terms.  What is meant by "distance"
in this funny geometry?  Well it turns out that within the context of
ordinary general relativity, space has a "metric" on it, which means
a rule assigning a length to any curve.  So distance is well-defined.

(Note:  Some people may be bothered by the fact that I'm talking about
space instead of spacetime, and not worrying about the fact that the
metric on spacetime can go negative.  For our purposes, I think it's OK
to just talk about individual spacelike slices through spacetime, in which
case each 3-dimensional "space" has an ordinary positive definite metric.)

How about "straight line?"  How should we define that?  There are two
ways people do it, and they turn out to be equivalent.  (1) You can say
that a straight line is a curve whose tangent vector can be parallel
transported along the curve.  This sounds obscure.  What it means is that
a straight line has the following property:  If you draw its tangent vector
at a particular point, and then imagine moving that tangent vector along
the curve, keeping it pointing in the same direction at all times, then
when you're done the vector will still be tangent to the curve.  This
is just another way of saying that a straight line "keeps pointing
the same direction" all the time:  That is, it's straight.

(2) You can also define a straight line by saying "It's the shortest curve 
connecting two points, as long as those points are sufficiently close 
together."  That last clause sounds a little bit strange.  Here's why it's
there.  Suppose you live on a sphere (which, to a good approximation,
you do).  Now suppose you start from some point on the equator and start
walking East.  You are walking in a straight line.  Imagine you keep
going until you've gone almost all the way around the sphere.  Then
your path is not the shortest distance between your starting point and
your finishing point, since a path that went the other way around the
sphere would be much shorter.  But your path is still considered a straight
line.  The "sufficiently close" clause in the definition is there to make
curves like that one count as straight lines:  If you divide your path
around the equator up into a bunch of little segments, each segment will
be the shortest path between its endpoints, even though the whole path
isn't.

Well now we have an answer to your question:  In non-Euclidean geometries,
it is possible to make a straight line which is not the shortest distance
between two points.  In fact, if the Universe is closed, then the
geometry of space at each instant of time is like the geometry of a 
three-dimensional sphere.  In that case, there would be paths like
the equator path described above, which are not the shortest path
between their endpoints.  However, the lines that have that property
would all be very long.  Lines of "reasonable" sizes (much smaller than
the size of the Universe) would all be the shortest paths between their
endpoints.

Of course, the Universe probably isn't closed anyway, and if it isn't, then
all straight lines would be shortest paths.

-Ted