From: zowie@daedalus.stanford.edu (Craig "Powderkeg" DeForest)
Subject: Re: Curved space?
Organization: Stanford Center for Space Science and Astrophysics
	<1992Dec21.052433.8033@hubcap.clemson.edu>
Date: 21 Dec 92 14:51:36

In article <bar> 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. 

It all depends on how you define `straight line'.  The most straightforward
way is to *define* a straight line to be the shortest distance between two
points, come Hell or high water.  Then the definition rests on your idea 
of what constitutes distance.  To construct a straight line, you start at
one endpoint and construct a random curve between the two points, integrating
the metric as you go, then adjust the curve until you get a minumum.

Now, we're used to dealing with 3D Euclidean space on a day-to-day basis, so
we're used to the Pythagorean metric ds = sqrt(dx^2 + dy^2 + dz^2) so that
the shortest distance between any two points is just the straight integral:
sqrt( (\delta x)^2 + (\delta y)^2 + (\delta z)^2).  

This particular choice of metric makes lines behave a certain way:  
triangles always have 180 degrees of internal angle; you can construct a new
straight line by displacing every point on an old one by the same distance 
and direction; etc.

In common usage, when we describe a line as straight, we are often referring
to these secondary attributes, rather than the strict minimization of 
distance.

When we consider the spaces that result from *other* metrics than the 
Pythagorean metric, we're frequently at a loss to describe the secondary 
behaviour of `straight' lines in the alternate space.  For example, consider
a 2-D space in which all distances are longer near the origin:

	ds = sqrt(dx^2 + dy^2) * (1 + 1/(1+x^2+y^2))

Here we've multiplied the normal metric by a term that is near 1 far from the
origin, and really big near the origin.  Far from the origin, this space
behaves a lot like Euclidean space.  However, near the origin strange things
happen.  For example, it's possible to make a closed figure with only two
endpoints.  I can't draw it in ASCII, but imagine an endpoint at (1,0) and
another at (-1,0).  If you draw a Euclidean straight line between them, along
the X-axis, it'll go through the origin, and be extra long because of the
penalty near the origin.  So the Euclidean line isn't `straight' in this
space.  You need to bend it around, either up or down, to get away from the
origin.  You're done when the amount of extra Euclidean distance you get by
going out of your way, is equal to the savings you get by not being near the
origin.  There are two such lines, one above and one below the origin.  If you
draw both of 'em, you get a diagram that looks like a flying saucer or a
galaxy seen edge on.  A circle around the origin, will have a different 
radius (longer) than in a Euclidean space.

So the figure looks curved on your diagram, even though it's got two straight
lines on it.  There're two reasons:  first, you've chosen `strange' coordinates
to describe the space (Cartesian coordinates probably aren't the most natural
for this space); and second, straight lines behave differently than you're
used to, because of the different metric.

One way of understanding curved spaces is to `embed' 'em in a larger space.
The idea is that (with exceptions), for a 2-D space, you can think of points 
in the space as being on a curved 2-D surface sitting in a 3-D space. 
You have the 3D Euclidean metric and an equation of constraint that describes
the surface.  Plugging-in the equation of constraint, into the metric, 
yields the 2-D `curved' metric.  For example, with the one above:

	ds = sqrt(dx^2 + dy^2 + (dx^2+dy^2)/(1+x^2+y^2))

we can write dz^2 = (dx^2+dy^2)/(x^2+y^2) and notice that our new expression
for ds is just the 3D-Euclidean metric, subject to the constraint above, which
defines a curved 2-D surface in the 3-D Euclidean space.

To figure out what it looks like, we examine the differential
constraint.  First notice that everything is radial, so we just need
to figure out how z varies with r.  Remembering that r = sqrt(x^2+y^2), 
you get dz = dr/(1+r), which integrates to z = ln(r+1).

So our non-Euclidean space can be described as a curved 2-D surface, embedded
in a larger 3-D space with the Euclidean metric (which we like).  Did you see
the movie, `The Black Hole'?  Remember those green line diagrams at the
beginning, with a flat plane and a big stretched hole in it?  That's what
we're thinkin' about, except with a floor at the bottom of the hole.

Straight lines in the 2-D space are not straight lines in the 3-D embedding
space. 

Oh yeah -- another simple 2-D curved space is described by the surface of the
Earth.  It's embedded in 3-D space as a sphere.  On small enough scales, 
it's flat, but on large enough scales, the curvature matters, so that, eg,
it's possible to draw a triangle on the surface of the Earth, with a 
total interior angle of 270 degrees.  But, within the space defined by the
Earth (ie walking on its surface), the shortest distance to, say, Sweden
*is* a straight line:  you just point yourself towards Sweden, and
walk that way.
--
DON'T DRINK SOAP! DILUTE DILUTE! OK!