5.4 Manifolds, Geodesics, Curvature, and Local Flatness (General Relativity)
Description
This article is from the Relativity and FTL Travel FAQ, by Jason W. Hinson jason@physicsguy.com with numerous contributions by others.
5.4 Manifolds, Geodesics, Curvature, and Local Flatness (General Relativity)
Before we discuss space-time in the presence of gravity, we need to
understand some basic geometric concepts which we will use. We will develop
these concepts by considering normal, spatial geometry which can be fully
grasped using common sense. Applying these concepts to space-time becomes
less intuitive (in part because we still aren't that used to thinking of
time as just another dimension); therefore, developing them using normal
spatial geometry will be beneficial.
First, we introduce the term "manifold". Basically, for our purposes,
you can think of a manifold as a fancy term for a space. The space around us
that you are used to thinking of can be called a three dimensional manifold.
The surface of a sheet of paper is a two dimensional manifold, as is the
surface of a cylinder or the surface of a sphere. Much of our focus on
manifolds will involve discussing their geometry. Understanding the geometry
of a manifold means understanding the relationships between various points
on the manifold and understanding various curves on the manifold as well as
knowing how to measure distances on the manifold. Thus, we want to define a
few specific notions which will help us understand and explain the geometry
of a manifold.
So, next we look at a particular type of path on a manifold called a
geodesic. A geodesic is essentially the path which takes the shortest
distance between two points on the manifold. On a piece of paper (a flat
manifold) the shortest distance between two points is found by following the
path of a straight line. However, for a sphere, the shortest distance
between two points would be traveled by following a curve known as a great
circle. If you imagine cutting a sphere directly in half and then putting it
back together, then the cut mark on the surface of the sphere would be a
great circle. If you move along the surface of a sphere between two points,
then the shortest path you could take would lie on a great circle. Thus, a
great circle on a sphere is basically equivalent to a line on a flat
manifold--they are both geodesics on their respective manifolds. Similarly,
on any other manifold there would be a path to follow between two points
such that you would travel the shortest distance. Such a path is a geodesic
on that manifold.
Next, we introduce the concept of the curvature of a manifold. There
are two different types of curvature: intrinsic and extrinsic. To
demonstrate the difference between the two, let's first consider a surface
which has only extrinsic curvature. Imagine taking a flat sheet of paper and
rolling it as if you were making a cylinder; however, don't let the two ends
touch to complete the cylinder. Now, while this two dimensional surface will
now look curved in our three dimensional perspective, the geometry of the
surface is still the same as the geometry of the flat sheet of paper from
which it was made. If you were a two dimensional creature confined to live
on this two dimensional surface, there would be no test you could perform to
prove you weren't on a flat sheet of paper rather than this cylinder-like
surface. Now if you did complete the cylinder, then a two dimensional
creature could tell that the global topology of the situation has changed
(for example, on a complete cylinder, he could follow a particular path
which would bring him around back to where he started). However, this
doesn't change the fact that throughout the cylinder, the internal geometry
is just like the geometry of a flat sheet of paper from which it was made.
So, for a two-dimensional cylinder, its curvature is only "visible"
when viewed from a higher dimensional space (our three-dimensional space).
We only say it is curved because a line on the 2-D cylinder can bend away
from a straight line in three dimensions. However, The cylinder has no
intrinsic curvature to its geometry, so its curvature is extrinsic.
Contrast this with the surface of a sphere. You cannot bend a flat
sheet of paper around a sphere without crumpling or cutting the paper. The
geometry on the surface of a sphere will then be different from the geometry
of a flat sheet of paper. To distinctly show this, let's consider a couple
of two dimensional creatures who are confined to the surface of a sphere.
Say that they stand facing the same direction at a given, small distance
apart from one another on the two dimensional surface, and then they begin
walking in the same direction parallel to one another. As they continue to
walk beside one another, each will continue in what seems to him to be a
straight line. If they do this--if each of them believes that he is
following a straight line from one step to the next--then each will follow
the path of a geodesic on the sphere. As we said earlier, this means that
they will each follow a great circle. But if they each follow a great circle
on the surface of a sphere, then they will each eventually notice that their
friend walking next to them is moving closer and closer, and eventually they
will meet. Now, they started out moving on parallel paths, and they each
believed that they were walking in a straight line, but their paths
eventually came together. This would not be the case if they performed this
experiment on a flat sheet of paper (or on a cylinder). Thus, creatures who
are confined to live on the two dimensional surface of a sphere could tell
that the geometry of their space was different from the geometry of a flat
piece of paper (even though they couldn't "see" the curvature because they
are trapped in only two dimensions). That intrinsic difference is due to the
intrinsic curvature of the sphere's surface.
This, then, is what we want to note about curvature: There are two
types of curvature, extrinsic and intrinsic. Extrinsic curvature is only
detectable from dimensions higher than the dimension of the manifold being
considered. Intrinsic curvature can be detected and understood even by
creatures who are confined to live within the dimensions of the manifold.
Thus, just because a manifold may looked "curved" in a higher dimension,
that doesn't mean that its intrinsic geometry is different from that of a
flat manifold (i.e. its geometry can still be flat--like the cylinder).
Thus, the test of whether a manifold is intrinsically curved does not have
anything to do with higher dimensions, but with experiments that could be
performed by beings confined on that manifold. (For example, if two parallel
lines do not remain parallel when extended on the manifold, then the
manifold possesses curvature). This is important to us in our discussion of
space-time in the presence of gravity. It means that the curvature of the
four dimensional manifold of space-time in which we live can be understood
without having to worry about or even speculate on the existence of any
other dimensions.
As a final note in this introduction to manifolds, I want to mention a
bit about local flatness. Note that even though a manifold can be curved, on
a small enough portion of that manifold, it will be fairly flat. For
example, we can represent a city on our curved Earth by using a flat map.
The map will be a very good representation of the city because it is a very
small piece of the curved manifold. Earlier I mentioned that over a small
enough piece of space-time in the presence of gravity, you can define a
frame of reference which is still very similar to an inertial reference
frame in special relativity. This gives an indication as to why the geometry
of space-time in special relativity is that of a flat manifold, while with
general relativity, space-time is said to be curved in the presence of
gravity. Still, the space-time of any observer being acted on only by
gravity is LOCALLY flat.
Later we will see how the concepts discussed here will help us in
explaining gravity and relativity. Next, however, we want to discuss another
property of manifolds which itself will tell us everything we want to know
about the geometry of a particular manifold. We will call this property the
invariant interval.
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