This article is from the Relativity and FTL Travel FAQ, by Jason W. Hinson jason@physicsguy.com with numerous contributions by others.

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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