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

Here we will basically be discussing distances on manifolds, and what

we can learn about a manifold based on how we calculate distances on that

manifold. We start by discussing the length of a random path on a manifold.

Consider a random path on a flat sheet of paper. We can use an x-y

coordinate system to specify any point on the paper and any point on the

path. With this coordinate system in place, how can we use it to measure the

length of that random path? One way is to break up the path into tiny parts,

each of which can be approximated with a straight line segment. Then, if we

know how to measure the length of a straight line, we can measure the length

of each line segment and add them up to find the approximate length of the

path. Now, since the random path doesn't have to be very straight, the line

segments we use might not be very good at approximating the path at some

point. However, if we break up the path into smaller pieces, then the

smaller line segments should do a better job of approximating the curve and

giving us the correct length for the path. The smaller we make the line

segments, the better our approximation of the path's length will be. The

ultimate result of this idea is to figure out what the calculated length

would be if we made the line segments infinitesimally small. That would give

us the actual length of the curve.

So, the next question is this: How do we calculate the length of a very

small (infinitesimal) line segment using our x-y coordinate system? Well,

each segment is made up of a component in the x direction (dx) and a

component in the y direction (dy) as shown in Diagram 5-8. These components

represent infinitesimal distances. The length of the infinitesimal line

segment (let's call the length ds) is then given by the following (using the

Pythagorean theorem):

(Eq 5:1)

ds^2 = dx^2 + dy^2

(Note that this is the length of a straight line--a geodesic on this

manifold--between an initial and a final position which are separated by a

distance dx in the x direction and dy in the y direction.)

Diagram 5-8

y | /. | / . | ds/ . | / .dy | / . | /...... | dx ----+------------->x |

This distance between two very-nearby points is what I call the

invariant interval. Why? Well, first I need to note that there are other

types of coordinate systems one could use to locate every point on a flat

surface, and that the equation for ds in terms of small changes in each

coordinate will depend on the coordinate system you use. However, though the

form of the equation will change, the actual distance between two points on

the manifold is a physical reality which won't change. The actual interval

is independent of the coordinate system you place on the manifold.

Now, Below, I will specifically use ds as defined here (in a flat, x-y

coordinate system) to make a comparison with an invariant interval defined

using a particular coordinate system on a curved manifold. However, all the

arguments I will make can also be made using any other coordinate system on

a flat manifold and any other coordinate system on a curved manifold. I

simply use two specific ones as solid examples.

So, to demonstrate how the equation for ds will tell us everything we

want to know about a manifold, we next need to consider a curved manifold.

We will use our old friend the sphere. Let's start by defining a coordinate

system on the sphere. Picture a sphere with a great circle drawn on it.

Let's call that great circle the equator. Next, consider a point on the

equator, and call that point our origin. We want to define two independent

coordinates which will allow us to locate any point on the sphere starting

from the origin (note: by "independent coordinates" I mean that you can

always change your position in one coordinate independent of any change in

the other). So, consider some other point on the sphere (call the point

"P"), and let's explain how to get to that point using two coordinates. We

start by moving either towards the "east" or "west" from our origin in the

general direction of "P" (you can define "east" and "west" however you

wish). We move along the equator until P is directly north or south of us,

and we call the distance we move "L" (L is positive if we move east). Next,

we need to move north or south on the sphere to reach P. The distance we

move north or south to reach P will be called "H" (H is positive if we move

north). That gives us our coordinate system. Every point on the sphere can

now be represented by an L-H coordinate pair. The "grid" on the surface of

the sphere which represents this coordinate system would be made of latitude

and longitude lines such as those on a globe.

Next, we need to figure out what infinitesimal distance (ds) would be

associated with moving a small distance in L (dL) and a small distance in H

(dH). For the sake of time, I'll just give the answer here. (Note, R is the

radius of the sphere we are considering):

(Eq 5:2)

ds^2 = dH^2 + [cos(H/R)]^2*dL^2

Remember what this represents. If you start at some point (L,H) on the

sphere, and you change your L coordinate by a small amount (dL) and your H

coordinate by a small amount (dH) then the shortest distance along the

sphere between your first position and your final position would be ds. Note

that this distance depends on your H position (because of the "cos(H/R)"

part of the equation). This is an interesting point because as soon as you

start moving from one position to the next, the equation for ds becomes

slightly different. We basically think of this difference as negligible as

long as dL is very small, but, in fact, the equation is only correct when dL

is truly "infinitesimal". Such concepts are generally covered in calculus,

and for our purposes, we will just claim that the equation is practically

true as long as dL is very small.

So now we come to an important statement to be made in this section:

THE FORM OF THE INVARIANT INTERVAL FULLY DEFINES THE INTRINSIC GEOMETRY OF A

MANIFOLD. For example, what if we tried to find another coordinate system on

the sphere using two independent coordinates (a and b) such that the

invariant interval on the sphere would be given by the following:

(Eq 5:3)

ds^2 = da^2 + db^2?

Well, because that invariant interval looks just like the formula for ds on

a flat sheet of paper (ds^2 = dx^2 + dy^2), then it should be impossible for

Equation 5:3 to be the invariant interval on the sphere (no matter how we

define "a" and "b"). If I drew a grid on a flat sheet of paper and labeled

the axes "a" and "b", then Equation 5:3 would appropriately describe the

relationship between every single point on that flat manifold given the "a"

and "b" coordinate system. Thus, if I define "a" and "b" to be independent

coordinates on a sphere, and I claimed that Equation 5:3 described the

invariant interval on the sphere given those coordinates, then I'd be saying

that Equation 5:3 describes the relationship between every single point on

the sphere given the "a" and "b" coordinate system. But that's saying that

by appropriately defining "a" and "b", I can make the relationship between

all points on the sphere be just like the relationship between every point

on a flat sheet of paper. We know that physically, this simply can't be

done, because there are intrinsic ways to tell the difference between the

geometry of a sphere and the geometry of a flat sheet of paper.

You might be looking back at Equation 5:2 and thinking, "but what if I

just define a new coordinate, L' such that dL'^2 = cos^2(H/R) dL^2? Then I

get ds^2 = dH^2 + dL'^2, which looks like the invariant interval for a flat

sheet of paper." Ah, but look at your definition for dL' and notice that it

involves your other coordinate, H. You see that H and L' are NOT independent

coordinates. To be valid in our discussion here, the coordinates you use on

a manifold must be independent.

So, Considering this example of a sphere and a flat sheet of paper,

let's make some general points: First, consider some manifold, M1. On M1, we

have some (valid) coordinate system, S1. Next we consider two very-nearby

points on M1 (call the points P and Q). If we know the distance between P

and Q along each of the coordinates (like dx and dy, for example), then we

can find some function for ds (the shortest distance on M1 between the

very-nearby points) using the coordinates in S1. Now, consider a second

manifold, M2. If a (valid) coordinate system, S2, can be defined on that

manifold such that ds has the same functional form in S2 as it did using the

S1 coordinate system on M1, then the geometry of the two manifolds must be

identical.

This indicates that the geometry of a manifold is completely determined

if one knows the form of the invariant interval using a particular

coordinate system on that manifold. In fact, starting with the form of the

invariant interval in some coordinate system on a manifold, we can determine

the curvature of the manifold, the path of a geodesic on the manifold, and

everything we need to know about the manifold's geometry.

Now, the mathematics used to describe these properties involves

geometric constructs known as tensors. In fact, the invariant interval on a

manifold is directly related to a tensor known as the metric tensor on the

manifold, and we will discuss this a bit later. First, I want to give a very

brief introduction to tensors in general.

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