5.5 The Invariant Interval (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.
5.5 The Invariant Interval (General Relativity)
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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