I write this note because the subject matter as treated on the web is all rather advanced. I want to start at an intuitive level but hopefully advance to explaining some mathematics necessary to understanding General Relativity.

I hesitate in this effort as I remember a book I studied as a teen ager. I learned the meaning of all the sentences but still did not understand tensors. I began to understand tensors after reading a short monograph which I cannot now locate by Roger Penrose on the Christoffel symbol Γijk.

The Manifold

Imagine a shiny yet irregular potato—so shiny that an ant can recognize its reflection. Its surface is a 2 dimensional (2D) manifold. The potato is 3D and its 2D surface is embedded in 3D. We will study manifolds without knowing if they are embedded in any higher dimension; we study thus their intrinsic geometry. We speak generally about manifolds of n dimensions for n > 0. We follow the historic route, mostly developed by the end of the 19th century. The Earth, imagined as a perfect ellipsoid, is a manifold that will provide many motivating examples.

To locate a point on a manifold takes n real numbers, just as locating a point in 3 dimensions requires 3 numbers, or 2 reals (latitude and longitude) locate a point on the Earth. Navigators flying near the poles of the Earth sometimes use a nearly rectangular grid near the poles in preference to polar coordinates. There is a patch near each pole where the rectangular grid is used. Mathematicians do the same trick in reasoning about manifolds so as to avoid a great deal of extra reasoning about what happens just at the pole. They speak of an atlas which consists of a set of charts. Each chart is a bijective map between a subset of the manifold and a hypercube in Rn. Each point on a chart corresponds to just one point on the manifold and every point on the manifold is mapped by at least one chart. It appears at most once on a given chart. Even with the polar grid, however, the X and Y coordinates on the Earth below are not quite perpendicular, especially as you get farther form the pole. The Earth is not flat.

Each of these charts will be flat n dimensional maps of a curved portion of the manifold. A chart comes with a set of n coordinates. When we imagine these coordinates on the manifold, as carried over by some particular chart, they may be distorted, just as lines of longitude are closer near the poles. It is worse; they are not in general even perpendicular to each other.

G.B.H. Riemann worked much of this out in about 1850. They were built on earlier ideas from Gauss but Gauss stuck to 2D surfaces embedded in 3D. If you hold n−1 of the coordinates constant and change the other a small amount you move along the surface of the manifold a short distance. If you fix each coordinate except xi, and you increase xi by d then you get a vector d ei. As d goes to 0 ei tends towards a limit. It is a shaggy dog story as to why we use superscripts to name the coordinates. But it is a very good reason.

If the manifold were flat for this chart then we could arrange that the vectors ei could be perpendicular and of length one. No such luck in general. For some mathematical purposes we can get this far with no notion of distance between even nearby points on the manifold. If we have such a notion then we collect for each combination of i and j the dot product of ei and ej and get the metric tensor
gij = ei • ej . gij will vary from point to point but it gives us the shape of this mapped portion of the manifold and along with such information from other maps, the shape of the entire manifold.

You might wonder what vector space this vector is in. Mathematicians speak of a tangent space, sort of as if it were a tangent to the manifold in some embedding space. If you study tensor bundles they make all this talk legitimate without finding an embedding space. We speak sloppily here as if the vector were in the manifold.

For the Earth the metric tensor (assuming a sphere) is
gij =
10
0sin2(latitude)

Some Formulae

Here is a famous formula:
ds2 = gijdxidxj
This is a mantra and we shall look closely at it. Imagine a curve in the manifold and a short length, ds, of the curve connecting the points A and B. A has coordinates <x0, x1, ... > and B has coordinates <x0+dx0, x1+dx1, ... >. In the equation ds2 is indeed the square of the length. There are two omitted summation signs on the right, omitted by Einstein’s summation convention. One says to sum over all n values of i, and the other over all values of j to produce the sum of n2 products. Note that if the coordinates were orthogonal to each other then gij=0 unless i=j and this would be exactly Pythagoras’ formula. The formulation in generalized coordinates is no worse than familiar coordinates. With this formula we can compute the length of a curve and thus speak of a geodesic that minimizes the length. We can compute the dot product of two vectors expressed as linear combinations of the e’s.
If p = pi ei and q = qi ei
then p•q = gij pi qj. This is all we need for small scale geometry.

Coordinates twist and turn

If we use polar coordinates, r and θ to draw a straight line we must account for the fact that the coordinates turn. If we begin the line by increasing only θ then soon we must start increasing r, lest we draw a circle. How much and how soon? The reader probably knows how to write a formula for r in terms of θ but if we were on an curved surface and had to discover how to navigate by enquiring about the value of gij at each point as we come to it what would we do? Suppose we are drawing a parametric curve with parameter λ. We begin at a known location xi with heading ∂xi/∂λ. How fast must we change the heading as expressed in our local coordinates? In other words how do we compute ∂2xi/∂λ2 which tells us how fast to turn relative to the coordinates, to keep from turning in reality?
Behold ∂2xi/∂λ2 = Γijk∂xj/∂λ ∂xk/∂λ
but what is this Γ thing?
I need some pictures! Otherwise I should just point them to Penrose.