The section above is still very generic and little of it depends on
whether the tensors are three or four or ten dimensional. We now need
to make them work for the *specific* geometry we are interested in,
which is one where we will ultimately be seeking transformations that
preserve the *invariant interval*:
(ds)^2 = (dx^0)^2 - (dx^1)^2 - (dx^2)^2 - (dx^3)^2
as this is the one that directly encodes an invariant speed of light.

From this point on, we must be careful not to confuse and , etc. Contravariant indices should be clear from context, as should be powers. To simplify life, algebraically indices are always greek (4-vector) or roman italic (3-vector) while powers are still powers and hence are generally integers.

Let us write this in terms of only contravariant pieces
. This
requires that we introduce a *relative minus sign* when contracting
out the components of the *spatial part of the differential only*.
We can most easily encode this requirement into a special matrix
(tensor) called the *metric tensor* as:
(ds)^2 = g_&alpha#alpha;&beta#beta; dx^&alpha#alpha;dx^&beta#beta;
The tensor
obviously satisfies the following property:
g_&alpha#alpha;&beta#beta; = g_&beta#beta;&alpha#alpha;
(that is, it is symmetric) because the multiplication in the Einstein
summation is ordinary multiplication and hence commutative. It is
called the **metric tensor** because it defines the way **length**
is **measured**.

At this point if we were going to discuss general relativity we would
have to learn what a *manifold*Manifold is. Technically,
a manifold is a coordinate system that may be curved but which is *locally* flat. By locally flat I mean very specifically that one can
cover the entire space with ``patches'' in the neighborhood of points
where the coordinate system is locally Euclidean (e.g. Cartesian). An
example of a curved space manifold is the surface of a sphere (think the
surface of the earth). When we look down at the ground beneath our
feet, it looks quite flat and we can draw triangles on it that appear to
have interior angles that sum to
and we can draw a map of (say)
our county that more or less accurately encodes distances on the ground
in terms of distances measured on the map. However, if we take too *big* a patch all of this breaks down. The angles in a triangle sum to
strictly *more* than
radians. Maps have to be distorted and
chopped into pieces to correctly represent distances on the ground as
distances on the flat 2-dimensional map. This is how a manifold works
- we can work with it in the *local neighborhood* of any point as
if it is flat, but if we go too far we have to work harder and correct
for its curvature, where ``too far'' is obviously defined in terms of
the scale of its curvature and some common sense.

General relativity introduces the hypothesis that gravitational
fields bend space-time. However, this bending is very, very slight
unless one is in a very *strong* gravitational field, and this
bending preserves a local smoothness of space-time so that space-time,
although it is no longer strictly Euclidean, is still a manifold and we
can do all sorts of transformations in a very general way as long as we
restrict the results to a locally flat patch.

In our discussion of *special* relativity we will assume from the
beginning that our space-time *is* flat and not bent by strong
gravitational fields. In this case the metric tensor can be expressed
in a very simple form. We will use the Lorentz metric (as opposed to
the Minkowski metric that uses
instead of
). Using our
definitions of the
coordinates,
in the
differentials above is just:
g_00 = 1, g_11 = g_22 = g_33 = -1
and we see that it is not just symmetric, it is diagonal.

The contravariant and mixed metric tensors for flat space-time are the same (this follows by considering the coordinate transformation matrices that define co- and contra-variance): g_&alpha#alpha;&beta#beta; = g_&alpha#alpha;^&beta#beta;= g^&alpha#alpha;&beta#beta; . Finally, the contraction of any two metric tensors is the ``identity'' tensor, g_&alpha#alpha;&gamma#gamma;g^&gamma#gamma;&beta#beta; = &delta#delta;_&alpha#alpha;^&beta#beta;= &delta#delta;_&alpha#alpha;&beta#beta; = &delta#delta;^&alpha#alpha;&beta#beta; .

Since we want
to be (to contract to) a scalar, it is clear
that:
x_&alpha#alpha;& = & g_&alpha#alpha;&beta#beta; x^&beta#beta;

x^&alpha#alpha;& = & g^&alpha#alpha;&beta#beta; x_&beta#beta;
or the metric tensor can be used to raise or lower arbitrary indices,
converting covariant indices to contravariant and vice-versa:
F^&mu#mu;&alpha#alpha;&nu#nu; = g^&alpha#alpha;&beta#beta; F^&mu#mu;&nu#nu;_&beta#beta;
This is an important trick! Note well that in order to perform a
contraction that reduces the rank of the expression by one, the indices
being summed *must* occur as a co/contra pair (in either order). If
both are covariant, or both are contravariant, one or the other must be
raised or lowered by contracting it with the metric tensor before
contracting the overall pair! We use this repeatedly in the algebra in
sections below.

Finally we are in a position to see how covariant and contravariant vectors
differ (in this metric). We have already seen that ``ordinary'' vectors must
linearly transform like contravariant vectors. Given a contravariant vector
we thus see that
A_0 = A^0, A_1 = -A^1, A_2 = -A^2, A_3 = -A^3
or
A^&alpha#alpha;= (A^0,**A**), A_&alpha#alpha;= (A^0,-**A**).
Covariant vectors are just spatially inverted contravariant vectors. Note
that this definition, together with our definition of the general scalar
product, reconstructs the desired invariant:
B ·A = B_&alpha#alpha;A^&alpha#alpha;= (B^0 A^0 - **B** ·**A**)

This tells us how ordinary quantities transform. However, we are also
interested in how tensor differentials transform, since these are involved in
the construction of a dynamical system. By considering the chain rule we see
that
^&alpha#alpha; = x^&beta#beta;^&alpha#alpha;
x^&beta#beta;
or, differentiation by a **contra**variant coordinate transforms like a **covariant vector operator**. This is more or less the definition of
covariant, in fact. Similarly, differentiation with respect to a **co**variant vector coordinate transforms like a **contravariant
vector operator**. This also follows from the above by using the metric
tensor,
x_&alpha#alpha; = g_&alpha#alpha;&beta#beta;
x^&beta#beta; .

It is tedious to write out all of the pieces of partial derivatives w.r.t.
various components, so we (as usual, being the lazy sorts that we are)
introduce a ``simplifying'' notation. It does, too, after you get used to it.
&part#partial;^&alpha#alpha;& = & x_&alpha#alpha; = ( x^0, -
)

&part#partial;_&alpha#alpha;& = & x^&alpha#alpha; = ( x^0, +
) .
Note that we have cleverly indicated the co/contra nature of the vector
operators by the placement of the index on the bare partial.

We cannot resist writing down the 4-divergence of a 4-vector:
&part#partial;^&alpha#alpha;A_&alpha#alpha;= &part#partial;_&alpha#alpha;A^&alpha#alpha;=
A^0x^0 + ·**A** = 1c
A^0t + ·**A**
which looks a *lot* like a continuity equation or a certain
well-known gauge condition. (Medidate on just what
would need
to be for either of these equations to be realized as a four-scalar).
Hmmmmmm, I say.

Even more entertaining is the 4-Laplacian, called the D'Lambertian operator:
& = & &part#partial;_&alpha#alpha;&part#partial;^&alpha#alpha;= ^2 x^02 -
&nabla#nabla;^2

& = & 1c^2 ^2 t^2 - &nabla#nabla;^2
which just happens to be the (negative of the) **wave operator**!
*Hmmmmmmmm!* By strange coincidence, certain objects of great
importance in electrodynamics ``just happen'' to be Lorentz scalars!
Remember that I *did* say above that part of the point of
introducing this lovely tensor notation was to make the various
transformational symmetries of physical quantities manifest, and this
appears to be true with a vengeance!

That was the ``easy'' part. It was all geometry. Now we have to do the messy part and derive the infinitesimal transformations that leave scalars in this metric invariant.