Recall that a great deal of simplification of the kinematics of
classical non-relativistic mechanics occurs when one considers the *group structure* of transformations with respect to the underlying
coordinates. Specifically, the group of **inversions, translations**
and **rotations** of a given coordinate system leave the **norm**
(length) of a given vector invariant. These transformations form the
Euclidean group in three dimensions, E
.

For those of you who led deprived childhoods, a **group**
is a
set of mathematical objects
with a rule of composition, or
group product,
such that:

- Every product of a pair of elements in the group is also in the
group. That is, if
then
.
This property is called
**closure**. - The group must contain a special element called the
**identity**such that for all . - Every element of the group
must have an
**inverse**, also in . If then such that . - The group product must be
**associative**. That is, .

If the group product **commutes** (
) the group
is said to be **Abelian**Abelian group; otherwise the group
is said to be **non-Abelian**, which is sensible enough. A **Lie**
group is a **continuous** groupLie group, such as the group
of infinitesimal transformations. It necessarily has an uncountable
infinity of elements. There are also discrete (but countably infinite)
groups, finite groups, and everything in between. There are also
``semi-groups'' (which do not, for example, contain an inverse).
Finally, one can construct ``non-associative'' structures like groups
from non-associative algebras like the octonions. Multiplication over
the reals forms a continuous Abelian group. Rotations form a
non-Abelian Lie group. Multiplication over rational numbers forms a
countably infinite group. The set of rotations and inversions that
leave a square invariant form a finite (point) group. The
``renormalization group'' you will hear much about over the years is not
a group but a semi-group -- it lacks an inverse.

However, our purpose here is not, however, to study group theory *per se*. One could study group theory for four years straight and still
only scratch the surface. It is somewhat surprising that, given the
importance of group theory in physics, we don't offer a single course in
it, but then again, it's not *that* surprising...

With that in mind, we can decide what we are looking for. We seek initially
the set of transformations in four dimensions that will leave
s^2 = x_0^2 - (**x** ·**x**)
invariant for a single event
with respect to a particular coordinate
origin. These transformations form a group called the **homogeneous
Lorentz group**. It consists of ordinary rotations in the spatial part,
the Lorentz transformations we have just learned that mix space and
time, and several discrete transformations such as space inversion(s)
and time inversion.

The set of transformations that leave the quantity
s^2(x,y) = (x_0 - y_0)^2 - { (x_1 - y_1)^2 + (x_2 - y_2)^2 + (x_3 -
y_3)^2) }
invariant form the **inhomogeneous Lorentz**Lorentz group,
or **Poincaré group**. It consists of the homogeneous group
(including the ``improper'' transformations that include spatial
reflection and time reversal) and *uniform translations of the
origin*. If anyone cares, the Lorentz group is the generalized
orthogonal group O(1,3). The proper subgroup of the Lorentz group (the
one that is simply connected spatially (no odd inversions) and contains
the identity) is SO(1,3) the special orthogonal group. If time's
direction is also preserved we add a +, SO
(1,3). This nomenclature
is defined here for your convenience but of course the wikinote
reference contains active links to a lot of this in detail.

We will define
to be the norm of relativistic space-time.
This quantity may be considered to be the invariant ``distance''
(squared) between two events,
and
, and of course is one of the
fundamental objects associated with the construction of differentials.
Since quantities that are unchanged by a geometric transformation are
called **scalars** it is evident that
is a 4-scalar. Since
the first postulate states that the laws of physics must be invariant
under homogeneous (at least) Lorentz transformations, they must
ultimately be based on Lorentz scalars. Indeed, the Lagrangian
densities upon which field theories are based are generally constructed
to be Lorentz scalars. This is a strong constraint on allowed theories.

These scalars are, however, formed out of 4-vectors (as we see above) or,
more generally, the contraction of 4-tensors. We must, therefore, determine
the general transformation properties of a tensor of arbitrary rank to
completely determine a theory. In the part of this book devoted to
mathematical physics is an entire chapter that discusses tensors, in
particular the definitions of covariant and contravariant tensors, how
to contract (Einstein sum) pairs of tensors to form tensors of lower
rank, and the role of the metric tensor in defining tensor coordinate
frames and transformations thereupon. We will *not repeat* this
review or introduction (depending on the student) and urge students to
at this time spend an hour or so working through this chapter before
continuing (even if you've seen it before).