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:
If the group product commutes ( ) the group is said to be Abelian16.1 otherwise the group is said to be non-Abelian, which is sensible enough. A Lie group is a continuous group16.2 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
The set of transformations that leave the quantity
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).