The Geometry of Space-Time

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$_3$.

For those of you who led deprived childhoods, a group ${\cal G}$ is a set of mathematical objects $(a,b,c \ldots)$ with a rule of composition, or group product, $(a \circ b)$ such that:

1. Every product of a pair of elements in the group is also in the group. That is, if $a,b \in {\cal G}$ then $c = a \circ b \in {\cal G})$. This property is called closure.
2. The group must contain a special element called the identity $I \in {\cal G}$ such that $a \circ I = a$ for all $a \in {\cal G}$.
3. Every element of the group ${\cal G}$ must have an inverse, also in ${\cal G}$. If $a \in {\cal G}$ then $\exists a^{-1} \in {\cal G}$ such that $a \circ a^{-1} = I$.
4. The group product must be associative. That is, $a \circ (b \circ c) = (a \circ b) \circ c, \forall a,b,c \in {\cal G}$.

If the group product commutes ( $a \circ b = b \circ a$) the group is said to be Abelian^16.1 otherwise the group is said to be non-Abelian, which is sensible enough. A Lie group is a continuous group^16.2 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

$$s^2 = x_0^2 - ({\bf x} \cdot {\bf x})$$ (16.1)

invariant for a single event $x$ 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

$$s^2(x,y) = (x_0 - y_0)^2 - \left\{ (x_1 - y_1)^2 + (x_2 - y_2)^2 + (x_3 - y_3)^2) \right\}$$ (16.2)

invariant form the inhomogeneous Lorentz^16.3 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 $s(x,y)$ to be the norm of relativistic space-time. This quantity may be considered to be the invariant ``distance'' (squared) between two events, $x$ and $y$, 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 $s(x,y)$ 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).