Transformations

In this course we shall only concern ourselves with coordinate transformations which transform an inertial frame into another inertial frame. This limits us to four classes of transformations: displacements of the coordinate axes, rotations of the coordinate axes, parity reversals (i.e., $x,y,z\rightarrow -x,-y,-z$), and Lorentz transformations. All of these transformations possess group properties. As a reminder, the requirements for an abstract multiplicative group are:
 
(i) The product of two elements is an element of the group.
(ii) The associative law $(ab)c=a(bc)$ holds.
(iii) There is a unit element $e$ satisfying $ae=ea=a$ for all $a$.
(iv) Each element $a$ possesses an inverse $a^{-1}$ such that $a^{-1}a = a a^{-1} = e$.

Consider Lorentz transformations (in the standard configuration). It is easily demonstrated that the resultant of two successive Lorentz transformations, with velocities $v_1$ and $v_2$, respectively, is equivalent to a Lorentz transformation with velocity $v=(v_1+v_2)/ (1+v_1 v_2/c^2)$. Lorentz transformations obviously satisfy the associative law. The unit element of the transformation group is just a Lorentz transformation with $v=0$. Finally, the inverse of a Lorentz transformation with velocity $v$ is a transformation with velocity $-v$. We can use similar arguments to show that translations, rotations, parity inversions, and general Lorentz transformations (i.e., transformations between frames which are not in the standard configuration) also possess group properties.

If we think carefully, we can see that the group properties of the above mentioned transformations are a direct consequence of the relativity principle. Let us again consider Lorentz transformations. Suppose that we have three inertial frames $S$, $S'$, and $S''$. According to $(i)$, if we can get from $S$ to $S'$ by a Lorentz transformation, and from $S'$ to $S''$ by a second Lorentz transformation, then it must always be possible to go directly from $S$ to $S''$ by means of a third Lorentz transformation. Suppose, for the sake of argument, that we can find three frames for which this is not the case. In this situation, the frame $S'$ could be distinguished from the frame $S''$ because it is possible to make a direct Lorentz transformation from $S$ to the former frame, but not to the latter. This violates the relativity principle and, therefore, this situation can never arise. We can use a similar argument to demonstrate that a Lorentz transformation must possess an inverse. The associative law and the requirement that a unit element exists are trivially satisfied.