In the previous section we saw that a combination of a scalar and a vector, a paravector, \(\v{A} = a_0 + \v{a}\), has an amplitude-squared with the same metric as that of spacetime and yet can be rotated like a normal vector such that the scalar part is unchanged, while the vector part rotates in space, as \(\v{\c{R}}\v{A}\v{R}\), where the rotor \(\v{R}\) is itself comprised of a scalar and a bivector, \(\v{R} = e^{-\v{xy}\frac{\theta}{2}}\).

We also saw that we can define a normalised paravector as \(\v{B} = e^{-\v{b}\frac{\phi}{2}}\), so that \(\v{B}\c{\v{B}} = 1\). In this section we will consider rotating the paravector \(\v{A}\) by a normalized paravector and see that the result, \(\v{B}\v{A}\v{B}\), is equivalent to the standard Lorentz transformation.

Similar to the spatial rotation, we can see that \(\v{A'} = \v{B}\v{A}\v{B}\) is some type of rotation of \(\v{A}\), as even though we are not using the Clifford conjugation of the rotor for the left-hand multilication, the amplitude squared is still unchanged:

\[ \begin{aligned} \v{\c{A'}A'} &= \c{\v{B}\v{A}\v{B}} \v{B}\v{A}\v{B} \\ &= \c{\v{B}}\c{\v{A}}\c{\v{B}} \v{B}\v{A}\v{B} \\ &= \v{\c{A}A} \end{aligned} \]

Evaluating \(\v{A'}\), we find

\[ \begin{aligned} \v{A'} &= \v{B}\v{A}\v{B} \\ &= e^{-\v{b}\frac{\phi}{2}}(a_0 + \v{a})e^{-\v{b}\frac{\phi}{2}} \\ &= a_0 e^{-\v{b}\phi} + e^{-\v{b}\frac{\phi}{2}} \v{a}e^{-\v{b}\frac{\phi}{2}} \end{aligned} \]

which is, in fact...

If we first look at the case where the vector being rotated is in the same direction as the rotation itself, ie. \(\v{a}\v{b} = \v{b}\v{a} = |\v{a}|\) and since \(\v{b}^2 = 1\), then \(\v{a} = |\v{a}|\v{b}\), we can further simplify the rotation:

\( \begin{aligned} \v{A'} &= a_0 e^{-\v{b}\phi} + e^{-\v{b}\frac{\phi}{2}} \v{a}e^{-\v{b}\frac{\phi}{2}} \\ &= a_0 e^{-\v{b}\phi} + \v{a}e^{-\v{b}\phi} \\ &= a_0 (\cosh{\phi} - \v{b} \sinh{\phi}) + \v{a}(\cosh{\phi} - \v{b} \sinh{\phi}) \\ &= (a_0 \cosh{\phi} - \v{ab}\sinh{\phi}) + (\v{a}\cosh{\phi} - a_0 \v{b} \sinh{\phi}) \\ &= (a_0 \cosh{\phi} - |\v{a}|\sinh{\phi}) + \v{b}(|\v{a}|\cosh{\phi} - a_0 \sinh{\phi}) \\ &= \cosh{\phi}(a_0 - |\v{a}|\tanh{\phi}) + \v{b}\cosh{\phi}(|\v{a}| - a_0 \tanh{\phi}) \\ \end{aligned} \)

Note that \(\tanh\phi\) has the property of ranging from -1 to 1 (see figure) and that \(\cosh\phi\) and \(\tanh\phi\) can be related by:

\[ \cosh\phi = \frac{1}{\sqrt{1-\tan^2\phi}} \]

So with the substitution of \(v = \tanh\phi\), we're left with the Lorentz transformation of \(\v{A} = a_0 + \v{a}\):

\[ \v{B}\v{A}\v{B} = \frac{1}{\sqrt{1-v^2}}(a_0 - |\v{a}|v) + \frac{1}{\sqrt{1-v^2}}(|\v{a}| - a_0 v)\v{x} \]

Next let us look at the case where the vector being rotated is perpendicular to the rotation, ie. \(\v{a}\v{b} = -\v{b}\v{a}\). In this case:

\[ \begin{aligned} \v{A'} &= a_0 e^{-\v{b}\phi} + e^{-\v{b}\frac{\phi}{2}} \v{a}e^{-\v{b}\frac{\phi}{2}} \\ &= a_0 e^{-\v{b}\phi} + \v{a}e^{+\v{b}\frac{\phi}{2}} e^{-\v{b}\frac{\phi}{2}} \\ &= a_0 e^{-\v{b}\phi} + \v{a} \\ \end{aligned} \]

So the vector perpendicular to the direction of the rotation is not affected by the transformation, just as with the Lorentz transformation.