In the previous article, we derived the Lorentz transformation using matrices. Matrix representation is not only formally beautiful but also has significant practical value, such as facilitating the search for various invariants. Once the Lorentz transformation is expressed in matrix form, many known theories from linear algebra can be applied. In this small sequel, we will attempt to elaborate on this idea.
In this article, we continue to set the speed of light c=1.
We have already obtained the matrix form of the Lorentz transformation: \left[\begin{array}{c} x\\t \end{array}\right]=\frac{1}{\sqrt{1-v^2}}\left[\begin{array}{c c}1 & v\\ v & 1 \end{array}\right]\left[\begin{array}{c}x'\\t' \end{array}\right]
Although time and space have been integrated into a single entity in relativity, time indeed possesses its own uniqueness. To reflect this uniqueness, we follow the consistent practice in theoretical physics of multiplying time by the imaginary unit i. Thus, the Lorentz transformation matrix becomes: \left[\begin{array}{c} x\\it \end{array}\right]=\frac{1}{\sqrt{1-v^2}}\left[\begin{array}{c c}1 & -iv\\ iv & 1 \end{array}\right]\left[\begin{array}{c}x'\\it' \end{array}\right]
This form has its own advantages. We denote the transformation matrix \frac{1}{\sqrt{1-v^2}}\left[\begin{array}{c c}1 & -iv\\ iv & 1 \end{array}\right] as \boldsymbol{A}, and denote \left[\begin{array}{c} x\\it \end{array}\right] and \left[\begin{array}{c} x'\\it' \end{array}\right] as vectors \boldsymbol{x} and \boldsymbol{x'}, respectively. That is: \boldsymbol{x}=\boldsymbol{A}\boldsymbol{x'}
Principle of Invariance of the Speed of Light
The matrix \boldsymbol{A} satisfies \boldsymbol{A}^T \boldsymbol{A}=I, which brings us great convenience because we can immediately discover an invariant quantity: \boldsymbol{x}^T \boldsymbol{x}=\boldsymbol{x'}^T \boldsymbol{A}^T \boldsymbol{A} \boldsymbol{x'}=\boldsymbol{x'}^T \boldsymbol{x'}
Translated, this means x^2-t^2=x'^2-t'^2. This is the manifestation of the principle of the invariance of the speed of light.
Of course, we could also take x^2-t^2=x'^2-t'^2 as the starting point to derive the Lorentz transformation matrix. This would yield two relational expressions for the components of a matrix \left[\begin{array}{c c}a & c\\ b& d \end{array}\right]. By combining this with the principle of relativity and relative motion, one can obtain the complete matrix form. Readers might want to try this for themselves.
Velocity Addition
What happens if we compose two Lorentz transformations? Suppose an object A moves at velocity v relative to B (in the same direction), and B moves at velocity w relative to C. What is the velocity of A relative to C? Let’s look at the composition of the two corresponding Lorentz transformation matrices \boldsymbol{A}_v and \boldsymbol{A}_w: \begin{aligned} \boldsymbol{x}=&\,\boldsymbol{A}_v \boldsymbol{A}_w \boldsymbol{x'} \\ =&\frac{1}{\sqrt{1-v^2}}\left[\begin{array}{c c}1 & -iv\\ iv & 1 \end{array}\right]\frac{1}{\sqrt{1-w^2}}\left[\begin{array}{c c}1 & -iw\\ iw & 1 \end{array}\right] \\ =&\frac{1}{\sqrt{(1-v^2)(1-w^2)}}\left[\begin{array}{c c}1+wv & -i(w+v)\\ i(w+v) & 1+wv \end{array}\right] \\ =&\frac{1+wv}{\sqrt{(1-v^2)(1-w^2)}}\left[\begin{array}{c c}1 & -i(\frac{w+v}{1+wv})\\ i(\frac{w+v}{1+wv}) & 1 \end{array}\right] \end{aligned}
Comparing the final matrix with \frac{1}{\sqrt{1-v^2}}\left[\begin{array}{c c}1 & -iv\\ iv & 1 \end{array}\right], it is easy to see that the resultant velocity is: \frac{w+v}{1+wv}
This is the relativistic velocity addition formula. (Of course, if you are skeptical about the term \frac{1+wv}{\sqrt{(1-v^2)(1-w^2)}}, you can verify it again.) For this type of treatment, these results are remarkably concise and clear, which is sufficient to fully demonstrate the immense charm of matrix-based research.
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