I admire Einstein more and more; his theory of relativity is a full manifestation of his genius. It was only after relativity was proposed that most phenomena and laws of macroscopic physics received a unified description. In Special Relativity, Einstein’s denial of our common sense regarding velocity addition already showed his great courage, while his brainstorm work—General Relativity—perfectly embodied his astonishing creativity. I was attracted to the mathematics of quantum mechanics, but for relativity, I was drawn by its beautiful logical system. Of course, there is also quite beautiful mathematics involved.
One of the core components of Special Relativity is what is known as the Lorentz transformation. This was derived by Lorentz before relativity was published, but he did not recognize its physical significance and thus did not initiate a physical revolution; that task was completed by Einstein. I saw the derivation of the Lorentz transformation a long time ago, which was solved by directly assuming a linear relationship. However, I always felt that such a derivation was not clear enough (perhaps it was my way of understanding), and it did not explicitly show how the two principles of Special Relativity are manifested. Therefore, after studying matrices, I tried to derive the Lorentz transformation using matrices. I found the results to be quite good, and I believe it reflects the symmetry in relativity.
Two Principles
1. Principle of Special Relativity: Physical laws have the same expression in all inertial frames. This is an extension of the principle of relativity in mechanics; it applies to all physical laws, and its essence is that all inertial frames are equal.
2. Principle of Constant Speed of Light: In all inertial frames, the speed of light in a vacuum is equal to c = 299,792,458 m/s, independent of the motion of the light source. The Michelson-Morley experiment is a strong proof of this.
Derivation Process
Below, we will start from these two principles and derive the Lorentz transformation through matrices. Similarly, to clarify the process, we only consider a two-dimensional spacetime with two variables x and t. For convenience, let the speed of light c=1, and let the x'\text{-}t' inertial frame move away from the x\text{-}t inertial frame at velocity v.
Principle of Special Relativity
What we need to understand is what each principle actually tells us. First is the Principle of Special Relativity. It says that all inertial frames are equal, which means that the transformation from one inertial frame to another does not change its essence. In this sense, the Lorentz transformation can only be a linear transformation. Of course, this is not a formal proof (proof would require more mathematical knowledge), but we believe this understanding is sufficient. That is, a linear transformation is a transformation that does not change the essence.
All linear transformations can be described by matrices, so we can set: \begin{bmatrix} x \\ t \end{bmatrix} = \begin{bmatrix} a & c \\ b & d \end{bmatrix} \begin{bmatrix} x' \\ t' \end{bmatrix}
In other words, the [x, t] vector in the ordinary Cartesian coordinate system becomes the [x', t'] vector in the matrix \begin{bmatrix} a & c \\ b & d \end{bmatrix}. The ordinary Cartesian coordinate system is the spacetime we are in (let’s call it the "stationary spacetime"), and \begin{bmatrix} a & c \\ b & d \end{bmatrix} is the spacetime where the mover is (moving spacetime). We measure [x, t] in our spacetime, while they measure [x', t'] in theirs. Regardless of the spacetime, their coordinate axes are straight, and none have become curvilinear coordinate systems; this is the geometric description of "not changing the essence."
Principle of Constant Speed of Light
The key next step is the Principle of Constant Speed of Light. This principle tells us that the measured speed of the same beam of light is the same in both stationary and moving spacetimes. Therefore, a point \begin{bmatrix} 1 \\ 1 \end{bmatrix} with velocity c=1 measured in the moving spacetime must also have a measured velocity of 1 in the stationary spacetime, i.e., \begin{bmatrix} \lambda_1 \\ \lambda_1 \end{bmatrix}. That is: \lambda_1 \begin{bmatrix} 1 \\ 1 \end{bmatrix} = \begin{bmatrix} a & c \\ b & d \end{bmatrix} \begin{bmatrix} 1 \\ 1 \end{bmatrix}
Similarly, the speed of light can be negative, so: \lambda_2 \begin{bmatrix} -1 \\ 1 \end{bmatrix} = \begin{bmatrix} a & c \\ b & d \end{bmatrix} \begin{bmatrix} -1 \\ 1 \end{bmatrix}
In matrix language, this means that \begin{bmatrix} 1 \\ 1 \end{bmatrix} and \begin{bmatrix} -1 \\ 1 \end{bmatrix} are eigenvectors of the matrix \begin{bmatrix} a & c \\ b & d \end{bmatrix}. This gives: a+c=b+d, \quad a-c=d-b Which is equivalent to: a=d, \quad b=c At this point, the transformation matrix becomes \begin{bmatrix} a & b \\ b & a \end{bmatrix}.
Principle of Special Relativity (Continued)
By now, more than half of the work is done. The rest involves discussing relative motion, i.e., the influence of velocity v. A point \begin{bmatrix} v \\ 1 \end{bmatrix} with velocity v in the stationary spacetime is undoubtedly \begin{bmatrix} 0 \\ t' \end{bmatrix} in the moving spacetime, which is the origin of the moving spacetime. That is: \lambda \begin{bmatrix} v \\ 1 \end{bmatrix} = \begin{bmatrix} a & b \\ b & a \end{bmatrix} \begin{bmatrix} 0 \\ t' \end{bmatrix}
We get: b=av. Now only one variable remains: \begin{bmatrix} x \\ t \end{bmatrix} = a \begin{bmatrix} 1 & v \\ v & 1 \end{bmatrix} \begin{bmatrix} x' \\ t' \end{bmatrix}
Next is the final step to victory! Again, derived from the Principle of Special Relativity. Because all inertial frames are equal, from the perspective of the stationary spacetime, the moving spacetime is a spacetime moving away from them at velocity v; while from the perspective of the moving spacetime, it is itself the stationary spacetime, and the stationary spacetime is a moving spacetime with velocity -v. Therefore, the transformation from [x', t'] to [x, t] should only require changing the velocity v to -v in the transformation from [x, t] to [x', t']. So: \begin{bmatrix} x' \\ t' \end{bmatrix} = a \begin{bmatrix} 1 & -v \\ -v & 1 \end{bmatrix} \begin{bmatrix} x \\ t \end{bmatrix} This necessarily implies: \left(a \begin{bmatrix} 1 & v \\ v & 1 \end{bmatrix}\right)^{-1} = a \begin{bmatrix} 1 & -v \\ -v & 1 \end{bmatrix}
Thus we obtain a^2 = \frac{1}{1-v^2}, which means a = \pm \frac{1}{\sqrt{1-v^2}}. A simple check shows we should take the positive value, i.e.: \begin{bmatrix} x \\ t \end{bmatrix} = \frac{1}{\sqrt{1-v^2}} \begin{bmatrix} 1 & v \\ v & 1 \end{bmatrix} \begin{bmatrix} x' \\ t' \end{bmatrix}
This is the matrix form of the Lorentz transformation!! Look, in this form and with these units, time and space are completely “entangled” together! Readers may now understand why time and space must become an inseparable whole. In the matrix \frac{1}{\sqrt{1-v^2}} \begin{bmatrix} 1 & v \\ v & 1 \end{bmatrix}, can you see anything special about time and space? They even have the exact same form! It’s truly surprising!!
Complete Form
Of course, if we do not set the speed of light c=1, the complete transformation matrix should be: \frac{1}{\sqrt{1-\frac{v^2}{c^2}}} \begin{bmatrix} 1 & v \\ \frac{v}{c^2} & 1 \end{bmatrix}
It can be seen that the Galilean transformation corresponds to the case where c \to \infty, i.e., the matrix: \begin{bmatrix} 1 & v \\ 0 & 1 \end{bmatrix}
For the general case of four-dimensional spacetime, we only need to perform a rotation on the two-dimensional spacetime. This is very easy to understand and handle using matrix ideas, and we will discuss this issue in the future.
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