Relativity Theory and Lorentz Transformation
Buildup
Relativity theory starts from the completion of electromagnetism. To say electromagnetism is complete means Maxwell finished the four partial differential equations for the electric field $\mathbf{E}$ and the magnetic field $\mathbf{B}$. From Maxwell’s equations, we learn that the speed of electromagnetic waves is equal to the speed of light. This leads us to the following two facts:
- Light is an electromagnetic wave.
- The speed of light is $\dfrac{1}{\sqrt{\epsilon_o \mu_o}}=300,000 \text{km/s}$.
However, this brings up issues that can’t be explained by classical physics:
- Light is a wave, and waves require a medium, but what is the medium for light?
- Why is the speed of light constant?
There were many efforts to explain this, and one of them is the theory of ether (which is different from the ether in ancient Greek philosophy). If we assume that ether fills the space of the universe, then ether would be the medium for light. If the speed of light is the speed relative to the ether, it also solves why the speed is constant.
However, as plausible as ether theory seemed, the Michelson-Morley experiment showed that there is no medium for light filling the space of the universe. And finally, in 1905, Einstein published the ’theory of relativity’ that solved this problem. Precisely, it was the theory of special relativity. Later, he also published the general theory of relativity. The word ‘special’ might lead one to think that the special theory of relativity is something more special, better, but it’s quite the opposite. The special theory of relativity means it only applies under certain special circumstances and does not explain everything.
Shockingly, Einstein did not try to fit new facts into existing theories. He began by assuming that light does not require a medium to propagate, and its speed is observed as $c=300,000km/s$ by any observer. That is, he started by accepting what was thought to be strange in classical physics as truth. Unlike us who learn the theory of relativity as true, this must have been a huge shock for physicists at the time.
If the speed of light is always $c$, then we cannot use the Galilean transform used in classical physics. However, it’s not that Galilean transforms are completely wrong. If the speed of a moving object is negligible compared to the speed of light, Lorentz transformation approaches the Galilean transform. That is, Galilean transform is a special case of Lorentz transformation and fits very well with our real world.
The reason why Lorentz transformation includes the name Lorentz is, of course, because Lorentz created it. However, Lorentz did not know about the theory of relativity and was just trying to create a coordinate transformation that fits well with Maxwell’s equations. In other words, Einstein did not directly create the Lorentz transformation, but acquired the idea of relativity theory from what Lorentz had made.
Definition
The Lorentz transformation between the inertial frame $A$ and the inertial frame $A^{\prime}$ moving at velocity $v_{0}$ in the direction of axis $x$ is as follows:
$$ \begin{pmatrix} ct^{\prime} \\ x^{\prime} \\ y^{\prime} \\ z^{\prime} \end{pmatrix} = \begin{pmatrix} \gamma_{0} & -\gamma_{0}\beta_{0} & 0 & 0 \\ -\gamma_{0}\beta_{0} & \gamma_{0} & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{pmatrix} \begin{pmatrix} ct \\ x \\ y \\ z \end{pmatrix} $$
Where $\gamma_{0}=\dfrac{1}{\sqrt{1-{\beta_{0}}^{2}}}$, and $\beta_{0} = \dfrac{v_{0}}{c}$.
Explanation
Let’s summarize some facts that can be known from the Lorentz transformation.
If the speed of the $A^{\prime}$ frame, which is $v_{0}$, is considerably small compared to the speed of light, which is $c$, then the Lorentz transformation approximates the Galilean transformation. This is the case for most of the realities we experience. This is why we still learn Newtonian mechanics.
The range of the Lorentz factor $\gamma_{0}$ is from $1\le \gamma_{0} \le \infty$. When $v_{0} \rightarrow 0$, $\gamma_{0}=1$ and it’s the minimum.mininum. When $v_{0} \rightarrow c$, $\gamma_{0}=\infty$.
An object cannot move faster than the speed of light, which is $c$. In Lorentz transformation, $\gamma_{0}=\dfrac{1}{\sqrt{1-\frac{{v_{0}}^{2}}{c^{2}}}}$, but if you look at the denominator, you can see it contains a root. The value inside the root must always be greater than or equal to $0$, therefore
$$ \begin{align*} && 1-\dfrac{{v_{0}}^{2}}{c^{2}} &\ge 0 \\ \implies && c^{2}-{v_{0}}^{2} &\ge 0 \\ \implies && c^{2} &\ge {v_{0}}^{2} \\ \implies && c &\ge v_{0} \end{align*} $$
The relative speed between inertial frames can never exceed the speed of light. Meaning, moving faster than light is impossible.