Many days have passed since the first article in this series. In fact, the content of this article was completed at the same time as the first one. Why has it taken so long to update? There are two reasons: first, with the arrival of spring, I have become somewhat lazy—quite decadent indeed; second, I have been reflecting on the problem of gauge transformations. According to the logic of Landau’s The Classical Theory of Fields, after developing the theory of particle mechanics, the next step is to develop field theory, such as electromagnetic and gravitational fields. However, there is something in field theory that puzzles me: the existence of "gauge invariance." According to the general view, we treat gauge invariance as a consequence of the electromagnetic field equations; that is, after deriving the equations for the electromagnetic field, we "discover" that they possess gauge invariance. But if we use the method described in this article—namely, assuming the field has this symmetry—we can then construct the field equations. However, I have not yet clearly understood why the field possesses gauge invariance in the first place. Based on the materials I have read, this invariance seems to be related to general invariance (this is true for the electromagnetic field as well, which seems to suggest that even the electromagnetic field theory in flat spacetime implies general invariance?). Furthermore, it seems that this invariance requires quantum field theory for a more satisfactory explanation, but that is still far beyond my current reach.
Anyway, let us first return to the derivation of relativistic mechanics.
Something from Nothing
In the previous article, we constructed the infinitesimal generator of relativistic mechanics and performed a prolongation. As I mentioned, only the infinitesimal form of the transformation is needed to construct the complete laws of relativistic mechanics (provided we make some "obvious" assumptions). This is a process of creating something almost from "nothing," which is the meaning behind the title of this article. On the other hand, this possibility of moving from the local to the global provides some inspiration: if the method is universal, then we can use it to construct the physical laws we need, including electromagnetic and gravitational field equations. (Of course, I am still some distance from that goal.)
Recalling the generator from the previous article: X^{(2)} = \frac{x}{c^2}\frac{\partial}{\partial t} + t\frac{\partial}{\partial x} + \left(1-\frac{\dot{x}^2}{c^2}\right)\frac{\partial}{\partial \dot{x}} - \frac{3\dot{x}\ddot{x}}{c^2}\frac{\partial}{\partial \ddot{x}}
Newton’s law of motion is: m\ddot{x} = f However, this equation does not satisfy relativity because: X^{(2)}(m\ddot{x}) = -\frac{3m\dot{x}\ddot{x}}{c^2} \neq 0 To satisfy the requirements of special relativity, we must modify this law. The force on the right side is a physical object, so we naturally cannot change it. We need to change the left side of the equation. To ensure the equation satisfies more symmetries (referring here to translational invariance in spacetime), we assume the left side of the equation only contains derivatives of x. Let the correct equation be: F(\dot{x}, \ddot{x}) = f We also have an assumption: at low speeds, F(\dot{x}, \ddot{x}) \to m\ddot{x}. After all, Newton’s laws of mechanics have passed extensive low-speed testing.
Thus, we must have: X^{(2)} F(\dot{x}, \ddot{x}) = 0 Which is: X^{(2)} F = \left(1-\frac{\dot{x}^2}{c^2}\right)\frac{\partial F}{\partial \dot{x}} - \frac{3\dot{x}\ddot{x}}{c^2}\frac{\partial F}{\partial \ddot{x}} = 0 This is a simple homogeneous linear partial differential equation. Its solution can be found using relevant PDE theory. We first solve the characteristic equation: \frac{d\dot{x}}{1-\dot{x}^2/c^2} = -\frac{d\ddot{x}}{3\dot{x}\ddot{x}/c^2} We obtain the general solution (note that we use prolongation to treat \dot{x} and \ddot{x} separately, so here \dot{x} and \ddot{x} are two independent variables): \frac{m\ddot{x}}{|1-\dot{x}^2/c^2|^{3/2}} = \text{Const.} Thus, the general solution of the above PDE is (taking the time-like solution): F = F\left(\frac{m\ddot{x}}{(1-\dot{x}^2/c^2)^{3/2}}\right) = F\left(\frac{dp}{dt}\right) Where: p = \frac{m\dot{x}}{\sqrt{1-\dot{x}^2/c^2}} (The form of relativistic momentum has emerged.)
From the perspective of satisfying special relativity alone, any law such as: F\left(\frac{dp}{dt}\right) = f would be acceptable. However, we must also consider two points:
1. It must be able to explain physical phenomena, which simply means it must reduce to Newton’s laws at low speeds;
2. While satisfying the first point, the physical law should be as simple as possible.
Therefore, we take the simplest case: F\left(\frac{dp}{dt}\right) = \frac{dp}{dt} Thus, we obtain the relativistic law of mechanics found in textbooks: \frac{dp}{dt} = f
Of course, from our current perspective, this is only one of the possible physical laws; its correctness remains to be tested by experiment. Naturally, the fact is that this law has already withstood a vast amount of empirical testing.
The Road Ahead?
So, what is our next task? We have completed the derivation of the laws of mechanics. There are two possible next steps: first, since our derivation is based on the simple case of two-dimensional spacetime, it is natural to think about generalizing it to general four-dimensional spacetime. However, we believe that the work in two dimensions has already allowed us to grasp the essentials of this idea; generalizing to four dimensions presents no fundamental or technical difficulty, only more complex mathematical calculations. Another possible and more meaningful task is to variationalize our "results." According to Feynman’s thinking, he believed that a law that does not satisfy some variational principle is "highly likely" to be wrong.
Therefore, in the next article, we will focus on understanding the symmetry of variational principles; that is, we will search for an action that remains invariant under a certain transformation! This is quite interesting and useful; it is similar to the symmetry of equations, yet somewhat different. It is also worth mentioning that describing things with a variational principle has another advantage: our usual "equations of motion" contain second-order derivatives of variables with respect to time, which means that in most cases, the action only contains first-order derivatives (the gravitational field being an exception). To some extent, this also simplifies our derivation.
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