I haven’t written an article for over half a month. Firstly, it’s because final exams are approaching and I’ve been busy; but of course, the main reason is that I’ve become lazy. Haha, while others steal leisure from busyness, I am stealing laziness from leisure.
In this article, I would like to share some knowledge about relativistic dynamics with everyone. We have already encountered the coordinate transformations of relativity. The next task should be to convert the dynamical laws of classical mechanics into their relativistic versions. This is clearly a necessary path for learning field theory—understanding how to construct relativistic versions of mechanical laws is the foundation for understanding how to construct relativistic fields. Just like in Landau’s Mechanics and The Classical Theory of Fields, our main thread is the “Principle of Least Action.” Let us recall that in classical mechanics, the action of a free particle is:
S_m = \int L dt = \int \frac{1}{2} m v^2 dt
Strictly speaking, we need to write out the upper and lower limits of the integral, but for simplicity, we omit them. Readers just need to remember that the right side is a definite integral, so S is a number! By varying S_m, we obtain the equation of motion for a free particle: m\dot{v}=0.
What does the relativistic version look like? I cannot give all the specific details here, as this article is not a textbook on relativity, so I don’t intend to write everything out. Assuming the reader already has basic knowledge of tensors, we know that to construct an S_m such that the resulting equations of motion are consistent with relativity, S_m must be a scalar. That is, Ldt must be a scalar. A scalar is a quantity that remains invariant under a Lorentz transformation. In relativity, time and space are on equal footing, but in the expression Ldt, time is clearly placed in a special position, which is obviously not allowed. We have the relativistic version of dt—the particle’s proper time d \tau = \sqrt{1-\frac{v^2}{c^2}}dt, therefore the relativistic version of the law should be rewritten as:
S_m = \int L d\tau = \int L \sqrt{1-\frac{v^2}{c^2}} dt
Next, we need to find an appropriate scalar L that can derive the relativistic version of the laws of motion. On the other hand, the relativistic equations must reduce to Newton’s equations when the velocity is very small. Therefore, we can expand \sqrt{1-\frac{v^2}{c^2}}:
\sqrt{1-\frac{v^2}{c^2}} \approx 1 - \frac{v^2}{2c^2}
Comparing this with the classical mechanics L = \frac{1}{2} m v^2, we find that v^2 has already appeared here (the constant term is not important!). Therefore, as the simplest guess, we can let L = mc^2, which gives:
S_m = -mc^2 \int d\tau = -mc^2 \int \sqrt{1-\frac{v^2}{c^2}} dt
In fact, this is correct; this is the relativistic equation of motion for a particle! However, according to our rules of the game, if we assume we know nothing about the relativistic laws of motion, this can only be considered a guess—the simplest guess. Some readers might ask, why do you take the simplest case? Isn’t it possible that there are more complex non-linear ones? The answer is simple: it is possible. But from our perspective of thinking, we always hope a theory is as simple as possible; and since we are making a guess now, of course we guess the simplest case first. Even if it’s incorrect, it can be improved slowly!
The case for a free particle has been obtained, but this is far from enough because we want to study the motion of particles under various forces. However, in relativity, the role of force is no longer so prominent; instead, fields are considered more, i.e., the motion of particles in various fields. Consider the simplest scalar field \phi, which describes the potential at every point in spacetime (classical mechanics has many analogies, such as electric potential and gravitational potential). We know that in classical mechanics, this term is:
S_{mf} = \int \alpha \phi dt
where \alpha is a constant related to the source of the field. For example, for electric potential, \alpha is the electric charge; for gravitational potential, \alpha is the mass. As mentioned before, this again puts time in a special position. But we adopt the previous method and replace dt with the relativistic version:
S_{mf} = \int \alpha \phi d\tau = \alpha \phi \sqrt{1-\frac{v^2}{c^2}} dt
This also satisfies the constraint of reducing to Newtonian mechanics at low speeds. Of course, this is also a guess. We combine the two actions to get a total S:
\begin{aligned} S &= -mc^2 \int \sqrt{1-\frac{v^2}{c^2}}dt - \alpha \phi \sqrt{1-\frac{v^2}{c^2}}dt \\ &= -\int (mc^2\sqrt{1-\frac{v^2}{c^2}} + \alpha \phi \sqrt{1-\frac{v^2}{c^2}})dt \end{aligned}
Through complex variation, the following equation of motion can be derived:
m\frac{du_i}{d\tau} + \frac{\alpha}{c^2}\frac{d(\phi u_i)}{d\tau} = \alpha\frac{\partial \phi}{\partial x^i}
where
\begin{aligned} u^i &= \frac{dx^i}{d\tau} \\ (x^0,x^1,x^2,x^3) &= (ct,x,y,z) \\ (x_0,x_1,x_2,x_3) &= (x^0,-x^1,-x^2,-x^3) = (ct,-x,-y,-z) \end{aligned}
As can be seen, the above is the relativistic version of the equation of motion for a particle in a scalar potential!
However, I played a joke on the readers: the formulas we spent so much effort on are actually not very useful. Of course, it’s not that we were wrong—there is no problem with the form—but because no single scalar potential field has been discovered in nature. The electromagnetic fields we encounter are already considered the simplest classical fields, and their potentials are vectors. Yes, I didn’t write it wrong; developing into relativity, not only is force a vector, but even the potential has become a vector!
As for why I still led everyone down this path, it is because this case most simply explains our line of thought in discovering relativistic laws: construct relativistic invariants, and ensure that the invariant reduces to Newtonian mechanics in low-speed situations! Whether it is special relativity, general relativity, or quantum field theory, this line of thought is consistently implemented throughout. Of course, just as Einstein originally broke the “iron laws” of Newtonian mechanics, perhaps in the future we will have to strive to break free from the constraints of relativity—but that is a story for another time, and I feel it is still very far from our era.
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