The articles on this blog actually reflect my studies and research during specific periods to some extent, so I find blogging to be a very pleasant activity; it records my journey of growth. Readers might have noticed that last semester I mentioned being very interested in quantum mechanics, and I managed to get my foot in the door. At the beginning of this semester, I expressed a strong interest in perturbation theory and studied it for a week or two. Later, I shifted my focus to relativity. And now? I am studying Landau’s The Classical Theory of Fields, primarily focusing on the electromagnetic field (electrodynamics) first.
Some readers might be at a loss for words: Why do you keep changing? Isn’t the value of learning in depth rather than breadth?
Actually, I would like to defend myself; this is a gradual process. Because I want to develop in the direction of theoretical physics, I hope to pursue the beauty of unity and harmony in physical laws. Therefore, when I studied quantum mechanics, I was not interested in questions like “how to solve the two-body problem in quantum mechanics,” but rather in how the Schrödinger equation is derived, for which I also studied path integrals for a while. In other words, I am interested in how physical laws are discovered, while I have little motivation for how they are applied. These two points might seem a bit contradictory, but this is my current learning style—I am in the mathematics department, not the physics department, so in learning physics, I must grasp the core and get straight to the point.
Since I hope to understand more general physical laws, relativity is something I must understand. My ultimate goal, of course, is to read and understand Einstein’s General Relativity, so naturally, Special Relativity comes first. There is also an interesting topic here, which is electromagnetic theory. We all know that electromagnetic theory was born before special relativity, but it is already consistent with special relativity. In classical mechanics, there is a striking fact: both Coulomb’s law and the law of universal gravitation are inverse-square laws. The difference is that Newton’s law of universal gravitation is considered to be in conflict with relativity. I find this very strange: if electromagnetic theory, including Coulomb’s law, can be consistent with relativity, why can’t the law of universal gravitation be modified into this form? What if we change the charge density in Maxwell’s equations to mass density and change the proportionality constant in Coulomb’s law to G?
Some readers might challenge me: How can such an analogy hold? You call it electromagnetic theory, but you only say the electric field is similar to the gravitational field; what then corresponds to the magnetic field?
Now I want to tell the readers a startling fact: The magnetic
field is entirely an “unfounded unnecessary” thing; it is
simply a relativistic effect of the electric field!! I will
introduce later that the magnetic field originates from moving charges,
which is related to length contraction; essentially, it is still the
attraction and repulsion between charges, and thus it is a relativistic
effect. This makes it easy to understand the consistency between
electromagnetic theory and relativity, as well as why no magnetic
monopoles have been found.
So, why can we only observe the relativistic effects of the electric field and not those of the gravitational field? Because compared to gravity, the Coulomb force is simply too large—for example, the ratio of the Coulomb force to the gravitational force between the nucleus and the electron in a hydrogen atom is about 10^{40} times. Thus, a tiny relativistic effect, once amplified by this factor, becomes observable.
In summary, it is entirely possible to establish such a gravitational theory that is similar to the electromagnetic field and consistent with relativity. In fact, this is just the linear approximation of General Relativity. In fact, the authors H.C. Ohanian [USA] and R. Ruffini [Italy] in the book Gravitation and Spacetime did exactly this: they first established a linear theory of the gravitational field and then generalized it to General Relativity. The “relativistic effects of the gravitational field” mentioned above do indeed exist; the perihelion precession of planetary orbits is one example. Yes, you read that correctly: even starting from General Relativity, one only needs to consider the first-order approximation to derive the solution for orbital precession. Therefore, this linear theory of gravity is sufficient for handling such problems. In fact, if you look at Landau’s The Classical Theory of Fields, the section “Motion in a Central Field” (specifically the Coulomb field case) contains this solution! (Post-note: This statement is an exaggeration; in fact, linear gravity cannot fully provide the true precession angle even under approximation.)
From this perspective, all theories regarding electromagnetism can have a corresponding “gravitational version.” For instance, the well-developed “Quantum Electrodynamics” could have a corresponding “Quantum Linear Gravidynamics.” This is perfectly fine. But why wasn’t it done this way historically? Quite simply, because Einstein was such a genius! He skipped the linear theory of the gravitational field entirely and, starting from the pursuit of beauty, arrived at an exquisitely beautiful General Relativity—a non-linear theory of the gravitational field. One can imagine that without a monumental figure like Einstein, the history of physics might have developed as I described: first producing a linear theory of gravity, then developing into quantum linear gravity alongside quantum electrodynamics, and finally, someone would have developed General Relativity—likely in the 1970s or 80s—rather than Einstein’s 1919! It is no exaggeration to say that Einstein’s ideas were 50 years ahead of his time!
Therefore, I am studying electromagnetic theory not to research electromagnetism, but simply to understand gravitational theory—I hope Maxwell won’t mind. ^_^.
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