In the previous article, I briefly discussed the necessity of field theory from my own perspective. This time, let us revisit this topic to gain a deeper understanding at a conceptual level.
Over the past week or two, I have been searching for materials, primarily on linear gravity, and I have discovered many interesting things that I would like to share with everyone. First, when I entered “linear gravity” into Google, I discovered a “miraculous book,” a true “magnum opus” in every sense of the word—Gravitation! A massive work of over 1,300 pages, co-authored by three “superstars”—Charles W. Misner, Kip S. Thorne, and John Archibald Wheeler. It is unlikely that any other book can rival its “all-star lineup.” The book is titled Gravitation. It covers the history of research and the development of gravity, and more importantly, it provides mathematical proofs for almost every historical milestone! Most importantly, the author Wheeler was a contemporary of Einstein, allowing us to truly experience the research of that era. Upon seeing this, I could not wait to buy it. For various reasons, it is difficult to purchase here, but I found an English version in the library and borrowed it immediately. Additionally, because I could not buy a physical Chinese version, I had to purchase an electronic version online and print it out. However, it is not very clear, and I personally feel the translation is not excellent (though it is sufficient for reading).
Why study linear gravity? Isn’t General Relativity a highly non-linear theory? There are deep reasons for this. General Relativity geometrizes gravity, resulting in a beautiful theory, but it is very difficult to reconcile with quantum theory. This is because, in essence, General Relativity is not a field theory but a geometric theory. Consequently, standard quantum theory (which is field theory) is incompatible with it. Physically speaking, quantum theory requires a fixed (absolute) spacetime background, whereas General Relativity denies the existence of an absolute reference frame. Thus, the two are naturally “as incompatible as fire and water.”
However, according to our intuition (though intuition is not always correct), gravity is indeed a real force. Explaining it using curved spacetime seems somewhat “absurd.” Therefore, some physicists developed gravity theories starting from a field theory perspective. What is both surprising and expected is that these two different paths lead to the same result—General Relativity. These physicists include Gupta, Feynman, Thirring, and Weinberg (note: Feynman appears once again!).
What is the core idea of field theory? We treat gravity as a real force. Forces follow the principle of superposition; therefore, the gravitational field should also follow the principle of superposition, meaning the gravitational field equations should be linear. But why is General Relativity so non-linear? Because the source of gravity is very special—mass. Yes, where there is mass, there is gravity, and E=mc^2 tells us that where there is energy, there is mass! A physical object creates a gravitational field “belonging to it.” As we briefly explained in the previous article, a field can store energy. Therefore, this gravitational field itself possesses a certain amount of energy, which in turn becomes an additional source of gravity. This source generates a secondary gravitational field that superimposes onto the original field, and then this secondary field generates a tertiary field, and so on. This results in an infinite series of gravitational fields. Although each step is linear, the total effect of this infinite series is non-linear! (This is somewhat like perturbation methods: a non-linear equation can be solved by superimposing infinitely many linear equations to solve it step-by-step.)
The advantage of the field theory approach is that it preserves flat spacetime. Yes, although the final result is still General Relativity, the two are identical in mathematical form but completely different in physical meaning. Einstein’s explanation is the curvature of spacetime, while the field theory idea is the superposition of infinite fields in flat spacetime. The important difference between the two is that a theory in flat spacetime is relatively easier to quantize (I am not entirely sure how easy it is; I have only heard so, and will revisit this view once I have studied it further ^_^), so we can obtain a quantum theory of gravity! Yes, I am not joking; Feynman actually derived a quantum theory of gravity. This is not particularly difficult; the difficulty lies in the fact that this theory is non-renormalizable!
Renormalization is a set of methods in quantum field theory for handling divergences. We can get a general impression of the concept of renormalization through the following example: I = \int_0^b \frac{1}{x}dx - \int_0^a \frac{1}{x}dx Both integrals on the right side are uncomputable, which is the divergence difficulty. However, in reality, I exists (in physical problems). The trick of renormalization is to add a small parameter \varepsilon: I = \int_{\varepsilon}^b \frac{1}{x}dx - \int_{\varepsilon}^a \frac{1}{x}dx Then it is renormalized as: I = \int_a^b \frac{1}{x}dx = \ln b - \ln a Of course, actual techniques are much more complex!
I will not discuss quantum gravity further here. In future articles, I will lean towards providing mathematical descriptions of these theories, hoping to guide everyone toward more advanced field theory and even frontier physics. However, one more person to mention is Feynman—the man who truly “rediscovered the whole of physics”! Previously, our impression of him was limited to quantum mechanics and path integrals, but he was also one of the masters of gravitational research. The linear gravity approach to General Relativity mentioned above owes much of its success to him. There is a book summarizing his research on gravity, titled Feynman’s Lectures on Gravitation, which is also an excellent resource! (PS: This book is available for download on Sina Share.)
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