Many failures and difficulty in understanding general relativity, quantum field theory, and conformal field theory
I tried to follow the video lectures of the MIT course 8.962 about general relativity (GR) back in September 2020 after its release on MIT/OCW. I only got to the point of Lecture 9 about Geodesics back then. The feeling wasn’t right. I thought I wasn’t prepared then.
I also took two quantum field theory (QFT) courses in the University of Tokyo during 2019 to 2020, besides reading and doing exercise of the Part I of Srednicki’s QFT book. Still I never felt I understood it.
At the same time, I had been spent much of my time thinking about one of my research topics, which is related to developing new renormalization group (RG) techniques to analyze critical systems. This unavoidably exposed the subject of conformal field theory (CFT) to me. Basically, a CFT is a QFT with more spacial symmetries, including the scale invariance. This is because the theory is now on an RG fixed point. It may sound quite technical. It is! One reason is that I even have more difficulty grasping CFT, let alone understanding its detailed relationship between RG and QFT.
Soon I realized all these difficulties I encountered are related to each other. For example, one of the key concepts in CFT is the stress-energy tensor, which is a concept key to any “field theory” with translational symmetry. It is usually also discussed a lot in a GR book. Also, a QFT is basically a combination of special relativity and quantum mechanics. In order to impose the locality in the theory, we use the format of a “field theory” to do such a thing.
Among all those things that I want to learn. I believe GR and special relativity is the right entry point, since it is well-developed and there are many great textbooks on the subjects. I believe it is the right time for me to really conquer GR. My plan is to start this journey by review Einstein’s special relativity as an appetizer.
The book I chose: Spacetime Physics by Taylor and Wheeler
When I was baffled by the first problem in the special relativity review chapter of Carroll’s GR textbook, I realized that I should relearn this whole subject for a second time.
Previously, I’ve taken a Coursera online course call understanding Einstein and followed an old book by A.P. French. I looked up the course material in the Coursera course and found a PDF for suggested further readings. Three books caught my attention: Spacetime Physics, Exploring Back Holes, and Black Holes and Time Warps. The first two is by Taylor and Wheeler, and the last one by Kip Thorne. I choose the first one since I want to first review special relativity.
This book by Taylor and Wheeler is really a masterpiece. It perfectly fits my unique tastes about physics. The story telling and narratives skill is perfect for me. The reading experience is so smooth that I finished reading this book and the odd-number exercises in less than one month, starting Chapter 1 from 24 August and finishing the chapter 9 on 18 September, 2022.
Focus on the central concepts
The focus of the textbook is on several key concepts and develop them in full details. They are only two key concepts: event and spacetime interval. The fundamental principle is the invariance of spacetime interval for different free-floating (or free-falling) frame observers. End of the story! These are the gist of the whole book and are clearly stated in the first chapter. Then, the rest of the book lays out all the curious thought experiments and paradoxes around these central concepts.
As a comparison, when I first learn relativity in my undergraduate general physics class, the focus is the Lorentz transformation, which makes this whole subject very confusing. In contrast, the Lorentz transformation is only a special topic in this book, not even put into its main story line.
Clarification of the subtlety of some "common-sense" concepts
Some concepts are so common sense that we almost forget to clarify them. However, a failure to appreciate their subtleties can lead to endless confusion. Below I gave my personal list of those concepts:
- Reference frames and observers. The discussion of these two concepts in section 2.6 and 2.7 makes me love this book! It is extremely important to think there is a clock every space and an observer as a whole collection of recording clocks associated with one free-float frame. I always tend to imagine an observer sitting at the origin of a free-float frame, but this causes me lots of confusion when analyzing problems.
- Free-float or free-fall frame. This defines what we mean by an inertial frame. In the Newtonian mechanics of high school, the inertial frame must be thought of some space at outer space, far out of our everyday life. However, in relativity, the concept of an inertial frame can be enlarged even if we are talking about space near the Earth, just the frame defined by a free-falling chamber.
- Synchronization of frame clocks in a given free-float frame. This procedure really defines the concept of time, which will lead to the baffling “relativity of simultaneity” and all the bizarre paradoxes in relativity.
Detailed analysis of paradoxes
Understanding the twin paradox is a land-marking point for understanding relativity. Besides, there are also many interesting paradoxes related to very long rigid body and Lorentz contraction of rigid body in relativity. You can find many interesting problems and though experiment in the text and exercises. I enjoyed them very much!
I love this book so much that I add another book about black hole by the same authors to my “must-read” list. The next step of my GR and field theory storyline is of course to embark and attach a real GR course and textbook. I will now return to the MIT/OCW’s GR course lecture videos and use Schutz and Carroll’s textbook at companion. I may write some updates about GR study in a later post.
A short review of the other two sources of special relativity I learned
As mentioned above, before learning Taylor and Wheeler’s book, I studied and Coursera course, understanding Einstein, and read a textbook by A.P. French. Here I want to speak a little about their features.
- The Coursera lecture videos are best for a first pass for any general audience who still remember his or her high school basic algebra. It provides a highly modern view of special relativity, synthesizing many interesting materials from different resources. Here you will get a board view of this whole subject. You will know the motivation behind special relativity and learn a central tool, spacetime diagram, to analyze problems in special relativity. As a comparison, Taylor and Wheeler’s book isn’t modern enough to develop the spacetime diagram technique in full details.
- A.P. French’s book has a very nice and detailed discussion about the historical development of the subject. For example, why do we dare to impose the invariance of the speed of light $c$ at the very beginning. In other sources, you may be asked to simply believe in this. But A.P. French discussed this.
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