From middle July 2022, I started studying general relativity (GR) again. I use three resource at the same time: 1) Prof. Scott Hughes’s MIT’s physics graduate course 8.962, 2) the textbook by Schutz and 3) Taylor, Wheeler and Bertschinger’s book: Exploring Black Holes. I’m happy that I finally finished all the video lectures of Prof. Hughes’s MIT course today. I talked about the first part of this course in my 2022 summary of physics post. After that, the second half of the course focuses on applications.
Weak-field limit and gravitational wave
I understand the weak-field approximation of GR as a sanity check of this theory, since it must match with Newtonian gravitational theory. The assumption is natural: the metric is nearly flat, and the speed is much less than the speed of light. It is a satisfactory experience to go through this matching with the old theory in details, since it makes one more comfortable working with the new spacetime formulation of gravitation. The technique details themselves are quite interesting, including the background Lorentz transformation and gauge transformation. Under these two transformations, the metric remains in the “nearly” Lorentz coordinate system.
The theory of gravitation wave is more intricate than I originally thought. The general idea is to use the geodesic deviation to detect the oscillation of spacetime. However, the connection to a real experiment is quite subtle but interesting to think about. The basic underlying physics is analogous to electromagnetic radiation. However, it is a pity that I haven’t followed everything in detail yet. The definition of the energy for the gravitational wave also looks quite intricate. I plan to defer it for my second study of GR.
Isotropic and homogenous solution in spatial part: cosmology
The second interesting application is when the spatial part of the spacetime is isotropic and homogenous, the so-called Robertson–Walker metric. It is related to the large-scale structure of the universe. Due to the symmetry, the Einstein field equation takes a simpler form here, called Friedmann equations.
Spherical compact sources and black holes
I’m most interested in this application, since it can be compared with the Kepler problem in Newtonian physics. What’s more, black holes are a fascinating theoretical model to study and ponder on. I have read a lot about physics of black hole in Hawking’s popular science book since high school. Now it is finally the time to make some of them precise. I have lots of puzzles regarding the physics near the horizon, one of which is about an extended object following through the horizon.
Intuitively, we know the situation inside and outside the event horizon is distinctively different. It is very hard to imagine what if an extended object has part inside and part outside the event horizon.
Prof. Hughes’s lecture videos didn’t discuss this. He focused on simpler situations like the orbital of a massive particle and null geodesics. And also a different coordinate system other than Schwarzschild coordinate, more suitable for understanding physics near event horizon, is discussed. However, I feel that I want to understand more physics before diving into these mathematical details. Probably I should first go through Taylor, Wheeler and Bertschinger’s book: Exploring Black Holes before working on Schutz’s exercise.
