Spring 2023, Deep Springs College, Prof. Brian Hill
In two momentous leaps, Einstein discovered that space and time are not what we thought them to be. They are intertwined. I cannot explain how they are intertwined in this overview. If I could, there would be no need for a course. You have never experienced this intertwining, because it only becomes apparent when either (a) you are moving at near the speed of light, or (b) you are in the vicinity of an object even heavier and denser than our Sun. The intertwining is highly counterintuitive. We call the intertwined fabric of space and time, “spacetime.”
More than anyone else since Einstein, John Archibald Wheeler has influenced our understanding of spacetime. He is an author or co-author on the best-known textbooks in the field and his students have gone on to advance the subject dramatically. Most notable of his students, perhaps, is Kip Thorne (one of the winners of the 2017 Nobel Prize)
We will work through Wheeler’s, Thorne’s and others’ descriptions of three things: (1) special relativity, which is how space and time are related in the absence of gravity, (2) the behavior of spacetime around a black hole, and (3) the behavior of the spacetime of the entire universe, which is the subject of cosmology.
How can we do an honest treatment of a subject so advanced? We are not going to solve Einstein’s equations. Instead, we will study the Schwarzschild solution for an uncharged, non-rotating black hole, and then the Friedman-Lemaître-Robertson-Walker solution for a spatially homogeneous and isotropic, expanding spacetime. By studying the solutions of Einstein’s equations rather than the equations themselves, we can bypass a lot of advanced mathematics that even Einstein needed help (from Hilbert) to understand.
I. Euclidean Space: Euclidean Metric, Euclidean Rotations, Galilean Relativity
II. Minkowski Space: Lorentz Metric, Lorentz Transformations, Special Relativity
III. Polar Coordinates in Flat and Curved Spacetime in 2+1 and 3+1 Dimensions
IV. Properties of the Schwarzschild Metric
IV. Conservation of Map-Energy in the Schwarzschild Metric, Infalling Light, Infalling Stones, Rain
V. Homogeneity, Isotropy, Hubble Plots, the Cosmic Microwave Background
VI. The Friedman-Robertson-Walker (FRW) Metric
VII. Matter, Radiation, Dark Matter, and Dark Energy in FRW Cosmology
Detailed daily schedules will be kept retrospectively:
We will use:
Additionally, we will draw from:
There will be a lot of handouts and our texts are PDFs that you will be printing out. To be organized, get a three-ring binder (or binders) to keep the texts, the handouts and problem sets organized. Assignments should be on 8 1/2 x 11 paper (and not torn out from a bound notebook). Multi-page assignments should be stapled. Corrections should be erased (if done in pencil) or recopied (if done in pen). To make nice diagrams and graphs, you will often need a ruler.
The College’s general policies on absences and late work are applicable. There was an email from the Dean on this September 8, 2022. The policies below are consistent with that email:
Whereas missed coursework affects both your classmates and professors by lowering the thinking and understanding you bring to a given class, and interrupts the course schedule that has been set up and is adjusted on an ongoing basis with substantial care. The same is true for absences — whereas a handful of absences might be “normal” at colleges with large lectures or less serious academics, at Deep Springs we expect students to miss no classes save for legitimate health issues or emergencies requiring also missing labor and governance obligations. For a student wishing to submit a course assignment past its required deadline, the student may request an extension on the assignment directly from the professor 48 hours in advance. Within 48 hours of the due date, the student must request an extension directly from the Dean. Exceptions will be granted by the Dean only if the student faces unforeseen and unforeseeable circumstances. A student who misses the deadline will be penalized an amount that is roughly equivalent to a letter grade for each day the assignment is late. Assignments cannot be turned in after solutions and graded assignments have been passed back, which generally happens one to two classes after they were turned in.