Notes on Problem Set 3 of Ph236a 2006-2007

As mentioned in my immediately prior post, I’m trying to put together discussion notes for General Relativity/Gravity. Included in the current update is my discussion and my attempt at working out (i.e. my solutions) Problem Set 3 of Ph236a General Relativity, taught by Prof. Kamionkowski at Caltech in 2006-2007.

Again, I am not typing onto wordpress all the LaTeX: I have to manually go in and type latex and reformat my LaTeX code to make it work in wordpress. If someone has an easy and completely automated and robust way of getting LaTeX, including all formats, and all possible ams packages, to show up on wordpress, let me know; otherwise, my pdf and LaTeX file is always available for download publicly on my Google Drive.

In my notes, in brief, on the Problem Set 3, I make these remarks on the problems.

1. Riemann normal coordinates, or a locally inertial frame. I try to show \partial_{\rho}g_{\mu \nu } = 0 in a manifestly covariant manner (i.e. coordinate-free), explain where the exponential map comes from, and emphasize the point of how Riemann normal coordinates resolve Mach’s Principle/Paradox with how Einstein did.

2. (Wald 3.3) Commutator of vector fields

We’re in the 21st century. Let’s try to do most tedious algebra with computers. I have the sympy/python code that I used in my notes.

3. (Wald 3.4) For part (b), again, I try to do everything in a manifestly covariant (i.e. coordinate-free) manner and it makes the geometry crystal clear.

4. (MTW, Problem 9.6) Practice with dual bases:
I use Sage Math, along with the sagemanifolds package to solve part (a). Code for part (a) is on my github
I want to emphasize, all my code and everything I use is open-source, as much as possible.

5. (MTW, Problems 9.7-8) Commutators for Euclidean space in spherical coordinates
I try to make these computations fun by doing them with Sage Math and the sagemanifolds package and the code I wrote is on my github for R3.sage.

6. (Wald 2.8b+) Rotating coordinates

To call the ds^2 thing a “line element”, in my opinion, is a misnomer. What actually is here is the metric in local form. And when going between coordinates, what you actually want to compute is the pullback of the metric, and pullback to the coordinates you want. It’s all in my notes and the code for computing Jacobians, inverse Jacobians, and the metric from the Jacobians is on my github: Min4d.sage

2 points:

I’ve implemented code using sympy, Sage Math, sagemanifolds, all built on Python, and all open-source. I want to use software that will remain open and public and transparent to anyone in the world(s). There are 2 points that shape what I do and gives me reason to believe that doing so is of utmost importance:

The more you could game-ify the process of learning, the better…
So to the degree that you could make somehow learning like a game, then it’s better. -Elon Musk

Khan Academy A Conversation with Elon Musk

Through a sequence of things that I’ll tell you about, I got really interested in seeing if we could make higher education available to everybody. I think it’s the most important that a human being could do, because human resources, people, are the most important resources on the planet
and everything that you can make and do comes through education.

For you this might be relatively trivial and obvious because you made it into Brown [University]
but if you’re in Africa, in India, and in the developing world, if you have medical problems, that might not be quite as obvious. -Sebastian Thrun

Online Learning: The Challenge Sebastian Thrun

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