Category: Science

Categories
Calculus Differential Equations General Relativity Math Multivariable Calculus Tensor Calculus

Carroll General Relativity: how you might have thought of the magic transform

I’m reading Spacetime and Geometry by Sean Carroll and working all the proofs as I go. I recorded this video after struggling to understand where a seemingly magic transform came from and why it simplified a complicated equation. If you’re working on his equation 5.139, this will help you understand how you might have thought of the transform and why the thing called m really is a mass.

Unfortunately, I discovered only after I recorded this that the whiteboard had been switched off for the introduction and, for reasons to tedious to mention, was unfixable with the tools at hand. So I provide here the equations I refer to. (Once I get launched on the explanation proper, you can see everything I’m writing.)

Here is the original time component of Einstein’s Equation, Carroll’s 5.139:

1r2e−2β(2r∂rβ−1+22β)=8πGρ\frac{1}{r^2} e^{-2\beta}(2r \partial_r\beta-1+2^{2\beta}) = 8\pi G\rho

Here is the transform he conjuries out of thin air, which I try to explain (Equation 5.142):

m(r)=12G(r−re−2β)m(r)=\frac{1}{2G}(r-r e^{-2\beta})

And here is the resulting simplified equation:

dmdr=4πr2ρ\frac{dm}{dr}=4\pi r^2\rho
Categories
General Relativity Tensor Calculus

Unpacking the Riemann Tensor

If you’re studying General Relativity or Tensor Calculus you’ve slammed into the Riemann tensor and its flurry-of-gammas-and-indices definition. But at its heart, the Riemann is just saying “move a vector around a tiny parallelogram first one way and then the other. Get a different result one way? Then the space is curved.” This video matches up the symbols and indices to that basic definition. I always remember something better when it makes sense. This video should help you remember the Riemann’s definition too.

Categories
Math Physics Pre-Calculus Proofs Proving Identities Science Trigonometry

Spin One Particle Rotates Like a Vector, y-axis

In The Feynman Lectures on Physics, Volume 3 Chapter 5, Feynman poses a challenge to the reader: show that a particular combination of plus, minus and zero states of a spin one particle transforms under a rotation just like a vector does. In this previous video I reviewed the details of proving it for rotations around the z-axis. Here I go over the more complicated proof for rotations around the y-axis.

Categories
Complex Numbers Math Physics Pre-Calculus Proofs Proving Identities Science Trigonometry

Spin One Particle Rotates Like a Vector, z-axis

In The Feynman Lectures on Physics, Volume 3 Chapter 5, Feynman poses a challenge to the reader: show that a particular combination of plus, minus and zero states of a spin one particle transforms under a rotation just like a vector does. Here I go over how to work that problem using only what was covered in the book up to that point.

Categories
Calculus Differential Equations Physics Science

Physics Pendulum Problem

Getting the Period of a Spring-Operated Physical Pendulum

A juicy physics problem that requires delving into the differential equation for pendulum motion. A rod of mass M on a pivot a distance r from the end is driven by a spring of constant k pulling the end back and forth to make an (admittedly stupid) physical pendulum. We have to show that its period T is given by T^2 = 4\pi \frac{M}{3kr^2} (L^2 + 3r^2 -3rl) .

Categories
Physics Science

A Rock on the Moon

A Flying Rock on the Moon — a Kinematics Problem

A nice tough physics kinematics problem: how high will a flying rock on the moon rise if all you know is how long it took to pass a viewport?

Categories
Chemistry Science

Chemistry: An Equilibrium Problem

This video walks you through the multifarious steps and guides you past all the gotchas in solving an equilibrium problem. I’ll assume you know what equilibrium problems *are* and just need some practice *doing* them.