Category: Differential Equations

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
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
Calculus Differential Equations Math ODE

What is a Differential Equation?

Lots of calculus students learn to solve differential equations without understanding what they really mean. Here’s a simple example of a differential equation so you can see how it works.