Category: Multivariable Calculus

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
Multivariable Calculus Proofs Special Functions

Fourier Transform of a Bessel Function

For those of you working on special functions like Bessel functions who know you learned Fourier Transforms at some point but need help untangling how they work.

Categories
Calculus Line Integrals Math Multivariable Calculus Vector Calculus

Find Potential Function Using Partial Integration

Once you’ve established that curl F = 0, you know that F is the gradient of some potential ⱷ. You can use the Fundamental Theorem to evaluate ∫F.dr — if only you could find the potential function. In this video I go over a straighforward method of finding the potential function using partial integration. (Note: the sound quality gets wonky near the end but you can still hear what I’m explaining perfectly well. Apologies and I’ll figure out eventually why that happens.)