Change of variable in 4D space-time

Question

I was reading Sidney Coleman's article "Fate of the false vacuum: Samiclassical theory" and i stumbled upon a change of variables that i can't seem to prove. The problem is this: trying to solve the equation (3.1)

$$\left( \frac{\partial^2}{\partial\tau^2}+\nabla^2\right)\phi = \frac{dU(\phi)}{d\phi}$$

he uses the variable $\rho = (\tau^2+|x|^2)^{1/2}$ to simplify the equation being $\phi$ invariant under four-dimensional Euclidian rotations. Under this change of variable the equation becomes

$$\frac{d^2\phi}{d\rho^2}+\color{red}{\frac{3}{\rho}}\frac{d\phi}{d\rho}.$$

I cannot get that red factor. Evaluating the change of variable as such

$$ \frac{\partial}{\partial\tau}=\frac{\partial\rho}{\partial\tau}\frac{\partial}{\partial\rho}\\ \frac{\partial}{\partial x} = \frac{\partial\rho}{\partial x}\frac{\partial}{\partial\rho} $$

and to the same extent, evaluating the second partial derivatives, gets me a factor $1/\rho$ instead of $3/\rho$.

Why is that? How should I evaluate that change of variables?

Answer 1

The chain rule for partial derivatives is rather more complicated than you seem to think. Your expressions for the change of coordinates are not correct. Look at a derivation of the Laplacian in three dimensions to see how the $(2/r)\partial_r$ arises there.

If you want the Laplacian in $n$ dimensions, it is easiest to start from the action
$$ \int r^{n-1} dr d\Omega \left(g^{\mu\nu}\partial_\mu \phi \partial_\nu \phi\right) $$ where $d\Omega$ refers to all the angular parts of the measure. The metric is $$ ds^2= dr^2+r^{n-1}(Angle-parts) $$ The $(n-1)/r$ in the Laplacian comes from the $n-1$ in the $r^{n-1} dr$.