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George Keeling

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- I have found a few different formulas for the Laplace operator. They don't seem to be the same. Or maybe they are and I just can't prove it.

I have found various formulations for the Laplacian and I want to check that they are all really the same. Two are from Wikipedia and the third is from Sean Carroll. They are:

A Wikipedia formula in ##n## dimensions:

\begin{align}

\nabla^2=\frac{1}{\sqrt{\left|g\right|}}\frac{\partial}{\partial x^i}\left(\sqrt{\left|g\right|}g^{ij}\frac{\partial}{\partial x^j}\right)&\phantom {10000}(2)\nonumber

\end{align}A Wikipedia formula in "in 3 general curvilinear coordinates ##(x^1,x^2,x^3)##":

\begin{align}

\nabla^2=g^{\mu\nu}\left(\frac{\partial^2}{\partial x^\mu\partial x^\nu}-\Gamma_{\mu\nu}^\lambda\frac{\partial}{\partial x^\lambda}\right)&\phantom {10000}(3)\nonumber

\end{align}And Carroll's formula (from exercise 3.4) :

\begin{align}

\nabla^2=\nabla_\mu\nabla^\mu=g^{\mu\nu}\nabla_\mu\nabla_\nu&\phantom {10000}(4)\nonumber

\end{align}The Wikipedia also gives a formula for the Laplacian in spherical polar coordinates:

\begin{align}

\nabla^2f&=\frac{1}{r^2}\frac{\partial}{\partial r}\left(r^2\frac{\partial f}{\partial r}\right)+\frac{1}{r^2\sin{\theta}}\frac{\partial}{\partial\theta}\left(\sin{\theta}\frac{\partial f}{\partial\theta}\right)+\frac{1}{r^2\sin^2{\theta}}\frac{\partial^2f}{\partial\phi^2}&\phantom {10000}(5)\nonumber\\

&=\frac{1}{r}\frac{\partial^2}{\partial r^2}\left(rf\right)+\frac{1}{r^2\sin{\theta}}\frac{\partial}{\partial\theta}\left(\sin{\theta}\frac{\partial f}{\partial\theta}\right)+\frac{1}{r^2\sin^2{\theta}}\frac{\partial^2f}{\partial\phi^2}&\phantom {10000}(6)\nonumber

\end{align}where ##\phi## represents the azimuthal angle and ##\theta## the zenith angle or co-latitude. So the metric will be

\begin{align}

g_{\mu\nu}=\left(\begin{matrix}1&0&0\\0&r^2&0\\0&0&r^2\sin^2{\theta}\\\end{matrix}\right)&\phantom {10000}(7)\nonumber

\end{align}I assumed that the coordinates are ordered ##r,\theta,\phi## although Wikipedia does not say that.

I want to prove that

A) (2), (3) and (4) both give (5) or (6) the Laplacian in spherical polar coordinates.

B) (4) is equivalent to (3) the general 3-dimensional expression.

C) (4) is equivalent to (2) the general ##n##-dimensional expression.

A was quite easy. B follows immediately from the formula for the covariant derivative. I could not prove C. I also tried it for a diagonal metric and failed.

A Wikipedia formula in ##n## dimensions:

\begin{align}

\nabla^2=\frac{1}{\sqrt{\left|g\right|}}\frac{\partial}{\partial x^i}\left(\sqrt{\left|g\right|}g^{ij}\frac{\partial}{\partial x^j}\right)&\phantom {10000}(2)\nonumber

\end{align}A Wikipedia formula in "in 3 general curvilinear coordinates ##(x^1,x^2,x^3)##":

\begin{align}

\nabla^2=g^{\mu\nu}\left(\frac{\partial^2}{\partial x^\mu\partial x^\nu}-\Gamma_{\mu\nu}^\lambda\frac{\partial}{\partial x^\lambda}\right)&\phantom {10000}(3)\nonumber

\end{align}And Carroll's formula (from exercise 3.4) :

\begin{align}

\nabla^2=\nabla_\mu\nabla^\mu=g^{\mu\nu}\nabla_\mu\nabla_\nu&\phantom {10000}(4)\nonumber

\end{align}The Wikipedia also gives a formula for the Laplacian in spherical polar coordinates:

\begin{align}

\nabla^2f&=\frac{1}{r^2}\frac{\partial}{\partial r}\left(r^2\frac{\partial f}{\partial r}\right)+\frac{1}{r^2\sin{\theta}}\frac{\partial}{\partial\theta}\left(\sin{\theta}\frac{\partial f}{\partial\theta}\right)+\frac{1}{r^2\sin^2{\theta}}\frac{\partial^2f}{\partial\phi^2}&\phantom {10000}(5)\nonumber\\

&=\frac{1}{r}\frac{\partial^2}{\partial r^2}\left(rf\right)+\frac{1}{r^2\sin{\theta}}\frac{\partial}{\partial\theta}\left(\sin{\theta}\frac{\partial f}{\partial\theta}\right)+\frac{1}{r^2\sin^2{\theta}}\frac{\partial^2f}{\partial\phi^2}&\phantom {10000}(6)\nonumber

\end{align}where ##\phi## represents the azimuthal angle and ##\theta## the zenith angle or co-latitude. So the metric will be

\begin{align}

g_{\mu\nu}=\left(\begin{matrix}1&0&0\\0&r^2&0\\0&0&r^2\sin^2{\theta}\\\end{matrix}\right)&\phantom {10000}(7)\nonumber

\end{align}I assumed that the coordinates are ordered ##r,\theta,\phi## although Wikipedia does not say that.

I want to prove that

A) (2), (3) and (4) both give (5) or (6) the Laplacian in spherical polar coordinates.

B) (4) is equivalent to (3) the general 3-dimensional expression.

C) (4) is equivalent to (2) the general ##n##-dimensional expression.

A was quite easy. B follows immediately from the formula for the covariant derivative. I could not prove C. I also tried it for a diagonal metric and failed.

**Does anybody have a view on how to calculate a Laplacian? I am new to them. Maybe it is possible to prove C. Is there some secret about the determinant of the metric which I have forgotten or don't know?**