Equations Using Comma-Goto-Semicolon Rule in Curved Spacetime

[Haorong Wu]

TL;DR Summary

What conditions should a physical equation satisfy so that the comma-goto-semicolon rule can be applied to it?

Recently, I am considering the wave equations of a light beam in curved spacetime. Here I have two approaches. Both start from the Helmholtz equation $\psi^{,\mu}_{~~,\mu}=\eta^{\mu\nu}\psi_{,\mu,\nu}=0$ in the Minkowski spacetime, and $\psi$ is assumed to be $T(x,y,z)e^{ik(z-t)}$.

In the first approach, I could impose the paraxial approximation on the Helmholtz equation yielding the scalar wave equation $2ik T_{,3}=\eta^{ij}T_{,i,j}$ where $i$ and $j$ run in the spatial coordinates. Then I write its counterpart in curved spacetime according to the comma-goto-semicolon rule yielding $2ik T_{;3}=g^{ij}T_{;i;j} .$

In the other approach, I would first use the comma-goto-semicolon rule on the Helmholtz equation to have $g^{\mu\nu}\psi_{;\mu;\nu}=0$. Then I express the covariant derivative by partial derivative. Along the process, the paraxial approximation is used to eliminate $T_{,3,3}$ term.

Now if I subscribe some metric to these two results, I will have inconsistent equations. In the second approach, $g^{33}\psi_{;3;3}$, which does not appear in the first approach, will introduce some new terms. I think the second approach is correct since the Helmholtz equation is more symmetric than the scalar wave equation. This makes me wonder what condition should a equation satisfy so that I could use the comma-goto-semicolon rule to turn it into a covariant form?

BTW, when I write covariant derivative, should I write $T(x,y,z)_{,i}$ or $T_{,i}(x,y,z)$? Also, if there are two covariant derivatives, should I write $T_{;i;j}$ or $T_{;ij}$?

Thanks!

 

[anuttarasammyak]

Also, if there are two covariant derivatives, should I write T;i;j or T;ij?

I observe $:i:j$ is used e.g. $T_{:i:j}$ in Dirac's text I have.

 

[ergospherical]

The "comma-to-semicolon"/$\partial$-to-$\nabla$ rule is not always clear-cut. To give another example, consider the Maxwell equation $\partial_{\mu} F^{\mu \nu} = 4\pi j^{\nu}$, which in terms of the vector potential reads\begin{align*}
\partial_{\mu} \partial^{\mu} A^{\nu} - \partial_{\mu} \partial^{\nu} A^{\mu} &= 4\pi j^{\nu} \ \ \ (\dagger) \\ \overset{\mathrm{curved \ spacetime}}{\implies} \nabla_{\mu} \nabla^{\mu} A^{\nu} - \nabla_{\mu} \nabla^{\nu} A^{\mu} &= 4\pi j^{\nu} \ \ \ (\mathrm{a})
\end{align*}On the other hand, since $\partial_{\mu} \partial^{\nu} = \partial^{\nu} \partial_{\mu}$ then one can re-write $(\dagger)$ as\begin{align*}
\partial_{\mu} \partial^{\mu} A^{\nu} - \partial^{\nu} \partial_{\mu} A^{\mu} &= 4\pi j^{\nu} \\ \overset{\mathrm{curved \ spacetime}}{\implies} \nabla_{\mu} \nabla^{\mu} A^{\nu} - \nabla^{\nu} \nabla_{\mu} A^{\mu} &= 4\pi j^{\nu} \ \ \ (\mathrm{b})
\end{align*}If one defines a "curvature" operator $\nabla_{\mu} \nabla^{\nu} - \nabla^{\nu} \nabla_{\mu} \equiv {R^{\nu}}_{\mu}$ then one can re-write this last equation as\begin{align*}
\nabla_{\mu} \nabla^{\mu} A^{\nu} - \nabla_{\mu} \nabla^{\nu} A^{\mu} + {R^{\nu}}_{\mu} A^{\mu} &= 4\pi j^{\nu} \ \ \ (\mathrm{c})
\end{align*}The equations $(\mathrm{a})$ and $(\mathrm{c})$ differ by this term ${R^{\nu}}_{\mu} A^{\mu}$.