I Equations Using Comma-Goto-Semicolon Rule in Curved Spacetime

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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!
 
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Haorong Wu said:
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.
 
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}##.
 
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