How can I adjust parenthesis on mathematics?

Question

How can I make the square brackets on the following equation the same size?

\begin{equation}
\begin{split}R^{\mu}_{\nu\rho\sigma} &= \partial_{\rho}\left[\bar{\Gamma}^{\mu}_{\nu\sigma} + \tfrac{1}{2}\,\bar{g}^{\mu\beta}\left(\bar{\nabla}_{\nu}h_{\sigma\beta}+\bar{\nabla}_{\sigma}h_{\nu\beta}+\bar{\nabla}_{\beta}h_{\nu\sigma}\right)\right] \\
& -\partial_{\sigma}\left[\bar{\Gamma}^{\mu}_{\nu\rho} + \tfrac{1}{2}\,\bar{g}^{\mu\alpha}\left(\bar{\nabla}_{\nu}h_{\rho\alpha}+\bar{\nabla}_{\rho}h_{\nu\alpha}+\bar{\nabla}_{\alpha}h_{\nu\rho}\right)\right]
\end{split}
\end{equation}

Edit: to clarify, I have uploaded the problem I was having with the above code:

Answer 1

Like some of the other commenters, I believe the square brackets in the code snippet you've posted already all have the same size. I would actually suggest that you make the round parentheses smaller -- more precisely, to leave them at their default size. This will allow for some more natural grouping of the "fences," from the inner parentheses to the outer brackets.

I'd also recommend that you make a few more changes, such as (i) aligning the two RHS expressions vertically and (ii) reducing the whitespace between \bar{\nabla} and the subscript terms:

\documentclass{article}
\usepackage{amsmath}
\begin{document}
\begin{equation}
\begin{split}
R^{\mu}_{\nu\rho\sigma} 
&= \phantom{-}\, % force extra bit of whitespace after equal sign
 \partial_{\rho}\bigl[ \bar{\Gamma}^{\mu}_{\nu\sigma} 
+ \tfrac{1}{2}\,\bar{g}^{\mu\beta}
(\bar{\nabla}_{\!\nu}h_{\sigma\beta}
+\bar{\nabla}_{\!\sigma}h_{\nu\beta}
+\bar{\nabla}_{\!\beta}h_{\nu\sigma} )\bigr] \\
&\phantom{{}=} 
-\partial_{\sigma}\bigl[ \bar{\Gamma}^{\mu}_{\nu\rho} 
+ \tfrac{1}{2}\,\bar{g}^{\mu\alpha}
(\bar{\nabla}_{\!\nu}h_{\rho\alpha}
+\bar{\nabla}_{\!\rho}h_{\nu\alpha}
+\bar{\nabla}_{\!\alpha}h_{\nu\rho} )\bigr]
\end{split}
\end{equation}
\end{document}

---

Addendum: Following @egreg's suggestion, here's an alternative form of displaying the same equation, one which deliberately doesn't attempt to perform a vertical alignment of the two RHS parts:

\documentclass{article}
\usepackage{amsmath}
\begin{document}
\begin{equation}
\begin{split}
R^{\mu}_{\nu\rho\sigma} 
&= \partial_{\rho}\bigl[\bar{\Gamma}^{\mu}_{\nu\sigma} 
+ \tfrac{1}{2}\,\bar{g}^{\mu\beta}
(\bar{\nabla}_{\!\nu}h_{\sigma\beta}
+\bar{\nabla}_{\!\sigma}h_{\nu\beta}
+\bar{\nabla}_{\!\beta}h_{\nu\sigma} )\bigr] \\
&\qquad{} -\partial_{\sigma}\bigl[\bar{\Gamma}^{\mu}_{\nu\rho} 
+ \tfrac{1}{2}\,\bar{g}^{\mu\alpha}
(\bar{\nabla}_{\!\nu}h_{\rho\alpha}
+\bar{\nabla}_{\!\rho}h_{\nu\alpha}
+\bar{\nabla}_{\!\alpha}h_{\nu\rho} )\bigr]
\end{split}
\end{equation}
\end{document}

Answer 2

You could also place a \vphantom{\bar{g}^{\mu\beta}} inside the second pair of square brackets. I've used it when I had to split an expression in parentheses into multiple lines using \left( \right. in the first line and \left. \right) in the last, and you want both the opening and the closing parenthesis to have the same height.