# Non-diagonal elements of the Schwarzchild metric

The Schwarzchild metric is the most general spherically symmetric, vacuum solution of the Einstein field equations.

I was wondering if there was a simple argument to explain why the Schwarzchild metric is diagonal in the spherical coordinate system, i.e. of the form

$$ds^2 = dt^2 + \cdots d\theta^2 + \cdots d\phi^2 + \cdots dr^2.$$

This Wikipedia article gives a really simple explanation which seems false.

(When you write the transformation law for $g_{\mu 4}$, it should be understood: $$g'_{\mu 4} (x') = \frac{\partial x^\alpha}{\partial x'^\mu}\frac{\partial x^\beta}{\partial x'^4}g_{\alpha\beta}(x)= -g_{\mu 4}(x) ,$$ while at the same time the invariance tells you that $$g'_{\mu 4}(x') = g_{\mu 4}(x') .$$ This leads to the conclusion that $$g_{\mu 4}(x') = -g_{\mu 4}(x) ,$$ but I fail to see how to go further in the reasoning without any aditionnal assumption.)

Other derivations either start from the diagonal form or are much more complicated.

It's probably a dumb question, but I fail to see a simple argument.

• I'm ok with the general transformation law part, but there's no reason to say that $g_{\mu,\nu}$ is invariant under those symmetries. Oct 17, 2016 at 10:07
• - I was wondering if there was a simple argument to explain why the Schwarzchild metric is diagonal in the spherical coordinate system - I may be missing something here, but isn't the fact that it just is diagonal an explanation in itself? And about spherical coordinates: it is rather due to the fact that the metric is of this particularly simple form in these coordinates that we call them spherical, not the other way around. Oct 21, 2016 at 12:14

## 1 Answer

OP is right: For a generic pseudo-Riemannian manifold $(M;g)$, there does not necessarily exist an open coordinate neighborhoods $U\subseteq M$, where the metric $g_{|U}$ is on diagonal form. Fermi normal coordinates always ensure a diagonal form along a geodesic $\gamma$ (but not necessarily in the ambient spacetime outside the geodesic). However, the Schwarzschild geometry has Killing symmetries that ensure that diagonal metrics exist in open neighborhoods.