# Stationary v/s Static

Blau, in his GR book, says that a stationary and spherically symmetric metric is automatically static. He says this easily follows from the fact that for a stationary metric, and in spherical symmetry, in coordinates $(t,r,\theta,\phi)$ suitable for expressing both these facts, the only allowed off-diagonal $g_{tk}$-term of the metric is $C(r)=g_{tr}(r)$, so that the $(t,r)$-part of the metric takes the form $$ds^2=-A(r)\,dt^2+B(r)\,dr^2+2C(r)\,dtdr.$$

He adds then $C(r)$ can be eliminated by a coordinate transformation $T(t,r)=t+\psi(r)$, and $\partial_t=\partial_T$ is thus orthogonal to the surfaces of constant $T$.

My question how can one know that $\partial_t=\partial_T$ is thus orthogonal to the surfaces of constant $T$?

• Are you supposed to pick a $\Psi (r)$ to make it work or did you forget to tell us what $\Psi (r)$ is? – Timaeus Jun 9 '15 at 22:45
• Beyond: I can't find any book by Blau online. There are lecture notes though. I strongly encourage you to precisely cite and link any mentioned work, so other users can easily find it. – magma Jun 10 '15 at 9:51

That's just a very basic concept of how you choose (curved) coordinates.

Maybe think first of the example of an Euclidean coordinate system. There your coordinate basis vector $\partial_x$ is also orthogonal to surfaces of constant $x$.

The same holds also true for curved coordinates.

If you like you can also parametrize your surface $T=const.$ as a function of $S_T=S_T(r,\theta,\phi)$, build the tangential vectors $\partial_r S_T, \partial_\theta S_T, \partial_\phi S_T$ and check that these are orthogonal to $\partial_T$ (use that the metric is diagonal).

Another easy exercise is just to build the normal vector to the surface: Characterize the surface as a scalar function of the coordinates $f(T,r,\theta,\phi)$ and the requirement $f=0$. Calculate the normal vector which is defined by: $$n^\nu=g^{\nu\mu}\partial_\mu f$$

(Of course both 'calculations' are equivalent.)

As I said in my comment above, I could not find any book by Blau. In his lecture notes at page 481 he makes the transformation you mention. In detail he starts with the metric in $(t,r)$ in eq 23.1 which has diagonal terms, makes the transformation 23.3 which introduces $T$, then he finds a suitable $\psi$, and finally he rewites the metric in (T and r) without the diagonal terms , but...- without saying this explicitly - instead of using the new $T$ he renames it again $t$ in eq 23.5. It is just a renaming operation using an old variable, nothing fancy. Not nice, but unfortunately common. These are notes for advanced students who are supposed to see this in a blink. He does this again a few lines later when he introduces $R(r)$ only to rename it again $r$ in the next equation.