I Static, Isotropic Metric: Dependence on x & dx

[davidge]

In Weinberg's book it is said that a Static, Isotropic metric should depend on $x$ and $dx$ only through the "rotational invariants" $dx^2, x \cdot dx, x^2$ and functions of $r \equiv (x \cdot x)^{1/2}$. It's clear from the definition of $r$ that $x \cdot dx$ and $x^2$ don't depend on the angular displacement. What I don't understand is why $dx^2$ is invariant under rotations, since it's the "pure" metric when written in spherical coordinates, and so it depends on the usual angles $\theta$ and $\varphi$.

 

[PeterDonis]

What I don't understand is why $dx^2$ is invariant under rotations

Because it's a distance (an infinitesimal one, but still a distance), and distances are invariant under rotations.

it depends on the usual angles $\theta$ and $\varphi$.

Formally, yes, but if you do a rotation of the coordinates, you will find that $\theta$ and $\varphi$ change in concert in such a way as to leave $dx^2$ invariant.

 

[davidge]

Thanks!