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!