Meaning of ds^2 according to Carroll

[vanhees71]

Of course you have to distinguish the symbols. I'm an advocate of the good old Ricci calculus, because there it's much more intuitive. What I used, however in my previous postings in this thread is Cartan calculus on manifolds for the special case of holonomous coordinates. There the symbol $\mathrm{d} q^{\mu}$ is a tangent-space dual vector, i.e., a one-form, i.e., a linear mapping from tangent space at a point on the manifold, parametrized with real generalized coordinates $q^{\mu}$. The holonomous tangent-space basis at the point defined by the tangent vectors of the coordinate lines, i.e., keeping all $q^{\mu}$ constant except one. This basis is denoted by $\partial_{\mu}$. By definition the dual basis $\mathrm{d} q^{\mu}$ maps these holonomous basis vectors as follows $\mathrm{d} q^{\mu}(\partial_{\nu})={\delta^{\mu}}_{\nu}$, where the latter symbol is the usual Kronecker $\delta$.

In the Ricci calculus the $\mathrm{d} q^{\mu}$ are simply infinitesimal increments of the coordinates and thus
$$\mathrm{d} s^2=g_{\mu \nu} \mathrm{d} q^{\mu} \mathrm{d} q^{\nu}.$$
I've given the derivation of this in the Cartan calculus also in my previous posting.

 

[guitarphysics]

Thanks for that reference to MTW, Peter.. It becomes clearer to me as time goes on why that book is the bible of GR.