0
$\begingroup$

The proof is often simplified by using the following theorem:

"If the metric tensor $(g_{ij})$ is positive definite, then, at the origin of a Riemannian coordinate system $(y^i)$, all $\partial g_{ij}/\partial y^k,\partial g^{ij}/\partial y^k, \Gamma_{ijk}$, and $\Gamma^i_{jk} $ are zero."

It follows that at $O$, the origin of normal coordinates: $$R_{ijkl}=\frac{\partial \Gamma_{lji}}{\partial y^k}-\frac{\partial \Gamma_{kji}}{\partial y^l}$$ (1.0)

The whole proof will not be shown here.

I understand that the above expression comes directly from the definition of the Riemann Tensor of the first kind $R_{ijkl}=\frac{\partial \Gamma_{lji}}{\partial y^k}-\frac{\partial \Gamma_{kji}}{\partial y^l}+\Gamma_{ilr}\Gamma^{r}_{jk}-\Gamma_{ikj}\Gamma^r_{jl}$ but if all the Christoffel symbols vanish at the origin of the Riemannian(normal) coordinates then why do the first two terms (i.e. the RHS of (1.0)) remain?

$\endgroup$
  • $\begingroup$ PS: I think my question deserves to be asked here at Physics SE since it is used so often in GR $\endgroup$ – gaugefixer Jun 9 at 15:02
4
$\begingroup$

For the same reason that given a function $f(x)$ which zero at some $x=x_0$, it does not imply that $f'(x_0)=0$.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.