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Onyx

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- TL;DR Summary
- Calculating the christoffel symbols of Hiscock coordinates.

The Hiscock coordinates read:

$$d\tau=(1+\frac{v^2(1-f)}{1-v^2(1-f)^2})dt-\frac{v(1-f)}{1-v^2(1-f)^2}dx$$

##dr=dx-vdt##

Where ##f## is a function of ##r##. Now, in terms of calculating the christoffel symbol ##\Gamma^\tau_{\tau\tau}## of the new metric, where ##g_{\tau\tau}=v^2(1-f)^2-1## and ##g_{\tau r}=0##, can I safely assume that ##\Gamma^\tau_{\tau\tau}=0##, since ##\frac{\partial g_{\tau\tau}}{\partial \tau}=\frac{\partial g_{\tau\tau}}{\partial r}\frac{\partial r}{\partial \tau}## (in the Jacobi matrix ##\frac{\partial r}{\partial \tau}=0)##?

$$d\tau=(1+\frac{v^2(1-f)}{1-v^2(1-f)^2})dt-\frac{v(1-f)}{1-v^2(1-f)^2}dx$$

##dr=dx-vdt##

Where ##f## is a function of ##r##. Now, in terms of calculating the christoffel symbol ##\Gamma^\tau_{\tau\tau}## of the new metric, where ##g_{\tau\tau}=v^2(1-f)^2-1## and ##g_{\tau r}=0##, can I safely assume that ##\Gamma^\tau_{\tau\tau}=0##, since ##\frac{\partial g_{\tau\tau}}{\partial \tau}=\frac{\partial g_{\tau\tau}}{\partial r}\frac{\partial r}{\partial \tau}## (in the Jacobi matrix ##\frac{\partial r}{\partial \tau}=0)##?

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