- #1
etotheipi
The general metric is a function of the coordinates in the spacetime, i.e. ##g = g(x^0, x^1,\dots,x^{n-1})##. That means that in the most general case we can't simplify an expression like ##\partial g_{\mu \nu} / \partial x^{\sigma}##. But, what about the special case of the flat spacetime metric$$\frac{\partial \eta_{\mu \nu}}{\partial x^{\sigma}} = \dots \,?$$can we simplify that? I thought it might be zero [since it is coordinate independent], but it is perhaps not the case. Also, I was under the impression that$$\frac{\partial x^{\rho}}{\partial x^{\sigma}} = \frac{\partial x_{\sigma}}{\partial x_{\rho}} = \delta^{\rho}_{\sigma}$$is this correct? Furthermore, in order to find what $$\frac{\partial x^{\rho}}{\partial x_{\sigma}} \quad \text{and} \quad \frac{\partial x_{\rho}}{\partial x^{\sigma}}$$are, it will be necessary to understand the answer to ##\partial {\eta}_{\mu \nu} / \partial x^{\sigma}##. Thanks!
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