In chapter 9.2 of "Tensors, Relativity and Cosmology", the contracted Christoffel symbol of the second kind as a function of the metric tensor was defined as: $$\Gamma_{nm}^m=\frac{1}{2}\left(g^{mk}\frac{\partial g_{kn}}{\partial x^m}+g^{mk}\frac{\partial g_{mk}}{\partial x^n}-g^{mk}\frac{\partial g_{nm}}{\partial x^k}\right). \tag{9.30}$$

Which simplified into $$\Gamma_{nm}^m=\frac{1}{2}g^{mk}\frac{\partial g_{mk}}{\partial x^n}. \tag{9.31}$$

Later on an alternative expression was obtained (the derivation was extremely lengthy so multiple steps ignored here) using $g=g_{mk}G^{mk}$ :$$\Gamma_{nm}^m=\frac{1}{2g}\frac{\partial g}{\partial x^n}=\frac{\partial \ln\sqrt g}{\partial x^n} \tag{9.36}$$

Combining the results (9.31) and (9.36) gives $$\frac{1}{2}g^{kn}\frac{\partial g_{kn}}{\partial x^l}\frac{1}{2g}\frac{\partial g}{\partial x^l}=\frac{\partial \ln\sqrt g}{\partial x^l} \tag{9.45}$$

However, I was not able obtain the RHS of (9.45) and it looks like the RHS of (9.36) remains unaltered upon the combination with (9.30)? Could anyone explain how the RHS of (9.45) was yielded?

  • $\begingroup$ $\dfrac {\partial }{\partial x} \ln \left( g^{1/2}\left( x\right) \right) =\dfrac {1}{2g} \dfrac {\partial g}{\partial x} $ $\endgroup$ – Eli Apr 21 '19 at 7:04
  • $\begingroup$ @Eli I did understand where this came from but the question is how multiplying 9.31 with 9.36 gives $\frac{\partial \ln\sqrt g}{\partial x^l}$ ? $\endgroup$ – Chern-Simons Apr 21 '19 at 8:29
  • $\begingroup$ Hi EXINT. Welcome to Phys.SE. If you haven't already done so, please take a minute to read the definition of when to use the homework-and-exercises tag, and the Phys.SE policy for homework-like problems. $\endgroup$ – Qmechanic Apr 21 '19 at 10:21
  • $\begingroup$ @Qmechanic♦ Hi, I'll have a read but the tag was actually added by another user who edited my question earlier on. $\endgroup$ – Chern-Simons Apr 21 '19 at 11:16
  • $\begingroup$ That user was me! $\endgroup$ – Qmechanic Apr 22 '19 at 22:02

I don't have this book (it would be good to add a better reference to it: I assume it is this), but I am almost certain that the (9.45) is wrong as given (in v2 / v4 of the question: I don't know what's in the book). Even without looking hard at it it's clear that it's syntactically dubious, at best: it contains a repeated index ($l$) on the LHS which can't sensibly be summed over and which corresponds to an unrepeated index on the RHS: that simply makes no sense in terms of the ESC.

I believe that (9.45) is simply missing an equals sign and should read:

$$ \frac{1}{2}g^{kn} \frac{\partial g_{kn}}{\partial x^l} = \frac{1}{2g} \frac{\partial g}{\partial x^l} = \frac{\partial\ln\left(\sqrt{g}\right)}{\partial x^l} $$

This is merely equating the expressions in (9.31) & (9.36).

I think a way of checking this would be to see what further use they make of (9.45): my guess is either 'none' or they use the corrected version above.

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