201902210304 Exercise 6.3.1

Fill in the gap between equations (6.18) and (6.19).

Roughwork.

Equation (6.16):

$g^{\mu\alpha}g_{\alpha\nu}=\delta^\mu_{\enspace\nu}$ and $g'^{\mu\beta}g'_{\beta\nu}=\delta^\mu_{\enspace\nu}$

Equation (6.9):

$g'_{\mu\nu}=\displaystyle{\frac{\partial x^\alpha}{\partial x'^\mu}} \displaystyle{\frac{\partial x^\beta}{\partial x'^\nu}}g_{\alpha\beta}$

Equation (6.11):

$g'^{\mu\nu}= \displaystyle{\frac{\partial x'^\mu}{\partial x^\alpha}} \displaystyle{\frac{\partial x'^\nu}{\partial x^\beta}}g^{\alpha\beta}$

$\begin{aligned} g'^{\mu\nu} g_{\mu\nu} & = \bigg( \displaystyle{\frac{\partial x'^\mu}{\partial x^\alpha}} \displaystyle{\frac{\partial x'^\nu}{\partial x^\beta}} g^{\alpha\beta} \bigg) \bigg( \displaystyle{\frac{\partial x^\gamma}{\partial x'^\mu}} \displaystyle{\frac{\partial x^\delta}{\partial x'^\nu}} g_{\gamma\delta} \bigg) \\ & = \bigg( \displaystyle{\frac{\partial x^\gamma}{\partial x^\alpha}} \displaystyle{\frac{\partial x^\delta}{\partial x^\beta}} \bigg) g^{\alpha\beta}g_{\gamma\delta} \\ & = \delta^\gamma_{\enspace \alpha} \delta^\delta_{\enspace \beta} g^{\alpha\beta} g_{\gamma\delta} \\ & = g^{\alpha\beta} g_{\alpha\beta} \end{aligned}$

M	T	W	T	F	S	S
				1	2	3
4	5	6	7	8	9	10
11	12	13	14	15	16	17
18	19	20	21	22	23	24
25	26	27	28

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