Physics Asked on December 22, 2020
In Hobsons’s General Relativity: An Introduction for Physicists, pg. 64,
he gave two different expressions for the affine connection $Gamma’^a_{bc}$ in a transformed coordinate basis $x’^a$ (the original coodinate basis is $x^a$).
The two expressions are
$$Gamma’^a_{bc}=frac{partial x’^a}{partial x^d}frac{partial x^f}{partial x’^b}frac{partial x^g}{partial x’^c}Gamma^d_{fg}+frac{partial x’^a}{partial x^d}frac{partial ^2x^d}{partial x’^c partial x’^b},$$
$$Gamma’^a_{bc}=frac{partial x’^a}{partial x^d}frac{partial x^f}{partial x’^b}frac{partial x^g}{partial x’^c}Gamma^d_{fg}-frac{partial x^d}{partial x’^b}frac{partial x^f}{partial x’^c}frac{partial^2 x’^a}{partial x^d partial x^f}.$$
Why are these two expressions the same?
Hobson said that the second expression is obtained by swapping derivative with respect to $x$ and $x’$ in the last term on the RHS of the first expression. I have a hard time seeing what he explicitly meant by that. The presence of a minus sign in the second expression confuses me.
The last term in second expression can found from first by following manipulation: $$frac{partial x'^a}{partial x^d}frac{partial ^2x^d}{partial x'^c partial x'^b}$$
$$=frac{partial}{partial x'^c}bigg(frac{partial x'^a}{partial x^d}frac{partial x^d}{partial x'^b}bigg)-frac{partial x^d}{partial x'^b}bigg[frac{partial}{partial x'^c}bigg]bigg(frac{partial x'^a}{partial x^d}bigg)$$
$$=frac{partial}{partial x'^c}bigg(frac{partial x'^a}{partial x'^b}bigg)-frac{partial x^d}{partial x'^b}bigg[frac{partial x^f}{partial x'^c}frac{partial}{partial x^f}bigg]bigg(frac{partial x'^a}{partial x^d}bigg)$$
$$=frac{partial}{partial x'^c}Big(delta^{'a}_{'b}Big)-frac{partial x^d}{partial x'^b}frac{partial x^f}{partial x'^c}frac{partial^2 x'^a}{partial x^d partial x^f}$$
$$0-frac{partial x^d}{partial x'^b}frac{partial x^f}{partial x'^c}frac{partial^2 x'^a}{partial x^d partial x^f}$$
There is a subtle point I have left in the above answer; try to figure it out. It's explained in the last.
Now for why he went for second expression. If you have probably tried to find how $partial_{alpha}V^{beta}$ transforms during a coordinate change you'll find there is an additional term that pops out, it's same as above term with opposite sign. This is the reason why define $partial_{alpha}V^{beta}+Gamma^beta_{alphagamma}V^gamma$ as some kind of derivative which transforms as a tensor during coordinate change. The nontensorial part cancels each other out. So the reason is just a coherence of expression, nothing more. Otherwise, both expressions are inherently fundamental.
! Missing point $$bigg[frac{partial}{partial x'^c}bigg]bigg(frac{partial x'^a}{partial x^d}bigg)$$ $$neqbigg[frac{partial}{partial x^d}bigg]bigg(frac{partial x'^a}{partial x'^c}bigg)$$ $$neq frac{partial}{partial x^d}delta^{'a}_{'c}$$ $$neq 0$$ The issue lies in the second line and our sloppiness in notation of partial derivative $$bigg[frac{partial}{partial x'^c(o)}bigg]_{mathrm{nhspace{2pt}ishspace{2pt}kepthspace{2pt}constant}}bigg(frac{partial x'^a(o)}{partial x^d(n)}bigg)_{mathrm{ohspace{2pt}ishspace{2pt}kepthspace{2pt}constant}}$$ where n stands for new coordinate, o for old coordinates.
Correct answer by aitfel on December 22, 2020
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