- #1
Bitometry
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- 0
I'm hoping someone can clarify for me, I have seen the following used:
[itex]\frac{\partial}{\partial g^{ab}}\left( g^{cd} \right) = \frac{1}{2} \left( \delta_a^c \delta_b^d + \delta_b^c \delta_a^d\right)[/itex]
I understand the two half terms are used to account for the symmetry of the metric tensor.
However, it appears to only half account for terms where [itex]a \ne b[/itex].
To demonstrate what I mean by example, consider a 2D metric where [itex](a,b,c,d = 1,2)[/itex]
[itex]\frac{\partial}{\partial g^{11}}\left( g^{11} \right) = \frac{1}{2} \left( \delta_1^1 \delta_1^1 + \delta_1^1 \delta_1^1\right) = 1[/itex]
but
[itex]\frac{\partial}{\partial g^{12}}\left( g^{12} \right) = \frac{1}{2} \left( \delta_1^1 \delta_2^2 + \delta_2^1 \delta_1^2\right) = \frac{1}{2}[/itex]
I expected (possibly incorrectly) the latter would also be 1.
Is the contribution of [itex]g^{cd}[/itex] evenly distributed between [itex]\frac{\partial}{\partial g^{ab}}[/itex] and [itex]\frac{\partial}{\partial g^{ba}}[/itex], and thus for [itex]a=b[/itex] the value is double the others?
[itex]\frac{\partial}{\partial g^{ab}}\left( g^{cd} \right) = \frac{1}{2} \left( \delta_a^c \delta_b^d + \delta_b^c \delta_a^d\right)[/itex]
I understand the two half terms are used to account for the symmetry of the metric tensor.
However, it appears to only half account for terms where [itex]a \ne b[/itex].
To demonstrate what I mean by example, consider a 2D metric where [itex](a,b,c,d = 1,2)[/itex]
[itex]\frac{\partial}{\partial g^{11}}\left( g^{11} \right) = \frac{1}{2} \left( \delta_1^1 \delta_1^1 + \delta_1^1 \delta_1^1\right) = 1[/itex]
but
[itex]\frac{\partial}{\partial g^{12}}\left( g^{12} \right) = \frac{1}{2} \left( \delta_1^1 \delta_2^2 + \delta_2^1 \delta_1^2\right) = \frac{1}{2}[/itex]
I expected (possibly incorrectly) the latter would also be 1.
Is the contribution of [itex]g^{cd}[/itex] evenly distributed between [itex]\frac{\partial}{\partial g^{ab}}[/itex] and [itex]\frac{\partial}{\partial g^{ba}}[/itex], and thus for [itex]a=b[/itex] the value is double the others?