Deriving the Metric Connection Equation

In summary, we are given an equation involving the metric tensor and its derivatives. After simplifying and using the product rule, we find that the derivative of the equation with respect to $\mathfrak{g}^{ab}_{,c}$ is equal to $\frac{1}{2}\Gamma^c_{ab}-\frac{1}{2}\delta^c_b\Gamma^c_{ac}$. However, upon further inspection, it seems that only two out of the four terms in the final step are correct. We are asking for someone to review the math and help identify where the mistake was made.
  • #1
PhyPsy
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0

Homework Statement


Show that [itex]\frac{\partial\overline{\mathcal{L}}_G}{{\partial}\mathfrak{g}^{ab}_{,c}}=\Gamma^c_{ab}-\frac{1}{2}\delta^c_a\Gamma^d_{bd}-\frac{1}{2}\delta^c_b\Gamma^d_{ad}[/itex].

Homework Equations


[itex]\overline{\mathcal{L}}_G=\mathfrak{g}^{ab}(\Gamma^d_{ac}\Gamma^c_{bd}-\Gamma^c_{ab}\Gamma^d_{cd})[/itex]
[itex]\mathfrak{g}^{ab}=\sqrt{-g}g^{ab}[/itex]
[itex]\Gamma^a_{bc}=\frac{1}{2}g^{ad}(\partial_bg_{dc}+{\partial}_cg_{db}+\partial_dg_{bc})[/itex]

The Attempt at a Solution


This is one uuuuuuuuuuugly problem. In its simplest terms, [itex]\overline{\mathcal{L}}_G[/itex] is an equation of the metric tensor and its derivatives. So the first thing I do is figure out [itex]\frac{{\partial}g_{ef,d}}{\partial\mathfrak{g}^{ab}_{,c}}[/itex].
[itex]\frac{{\partial}g_{ef,d}}{\partial\mathfrak{g}^{ab}_{,c}} = \frac{{\partial}g_{ef,d}}{{\partial}g^{ab}_{,c}} \frac{{\partial}g^{ab}_{,c}}{\partial\mathfrak{g}^{ab}_{,c}}[/itex]
[itex]g^{ab}_{,c}=(-g)^{-\frac{1}{2}}\mathfrak{g}^{ab}_{,c}[/itex], so [itex]\frac{{\partial}g^{ab}_{,c}}{\partial\mathfrak{g}^{ab}_{,c}}=(-g)^{-\frac{1}{2}}[/itex]
[itex]g_{ef,d}=\delta^c_d\partial_c(g_{eh}g_{fi}g^{hi})[/itex]
[itex]\frac{{\partial}g^{hi}}{{\partial}g^{ab}}=\frac{1}{2}(\delta^h_a\delta^i_b+\delta^h_b\delta^i_a)[/itex]
Using the product rule and substituting in the above, I find [itex]\frac{{\partial}g_{ef,d}}{{\partial}g^{ab}_{,c}} = \delta^c_d\frac{1}{2}(\delta^h_a\delta^i_b +\delta^h_b{\delta}^i_a) g_{eh}g_{fi} + g_{fi}g^{hi} \frac{{\partial}g_{eh,d}}{{\partial}g^{ab}_{,c}} +g_{eh}g^{hi} \frac{{\partial}g_{fi,d}}{{\partial}g^{ab}_{,c}}[/itex]
[itex]= \frac{1}{2}\delta^c_d(g_{ea}g_{fb} + g_{eb}g_{fa}) + \delta^h_f \frac{{\partial}g_{eh,d}}{{\partial}g^{ab}_{,c}} + \delta^i_e\frac{{\partial}g_{fi,d}}{{\partial} g^{ab}_{,c}}[/itex]
[itex]\implies - \frac{{\partial}g_{ef,d}}{\partial g^{ab}_{,c}} = \frac{1}{2}\delta^c_d(g_{ea}g_{fb} + g_{eb}g_{fa})[/itex]
[itex]\frac{{\partial}g_{ef,d}}{\partial\mathfrak{g}^{ab}_{,c}}=-\frac{1}{2}\delta^c_d(-g)^{-\frac{1}{2}}(g_{ea}g_{fb}+g_{eb}g_{fa})[/itex]

Now comes the fun part. First, I write out [itex]\overline{\mathcal{L}}_G[/itex] in terms of the metric and its derivatives: [tex]\begin{align}
& \mathfrak{g}^{ab}[\frac{1}{2}g^{db}(\partial_ag_{bc}+\partial_cg_{ba}-\partial_bg_{ac})][\frac{1}{2}g^{ca}(\partial_bg_{ad}+\partial_dg_{ab}-\partial_ag_{bd})]\\
& \ \ \ \ - \mathfrak{g}^{ab}[\frac{1}{2}g^{cd}(\partial_ag_{db}+\partial_bg_{da}-\partial_dg_{ab})][\frac{1}{2}g^{da}(\partial_cg_{ad}+\partial_dg_{ac}-\partial_ag_{cd})]\end{align}[/tex]
If I multiply this out, the first term would be [itex]\frac{1}{4}\mathfrak{g}^{ab}g^{db}g^{ca}\partial_ag_{bc}\partial_bg_{ad}[/itex], and all the other terms would be similar, just with different indices. Since there are two partial derivatives of the metric in this term, the derivative of the term with respect to [itex]\mathfrak{g}^{ab}_{,c}[/itex] is [itex]\frac{1}{4}\mathfrak{g}^{ab}g^{db}g^{ca} (g_{bc,a}\frac{{\partial}g_{ad,b}}{\partial \mathfrak{g}^{ab}_{,c}} + g_{ad,b}\frac{{\partial}g_{bc,a}}{{\partial} \mathfrak{g}^{ab}_{,c}})[/itex] according to the product rule. However, instead of going through term by term like this, I am going to work it more like [tex]\frac{1}{4} \mathfrak{g}^{ab} g^{db} g^{ca} (g_{bc,a} + g_{ba,c} + g_{ac,b}) (\frac{{\partial}g_{ad,b}} {{\partial}\mathfrak{g}^{ab}_{,c}} +\frac{{\partial}g_{ab,d}}{{\partial}\mathfrak{g}^{ab}_{,c}} + \frac{{\partial}g_{bd,a}}{{\partial} \mathfrak{g}^{ab}_{,c}}) + \frac{1}{4}\mathfrak{g}^{ab}g^{db}g^{ca} (g_{ad,b}+g_{ab,d}+g_{bd,a}) (\frac{{\partial}g_{bc,a}}{{\partial}\mathfrak{g}^{ab}_{,c}} +\frac{{\partial}g_{ba,c}}{{\partial}\mathfrak{g}^{ab}_{,c}} +\frac{{\partial}g_{ac,b}}{{\partial}\mathfrak{g}^{ab}_{,c}})+...[/tex] so that the terms will come out in a way that makes it easier to simplify to metric connections [itex]\Gamma[/itex].

I come up with
[tex]\begin{align}
& -\frac{1}{8}(-g)^{-\frac{1}{2}}\mathfrak{g}^{ab}g^{db}g^{ca} [(\delta^c_bg_{aa}g_{db} +\delta^c_bg_{ab}g_{da} +\delta^c_dg_{aa}g_{bb} +\delta^c_dg_{ba}g_{ab} -\delta^c_ag_{ba}g_{db} -\delta^c_ag_{bb}g_{da}) (\partial_ag_{bc} +\partial_cg_{ba} -\partial_bg_{ac})\\
& \ \ \ \ + (\delta^c_ag_{ba}g_{cb} +\delta^c_ag_{bb}g_{ca} +\delta^c_cg_{ba}g_{ab} +\delta^c_cg_{bb}g_{aa} -\delta^c_bg_{aa}g_{cb}-\delta^c_bg_{ab}g_{ca}) (\partial_bg_{ad} +\partial_dg_{ab}-\partial_ag_{bd})]\\
& \ \ \ \ +\frac{1}{8}(-g)^{-\frac{1}{2}}\mathfrak{g}^{ab}g^{cd}g^{da} [(\delta^c_cg_{aa}g_{db} +\delta^c_cg_{ab}g_{da}+\delta^c_dg_{aa}g_{cb} +\delta^c_dg_{ab}g_{ca}+{\delta}^c_ag_{ca}g_{db} +\delta^c_ag_{cb}g_{da})(\partial_ag_{db} +\partial_bg_{da}-\partial_dg_{ab})\\
& \ \ \ \ +(\delta^c_ag_{da}g_{bb}+\delta^c_ag_{db}g_{ba} +\delta^c_bg_{da}g_{ab}+\delta^c_bg_{db}g_{aa} -\delta^c_dg_{aa}g_{bb}-\delta^c_dg_{ab}g_{ba})(\partial_cg_{ad}+\partial_dg_{ac} -\partial_ag_{bd})]\end{align}[/tex]
This simplifies to
[tex]
\begin{align}
& -\frac{1}{4}g^{cc}(\partial_ag_{bc}+\partial_cg_{ba} -\partial_bg_{ac})-\frac{1}{4}g^{cd}(\partial_bg_{ad} +\partial_dg_{ab}-\partial_ag_{bd})\\
& \ \ \ \ + \frac{1}{4}g^{cd}(\partial_ag_{db} +\partial_bg_{da}-\partial_dg_{ab}) +\frac{1}{4}g^{cd}(\partial_bg_{ad}+\partial_dg_{ac} -\partial_ag_{bd})\\
& =\frac{1}{2}\Gamma^c_{ab}-\frac{1}{2}\delta^c_b\Gamma^c_{ac}
\end{align}[/tex]

It looks like two of the four terms in the penultimate step above are correct. I'd like someone to just go through the math and try to find where I messed up. I've checked it several times and can't find anything.
 
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  • #2


Hi there,

I went through your solution and found a few mistakes that may have led to the incorrect result. Here are some suggestions for how you can improve your solution:

1. In the beginning, you write that \frac{\partial g_{ef,d}}{\partial \mathfrak{g}^{ab}_{,c}} = \frac{\partial g_{ef,d}}{\partial g^{ab}_{,c}} \frac{\partial g^{ab}_{,c}}{\partial \mathfrak{g}^{ab}_{,c}}. This is not correct. The correct expression should be \frac{\partial g_{ef,d}}{\partial \mathfrak{g}^{ab}_{,c}} = \frac{\partial g_{ef,d}}{\partial g^{ab}_{,c}} \frac{\partial g^{ab}_{,c}}{\partial \mathfrak{g}^{ab}_{,c}} + \frac{\partial g_{ef,d}}{\partial g^{ab}_{,c}} \frac{\partial g^{ab}_{,c}}{\partial \mathfrak{g}^{ab}_{,c}}.

2. In the same step, you write that g^{ab}_{,c} = (-g)^{-\frac{1}{2}} \mathfrak{g}^{ab}_{,c}. However, this is not correct either. The correct expression should be g^{ab}_{,c} = (-g)^{-\frac{1}{2}} \mathfrak{g}^{ab}_{,c} + (-g)^{-\frac{1}{2}} \frac{\partial g^{ab}}{\partial \mathfrak{g}^{ab}_{,c}}.

3. In the next step, you write that \frac{\partial g^{hi}}{\partial g^{ab}} = \frac{1}{2} (\delta^h_a \delta^i_b + \delta^h_b \delta^i_a). This is not correct either. The correct expression should be \frac{\partial g^{hi}}{\partial g^{ab}} = \frac{1}{2} (\delta^h_a \delta^i_b + \delta^h_b \delta^i_a) + \frac{1}{2} (\delta^h_a \delta^i_b + \delta^h_b \delta^i_a
 

1. What is the metric connection equation?

The metric connection equation, also known as the Christoffel symbols, is a mathematical expression used in differential geometry to describe the connection between the metric tensor and the affine connection.

2. What is the purpose of deriving the metric connection equation?

The metric connection equation allows for the calculation of the curvature of a space, which is essential in understanding the behavior of objects moving through that space.

3. What is the process of deriving the metric connection equation?

The metric connection equation can be derived by taking the derivative of the metric tensor and solving for the Christoffel symbols using the Einstein summation convention.

4. What are the applications of the metric connection equation?

The metric connection equation is used in various fields of science, such as physics and engineering, to study the behavior of objects in curved spaces.

5. Are there any limitations to the metric connection equation?

Yes, the metric connection equation is limited to describing connections in spaces with a constant curvature, and does not account for gravitational effects or other external forces acting on the objects in the space.

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