The contracting relations on the Christoffel symbols

Arman777
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Homework Statement
The contracting relations on the Christoffel symbols
Relevant Equations
Tensor Equations
I am trying to find $$\Gamma^{\nu}_{\mu \nu} = \partial_{\mu} log(\sqrt{g})$$ but I cannot.

by calculations, I manage to find

$$\Gamma^{\nu}_{\mu \nu} = \frac{1}{2}g^{\nu \delta}\partial_{\mu}g_{\nu \delta}$$

and from research I have find that $$det(A) = e^{Tr(log(A))}$$ but still I cannot make the connection. Any ideas?

where ##g = det(g_{\alpha \beta})##
 
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First of all you must always have ##\sqrt{-g}##, because ##g<0## due to the signature of the Lorentzian manifold.

Further think about, how (the matrix) ##g^{\mu \nu}## is related to (the matrix) ##g_{\mu \nu}## and what this has to do with determinants!
 
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vanhees71 said:
First of all you must always have ##\sqrt{-g}##, because ##g<0## due to the signature of the Lorentzian manifold.

Further think about, how (the matrix) ##g^{\mu \nu}## is related to (the matrix) ##g_{\mu \nu}## and what this has to do with determinants!
I ll try and get back to you
 
vanhees71 said:
First of all you must always have ##\sqrt{-g}##, because ##g<0## due to the signature of the Lorentzian manifold.

Further think about, how (the matrix) ##g^{\mu \nu}## is related to (the matrix) ##g_{\mu \nu}## and what this has to do with determinants!
I can see that ##det(g^{\mu \nu })= 1/g##
 
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Then further note that
$$\mathrm{det} (g_{\mu \nu})=g=\frac{1}{4!} \Delta^{\mu_1\ldots \mu_4} \Delta ^{\nu_1 \ldots \nu_4} g_{\mu_1 \nu_1} \cdots g_{\mu_4 \nu_4}.$$
Here and in the following I use the notation
$$\Delta^{\mu_1 \ldots \mu_4}=\Delta_{\mu_1 \ldots \mu_r}$$
for the Levi-Civita symbols (which are NOT tensor components), i.e., the totally antisymmetric symbols with ##\Delta^{0123}=\Delta_{0123}=+1##.

Now take the derivative ##\partial_{\mu}## which gives
$$\partial_{\mu} g = \frac{1}{3!} (\partial_{\mu} g_{\mu_1 \nu_1}) \Delta^{\mu_1\ldots \mu_4} \Delta ^{\nu_1 \ldots \nu_4} g_{\mu_2 \nu_2} \cdots g_{\mu_4 \nu_4}=:\tilde{g}^{\mu_1 \nu_1} \partial_{\mu} g_{\mu_1 \nu_1}.$$
But now let's think about
$$\tilde{g}^{\mu_1 \nu_1} = \frac{1}{3!} \Delta^{\mu_1\ldots \mu_4} \Delta ^{\nu_2 \ldots \nu_4} g_{\mu_2 \nu_1} \cdots g_{\mu_4 \nu_4}$$
is. To that end we evaluate
$$g_{\alpha \mu_1} \tilde{g}^{\mu_1 \nu_1}= \frac{1}{3!} \Delta^{\mu_1\ldots \mu_4} \Delta ^{\nu_1 \ldots \nu_4} g_{\mu_1 \nu_1} \cdots g_{\mu_4 \nu_4} g_{\mu_1 \alpha}.$$
In the last step I used the symmetry of the metric, ##g_{\mu_1 \alpha}=g_{\alpha \mu_1}##. Now do the contraction wrt. the ##\mu_k##, which leads to
$$g_{\alpha \mu_1} \tilde{g}^{\mu_1 \nu_1}=g \frac{1}{3!} \Delta_{\alpha \nu_2\cdots \nu_1} \Delta ^{\nu_1 \ldots \nu_4} = g \delta^{\nu_1}_{\alpha}.$$
From this you have
$$\tilde{g}^{\mu_1 \nu_1}=g g^{\mu_1 \nu_1}.$$
So you get
$$\partial_{\mu} g=g g^{\mu_1 \nu_1} \partial_{\mu} g_{\mu_1 \nu_1}.$$
So finally
$$\partial_{\mu} \ln (\sqrt{-g})=\frac{1}{2} \partial_\mu \ln(-g)=\frac{1}{2g} \partial_{\mu} g=\frac{1}{2} g^{\alpha \beta} \partial_{\mu} g_{\alpha \beta},$$
which is what you wanted to prove.
 
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vanhees71 said:
Then further note that
$$\mathrm{det} (g_{\mu \nu})=g=\frac{1}{4!} \Delta^{\mu_1\ldots \mu_4} \Delta ^{\nu_1 \ldots \nu_4} g_{\mu_1 \nu_1} \cdots g_{\mu_4 \nu_4}.$$
Here and in the following I use the notation
$$\Delta^{\mu_1 \ldots \mu_4}=\Delta_{\mu_1 \ldots \mu_r}$$
for the Levi-Civita symbols (which are NOT tensor components), i.e., the totally antisymmetric symbols with ##\Delta^{0123}=\Delta_{0123}=+1##.

Now take the derivative ##\partial_{\mu}## which gives
$$\partial_{\mu} g = \frac{1}{3!} (\partial_{\mu} g_{\mu_1 \nu_1}) \Delta^{\mu_1\ldots \mu_4} \Delta ^{\nu_1 \ldots \nu_4} g_{\mu_2 \nu_2} \cdots g_{\mu_4 \nu_4}=:\tilde{g}^{\mu_1 \nu_1} \partial_{\mu} g_{\mu_1 \nu_1}.$$
But now let's think about
$$\tilde{g}^{\mu_1 \nu_1} = \frac{1}{3!} \Delta^{\mu_1\ldots \mu_4} \Delta ^{\nu_2 \ldots \nu_4} g_{\mu_2 \nu_1} \cdots g_{\mu_4 \nu_4}$$
is. To that end we evaluate
$$g_{\alpha \mu_1} \tilde{g}^{\mu_1 \nu_1}= \frac{1}{3!} \Delta^{\mu_1\ldots \mu_4} \Delta ^{\nu_1 \ldots \nu_4} g_{\mu_1 \nu_1} \cdots g_{\mu_4 \nu_4} g_{\mu_1 \alpha}.$$
In the last step I used the symmetry of the metric, ##g_{\mu_1 \alpha}=g_{\alpha \mu_1}##. Now do the contraction wrt. the ##\mu_k##, which leads to
$$g_{\alpha \mu_1} \tilde{g}^{\mu_1 \nu_1}=g \frac{1}{3!} \Delta_{\alpha \nu_2\cdots \nu_1} \Delta ^{\nu_1 \ldots \nu_4} = g \delta^{\nu_1}_{\alpha}.$$
From this you have
$$\tilde{g}^{\mu_1 \nu_1}=g g^{\mu_1 \nu_1}.$$
So you get
$$\partial_{\mu} g=g g^{\mu_1 \nu_1} \partial_{\mu} g_{\mu_1 \nu_1}.$$
So finally
$$\partial_{\mu} \ln (\sqrt{-g})=\frac{1}{2} \partial_\mu \ln(-g)=\frac{1}{2g} \partial_{\mu} g=\frac{1}{2} g^{\alpha \beta} \partial_{\mu} g_{\alpha \beta},$$
which is what you wanted to prove.
Just one question why ##\partial_{\mu} g \neq 0## ? or can we say that ##\partial_{\mu}\partial^{\mu} g = 0##
 
Why should ##\partial_{\mu} g=0##? Of course if you are in SR you can always choose a global Minkowski-orthonormal basis. Then ##g_{\mu \nu}=\eta_{\mu \nu}=\mathrm{diag}(1,-1,-1,-1)=\text{const}## and ##\partial_{\rho} g_{\mu \nu}=0## and thus also ##\partial_{\mu} g=0##. In this case of course the covariant derivative coincides with the partial derivative, because all Christoffel symbols vanish too.
 
vanhees71 said:
In this case of course the covariant derivative coincides with the partial derivative, because all Christoffel symbols vanish too.
So its true that ##\partial_{\mu}\partial^{\mu}g = 0##
 
How do you come to this conclusion?
 
  • #10
vanhees71 said:
How do you come to this conclusion?
I did not actually :p Well I was doing another mathematical calculation about the ##\nabla_{\mu}\nabla^{\mu} \Phi## ,where ##\Phi## is a scalar function. In order to obtain the desired result it seems that ##\partial_{\mu}\partial^{\mu}g## must be 0 ? If you want I can share a separate question about it as well.
 
  • #11
This we can answer pretty easily with what we've derived above so far. First take an arbitrary vector field ##V^{\mu}## and calculate the "divergence". We have
$$\nabla_{\nu} V^{\mu} = \partial_{\nu} V^{\mu} +{\Gamma^{\mu}}_{\nu \rho} V^{\rho}.$$
From that we find taking the trace
$$\nabla_{\mu} V^{\mu} = \partial_{\mu} V^{\mu} + {\Gamma^{\mu}}_{\mu \rho} V^{\rho}.$$
Now we use our formula for the contraction of the Christoffel symbol
$$\nabla_{\mu} V^{\mu} =\partial_{\mu} V^{\mu} + (\partial_{\rho} \ln (\sqrt{-g})] V^{\rho} = \partial_{\rho} V^{\rho} + \frac{1}{\sqrt{-g}} V^{\rho} \partial_{\rho} \sqrt{-g} = \frac{1}{\sqrt{-g}} \partial_{\rho} (\sqrt{-g} V^{\rho}).$$
Further we have
$$V_{\mu}=\partial_{\mu} \Phi \; \Rightarrow \; V^{\rho}=g^{\rho \mu} \partial_{\mu} \Phi.$$
From this we get,
$$\nabla_{\rho} \nabla^{\rho} \Phi=\frac{1}{\sqrt{-g}} \partial_{\rho} (\sqrt{-g} g^{\rho \mu} \partial_{\mu} \Phi).$$
So you get the so-called Laplace-Beltrami operator.
 
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  • #12
In my calculations I did something like this
1618137971515.png


but it seems wrong ... But I see your point.
 
  • #13
I don't understand, how you get from the 2nd to the 3rd expression. Of course ##\nabla_{\nu} \Phi=\partial_{\nu}\ Phi##, and then you get the correct expression, the Laplace-Beltrami operator.
 
  • #14
vanhees71 said:
2nd to the 3rd expression.
how so ?
By using this I am getting

$$\nabla_{\mu}(g^{\mu \nu}\nabla_{\nu}\Phi) = \frac{1}{\sqrt{g}}\partial_{\mu}[g^{\mu \nu}\partial_{\nu}(\sqrt{g}\Phi)]$$

but you are writing

$$\nabla_{\mu}(g^{\mu \nu}\nabla_{\nu}\Phi) = \frac{1}{\sqrt{g}}\partial_{\mu}[g^{\mu \nu}\sqrt{g}\partial_{\nu}(\Phi)]$$

there is a ##\sqrt(g)## difference. In my its inside the partial and in yours its outside..
 
  • #15
As I said, I don't understand your step. Can you prove it?
 
  • #16
vanhees71 said:
As I said, I don't understand your step. Can you prove it?
I have just put the definiton of the ##\nabla_{\nu}\Phi = \frac{1}{\sqrt{g}} \partial_{\nu}(\sqrt{g}\Phi)## inside the paranthesis ?
 
  • #18
Arman777 said:
I have just put the definiton of the ##\nabla_{\nu}\Phi = \frac{1}{\sqrt{g}} \partial_{\nu}(\sqrt{g}\Phi)## inside the paranthesis ?
No! The definition of ##\nabla_{\mu} \phi=\partial_{\mu} \Phi##. ##\Phi## is a scalar field!
 
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  • #19
oh shi*t. I understand it now okay. I was also saying there's something wrong but could not figure out why. Okay some things make more sense now after your approach in #11. Thanks a lot
 
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