Derivative of metric and log identity

In summary: QUOTE]In summary, the identity states that the covariant derivative of the metric is zero in general relativity because the connection is chosen 'metric compatible'.
  • #1
robousy
334
1
Has anyone seen this identity:

[tex]g^{ab}\nabla g_{ab}=\nabla ln|g|[/tex]

I've seen it used, but want to figure out where it comes from.

Does anyone know a name or have any ideas??
 
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  • #2
You will have to show the exact formula. The way it's written, it doesn't make sense - covariant derivative of the metric is zero in GR because the connection is chosen 'metric compatible'.
 
  • #3
ok, thanks, I'll check the paper when I get back to my office tomorrow and post.

Rich
 
  • #4
robousy said:
Has anyone seen this identity:

[tex]g^{ab}\nabla g_{ab}=\nabla ln|g|[/tex]

I've seen it used, but want to figure out where it comes from.

Does anyone know a name or have any ideas??

I think you mean

[tex]g^{\alpha \beta} \partial_\mu g_{\alpha \beta} = \partial_\mu \ln \left|g\right|.[/tex]

See pages 12-13 of Poisson.
 
  • #5
Thanks Greorge. I Don't have that book but I'll see if I can find someone with it. And thanks for pointing out the correction.
 
  • #6
Lots of have books probably have this, but I won't be able to tell you any others until Monday.

Maybe Carroll.
 
  • #7
robousy said:
Has anyone seen this identity:

[tex]g^{ab}\nabla g_{ab}=\nabla ln|g|[/tex]

I've seen it used, but want to figure out where it comes from.

Does anyone know a name or have any ideas??

Differentiate the matrix identity

ln[(det.M)] = Tr[(ln M)]

and put

[tex] M = g_{\mu \nu}[/tex]


regards

sam
 
  • #8
I'm not seeing it Sam.

[tex]\partial_\mu ln|g^{ab}|=\partial_\mu ln[(det.g^{ab})]=\partial_\mu Tr[ln g^{ab} ] =
\partial_\mu (ln g^{00}+lng^{11}+...)

[/tex]
 
  • #9
robousy said:
I'm not seeing it Sam.

[tex]\partial_\mu ln|g^{ab}|=\partial_\mu ln[(det.g^{ab})]=\partial_\mu Tr[ln g^{ab} ] =
\partial_\mu (ln g^{00}+lng^{11}+...)

[/tex]


[tex]\partial (ln |G|) = Tr ( \partial ln G) = (G^{-1} \partial G)^{\mu}_{\mu}[/tex]

thus

[tex]\partial (ln|g|) = g^{\mu\nu}\partial g_{\mu\nu}[/tex]
 
  • #11
samalkhaiat said:
robousy said:
[tex]\partial (ln |G|) = Tr ( \partial ln G) = (G^{-1} \partial G)^{\mu}_{\mu}[/tex]

thus

[tex]\partial (ln|g|) = g^{\mu\nu}\partial g_{\mu\nu}[/tex]

Great, got it! Thanks a bunch.
 
  • #12

Related to Derivative of metric and log identity

1. What is the derivative of a metric?

The derivative of a metric is a mathematical operation that calculates the rate of change of the metric with respect to another variable. It is represented by the symbol "d" and is commonly used in calculus and physics.

2. How is the derivative of a metric calculated?

The derivative of a metric is calculated by taking the limit of the change in the metric over the change in the other variable as the change in the other variable approaches zero. This can be written as "d(metric)/d(other variable)" or "d(metric)/d(x)".

3. What is the log identity in mathematics?

The log identity in mathematics is a logarithmic property that states that the logarithm of a product is equal to the sum of the logarithms of each individual factor. It can be written as "log(ab) = log(a) + log(b)".

4. How is the derivative of a log identity calculated?

The derivative of a log identity can be calculated using the chain rule in calculus. This involves taking the derivative of the outer function (log) and multiplying it by the derivative of the inner function (product of the individual factors). The resulting derivative will be equal to the sum of the derivatives of each individual factor.

5. Why is the derivative of a log identity important?

The derivative of a log identity is important because it allows for the calculation of the rate of change of logarithmic functions. This is useful in many fields, such as finance, physics, and engineering, where logarithmic relationships are commonly used to model real-world phenomena.

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