# Derivative of metric and log identity

1. Feb 29, 2008

### robousy

Has anyone seen this identity:

$$g^{ab}\nabla g_{ab}=\nabla ln|g|$$

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

Does anyone know a name or have any ideas??

2. Feb 29, 2008

### smallphi

You will have to show the exact formula. The way it's written, it doesnt make sense - covariant derivative of the metric is zero in GR because the connection is chosen 'metric compatible'.

3. Mar 1, 2008

### robousy

ok, thanks, I'll check the paper when I get back to my office tomorrow and post.

Rich

4. Mar 1, 2008

### George Jones

Staff Emeritus
I think you mean

$$g^{\alpha \beta} \partial_\mu g_{\alpha \beta} = \partial_\mu \ln \left|g\right|.$$

See pages 12-13 of Poisson.

5. Mar 1, 2008

### robousy

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. Mar 1, 2008

### George Jones

Staff Emeritus
Lots of have books probably have this, but I won't be able to tell you any others until Monday.

Maybe Carroll.

7. Mar 1, 2008

8. Mar 1, 2008

### robousy

I'm not seeing it Sam.

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

9. Mar 1, 2008

### samalkhaiat

10. Mar 1, 2008

### robphy

11. Mar 1, 2008

### robousy

12. Mar 1, 2008