Homework Help: Variation of the metric tensor determinant

1. Dec 12, 2012

InsertName

1. The problem statement, all variables and given/known data

This is not homework but more like self-study - thought I'd post it here anyway.

I'm taking the variation of the determinant of the metric tensor:

$\delta(det[g\mu\nu])$.

2. Relevant equations

$\delta(det[g\mu\nu])$ =det[g\mu\nu] g$\mu\nu$ $\delta$g$\mu\nu$

Here, g$\mu\nu$ is the metric tensor, [g$\mu\nu$] is the matrix of the components of the metric tensor, and $\delta$ is a variation.

3. The attempt at a solution

I have managed to get close to the answer, I hope, with
$\delta(det[g\mu\nu])$
=det[g\mu\nu] tr([g\mu\nu]-1$\delta$([g\mu\nu]))

The problem, in my view, is the trace. I cannot see how to remove it.

Also, if someone could kindly describe how to tidy the LaTeX up, I will do that.

Thank you!

2. Dec 13, 2012

clamtrox

You can't use the tags in latex. Instead, write g_{\mu \nu}, g^{\mu \nu} etc.

This is a little bit tricky, as you dont really keep track of how your matrices are multiplied together. If you do it with more care, you should find something like
$$\delta \det(g) = \det(g) \mathrm{Tr}( g \cdot \delta(g) ),$$
with $(g \cdot \delta(g))_{\mu \nu} = g_{\mu \lambda} \delta ({g^\lambda}_\nu)$
Now it's easy to see what the trace is:
$$\mathrm{Tr}( g \cdot \delta(g) ) = (g \cdot {\delta(g))_\mu}^{\mu} = g_{\mu \lambda} \delta (g^{\lambda \mu})$$

3. Dec 13, 2012

andrien

use
g=(1/4!)εαβγδεμvρσgαμgβvgγρgδσ
ggαμαβγδεμvρσgβvgγρgδσ
now when you apply δ on g you can use the second of the above to get whole result.(i am not going to do it in full,because of the requirement of homework section)

4. Dec 14, 2012

andrien

Hmm,may be tough.
so use Tr (ln M)=ln(det M)
variation of it yields,
Tr(M-1 δM)=(1/det M)δ(det M)
JUST PUT M=gμv