# EFE fully expanded in terms of g

1. Mar 9, 2014

### HomogenousCow

Does anyone have a link to a version of the EFE fully expanded in terms of the metric?

2. Mar 9, 2014

### HomogenousCow

I'm trying to do this on a large piece of paper, finding myself running out of greek indices

3. Mar 9, 2014

### stevendaryl

Staff Emeritus
It's not so bad in terms of the connection coefficients $\Gamma^\mu_{\nu \lambda}$, which can be expanded in terms of the metric and its derivatives. Why do you want it fully expanded? Just so you can bask in its fully glory?

4. Mar 9, 2014

### HomogenousCow

Pretty much, thought it would be interesting to see the EFE in it's "full glory".

5. Mar 9, 2014

### pervect

Staff Emeritus
Raising the display limit to the 2.3 million words required to display it, I can compute the expression for the components of the Einstein tensor for a general metric in GrTensor in under a minute of CPU time.

It's a bit impractical to cut and paste the result here, though, due to its extreme length, and it wouln't really serve any purpose except to visually illustrate how messy it is.

6. Mar 9, 2014

### bcrowell

Staff Emeritus
7. Mar 10, 2014

### stevendaryl

Staff Emeritus
Once, I tried to see if it was possible to get the Einstein Field Equations by starting with the non-covariant differential equation:

$g^{\alpha \beta} \partial_\alpha \partial_\beta g_{\mu \nu} = K T_{\mu \nu}$

and then adding correction terms in order to make it have the same form in any coordinate system. I quickly became lost in a sea of terms and indices. It's probably possible, but not a very efficient way to derive it.