Second Order Metric: Manipulating & Calculations for Einstein Equations

[Vrbic]

TL;DR Summary

How to work with metrics when solving a problem that is inherently limited to the second order in one parameter?

I use metric, which describes spacetime upto second order terms in rotation. It is solution of Einstein equations expanded upto second order. My query is, how to manipulate with such metric during calculations? Concetrly I make inverse metric, produce effective potential (ie, multiplying, makeing square root of coefficients etc.). Finally, I use it in numerical solution of Hamilton equations. What is appropriate procedure to hadle with this problem? Shall I expand upto second order all function after every calculation? Or...?

Thank you for any suggestion.

 

[PeroK]

Summary:: How to work with metrics when solving a problem that is inherently limited to the second order in one parameter?

I use metric, which describes spacetime upto second order terms in rotation. It is solution of Einstein equations expanded upto second order. My query is, how to manipulate with such metric during calculations? Concetrly I make inverse metric, produce effective potential (ie, multiplying, makeing square root of coefficients etc.). Finally, I use it in numerical solution of Hamilton equations. What is appropriate procedure to hadle with this problem? Shall I expand upto second order all function after every calculation? Or...?

Thank you for any suggestion.

I can see you are finding this quite hard to explain in English. You should try posting the actual equations - which should be the same in any language.

 

[Vrbic]

I can see you are finding this quite hard to explain in English. You should try posting the actual equations - which should be the same in any language.

Ok :-)
I have axially symmetric metic (exist $g_{14}$ and $g_{41}$ components) in this form: $g_{ab}=g0_{ab}+g1_{ab}*P+\frac{g2_{ab}}{2}*P^2 + (O^3)$.

For example:
I wants Hamilton equations.
I need Hamitlonian $H=g^{ab}p_ap_b$ so I need inverse of $g_{ab}$.
My query is: Shall I immediatelly expand $g^{ab}$ when I get inverse of $g_{ab}$, or shall I construct Hamilton equations and expand at the end?
Are there some rules for working with such "expanded" functions?

Thank you for your time.