Weak field approximation and inverse metric expansion

Question

Dear Dr. Peeters,

I am trying to calculate the Riemann metric for weak-field approximation. In fact, I would like to consider this approximation up to the third order of the expansion. I mean, as it is well-known,

g_{\mu\nu}= n_{\mu\nu}+ k h_{\mu\nu}

where "k" is an infinitesimal constant. Now I am interested in the inverse metric, g^{\mu\nu}, up to k^3. Would you please give me some hints?

Regards

Answer 1

Not a concrete answer, but some ideas that might help you

Use the `weight` property

See for example Sec. 10.1 of Leo Brewin's paper ---Notice that from arXiv you can even get the source files---, all the necessary properties and algorithms are already functional in cadabra2.

Use the trick `k**4 -> 0`

If you have a function depending on k, you can use in the post_process algorithm the substitution:

def post_process(ex):
    expand_power(ex)
    substitute(ex, $k k k k -> 0$)

along with your extra post_process manipulations

Use another software

You can always try to compute the quantity you need using another software, and then manually input the result on Cadabra2.

Notice that for example you can use python functions within Cadabra2, see my question

Cheers.

Comments

Nice Ideais! Thanks very much for your kind attention.