Determining the spin connection

Question

Given a metric, one can extract the vielbein 1-forms, which must then satisfy the Maurer-Cartan equations, which in the case of a torsionless space, are of the form

$$de^{a} + {\omega^a}_{b}\wedge e^{b} = 0 $$

where all indices are flat and ${\omega_\mu}^{ab}$ is the spin connection (with $\mu$ being a curved index).

Defining a vielbein 1-form as an object with just flat indices in Cadabra (until the Vielbein function is re-introduced), I would like to solve for ${\omega^a}_{b}$, the spin connection 1-forms. As I understand, I need the following ingredients:

1. The wedge product.
2. Handling differential forms, specifically introducing an operator $d$ which satisfies $d^2 = 0$ and $d(A_p \wedge B_q) = dA_p \wedge B_q + (-1)^p A_p \wedge dB_q$.
3. The ability to go back and forth between coordinate differentials in the "curved" space, and vielbein 1-forms.
4. Actually solving the Maurer-Cartan equatons.

Of course, one might argue that since there's an explicit formula for the spin connection components, why not just use it? I feel like it might be more efficient to get the 1-forms in flat space, rather than the components themselves. Any suggestions?

Also, is there any faster way in Cadabra to get the spin connection 1-forms? In the practical application I'm interested in, the metric splits into 3 parts.

Answer 1

I have just merged some logic for differential forms into the master branch at github. See the 'examples/exterior.cnb' notebook for some rudimentary examples. You can do e.g.

A::DifferentialForm(degree=2);
B::DifferentialForm(degree=3);
d{#}::ExteriorDerivative;
ex:= d( B ^ A + d( A ^ A ) );
distribute(_)
product_rule(_);

to get

$$d B \wedge A - B \wedge d A $$

This does not yet really solve your point 3 and 4, mainly because the differential form logic has not been tied into the component engine yet.