Relativistic constant acceleration in the framework of general relativity

Regarding the constant acceleration of a spacecraft towards relativistic velocities, the general equations are found at the following link: Baez

However, if the distance covered is in excess of 1 billion light-years, general relativity would have to be used to compute the distance.

So an attempt to find the corresponding equations was made in this post: post for the framework of GR

Can someone actually give a numerical example of the calculations involved in the latter link?

Also no description was given for the calculation of $a(t)$.

Thanks