This topic is in response to some questions received in the 'Why is relativity so hard to learn?' topic, but which were deemed too technical to be dealt with there.

Because of the Euclidean nature of space-propetime (SPT) diagrams, initial acceleration can be closely approximated as a rotation at a radius R around some fixed point, which I call the 'center of curvature' below. In the appropriate dimensions, the acceleration is inversely proportional to the radius R.

Because it this is not Euclidean space that we are dealing with, but rather SPT, the radius R is variable and the center of curvature's position shifts during acceleration, as indicated below.

Note that is the Epstein angle of the velocity vector from the axis in SPT diagrams and . Acceleration is indicated by , because the attachments come from the eBook Relativity-4-Engineers, where this is the normal symbol.

Below is a simple algorithm for solving the curve through numerical integration (engineers simply love this method...)

It is largely self-explanatory, so just a few comments are in order, I hope. The first 2 lines of code after the 'output' line simply rotates the present center of curvature to temporarily sit at a more convenient spot for calculation. The simpler calculation is performed and then the values are rotated back so that the new x,tau coordinate sits in the the correct place. One can't use zero acceleration, but I stick in a value of 1E-12 as close enough to zero for all practical purposes.

This same thing can be done in Rindler coordinates and the point by point results then transformed to SPT coordinates. I find that pretty complex and not very helpful for insight into the problem.

This was very brief, but I'll be happy to provide more information as required.

PS: without me noticing, the spell-checker corrected an error in my spelling of "position" in "center of curvature's position" to "center of curvature's "postillion". :)

Def. of postillion: a person who rides the leading left-hand horse of a team or pair drawing a coach or carriage, especially when there is no coachman. LOL.

On a more serious note, the very first diagram can be a little confusing. That should have been next to the curved path to indicate that it means the SPT path length and not the time of the accelerating frame. The second diagram indicates it correctly.