Hello, Mr/Ms WannabeNewton. Thank you for your mention.
As you said, it is completely right to define an affine parameter as one such that the geodesic equation in that form holds.
However, it seems to me that Mr/Ms stevendaryl, who is the original questioner, wants to get physical...
Hello. I found a tetrad formalism of relativistic point particles, which might be useful for your question. Partial information about this formalism can be found in the textbook by Polchinski, "String Theory."
This formalism is useful in that it is valid also for massless particles, and you...