Modifying Lorentz Transformations for Invariant Length

[unchained1978]

I've been reading up on the idea of an invariant length (planck length possibly) and have tried my hand at modifying the lorentz transformations of SR such that they accommodate an invariant length. Unfortunately I haven't been able to find a way to do it. Does anyone know how one would go about this?

The Planck length is a 3D length as far as I understand, so I don't see it fit to incorporate this invariance into the line element of SR, which is a 4D length. Any help would be appreciated.

 

[bcrowell]

http://en.wikipedia.org/wiki/Doubly_special_relativity