Lorentz transformations: 1+1 spacetime only

[PeterDonis]

1+1 dimensions, where we are talking about 2 independent observers. I struggle in 2+1 dimensions, where we have three independent observers. In 3+1 dimensions, with four independent observers

This makes no sense to me. What does the number of spacetime dimensions have to do with the number of observers?

 

[PeterDonis]

from the one-dimensional Lorentz transformations it is impossible to infer what the two-dimensional group is. Therefore there is a physical assumption going into the process somewhere.

I would assume that the added assumption has to do with invariance under purely spatial rotations, which don't exist in 1+1 spacetime but do exist when there are two or more spatial dimensions. We know that spatial rotations form a group, so then we just have to figure out what larger group includes spatial rotations as a subgroup and also includes Lorentz boosts. The usual definition of "Lorentz transformations" includes spatial rotations (I have seen the group of such transformations referred to as "spacetime rotations") because that group (the proper orthochronous Lorentz group, as I referred to it in a previous post) is the smallest one that includes both spatial rotations and boosts (but, as already noted by @vanhees71, boosts only form a subgroup if we restrict to boosts in a single direction).

 

[robwilson]

The OP considers that this thread has outlived its usefulness. I thank those people who provided useful responses. Those from physicists and engineers trying to teach a group theorist how to do group theory were not useful, and were frankly insulting. Those from people unwilling to distinguish between physical reality and our current best fit mathematical models of reality merely muddied the waters. I notice that many of you have visited my blog, in which I explain the problem as I see it, and the wider context, and perhaps some of you have looked at my three recent papers on the arXiv, where I explain the group theory in detail. I do not claim to have any answers, but I do claim to be asking the right questions.

 

[PeterDonis]

my three recent papers on the arXiv, where I explain the group theory in detail. I do not claim to have any answers, but I do claim to be asking the right questions.

In any case, you are pursuing your research, and time will tell whether it is useful. Best of luck to you in your endeavors.

Thread closed.