- #1

Enrico

- 36

- 7

*[Moderator's note: Thread spun off from*

*previous one*

*due to topic shift.]*

Please forgive my ignorance, I've never studied group theory systematically up to now, so I'm not aware of all the concepts and symbols that have been used up to now. Yet, I'm interested in the derivation of the Lorentz transformations, and I think I quite grasp the concept expressed in the last post (#16). A few years ago I made a bit of bibliographic research on the "1-postulate" derivations of the Lorentz transformations, and there are a couple of things I'm trying to clarify within that framework. These are related to the group structure of the set of the transformations.

Let me express my thoughts in a very humble way, without invoking group theory from the outset. Let's consider a single spatial dimension. So, we are dealing with linear transformations from ##\mathbb{R}^2## into itself, i.e. with ##2\times2## matrices. As far as I know, all existing "1-postulate" derivations assume isotropy of space and the group structure for the set of transformations (matrices), and also a couple of further "technical" things. The first is that the coefficients of the transformation (elements of the matrix) be (at least) continuous functions of the group parameter. The second is a "causality" or "precausality" assumption, stating that "if two events occur at the same place in some frame of reference, their time order must be the same in all other frames of reference" (or the equivalent condition that by composing two positive velocities one must obtain a positive velocity).

Let's focus first on the second of these two assumptions, the "precausality" condition. Is it really needed? From my present understanding, this condition is necessary if we impose group structure on ##\mathcal{L}_+##. If we impose group structure on ##\mathcal{L}_+^\uparrow## instead, I think it is not necessary. This is related to the choice of the group parameter. If the latter is ##v##, the relative velocity between the two frames of reference that we are transforming (which amounts to say that, given a frame, the other one is completely defined once ##v## be fixed), we are actually considering ##\mathcal{L}_+^\uparrow##, so precausality is not needed.

Now, what about the other assumption, the one about continuity of the coefficients as functions of the group parameter (##v##)? I've been trying to do some calculations by hand (i.e. without any tool from group theory) in the course of the last week, and I have an idea that this assumption is not needed actually.

Just to get a feeling of the general opinion on these topics. Again, please forgive my present ignorance on group theory.

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