Lorentz-like transformations of 1+1 dimensional Minkowski space

Question

Alice lives in 1+1 dimensional Minkowski spacetime. Bob travels at a constant velocity $v$ with respect to Alice. When Alice assigns coordinates $(t,x)$ to an event, Bob assigns coordinates $F(t,x)$ to that event. I'll call $F$ allowable if it leads to physics that satisfies Einstein's postulates. Obviously the Lorentz transformation $L$ is allowable.

Light-clock style arguments convince me of this:

If $F$ extends to an allowable transformation on 2+1 dimensional Minkowsi spacetime, then the restrictions of $F$ and $L$ to the inside of Alice's future lightcone must agree.

This leads to two questions:

Question 1: If I don't require $F$ to extend to a higher-dimensional spacetime, must it still agree with $L$ inside Alice's future lightcone?

Question 2: If $F$ does agree with $L$ inside Alice's future lightcone, must it agree with $L$ everywhere?

Some notes:

• I care about Question 1 because I want to derive the Lorentz transformation for students who have already been told that we will restrict our attention to 1+1 dimensional spacetime for simplicity. I'd therefore prefer not to invoke an additional spatial dimension. (Yes, there are all sorts of ways to justify this, including the fact that there is obviously more than one spatial dimension in the real world. But that still seems at least mildly jarring given the initial setup.)
• Question 2 of course becomes trivial if we assume that $F$ must be linear, or even just rational. But I'd prefer not to make extra assumptions.
• For each question, if the answer is "yes", my followup question is: "Is there an easy way to prove this with arguments at about the level of the light-clock stuff?" If the answer is "no", then my followup question is "What are some minimal assumptions that could convert the answer to "yes"?
• I've held Bob's velocity fixed and asked for a transformation $F$. I could instead ask for a family of transformations $F(v)$ depending on velocity. This might or might not change the answers.
• Maybe there's enough mathematical vagueness in "Einstein's postulates" that these questions admit multiple answers. I'll still be interested in those answers.

Answer 1

While my answer doesn't really answer your Questions 1 or 2, these may help:

• For a formulation of 1+1-Minkowski without appealing to a second spatial dimension (e.g. a transverse light clock), one could use the Bondi k-calculus (which uses radar methods with light-signals).
  
  One could also use a "longitudinal light clock" and my related "rotated graph paper" method.
  
  Both are described in my answer https://physics.stackexchange.com/a/689291/148184 to Time dilation clock experiment: what would happen if the clock were flipped 90 degrees?
  
  
  

• It seems that your questions are seeking an axiomatic approach, with emphasis on what assumptions are progressively needed to Minkowski spacetime. The following may be enlightening:

  • https://en.wikipedia.org/wiki/Alfred_Robb (assorted works involving causal-ordering)

  • "On Lorentz transformations", A.D. Alexandrov, Uspehi Mat. Nauk,5, 3 (37) (1950), p. 187.

  • "A Contribution to Chronogeometry" A.D. Alexandrov (1967). Can. J Math, 19, 1119-1128. https://doi.org/10.4153/CJM-1967-102-6

  • "Mappings of spaces with families of cones and space-time transformations", A.D. Alexandrov (1975), Annali di Matematica Pura ed Applicata volume 103, pages 229–257 https://doi.org/10.1007/BF02414157

  • "Causality Implies the Lorentz Group" E.C. Zeeman, J.Math.Phys. Vol 5, April 1964, p.490-493 https://doi.org/10.1063/1.1704140

  • found in a google search...
    "On Finslerian extension of special relativity", A.E. Sagaydak, Z.K. Silagadze, Mod. Phys. Lett. A 37N17 (2022), 2250106 https://doi.org/10.1142/S0217732322501061 and https://arxiv.org/abs/2201.12279