If all one would want to do is label "hyperplanes of simultaneity"
(this hyperplane is "all space at t=0", that hyperplane is "all space at t=1", ...)
and thus only care about the "elapsed time between events"
then I could see why you may be saying that
you need the "time-basis vector" in SR (since the hyperplanes depend on inertial observer)
you don't need the "time-basis vector" in the Galilean case (since the hyperplanes are independent of inertial observer).
Here you seem to be talking about "elapsed time between events" that are "simultaneous", even if separated by distance, so elapsed time between them is 0. You note that, since simultaneity is absolute in Galilean relativity, there is nothing here related to time that is observer-dependent. So no need for a "time-basis vector" for this set of events. I cannot agree more.
Here you talk about events with "spatial separation" but happening "at different times" and you say that this "depends on observer". But what "depends on observer"? Let us scrutinize each element of the situation to check if anything related to time is observer-dependent.However, we generally want to do more than just label hyperplanes of simultaneity.
Implicit in what I said above,
we would also want to identify (e.g.)
the set of events that correspond to x=0 (,y=0, z=0) according to Alice, for each reading on Alice's wristwatch,
which is different from
the set of events that correspond to x=0 (,y=0, z=0) according to Bob, for each reading on Bob's wristwatch,
Thus, we also care about the "spatial-separation between events at different times",
(so that we can answer "Is that particle at rest in my frame? and if not, how fast is it traveling in my frame?")
and this depends on observer.
- The spatial separation certainly is frame-dependent, but this is the x coordinate, it is not due to the time period elapsed between the two events, which is the same for all observers, no matter if they measure with a wristwatch present at the two events or with distant clocks.
- The speed of any object traveling between the two events is also frame-dependent, but this is an automatic consequence of the previous statement: it is due to the distance traversed being relative, while the time period remains absolute.
- Certainly, if you look at Alice and Bob, as you propose, who are the observers located at the origin of each frame, it is clear that their wristwatches get progressively apart from each other... Does it mean anything in terms of time? No, because the watches keep ticking in perfect sync, as could be checked through the rigid rod with infinite sound speed which we identified as the eigenvector of the Galilean transformation. And if they stopped to do so, we would recalibrate them with that magical rod so that they keep ticking in sync. Obviously, you don't mean that the mere fact that the watches are getting apart from each other makes time observer-dependent, just like if the watches were of different colors, this would not make their time readings color-dependent.
- You say the "set of events" for Alice and Bob (from their respective vantage points, x=0 and x'=0) are "different". "Different" in what sense? If an event is characterized by an x and a t coordinate, the x coordinate is different but the t coordinate is not.
So far I did not spot here, either, anything related to time that is observer dependent.
So, we need the "time basis vector" ("a 4-velocity vector") in both the SR and Galilean case.
4-velocity seems to be a key concept in your argument. Do you mean that, for example, the 4-velocity of a particle traveling between the two events would be different in each frame? But let us open up this concept.
In SR it means 4 observer-dependent coordinates (coordinate time and the 3 spatial coordinates) divided by proper time measured from the frame at rest with the particle, which is an invariant concept. In SR coordinate time and proper time are different, except in the frame of the particle in question.
But in Galilean relativity, time is absolute, so coordinate time is always the same as proper time. Hence the first coordinate is always 1 in any frame. In turn, the three spatial coordinates are observer-dependent, but they are not related to time. Thus the concept is just 3-velocity to which you have added a value related to time, which is invariantly 1. And you conclude that this way you have found a "time-basis vector"?
Honestly, this looks to me contrived: you have artificially mixed, under an ad hoc concept (Galilean 4-velocity), absolute time separation between two events with the relative spatial distance between them, but through this forced cohabitation of one with the other you are not managing to make time observer-dependent in any sense.
Every inertial observer would like to draw her spacetime diagram
with her worldline vertically upwards (in the time-upward convention),
as granted by the relativity principle.
What prevents it? Why do you need a time-basis vector for this? Do you rather mean that if I draw my vertical line perpendicular to my X axis, then yours must be inclined? Sure, but this has nothing to do with the time axis. You don't need to attribute this different inclination to any observer dependency in terms of time. You just have to admit that the different evolution over time of the origins of the frames reflects their progressive spatial separation.
See this image which I posted before, now improved on the basis of our discussion on eigenvectors, where the vertical (absolute time) and horizontal (absolute length) orange axes reflect the common scale arising from eigenvalue = 1.