I will elaborate on what others, esp. Ibix, have said ("distance" is here used as an analogy, not an identity, so do not simply automatically rely on your Euclidean understanding of the word), but take my ideas with a grain of salt, as I am just a student of special relativity.
Whenever you say...
False. As you can see below, I received the information that the tangent space is composed of things other than 4-velocity, I accepted it as good knowledge, and it simply happens that I returned to the question that interested me, because it is related to the object of the thread: where is the...
I don't have a problem with that. How could I? It is obvious that, if you scale a unit vector, you get that vector scaled by whatever scalar you have multiplied it by, i.e. not a unit vector anymore.
However, the statement by Wikipedia is the following: "addition of two four-velocities does...
This proves that you don't understand the question.
Addition of two vectors of the same magnitude but pointing in different directions is obviously not the same as scaling any of the two vectors by the real number 2.
Addition of two vectors of the same magnitude and being colinear obviously...
There may be a misunderstanding here. The fact that the 4-velocities do not form by themselves a vector space is another issue. My "that" refers to the part in red: if you cannot add two four-velocities it means that you cannot multiply one four-velocity by the real number two, i.e. "scale" it...
I looked at Wikipedia entry for 4-velocity and it does say (at a note) that a tangent space is a vector space (like the entry about tangent space itself), but the thing is that it also says this with regard to 4-velocity: "addition of two four-velocities does not yield a four-velocity: the space...
Well-defined? Yes, Galilean 4-velocity is 3-velocity to which one adds a coordinate time over proper time component which is 1 in all reference frames, since in this context both things coincide. What is the added value of this concept, i.e. its practical use? If it were none, it would fit in...
If you told me that you can paint it there, overlapping the worldlines, well, the paper accepts anything. But if you affirm that this has a “heuristic“ sense, which one is it? Is it that Bob’s wristwatch is at rest with Bob, so it is always spatially located by the origin of his frame? But that...
Tangent spaces in general, yes. But this particular tangent space is supposed to be made of 4-velocity vectors, right? And if I look at the first axiom of a vector space and try to add one 4-velocity vector to another, I have doubts. Can you do that?
Thanks, this is enlightening, as it does explain the units issue. Can we thus say that we have two options to define the "time basis vector" in SR: (i) one in spacetime (the "object space"?), where the time basis vector would be a 4-displacement vector with time component = 1 unit of time or...
Please feel free to leave these questions aside, while you discuss your more advanced clarifications.
I was asking if you were identifying the time basis vector with Galilean spatial velocity because in SR you identify it with 4-velocity and you said that you did not need 4-velocity at...
@robphy please let me come back for a moment to my prior question.
I had specifically asked:
Your comments are:
Do you mean that the answer is "somehow both", in the sense that your "unit tangent vector" (spatial velocity in Galilean relativity, 4-velocity in SR) is the unit vector of the...
Same for me! The priority is to check if I understand this point of yours, so let us make it clear before anything else.
When I say "this point", I mean the link that you establish between the unit tangent vector, relative velocity btw the two frames (and their clocks) and the time basis...
Sorry if I sound sometimes too radical, it is just that I am struggling with the ideas!
Let us check if I can follow you.
Say event A is when Bob and Alice meet ( t = 0), event B when Bob's wristwatch ticks 1s and event C when Alices' wristwatch also ticks 1s.
A-B is different from A-C.
The...
@robphy
Since you know more than me and have been patient enough to keep objecting me, I have made more effort to converge to you.
I understand that you are telling me that one Galilean frame sees that the other’s time-basis vector is dilated, despite measuring the same time, because it...