Can time be another basis vector under Galilean relativity?

[PeterDonis]

We are here to learn, not to play childish ego games.

a surprisingly lengthy and repetitive way of reiterating the ad hominem attacks.

Should you prefer to keep sabotaging me

These comments are uncalled for and further indicate that you do not understand the points I have been trying to make. They are also likely to get you a warning if you continue along these lines.

I would like to report the situation

You are always free to use the Report button.

 

[PeterDonis]

As I repeatedly said, I don't think that in Galilean spacetime a four-vector formalism makes too much sense

I would kindly request you to engage in a discussion about that

A discussion about that is fine if you frame a question based on it.

The subthread you are complaining about started with your incorrect assertion in post #126 that the tangent space is only composed of 4-velocity vectors (i.e., unit timelike vectors). It's not. Instead of realizing that it's not and rethinking your position in the light of that new knowledge, you have continued to make further erroneous claims, which I have been correcting (see my post #140 for my latest corrections).

Bear in mind that this thread is not just for you, even though you're the OP. People come to PF all the time to read threads, which means when incorrect statements are made in PF threads, we try to correct them.

 

[Saw]

You are the one who keeps making an issue of the fact that adding two 4-velocity vectors gives a vector that is not a unit vector.

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 not yield a four-velocity". That seems to be a problem because it implies that the addition gives off a different object (something that is not a 4-velocity vector), not the same object (a 4-velocity vector) with a coefficient other than 1.

It seems that your ace in the sleeve is that a 4-velocity's magnitude is always 1. You don't need to keep it, I am saying it right now. But if so, what is then the object that yields the addition of two colinear 4-velocities or in general the scaling thereof by some coefficient? What is its name and what kind of object is it?

 

[PeterDonis]

the statement by Wikipedia is the following: "addition of two four-velocities does not yield a four-velocity".

Yes, that's correct. Adding two unit vectors does not give another unit vector. That is what Wikipedia is saying, though of course it's Wikipedia and you should not expect it to say things rigorously.

That seems to be a problem because it implies that the addition gives off a different object

No, it doesn't. You are adding two vectors and getting another vector.

It seems that your ace in the sleeve is that a 4-velocity's magnitude is always 1

It's not an "ace in the sleeve", it's part of the definition of a 4-velocity. It's a unit timelike vector tangent to an object's worldline.

what is then the object that yields the addition of two colinear 4-velocities or in general the scaling thereof by some coefficient? What is its name and what kind of object is it?

A vector. I have already said this multiple times. As above, you are adding two vectors and getting another vector. Or you are multiplying a vector by some scalar and getting another vector. This is vector spaces 101. If you seriously don't understand how this works, please take some time to learn it from a textbook or similar source on the basics of vector spaces.

Your issue seems to be that "4-velocity" or "unit vector" is somehow some "different" type of object from other vectors. As far as vector addition and scalar multiplication is concerned, it's not. It's just a vector. The fact of having unit norm is useful for some other purposes, but it has no significance as far as the basics of vector spaces are concerned.

 

[Saw]

The subthread you are complaining about started with your incorrect assertion in post #126 that the tangent space is only composed of 4-velocity vectors (i.e., unit timelike vectors). It's not. Instead of realizing that it's not and rethinking your position in the light of that new knowledge, you have continued to make further erroneous claims, which I have been correcting (see my post #140 for my latest corrections).

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 time basis vector in Galilean relativity as compared to SR, which is what we are discussing now.

I guess you refer to other derivatives, like 4-force or 4-acceleration... But focusing again on 4-velocity, which is our object of interest because it i said to be at least one option for the time basis vector, I would come back to the issue mentioned before...

I am just trying to understand something that looks weird to me, that the 4-velocity is the time basis vector and I do think that others may have this legitimate lack of understanding. And I do find puzzling this twisted way of mentoring where the objective seems to be, not answering things, but convincing the asker that she should not ask. And I guess there may be many askers who have suffered these methods and may be thinking the same. And this is a shame for PhysicsForums and it should be stopped because it is a most valuable place that does not deserve to gain this ugly image.

 

[PeterDonis]

I am just trying to understand something that looks weird to me, that the 4-velocity is the time basis vector

The 4-velocity is not necessarily the time basis vector. A particular object's 4-velocity can be the time basis vector, if you choose to work in that object's rest frame and you want to use unit vectors as your basis vectors (which is usually a good choice but is not a requirement).

I still do not understand what the issue is. In the previous post of yours that you quoted from, you cut off your quote; here is the rest of that paragraph of your post:

a basis vector, in a normalized basis, must be unitary but then its role is being scaled by a certain coefficient...

I don't see what the issue is here. Yes, an arbitrary vector in a vector space can be written, given a choice of basis, as a linear combination of the basis vectors, i.e., with each basis vector multiplied by some scalar coefficient. As I said before, this is vector spaces 101. I cannot understand what the problem is.

(Btw, "unitary" does not mean the same thing as "unit vector", which is what basis vectors will be in a normalized basis. "Unitary" is a term that applies to operators and means something quite different.)

this twisted way of mentoring where the objective seems to be, not answering things, but convincing the asker that she should not ask

As I have already said, if people make incorrect statements here, we try to correct them. You have made a number of them in this thread, which I have corrected. All of that is a separate issue from what your actual question is (as above, I still don't understand what it is) and getting that question answered.

 

[berkeman]

Thread is closed temporarily for a Mentor discussion...