Vector

[A23]

Could somebody help me to remember :

 

a) If a quantity [math]x_\nu=(ct, x) [/math] is a vector,

 

is [math]\partial_t x_\nu=(c, v)[/math]

 

a vector too ?

 

b) is not [math]B_{\mu \nu}=\partial_\mu x_\nu [/math] a tensor of order 2, and hence [math]C_\mu=g_\mu^{\quad\nu} B_{\mu\nu}[/math] a vector which come at the same as the 1st ?

[Ben]

Could somebody help me to remember :

 

a) If a quantity [math]x_\nu=(ct, x) [/math] is a vector,

Well, this not a vector, in fact your notation makes no sense. So it seems like you are trying to run before you can walk.

 

Let me help you take your first tottering steps

 

Consider the [math]x,y[/math] plane. Here we may have a directed line segment with a particular start point and a particular end point, each referred to the [math]x,\,\,y[/math] coordinates, which will give us information about the length and direction of our line segment.

 

But we may find an infinity of line segments with the same length and same direction, but different start and end points. Let's call the class of all such segments with the same length and direction an equivalence class (there is a technical definition for this, but we don't need it).

 

So the equivalence class of all directed line segments with the same length and direction, regardless of their start of end points, is called a vector. Obviously then, it makes no sense to try and refer to a vector so defined to the [math]x,\,\,y[/math] coordinates, since each element in the equivalence class refer to these coordinates in a unique way.

 

What is less obvious that the arithmetic sum of two such equivalence classes/vectors is another equivalence class/vector and that arithmetic multiplication by a number, a scalar, is also another equivalence class/vector. However it is true, and what is also true is that these operations "combine" as follows:

 

Suppose [math]v,\,\,w[/math] are vectors as defined, and that [math]\alpha[/math] is a scalar. Then we must have that [math]\alpha (v+w)=\alpha v+\alpha w[/math] as another vector, and one or two other tricks. This is called linearity.

 

So. If our vectors cannot be referred to our coordinates, what can they be referred to? By the above, the linear combination of any two vectors is a third vector, though it might be the zero vector, but the converse is false: there are vectors which are not the linear combination of any other vectors, and these are called basis vectors.

 

Specifically in our case, for the basis vectors [math]e_1,\,\,e_2[/math] there is no scalar such that [math]\alpha e_1 = e_2[/math] (and topsy-turvy).

 

Then you go on and ask about the differential operator. This is a fish in a different kettle entirely. Ask if you want to know more.....

[A23]

If I would have written : [math]x_\mu=ct\vec{e}_t+x\vec{e}_x [/math] would have this have made any sense ?

[Ben]

No. What is [math]\vec{e}_t[/math]? What is [math]\vec{e_x}[/math]? Specifically - to what do the subscripts refer? They should be natural numbers - are they?

 

Likewise, what are the co-factors [math]ct[/math] and [math]x[/math]? They should be elements in some scalar field. Are they? How can [math]x[/math], say, be both a natural number and also an element in some scalar field? Hint: the natural numbers are not a field.

 

Ask if you have a question.

[Pyrotex]

No. What is [math]\vec{e}_t[/math]? What is [math]\vec{e_x}[/math]? ...

 

[math]\vec{e}_t[/math] is a Unit Vector, defined to point in the "direction" of time.

 

[math]\vec{e_x}[/math] is another Unit Vector (by definition, a Unit Vector has length 1), defined to point in the direction of the X axis.

 

If you re-define [math]\vec{e}_t[/math] as i [math]\vec{e}_y[/math] then you have the Complex Plane, with time along the Y axis. And your vector math is the same as complex algebra. Convenient.

[A23]

If: [math]ct,x\in\mathbb{R}[/math], they then are in a field ?

[A23]

I don't know if it is authorised, the indices used above are like the convention sometimes found for spherical coordinates, but don't know if it is 'allowed' (neither by which mathematical authority) : [math]\vec{e}_r,\vec{e}_\theta,\vec{e}_\phi[/math] ?

 

Here I suppose it is the symbol, 'r' for example, and not the numerical value of 'r', in fact i do not know exactly this convention work, seem complicated after all.

 

Else, if for example [math] r=\theta=1.3[/math] then [math]\vec{e}_r=\vec{e}_{1.3}<>\vec{e}_\theta\textrm{ even if }=\vec{e}_{1.3}[/math]

 

~Like in C, int *r;

the difference between r and *r

[Ben]

If you re-define [math]\vec{e}_t[/math] as i [math]\vec{e}_y[/math]

So time is imaginary after all!

 

If: [math]ct,x\in\mathbb{R}[/math], they then are in a field ?

Look I am finding this weird notation rather confusing.

 

I assume that [math]c[/math] is a constant - light speed specifically. So that for any fixed value of [math]t[/math], say [math]t_n[/math], then [math]ct_n[/math] is the distance travelled by light in this time.

 

Now a field has, among its other axioms, that all additive inverses exist. So what is meant by "negative distance"? On other words, are you sure that [math]ct_n[/math] is an element in some field?

 

Or is there some subtlety that physicists are aware of and I am not?

[A23]

Distance in 1D, could need a "sense" as a supplemental data, i.e. right or left (like : +c and -c ?). Could we find in this way an inverse ?

 

This is a good remark, since, if ct is a distance, it should be along the x-axis.

 

With the imaginary time, c with units m/s achieve a rotation of -90°, which seems more consistent.

 

So in the first case, can we deduce c should in fact be a 2x2 matrix ?

[Ben]

Distance in 1D, could need a "sense" as a supplemental data, i.e. right or left (like : +c and -c ?). Could we find in this way an inverse ?

 

This is a good remark, since, if ct is a distance, it should be along the x-axis.

 

With the imaginary time, c with units m/s achieve a rotation of -90°, which seems more consistent.

 

So in the first case, can we deduce c should in fact be a 2x2 matrix ?

This is complete gibberish; I thought you were looking for some guidance about vector spaces as they are defined in mathematics. I tried my best.

 

It seems my words, which were intended to be helpful, fell on deaf ears.

 

So I guess I'll see you around....

[A23]

Anyhow this theory was understood by a handful only of people, and it were pretentious or for very long lasting winter evening. Like we see sometimes on t-shirts : No, really God made of speeds in 1+1 special relativity 2x2 matrices and the imaginary became real, (BTW this is the same why God created some quantum elements), and saw that this was right and good.

 

I was looking if there were not another 'definition' in physics, functions that are not forcedly linear [math] A_i(x_j)[/math] ?