A six-vector is related to a 2nd rank antisymmetric tensor in Minkowski Spacetime?

[grzz]

I read that a six-vector is related to a 2nd rank antisymmetric tensor in Minkowski Spacetime. But what is a six-vector? Any example of such a vector?
Thanks for any help.

 

[Vanadium 50]

I read that a six-vector

Where?

 

[Ibix]

Where did you read this? That might help...

 

[Orodruin]

Antisymmetric rank 2 tensors form a 6-dimensional vector space. I have never heard or seen this called a 6-vector and I have been teaching relativity for 20 years.

So I’ll join the chorus: Where did you read/hear this?

 

[grzz]

Where did you read this? That might help...

Thanks for your help.
I tried to read E. T. Whittaker's 'On the relation of tensor calculus with spinor calculus' of 1936.
He mentions six-vectors and he is using what, I think are components of an antisymmetric 2nd rank tensor R. Hence I assumed that these are some of the components of the six-vector that he is mentioning.

 

[Sagittarius A-Star]

Arnold Sommerfeld described the electromagnetic field by a six-vector:
https://en.wikisource.org/wiki/Tran...nal_Vector_Algebra#§_1._Four-_and_Six-vectors.

 

[grzz]

Arnold Sommerfeld described the electromagnetic field by a six-vector:
https://en.wikisource.org/wiki/Translation:On_the_Theory_of_Relativity_I:_Four-dimensional_Vector_Algebra#§_1._Four-_and_Six-vectors.

This is interesting!
I will try to go through it.
Thanks for your help.

 

[Orodruin]

E. T. Whittaker's 'On the relation of tensor calculus with spinor calculus' of 1936

Arnold Sommerfeld described the electromagnetic field by a six-vector:
https://en.wikisource.org/wiki/Translation:On_the_Theory_of_Relativity_I:_Four-dimensional_Vector_Algebra#§_1._Four-_and_Six-vectors.

So these sources have one thing in common: They are archaic. On top Sommerfeld is using the dreaded and long since defunct ict variant of the time component of 4-vectors.

I strongly suggest not using such ancient sources and instead going for a modern textbook on relativity. If nothing else to gain 100 years of development in the approach and didactics of the subject as well as modern nomenclature so that you can actually make yourself understood.

 

[grzz]

So these sources have one thing in common: They are archaic. On top Sommerfeld is using the dreaded and long since defunct ict variant of the time component of 4-vectors.

I strongly suggest not using such ancient sources and instead going for a modern textbook on relativity. If nothing else to gain 100 years of development in the approach and didactics of the subject as well as modern nomenclature so that you can actually make yourself understood.

Perhaps I should have quoted from the very start the source of my first post. But trying to read old Physics papers remains interesting.

 

[Vanadium 50]

as well as modern nomenclature so that you can actually make yourself understood.

Verily and forsooth, my good man! Odds bodkins!

 

[Klystron]

Perhaps I should have quoted from the very start the source of my first post. But trying to read old Physics papers remains interesting.

Agree. I learned a lot by reading my father's pre-WWII physics textbooks. Chemistry texts were not as useful due to major paradigm shifts. Some 19th C. math books are very interesting.

Language, censors and 20th politics limit usefulness of some old science publications. Time was much new physics (and math) was published in German with English translations not immediately available. After 1918, the Soviet Union restricted science and knowledge exchange with perceived enemies into the 1990s.

 

[Orodruin]

Perhaps I should have quoted from the very start the source of my first post. But trying to read old Physics papers remains interesting.

Interesting, yes. Although as a good source for learning modern physics, usually less useful. It certainly helps to be aware of the modern understanding of the theory before delving in.

 

[Dale]

Interesting, yes. Although as a good source for learning modern physics, usually less useful. It certainly helps to be aware of the modern understanding of the theory before delving in.

I agree. It is great to dive in to original and outdated sources, but should be done after learning the modern theory from current sources.

 

[Histspec]

I read that a six-vector is related to a 2nd rank antisymmetric tensor in Minkowski Spacetime. But what is a six-vector? Any example of such a vector?
Thanks for any help.

As the electromagnetic field tensor consists of six independent components, three electric ($E_x$, $E_y$, $E_z$) and three magnetic ($B_x$, $B_y$, $B_z$) ones, it was initially represented by Sommerfeld as a “six vector” in terms of Plücker (or Grassmann) line coordinates in order to avoid Minkowski's matrix formulation (which happens to be equivalent to Plücker matrices anyway) - see this historical overview from Grassmann/Plücker to Einstein (yes, even Einstein used that expression).
Another six-vector representation of the EM field, which is still be used, is the Weber (or Riemann–Silberstein) vector.

 

[Sagittarius A-Star]

Interesting, yes. Although as a good source for learning modern physics, usually less useful.

I agree. However, I think that Sommerfeld's six-vectors are not only of historical interest. They may have still a didactic value. I think they gave me more intuition to the EM field-tensor, when I read about it while posting in this thread.

Feynman built a geometric intuition to an antisymmetric 3-tensor by discussing the torque as an axial 3-vector and as a tensor. Source (see chapter "31–5 The cross product"):
https://www.feynmanlectures.caltech.edu/II_31.html

Sommerfeld did a similar thing in 4D-space: The vectors of first kind have four components, "four-vectors", those of second kind (axial vectors) have six components, "six-vectors". To my understanding (as also written by Feynman in the link above) that is because there are 6 unique planes, to which coordinate-axes of 4D-space are perpendicular (zy, zx, xy, xt, yt, zt). That are positions in the EM field tensor with different independent components of the EM field.

Source:
https://en.wikisource.org/wiki/Translation:On_the_Theory_of_Relativity_I:_Four-dimensional_Vector_Algebra#§_1._Four-_and_Six-vectors

The EM field can be described as an axial vector, because it's "4D-divergence" is the 4-current.
Another argument is, that the EM field is the "4D-curl" of the 4-potential.

Source:
https://en.wikisource.org/wiki/Translation:On_the_Theory_of_Relativity_II:_Four-dimensional_Vector_Analysis#§_5._The_differential_operations_of_four-dimensional_vector_analysis.

In "Electrodynamics: Lectures on theoretical physics, Vol III" from 1948, Sommerfeld described the EM field as six-vector and also as antisymmetric tensor.

Source (see part III, §26 B, page 214):
https://www.amazon.com/-/de/dp/B0CCCJD2HF/?tag=pfamazon01-20

Minkowski wrote in 1908:

We shall define a space-time Vector of the 2nd kind as a system of six-magnitudes ...

Source:
https://en.wikisource.org/wiki/Tran...or_Electromagnetic_Processes_in_Moving_Bodies

 

[Orodruin]

Yes and no. That identification risks confusion with the actual 4-dimensional pseudo-vector representation. Even if there are similarities, making the connections using the language of differential forms seems significantly more intuitive and less prone to misunderstandings to me.

 

[PeterDonis]

To my understanding (as also written by Feynman in the link above) that is because there are 6 unique planes, to which coordinate-axes of 4D-space are perpendicular (zy, zx, xy, xt, yt, zt).

If you unpack this further I think you end up with the representation in terms of differential forms that @Orodruin referred to.

 

[Orodruin]

If you unpack this further I think you end up with the representation in terms of differential forms that @Oroduin referred to.

I mean, yes. It is essentially what 2-forms are. And to me that language just plays out more naturally. The vector space of two-forms in spacetime are spanned by
$$
dt\wedge dx,\
dt\wedge dy,\
dt\wedge dz,\
dx\wedge dy,\
dy\wedge dz,\
dz\wedge dx
$$

 

[Orodruin]

@Oroduin

Also

 

[PeterDonis]

Also

I wondered why it wasn't highlighting. Fixed now.

 

[Sagittarius A-Star]

Yes and no. That identification risks confusion with the actual 4-dimensional pseudo-vector representation. Even if there are similarities, making the connections using the language of differential forms seems significantly more intuitive and less prone to misunderstandings to me.

Thanks for referring to differential forms. I found a good video about this topic (via playlist):



A book referenced under the video is David Bachman: "A Geometric Approach to Differential Forms".
https://www.amazon.com/-/de/dp/0817683038/?tag=pfamazon01-20

An older version of the book text can be found online in a paper:
https://arxiv.org/abs/math/0306194v1

 

[Sagittarius A-Star]

Derivation, that the EM field must have six independent components

The 4-potential can be written as 1-form:
$\mathbf A = - \Phi\ dt + A_1\ dx + A_2\ dy + A_3\ dz$

Show, that the Faraday is a 2-form in 4D space and therefore must have six independent components:

$\mathbf F = d\mathbf A =$
$\require{color}{\partial \over \partial t}(- \Phi\ \color{red}dt \wedge dt\color{black}+ A_1\ dt \wedge dx+ A_2\ dt \wedge dy+ A_3\ dt \wedge dz )$
$\require{color}+{\partial \over \partial x}(- \Phi\ dx \wedge dt+ A_1\ \color{red}dx \wedge dx\color{black}+ A_2\ dx \wedge dy+ A_3\ dx \wedge dz )$
$\require{color}+{\partial \over \partial y}(- \Phi\ dy \wedge dt+ A_1\ dy \wedge dx+ A_2\ \color{red}dy \wedge dy\color{black}+ A_3\ dy \wedge dz )$
$\require{color}+{\partial \over \partial z}(- \Phi\ dz \wedge dt+ A_1\ dz \wedge dx+ A_2\ dz \wedge dy+ A_3\ \color{red}dz \wedge dz\color{black} )$

With anti-commutativity of the wedge-product follows:
$\mathbf F =$
$(-{\partial \Phi \over \partial x} - {\partial A_1 \over \partial t}) dx \wedge dt+(-{\partial \Phi \over \partial y} - {\partial A_2 \over \partial t}) dy \wedge dt+(-{\partial \Phi \over \partial z} - {\partial A_3 \over \partial t}) dz \wedge dt$
$+({\partial A_3 \over \partial y} - {\partial A_2 \over \partial z}) dy \wedge dz+({\partial A_1 \over \partial z} - {\partial A_3 \over \partial x}) dz \wedge dx+({\partial A_2 \over \partial x} - {\partial A_1 \over \partial y}) dx \wedge dy$

$\mathbf F = E_1\ dx \wedge dt+E_2\ dy \wedge dt+E_3\ dz \wedge dt+B_1\ dy \wedge dz+B_2\ dz \wedge dx+B_3\ dx \wedge dy$

 

[Sagittarius A-Star]

Another six-vector representation of the EM field, which is still be used, is the Weber (or Riemann–Silberstein) vector.

A similar six-vector representation of the EM field is used in modern geometric algebra.

...

...

The $i$ in equation (9) represents the unit pseudo-scalar with $i^2=-1$.

The equation (11)
$$\nabla F=J$$contains all of Maxwell's equations, including the inhomogeneous and homogeneous ones.

So this notation is more compact than the usual tensor notation.

Source:
https://geometry.mrao.cam.ac.uk/wp-content/uploads/2015/02/ImagNumbersArentReal.pdf
via:
https://geometry.mrao.cam.ac.uk/199...-not-real-the-geometric-algebra-of-spacetime/