Understanding 4-Tensors: Demystifying the Misconceptions

[pmb]

what is a "4-tensor"?

Has anyone here thought at one time that the term "4-tensor" (aka "four tensor") was referring to a 4th rank tensor? Somone made this mistake and I'm, of course, curious as to how wide spread this misconception is.

Thanks

Pete

 

[chroot]

I don't know that I've ever made the mistake, or even used the term "4-tensor." Is a 4-tensor a tensor whose indices range over four values?

- Warren

 

[pmb]

Originally posted by chroot
I don't know that I've ever made the mistake, or even used the term "4-tensor." Is a 4-tensor a tensor whose indices range over four values?

- Warren

Yes. It's a use which is similar to that of "4-vector" in that the tensor is defined on a 4d manifold. For examples of usage see

For online notes for details see --
www.wikipedia.org/wiki/Maxwell's_equations
farside.ph.utexas.edu/~rfitzp/teaching/jk1/lectures/node10.html
farside.ph.utexas.edu/~rfitzp/teaching/jk1/lectures/node13.html
farside.ph.utexas.edu/~rfitzp/teaching/jk1/lectures/node23.html
farside.ph.utexas.edu/teaching/jk1/relativity.pdf
cosmos.astroscu.unam.mx/~sergio/phdthesis/phdlatex2html/node17.html
www.hep.princeton.edu/~mcdonald/examples/fieldmomentum.pdf



Pete

 

[pmb]

Originally posted by chroot
I don't know that I've ever made the mistake, or even used the term "4-tensor." Is a 4-tensor a tensor whose indices range over four values?

- Warren

There are some interesting comments inThorne and Blanchard's new text. From Chapter 1: Physics in Flat Spacetime: Geometric
Viewpoint - page 38-39
(http://www.pma.caltech.edu/Courses/ph136/yr2002/chap01/0201.2.pdf)

Evidently E is the electric field and B the magnetic field as measured in our chosen Lorentz frame.

This may be familiar from standard electrodynamics textbooks, e.g. Jackson(1999).

Not so familiar, but quite important, is the following geometric
interpretation of the electric and magnetic fields: E and B are spatial vectors as measured in the chosen inertial frame. We can also regard these quantities as 4-vectors that lie in the 3-surface of simultaneity t = const. of the chosen frame, i.e. that are orthogonal to the 4-velocity (denote it ~w) of the frame's observers (cf. Fig. 1.10).

If anyone knows of another source which discusses the E and B fields as 4-vectors can you please let me know - references etc.?

Thanks

Pete