Understanding 4-Tensors: Demystifying the Misconceptions

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Discussion Overview

The discussion centers around the concept of "4-tensors," exploring potential misconceptions regarding the term and its implications in the context of tensor indices and dimensionality. Participants examine the definition and usage of 4-tensors, particularly in relation to 4-vectors and their application in physics, including electrodynamics.

Discussion Character

  • Exploratory
  • Conceptual clarification
  • Debate/contested

Main Points Raised

  • Some participants express curiosity about whether the term "4-tensor" is commonly misunderstood as referring to a 4th rank tensor.
  • One participant questions if a 4-tensor is simply a tensor whose indices range over four values.
  • Another participant clarifies that the term "4-tensor" is analogous to "4-vector," indicating that it is defined on a 4-dimensional manifold.
  • References to various educational materials and texts are provided to illustrate the concept and usage of 4-tensors and their relation to electric and magnetic fields as 4-vectors.
  • A request for additional sources discussing the geometric interpretation of electric and magnetic fields as 4-vectors is made.

Areas of Agreement / Disagreement

Participants do not appear to reach a consensus on the commonality of the misconception regarding 4-tensors, and there are differing views on the clarity of the term's definition and its implications in physics.

Contextual Notes

Some limitations include the potential ambiguity in the term "4-tensor" and its relationship to rank and dimensionality, as well as the dependence on specific definitions and contexts in physics.

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
 
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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
 
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
 
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
 

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