# Understanding 4-Tensors: Demystifying the Misconceptions

• pmb
In summary, a "4-tensor" refers to a fourth rank tensor with indices ranging over four values. It is similar to a "4-vector" in that it is defined on a 4-dimensional manifold. The terms "E" and "B" refer to the electric and magnetic fields, respectively, and can be interpreted as spatial vectors or 4-vectors in the 3-surface of simultaneity of a chosen inertial frame. Some sources, such as Thorne and Blanchard's new text, discuss the E and B fields as 4-vectors, but it is not a commonly used terminology.
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

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

## 1. What is a 4-tensor?

A 4-tensor is a mathematical object that has 4 indices and represents a multilinear mapping between four vector spaces. In simpler terms, it is a way to describe the relationships between four different quantities or variables.

## 2. How is a 4-tensor different from a scalar, vector, or matrix?

A scalar is a single value, a vector has 1 index and represents a quantity with direction, and a matrix has 2 indices and represents a linear transformation. A 4-tensor has 4 indices and represents a multilinear mapping between four vector spaces, making it a more complex mathematical object.

## 3. What are some common misconceptions about 4-tensors?

One common misconception is that 4-tensors are always represented by a 4x4 matrix, but in reality, the size of the matrix representation depends on the dimensions of the vector spaces involved. Another misconception is that 4-tensors are only used in physics, but they have applications in many fields such as engineering, computer science, and economics.

## 4. How can understanding 4-tensors be useful in scientific research?

4-tensors are a powerful tool for modeling and analyzing complex systems with multiple variables. They can help researchers understand the relationships between different quantities and how they affect each other. They are also useful for solving equations and making predictions in fields such as physics, mechanics, and electromagnetism.

## 5. Are there any resources available for learning more about 4-tensors?

Yes, there are many textbooks, online courses, and video tutorials available for learning about 4-tensors and their applications. Some recommended resources include "Tensor Calculus for Physics" by Dwight E. Neuenschwander and "Introduction to Tensor Calculus and Continuum Mechanics" by J. H. Heinbockel. Additionally, many universities offer courses on tensor calculus as part of their math or physics curriculum.

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