# Anitsymmetric tensor/switching indices problem

• wasia
In summary, the conversation discusses the antisymmetry of a non-operator tensor with only numbers as its components. The conversation also mentions a possible error in the derivation of the tensor's antisymmetry and asks for clarification on any potential mistakes.
wasia
Let's say that some non-operator (having only numbers as it's components) tensor is antisymmetric:

$$\omega^{\sigma\nu}=-\omega^{\nu\sigma}$$
and
$$\omega_{\sigma\nu}=-\omega_{\nu\sigma}$$,

however, I have read in the Srednicki book that it is incorrect to say that the same tensor with one index down and one up would be antisymmetric as well.

Could you please point out, where and what are the errors of the derivation? Should I read something before asking such questions? g here is the Minkowski metric:

$$\omega^{\nu}\,_{\sigma}=\omega^{\nu\beta}g_{\beta\sigma}=-\omega^{\beta\nu}g_{\beta\sigma}=-\omega_{\sigma}\,^{\nu}$$

Thank you.

wasia said:
Could you please point out, where and what are the errors of the derivation? Should I read something before asking such questions? g here is the Minkowski metric:

$$-\omega^{\beta\nu}g_{\beta\sigma}=-\omega_{\sigma}\,^{\nu}$$

Shouldn't there be another step in between?

wasia said:
Could you please point out, where and what are the errors of the derivation? Should I read something before asking such questions? g here is the Minkowski metric:

$$\omega^{\nu}\,_{\sigma}=\omega^{\nu\beta}g_{\beta\sigma}=-\omega^{\beta\nu}g_{\beta\sigma}=-\omega_{\sigma}\,^{\nu}$$

This should be fine. Notice that you switched which index was up and which was down.

xboy, the steps in between might be something like
$$-\omega^{\beta\nu}g_{\beta\sigma}=-\omega^{\beta\nu}g_{\sigma\beta}=-g_{\sigma\beta}\omega^{\beta\nu}=-\omega_\sigma\,^\nu$$
I assume they are valid, as g is undoubtly symmetric (at least in my case) and also g commutes with omega, as they contain only numbers.

Ben Niehoff, I do have noticed, that positions have changed.

However, if anyone could point out a mistake, or tell if that's correct, as Ben says, please do it.

Last edited:
Walia, your derivation seems correct to me. I can't think of a case of your derivation being invalid except for the metric being non-symmetric. But I don't know if the metric can be non-symmetric at all.

## 1. What is an antisymmetric tensor?

An antisymmetric tensor is a type of mathematical object that is used to represent multilinear maps between vector spaces. It is defined by its components, which satisfy a specific set of equations that give it its antisymmetric properties.

## 2. What are the applications of antisymmetric tensors?

Antisymmetric tensors have a wide range of applications in physics and engineering. They are commonly used to describe electromagnetic fields, fluid dynamics, and elasticity. They are also used in mathematical models for computer graphics and image processing.

## 3. What does it mean to switch indices in an antisymmetric tensor?

Switching indices in an antisymmetric tensor means interchanging the order of its components. This is typically done to simplify calculations or to transform the tensor into a different coordinate system. It does not change the overall properties or behavior of the tensor.

## 4. How do you solve the switching indices problem for an antisymmetric tensor?

The switching indices problem for an antisymmetric tensor can be solved by using the properties of antisymmetry and the rules of index notation. It involves manipulating the components of the tensor according to these rules to obtain the desired result.

## 5. Are there any real-life examples of the antisymmetric tensor/switching indices problem?

Yes, the antisymmetric tensor/switching indices problem is encountered in many real-life scenarios. One example is in the study of fluid dynamics, where antisymmetric tensors are used to represent the rotation of a fluid. Another example is in the analysis of stress and strain in solid materials, where the switching indices problem is used to transform the tensor into a more convenient coordinate system.

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