# Questions tagged [tensor-calculus]

Tensor calculus (tensor analysis) is a systematic extension of vector calculus to multivector and tensor fields in a form that is independent of the choice of coordinates on the relevant manifold, but which accounts for respective sub-spaces, their symmetries, and their connections.

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### The dual of a surface element in 4-space

In reading the classic text, "The Classical Theory of Fields", Third Edition, by Landau and Lifschitz, I found an "obvious" statement not so obvious to me. It is on p.19, the statement of the ...
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### Rewriting a lagrangian in terms of Hodge duals?

Spinors have been found to have some interesting applications in general relativity (such as Wittens positive energy proof). Recently I'd come across a series of papers 1 2 3 (there are many more ...
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### Understanding Riemann Curvature Tensor in Misner, Thorne and Wheeler's Gravitation

I'm trying to understand section 11.4 of Misner, Thorne and Wheeler's Gravitation textbook, which explains how the output of the Riemann Curvature Tensor $Riemann(...,A,u,v)$ is a vector describing ...
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### Doubt about the history of the tensor object in physics

The word "tensor" (or, maybe the whole physical idea of this object) could be traced date to 1898 on the work on Crystals due to Voigt . It seems that the references  and , introduce the idea ...
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### Transformations of contravariant and covariant tensor operators

I've been able to convince myself that a set of contravariant tensor operators $\hat{O}^{x}$ for $x=1,2,...,n$ respond to a small transformation $\hat{A}$ as, \begin{equation} [\hat{A},\hat{O}^{x}]=-(\...
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### Momentum in capacitor field; How can an EM field have zero momentum density but non-zero momentum flux?

Consider the case of a simple, stationary parallel plate capacitor oriented with its plates lying in the x-y plane. The E-field is simply given by: $$\vec{E} = \frac{Q}{\epsilon_0A}\hat{z}$$ with ...
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### A question from cosmological perturbation theory

We consider the following scalar perturbation on the FRW metric $$ds^2=-(1+2\Phi)dt^2+2a(\partial_iB)dx^idt+a^2[(1-2\Psi)\delta_{ij}+2\partial_{ij}E]dx^idx^j,$$ where $\Phi$, $B$, $\Psi$ and $E$ are ...
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### What is the spin of an operator in QFT?

Operators in quantum field theory with $n$ Lorentz indices that are symmetrized and traceless are referred to as spin-$n$ operators. For example, a spin two operator would be \begin{equation} \bar{\...
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### Why is a spinor not a tensor?

The title says it. why is a spinor not a tensor? I know the transformation rules for a spinor but I cant see why it is not a tensor?
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### Rindler Coordinate Transformation (Cylindrical Coordinates)

How might one go about transforming from cylindrical flat space: $$ds^2=-dt^2+d\rho^2+dz^2+\rho^2d\phi^2,$$ to Rindler space represented by: $$ds^2=-\rho^2dt^2+d\rho^2+dz^2+d\phi^2?$$ The normal ...