# 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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### 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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### 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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### How to determine if a tensor is covariant or contravariant?

In special relativity, the coordenates of a event are in general written using a 4-vector: $$x^{\mu} = \binom{ct}{\textbf{x}}$$ where $\textbf{x} = (x,y,z)$ are the spacial coordenates. This is a ...
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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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### Riemann curvature in orthonormal frame and Lorentz transformations

I have problem with understading how Riemann tensor in orthonormal frame transforms using Lorentz transformation of frames. I was reading Morris Thorne paper from 1988 (American Journal of Physics 56, ...
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### I have been reading Mathematical methods for physicists (sixth edition) by Arfken and Weber and got stuck in section 2.9 Pseudo Tensors,Dual Tensors

In Mathematical methods for physicists (sixth edition) by Arfken and Weber the triple scalar product was defined as (in page 147-148): For $\vec A, \vec B, \vec C$ with components $A^i, B^j, C^k$ and ...
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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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### 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 ...
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### How can I convert an action in terms of differential forms to tensors?

I have an action of a gravitational theory wich is in terms of differential forms. Now, I need to transform this action (including wedge product and exterior derivative of tetrad and metric ) to an ...
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### How to represent a axisymmetric, stationary metric in a coordinate independent way?

A classic example of a stationary, axisymmetric metric in GR is the Kerr metric. In Boyer-Lindquist coordinates $(t,r,\theta,\phi)$ it is obvious that the metric is independent of $t,\phi$ and so is ...
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### Shape of 3-order tensors in $O_h, O, T_d$ and $D_3$ point gruops

How does one calculate the shape of higher order $(Dimension>2)$ tensors in respect of point group symmetry? I understand that you have to use transformation matrix corresponding to a symmetry ...