# 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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### What's the most common convention for torsion and contorsion tensor index position?

In Einstein-Cartan theory, the torsion tensor is usually defined as the antisymetric part of the connection: \begin{gather} \nabla_{\mu} \, A^{\lambda} = \partial_{\mu} \, A^{\lambda} + \Gamma_{\mu \...
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### Raising and lowering indices and tensor contraction

I'm really confused by the notation of raising and lower indices in tensors when mixed with einstein summation notation and referencing the metric tensor. I need help separating several conflicting ...
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### Why are isotropic tensors not considered scalars?

In introductory textbooks (Griffiths, Shankar, Boas) a tensor is introduced as a mathematical objects which transform in a specific manner under changes of basis (i.e. changes of the coordinate system)...
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### What is the magnitude of a tensor property in a fixed direction?

If I have a physical property represented by a $3 \times 3$ tensor, how can I find its magnitude in a particular direction, say $(\phi, \theta)$ in spherical coordinate system?
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### Contracting Riemann Tensor Troubles

It has been several years since I looked at General relativity, and I am trying to brush up on it because it was always interesting and I am in need of it for my research. Specifically, I am looking ...
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### General relativity: Principle of minimal coupling computations

I have a question about computations in general relativity and transition from a Lorentz frame to a general fame just by substituting the flat metric with a general one and ordinary derivatives with ...
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Background I'm familiar enough with the Faraday tensor $F_{\alpha\beta}$ to know that it's is a 2-form. Hence, at each point $P$ in spacetime $V$, it's a multilinear map $$F: T_PV\times T_pV\to\mathbb{... 2answers 82 views ### Are Lagrange's equations physical laws? Well, I studied that a physical law is an equation between tensors that are function of position and time because when the frame is changed tensors change in order to leave the equation true:$$T_1(\...
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In special relativity, a four-vector $\mathbf{x}$ in an inertial frame is related to $\mathbf{\overline{x}}$ through a Lorentz transformation $\mathbf{\Lambda}$: \begin{align} \overline{\mathbf{x}}...
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### Timelike and spacelike projections in General Relativity and associated conservation laws

For any timelike curve $p_\mu$ in General Relativity (section 3 of this review), we can project this into its timelike and spacelike components. Further, these projections are associated with ...
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### Scalar coupled to Gauss-Bonnet invariant vs Horndeski theory

So here it is a somewhat tormenting question. The first statement will be a little specific but then I will make clear what the jargon indicates. How can we show that a Lagrangian made of a scalar ...
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### Indices misprint in Sean Carroll's Spacetime and Geometry?

To my knowledge, 3 or more indices may not appear in a given term, as I've found in a video produced by "Faculty of Khan": However, on page 30, Sean Carroll writes: As obvious, the indices 0 and 1 ...
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### Is $X\otimes X$ not the simultaneous position operator?

I had thought that $X\otimes X$ would be the operator on $H_1\otimes H_2$ to simultaneously measure the x-positions of two particles. But there seems to be something wrong with this -- for a given ...
The compact form of Maxwell's equations: $$\boxed{\square\, \boldsymbol{\mathsf{F}}=\mu_0 \boldsymbol{\mathcal{J}}} \tag{1}$$ where the current density quadrivector is given by the relation \$\...