# 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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### Is the spacetime interval a tensor?

Tensors are objects that are invarient under a change of basis representation and whose coordinates change predictably. The spacetime interval is invarient under a change of coordinate representation ...
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### Tensor Method $SU(N)$

I'm working out the $SU(N)$ tensor method and reading Cheng-Li page 102, 103 (Sec. 4.3). I'm following the definition (4.94) which are $\psi^i=\psi_i^*$, $U_i^{.j}=U_{ij}$ and $U^i_{.j}=U_{ij}^*$ ...
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### Why do we write this tensor notation of space-time gradient contravariant tensor?

Why is $\partial^\mu=\frac{\partial}{\partial x_{\mu}}$ the contravariant component of space-time gradient four vector instad of $\partial^{\mu}=\frac{\partial}{\partial x^{\mu}}$?
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### Scale transformation of $\epsilon$ and $\partial$

I got confused about scale transformation. Under \begin{align} \tilde{g}_{ab} = \Omega^2 g_{ab} \end{align} And consider scalar $\Phi$ and levi-civita symbol $\epsilon$, then how \begin{align} \...
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### Young Tableaux and Tensors

We can represent a tensor with $(n, m)$ where $n$ are the upper indices and $m$ the lower ones. If i get the direct product of $(n,m)\otimes (n', m')$ then i will have irreducible representation. Let'...
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### How to interprete this singularity? [closed]

I am calculating the Kretschmann scalar for the Schwartzchild metric. This is the graphic I get: Where $R$ is the radial coordinate and $x=\cos(\theta)$. So, there is the singularity at $R=0$ as it ...
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### Tensor indices from Ramond [closed]

I'm working through Pierre Ramond's "Field Theory: A Modern Primer". I can't connect the steps in I.2. Eq 2.6 (p. 7) gives a property of two linear transformations in relation to the Minkowski metric,...
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### Question about Einstein summation convention

I'm dealing with the following: $$\eta^{\alpha \mu} \eta_{\alpha \nu} \phi,_{\beta \mu}$$ $$\eta^{\alpha \beta} \phi,_{\alpha \beta}$$ where $\eta$ is the Minkowski metric and $\phi$ is a function ...
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### What is a three dimensional irrep ${\bf 3}$ of $SO(3)$?

What is three dimensional irreducible representation of $SO(3)$ denoted by ${\bf 3}$? Are they vectors or antisymmetric tensors of rank two each of them has three independent components. Also when ...
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### Deriving $\nabla_\mu \nabla_\sigma \mathcal{K}^\rho=R^\rho_{\sigma\mu\nu}\mathcal{K}^\nu$

I want to derive this equation from Carroll's book. $$\nabla_\mu \nabla_\sigma \mathcal{K}^\rho=R^\rho_{\sigma\mu\nu}\mathcal{K}^\nu$$ We know that $\mathcal{K}^\nu$ is a killing vector and ...
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### Confusion about expressing an inner product using the Einstein summation convention

I think this likely comes down to the following expression, $$g’^{ab}e’_a e’_b = \delta ^a_b$$ Is this in agreement with the Einstein summation convention? Because even though the two indices are ...
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### Are dual bases and the Hodge dual “entirely distinct” uses of the word “dual”, as per MTW

NB: Basis one-forms and contravariant basis vectors (which, following Menzel, I am calling reciprocal) are the same thing. See, for example, the Mathematical Appendix to Gravitation and Inertia, by ...
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### Difference between $g^{\alpha\beta}$ and $g^\alpha_{\space\space\beta}$

I'm working out a problem where at some point get the following product of metric tensors and momenta: $$g^{\mu\beta}g^\nu_{\space\space\alpha}(2k+\frac{q}{2})^\alpha(\frac{q}{2}-k)_\beta$$ How can I ...
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### In electromagnetism, how do we know that either $F^{\mu\nu}$ or $A^\mu$ is a tensor?

In special relativity the partial derivative $\partial_\mu$ is a tensor. Now if some function $A^\mu$ was a tensor, then also the quantitiy $F^{\mu\nu}=\partial^\mu A^\nu - \partial^\nu A^\mu$ would ...
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### Is the moment of inertia calculated about an axis, or about a point? And must the point be at the center of mass?

I know that, $$L=I\omega$$ where $L$ is the angular momentum vector, $I$ is the inertial tensor, and $\omega$ is the angular velocity. Now here are my doubts :- Before I was taught the moment of ...
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### How to calculate and express $*d*F$ in index notation? [closed]

Gravitation by Charles W. Misner, Kip Throne and John Wheeler page 120 Box 4.4 Duality Plus Exterior Differentiation, related exercise 3.13, 3.17 I want to calculate the component of $*d*F$ in index ...
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### Is every Lorentz invariant a Lorentz scalar?

All examples of lorentz invariant quantities that I have come across seem to be scalars: rest mass, proper time, spacetime interval,dot product of two 4 vectors etc. Another thing is that these are ...
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### Does completely antisymmetric tensor act on a tensor always produce a tensor or not?

So completely antisymmetric tensor $\epsilon$ act on a tensor can produce a new object. i.e. $G_{\alpha\beta}=\frac{1}{2}\epsilon_{\alpha\beta\mu\nu}F^{\mu \nu}$. However, According to Landau's ...
The covariant derivative of a contravector field is given by: \begin{equation} D_{k} A^{i} \equiv A^{i}_{\parallel k} = A^{i}_{\mid k} + \Gamma^{i}_{kp} A^{p} \end{equation} With $A^{i}_{\mid k} = \... 2answers 85 views ### The definition of quantities in special relativity as upper-index or lower-index My question is for Minkowski metric$\eta_{\alpha\beta}=\mathrm{diag}(1,-1,-1,-1)$. While defining quantities like the four potential, four momentum or even space-time interval for that matter, why do ... 1answer 96 views ### Motivation for tensor theory of gravity In class we were shown that $$\rho = \frac{dm}{dV}$$ has the transformation properties of the 00 component of a rank 2 tensor. So we'd like to turn the classical Poisson equation for gravity into a ... 0answers 47 views ### How does vectorization affect$\nabla$? The homework and exercise was to prove$\nabla \times {A}$transform as a vector, and I've solved it thorough hard algebra. However, something occurred to my mind and I have a hard time to resolve it.... 1answer 55 views ### Co-spinors and contra-spinors As i was reading my teacher's notes on$SU(2)$and$SO(3)$, i have had this question. Why do co-spinors transform differently under a rotation than contra-spinors? 1answer 52 views ### Are rotation matrices tensors? Are rotation matrices tensors? If no, why? I'm not sure about it, for example considering the$z$-axis rotation matrix when I rotate the coordinate system the rotation matrix around the old$z$-axis ... 1answer 56 views ### Why covariant derivative is partial derivative in SR? I'm new in these topics and i've been confuse at some relations between the limit of SR for GR. In cartesian coordinates, basis do not change, so \begin{equation}\Gamma^{\mu}_{\alpha\beta}=0 \quad \... 3answers 161 views ### A form$F$is simple if and only if$F\wedge F=0$? Gravitation by Charles W. Misner, Kip Throne and John Wheeler page 93 Box 4.1 point 5 b. Applications: a. "In four dimensions, all 0 forms, 1- forms, 3-forms, and 4-forms are simple. A 2-form$F$is ... 1answer 28 views ### How to find the mixed tensor, contravariant tensor and tensor trace of$F$I have a question in particle physics that ask me to find the mixed tensor, contravariant tensor and tensor trace of$F$: Our professor didn't teach us that much about the math of tensor, which makes ... 0answers 26 views ### Metric (in)dependence of the electromagnetic field strength In GR, the vector potential is defined as$A^\mu$which is a contravariant vector. Then lowering the indices requires the metric$A_\mu =g_{\mu\nu} A^\nu$, using this vector one defines the field ... 0answers 37 views ### Given no strain and uniform angle of rotation, can the displacement field be expressed as a uniform rotation? Consider displacement vector$\boldsymbol{\xi}$where$\boldsymbol{\mathbf{\xi}}=\boldsymbol{\mathbf{\phi}} \times \mathbf{x}$.$\boldsymbol{\phi}$is an angle vector along the axis of rotation and$\...
One traditional representation of the stress tensor among relativists is a rank-2 fully contravariant tensor, associating a contravariant force per unit area $t^i$ to a unit normal $n_j$ defined on a ...