# 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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### Non-zero Riemann tensors for the BTZ solution without charge or angular momentum

Can someone tell me which are the non-zero components of the Riemann tensor for the BTZ solution with the absence of charge and angular momentum? I tried searching it up but couldn't find any ...
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### Why can integrals be written as $I=\int \phi(x) \epsilon$?

Carroll's book Spacetime and Geometry gives this new way to think about integrals: $$I=\int \phi (x) \epsilon\tag{2.98}$$ $\epsilon$ is the Levi - Civita tensor (2.96). I don't see how the RHS equals ...
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### Electromagnetic tensor in a FRW metric

In some papers [like https://arxiv.org/pdf/2204.06883.pdf, eq. (31) ], I see that the Electromagnetic tensor field, for a FRW metric (written in a conformal way) \begin{equation} ds^{2} = a^{2}(\tau) \...
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### Thinking of a linear operator as a (1,1) tensor

I am reading that a linear operator $A$ can be thought of as a (1,1) tensor [where $(r,s)$ corresponds to $r$ vectors and $s$ dual vectors]. This can be done by saying $$A(v,f) \equiv f(Av)$$ where $v$...
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### Index position of spherical harmonic function and tensorial properties

The spherical harmonics functions are denoted as $Y_l^m$ or $Y_{lm}$in https://en.wikipedia.org/wiki/Spherical_harmonics e.g., \begin{align} Y_{lm} &= \dfrac{i}{\sqrt{2}} \left(Y_\ell^{m} - (-1)^...
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### How to sum with $\epsilon_{ijk}$?

How does one find the sum with $\epsilon_{ijk}$? For example, $\sum\limits_{i,j,k=1}^3 \epsilon_{ijk} \frac{a_i b_j}{c_k} T_k$? Here $a_i, b_j$ and $c_k$ are scalars and $T_k$ is an operator.
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### How to tell if the covariant derivative of something is timelike or spacelike

How can I tell if the covariant derivative of something is timelike or spacelike?
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### Doubt on transformation laws of tensors and spinors using standard tensor calculus and group theory

1) Introduction From standard tensor calculus, here restricted to Minkowski spacetime, we learned that: A scalar field is a object that transforms as: $$\phi'(x^{\mu'}) = \phi(x^{\mu})\tag{1}$$ A ...
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### Tensor symmetry: A symmetric mixed $(1,1)$ tensor is necessarily a multiple of the identity tensor

I found this observation in a book. "It is not possible to have an invariant definition of symmetry in one contravariant and one covariant index". That's all right, my problem is that to ...
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### Determinant of the inverse of a Lorentz transformation

In many text book (Ashok Das Quantum Field Theory) $$(\Lambda^T)_\nu{}^\mu=\Lambda^\mu{}_\nu$$ that gives $\Lambda^T$ = $\Lambda^{-1}$, where $\Lambda$ is Lorentz Transformation matrix. However, this ...
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### How to contract spinor indices?

In normal vector representation, vectors can be contracted as follows: $$v^\mu v_\mu$$ with one covariant and one contravariant index. But in spinor representation, there are 4 possible type of ...
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### What unit of measurement are used for 𝑀 and 𝑟 in the Schwarzschild metric?

So I understand that 𝑀 in any metric solution stands for mass. But what unit of measurement do you use to define 𝑀?
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### Can you convert the Christoffel Symbol to the form of a scalar?

Given some tensor $T_{\mu v}$, you can use the metric tensor to contract its indices, converting it into the form of a scalar: $$g^{\mu v}T_{\mu v}=T$$ Even though the Christoffel Symbol is not a ...
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### Taking the covariant derivative of the derivative of the metric tensor

Does the covariant derivative of the derivative of the metric tensor exist? if so, how do you evaluate it? $$\nabla_a (\partial_b g^{\mu v})=?$$ It would seem natural to assume that this cannot exist, ...
1 vote
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### What are some good textbooks to learn EM with differential forms? (With focus on condensed matter)

As the title suggests, does anyone have recommendations for learning EM with differential forms? Both undergraduate and graduate level textbook suggestions are welcomed. I know there's Lindell's ...
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### Tensor manipulations in Landau & Lifschitz "Classical theory of fields"

Landau & Lifshitz "Classical theory of fields" section 6 p. 19 define: $$df^{ik} = dx^i dx'^k - dx^k dx'^i$$ and $$df^{*ik}=\frac{1}{2} \; \epsilon^{iklm}df_{lm} \tag{6.11}$$ and ...
1 vote
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### Reading "A Convex Darboux Theorem" by Chiappori & Ekeland

This is cross-posted from https://math.stackexchange.com/q/4452087/ but got no reply I am reading "A Convex Darboux Theorem" by Chiappori and Ekeland . I do not pretend to understand the ...
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### Is it possible to derive the Weyl tensor from the Ricci tensor or Ricci scalar? If so, how?

I understand that the Ricci tensor is derived from the Riemann tensor, but I know that the Weyl tensor is also derived from the Riemann tensor. So, is it possible to calculate the Weyl tensor from the ...