# 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 is the connection between the Ricci tensor and the metric flatness?

Actually, this question was answered by Lawrence B. Crowell, but I would like to explore this topic further. Can anyone give me please references on where I can find it?
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### Breaking product of three vectors into symmetric and anti-symmetric vectors [closed]

Let's consider we have three arbitrary vectors A, B and C. We have the quantity $A_{\mu}B_{\nu}C_{\rho}$. Is it possible to break the above quantity into sum of symmetric and anti-symmetric vectors in ...
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### Confusion regarding Riemann Tensor and Ricci Tensor

Ricci Tensor is the contraction of the Riemann Tensor. Even if all the components of the Ricci Tensor is zero, that doesn't mean that the spacetime is flat. If all the components of the Riemann Tensor ...
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### How to express symmetry of a mixed (1,1) tensor with upper and lower index?

In the context of general relativity, I am working with the energy-momentum tensor $T$, which is a rank-2 tensor whose components are usually denoted by $T^\mu_{\ \ \ \ \nu}$. However, I am unsure of ...
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### Dimension of a vector space of all tensors of rank $(k,l)$ in 4D

Dual vector space is the set of all linear functionals defined on a given vector space. The vector space and dual vector space is isomorphic and hence have the same dimension. A rank $(k,l)$ tensor is ...
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### Taking the derivative of the integral of a tensor [migrated]

This is a bit of a technical question. Say I had a tensor of some arbitrary rank, for the sake of this example I'll use a vector $V_i$. If I were to take the following integral: $$\int V_i dx^i$$ And ...
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### Is there a Lorentz invariant electromagnetic quadrupole moment tensor?

I'm familiar with the electric and magnetic quadrupole moment tensors. However, I'm bothered that these objects are tensors only in the sense of spatial rotations. After all, Maxwell's equations and ...
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### How to derive the energy tensor invariantly?

On a (pseudo-)Riemannian manifold $(M, g)$ I can define the following action for any $\phi \in C ^{\infty}(M)$: $$\mathcal{S}(\phi) = \int_M g(\text{grad }\phi, \text{grad }\phi) \mathrm{d} V.$$ ...
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### Notational meaning of $\nabla_{\lambda}V^{\rho}$ and $\nabla_{\mu}\nabla_{\nu}V^{\rho}$

This question is related to Reconciling different expressions for Riemann curvature tensor, but it's different since it asks for some notational clarification arising out of calculations I did. To not ...
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### Bilinear covariants of Dirac field

In the book "Advanced quantum mechanics" by Sakurai there is a section (3.5) about bilinear covariants, however i can't really find a definition of these objects, neither in the book nor ...
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### Calculations with tensors give two different results from seemingly equivalent paths:

$\require{cancel}$ I want to calculate some operators in the context of Metric-Affine-Gravity: specifically spinors. I will work in tangent space, where all greek ($\mu,\nu,..$) indices are cast into ...
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### Exterior derivatives Leibniz rule

I want to prove Sean Carroll's "spacetime and geometry"'s eq.(2.78): $$\mathrm{d}(\omega \wedge \eta)=(\mathrm{d} \omega) \wedge \eta+(-1)^p \omega \wedge(\mathrm{d} \eta) \tag{2.78}$$ ...
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### Contraction of Lorentz indices in gluon propagator of QCD

In QED, the photon propagator has a factor of $g^{\mu \nu}$, and both $\mu$ and $\nu$ contract with the $\gamma$ matrix indices, which come from the fermion antifermion photon vertices on either end ...
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### What is the connection between a mathematician and physicist's definition of a tensor?

I study mathematics but I have a deep interest in physics as well. I have taken a course in smooth manifolds where a tensor is defined as an alternating multilinear function. Recently I have learned ...
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### Reconciling different expressions for Riemann curvature tensor

[Note: I think this question is more suited to Physics SE rather than Math, since it refers to Carroll's notes and some equations might have inherent Physics-related assumptions EDIT: This question ...
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### How many independent equations are contained in $R_{rsmn}=0$ in consideration of the Bianchi identity?

In $d$ dimensions, how many independent equations are contained in $R_{rsmn}=0$ in consideration of the Bianchi identity $\nabla_{[a}R_{bc]de}=0$? This discussion reveals the independent equations ...
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### Is there a physical interpretation of why Christoffel symbols do not transform like a tensor? [duplicate]

I understand mathematically why they don’t, but I was hoping someone could provide a physical interpretation to this. Is there a physical consequence of this fact?
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### Can either the covariant or the contravariant version of a physical tensor be more fundamental?

This question may be too subjective, but here goes: Essentially any physically interesting quantity can be represented by a tensor on an inner product space or by a tensor field on a pseudo-Reimannian ...
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### Continuity equation for the conservation of energy from the conservation of the energy-momentum tensor

I am working through the book Cosmology by Daniel Baumann, and in the subsection that covers the continuity equation (part of section 2.3.1 on perfect fluids) the author makes a claim that confuses me....
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### Transpose of a bilinear in Einstein notation

In Einstein notation we can take generic 1-vectors $x, y$ and (1,1) tensor $M$. As we know $x_{\mu}$ represents $x^{T}$, i.e. row vector (a co-vector), while $x^{\mu}$ is a column vector. So we can ...
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### Four-vector and Notation significance [closed]

As the title suggest, this has to do, on the most part, with four vector notation. I have a series of questions, the majority, related to this topic: 1- If we assume a lorentz boost in the x direction ...
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### Trace and index manipulation

Imagine that I have a quantity $F_{ab}$ multiplying the stress tensor $T^{ab}$: \begin{equation} F_{ab} T^{ab}. \end{equation} There is also a metric, say $h_{ab}$. If I want to write the above ...
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### Problem with proving the invariance of dot product of two four vectors

I am having a spot of trouble with index manipulation (its not that I am very unfamiliar with this, but I keep losing touch). This is from an electrodynamics course - we're just getting started with 4 ...
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### Which finite-dimensional representations of the Lorentz group do $p$-forms correspond to?

On the Wikipedia article about the representation theory of the Lorentz group, the finite-dimensional representations $(1,0)$ and $(0,1)$ are referred to as "$2$-form" representations. On ...
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The Lagrangian density of the Maxwell equations in vacuum is $$\mathcal{L} = - \frac{1}{4} F_{\mu\nu}F^{\mu\nu} . \tag{1}$$ My question is, $F$ is a tensor, namely  F = \frac{1}{2}F_{\mu\nu} dx^{\...
Suppose I model an atom as a two-level system with states $|g \rangle$ and $|e\rangle$, with eigenvalue equations $\hat{H_1}|g\rangle = g|g \rangle$ and $\hat{H_1}|e\rangle = e|e \rangle$, and an ...